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 <title>FLOW Lab</title>
 <link href="http://flow.byu.edu/atom.xml" rel="self"/>
 <link href="http://flow.byu.edu/"/>
 <updated>2025-01-08T17:29:37-07:00</updated>
 <id>http://flow.byu.edu</id>
 <author>
   <name>Andrew Ning</name>
   <email></email>
 </author>

 
 <entry>
   <title>Optimization Book Available</title>
   <link href="http://flow.byu.edu/posts/opt-book"/>
   <updated>2021-10-15T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/opt-book</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:share:6853387400052391936&quot; height=&quot;698&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>BEM Paper</title>
   <link href="http://flow.byu.edu/posts/bem-paper"/>
   <updated>2021-07-30T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/bem-paper</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:share:6825450879324434432&quot; height=&quot;622&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>Eduardo Research Update</title>
   <link href="http://flow.byu.edu/posts/eduardo-latest"/>
   <updated>2021-03-01T00:00:00-07:00</updated>
   <id>http://flow.byu.edu/posts/eduardo-latest</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:ugcPost:6779795433318420480&quot; height=&quot;1142&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>Reformulated VPM</title>
   <link href="http://flow.byu.edu/posts/reformulated-vpm"/>
   <updated>2021-02-01T00:00:00-07:00</updated>
   <id>http://flow.byu.edu/posts/reformulated-vpm</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:ugcPost:6767685749157588992&quot; height=&quot;990&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>Optimization Book Announcement</title>
   <link href="http://flow.byu.edu/posts/optimization-book"/>
   <updated>2021-01-20T00:00:00-07:00</updated>
   <id>http://flow.byu.edu/posts/optimization-book</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:share:6758135437300154368&quot; height=&quot;679&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>PJ Defense</title>
   <link href="http://flow.byu.edu/posts/pj-defense"/>
   <updated>2020-09-30T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/pj-defense</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:share:6714312529692237824&quot; height=&quot;641&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>VPM Paper</title>
   <link href="http://flow.byu.edu/posts/vpm-paper"/>
   <updated>2020-08-30T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/vpm-paper</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:ugcPost:6711406263634419712&quot; height=&quot;857&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>Airborne Wind with Vortex Particle Method</title>
   <link href="http://flow.byu.edu/posts/wind-harvesting"/>
   <updated>2020-08-16T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/wind-harvesting</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:ugcPost:6694076731474739200&quot; height=&quot;949&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>Eduardo's Past Three Years</title>
   <link href="http://flow.byu.edu/posts/eduardo-three-years"/>
   <updated>2020-08-10T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/eduardo-three-years</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:ugcPost:6689985497042944000&quot; height=&quot;797&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>FLOWUnsteady in Google Drive</title>
   <link href="http://flow.byu.edu/posts/google-drive-vpm"/>
   <updated>2020-08-02T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/google-drive-vpm</id>
   <content type="html">&lt;iframe src=&quot;https://www.linkedin.com/embed/feed/update/urn:li:share:6689003185174118400&quot; height=&quot;1094&quot; width=&quot;504&quot; frameborder=&quot;0&quot; allowfullscreen=&quot;&quot; title=&quot;Embedded post&quot;&gt;&lt;/iframe&gt;
</content>
 </entry>
 
 <entry>
   <title>Takeoff and Performance Tradeoffs of Retrofit Distributed Electric Propulsion for Urban Transport</title>
   <link href="http://flow.byu.edu/posts/electric-stol"/>
   <updated>2019-09-05T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/electric-stol</id>
   <content type="html">&lt;p&gt;&lt;img src=&quot;https://byu.box.com/shared/static/2ilze6i6aq4srqita1xqdu2sd6m62cqb.gif&quot; alt=&quot;Electric Aircraft Optimization&quot; /&gt;&lt;/p&gt;

&lt;p&gt;A trade study investigating the use of distributed electric propulsion on fixed-wing aircraft to shorten takeoff distance. This aircraft configuration could potentially compete with VTOL aircraft in urban air transportation.  The trade study was done using a MDO framework including blade element momentum method, vortex lattice method, linear-beam finite element analysis, classical laminate theory, composite failure, empirically-based blade noise modeling, motor and motor-controller mass models, and gradient-based optimization.&lt;/p&gt;

&lt;p&gt;Summary:&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;Validation of propeller-on-wing modeling, propeller performance, noise, and motor performance&lt;/li&gt;
  &lt;li&gt;New method for smoothly evaluating propeller effects on discrete VLM&lt;/li&gt;
  &lt;li&gt;Retrofit application maintaining the P2006T airframe, maximum takeoff weight, and balance&lt;/li&gt;
  &lt;li&gt;Pareto fronts of takeoff distance for varying cruise speeds and noise levels&lt;/li&gt;
  &lt;li&gt;System performance tradeoff evaluation including augmented lift, thrust, and takeoff profile&lt;/li&gt;
&lt;/ul&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;Moore2019&quot;&gt;Moore, K., and Ning, A., “Takeoff and Performance Tradeoffs of Retrofit Distributed Electric Propulsion for Urban Transport,” &lt;i&gt;Journal of Aircraft&lt;/i&gt;, Vol. 56, No. 5, pp. 1880–1892, Sep. 2019. doi:10.2514/1.C035321&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;Moore2019-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.2514/1.C035321&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=4204&amp;amp;context=facpub&quot;&gt;[PDF]&lt;/a&gt;  

&lt;pre id=&quot;Moore2019-bibtex&quot; class=&quot;bibtex&quot;&gt;@article{Moore2019,
  author = {Moore, Kevin and Ning, Andrew},
  doi = {10.2514/1.C035321},
  journal = {Journal of Aircraft},
  month = sep,
  number = {5},
  pages = {1880-1892},
  title = {Takeoff and Performance Tradeoffs of Retrofit Distributed Electric Propulsion for Urban Transport},
  volume = {56},
  year = {2019}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;
</content>
 </entry>
 
 <entry>
   <title>Multirotor - Vortex Particle Method</title>
   <link href="http://flow.byu.edu/posts/multirotor-vpm2"/>
   <updated>2019-07-01T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/multirotor-vpm2</id>
   <content type="html">&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube.com/embed/SLpnVIBpkps&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture&quot; allowfullscreen=&quot;&quot;&gt;&lt;/iframe&gt;

&lt;p&gt;The viscous vortex particle method is a mesh-free Lagrangian CFD method for the numerical solution of Navier-Stokes in their vorticity form. It’s about two orders of magnitude faster than mesh-based CFD (RANS or LES). We are exploring its potential for modeling the complicated aerodynamics interactions in eVTOL aircraft design. All codes are in-house developed at BYU’s FLOW Lab,&lt;/p&gt;

&lt;p&gt;Slides available &lt;a href=&quot;https://lnkd.in/gEDYPch&quot;&gt;here&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;SUMMARY:&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;Validation of predicted multirotor interactions in both hover and forward flight, at both low and high Reynolds numbers.&lt;/li&gt;
  &lt;li&gt;We ran a parametric study with 1152 simulations taking ~3.5 days in a desktop computer (~4 mins per simulation).&lt;/li&gt;
  &lt;li&gt;We built response surfaces of unsteady loading and performance drop during rotor-on-rotor interactions of the APC 10x7 propeller.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;FUTURE WORK&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;Coupling with noise code for prediction of acoustic interactions.&lt;/li&gt;
  &lt;li&gt;Automatic derivatives for gradient-based design optimization.&lt;/li&gt;
&lt;/ul&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;Alvarez2019&quot;&gt;Alvarez, E. J., and Ning, A., “Modeling Multirotor Aerodynamic Interactions Through the Vortex Particle Method,” &lt;i&gt;AIAA Aviation Forum&lt;/i&gt;, Dallas, TX, Jun. 2019. doi:10.2514/6.2019-2827&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;Alvarez2019-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.2514/6.2019-2827&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=4145&amp;amp;context=facpub&quot;&gt;[PDF]&lt;/a&gt;  

&lt;pre id=&quot;Alvarez2019-bibtex&quot; class=&quot;bibtex&quot;&gt;@inproceedings{Alvarez2019,
  address = {Dallas, TX},
  author = {Alvarez, Eduardo J. and Ning, Andrew},
  booktitle = {{AIAA} Aviation Forum},
  doi = {10.2514/6.2019-2827},
  month = jun,
  title = {Modeling Multirotor Aerodynamic Interactions Through the Vortex Particle Method},
  year = {2019}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;
</content>
 </entry>
 
 <entry>
   <title>Development of a Vertical-Axis Wind Turbine Wake Model</title>
   <link href="http://flow.byu.edu/posts/vawt-wake"/>
   <updated>2019-06-05T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/vawt-wake</id>
   <content type="html">&lt;p&gt;&lt;img src=&quot;/posts/vawtV.gif&quot; alt=&quot;Wake Visualization&quot; /&gt;&lt;/p&gt;

&lt;p&gt;Wind farm designers use mathematical models to describe turbine wakes, and through optimization, turbines can be positioned to minimize the negative wake interference of the wind farm. This method has been conducted by numerous researchers for the well-known horizontal-axis wind turbine, but the same type of research has been much more limited for a vertical-axis wind turbine (VAWT), as shown below. VAWT wake behavior is quite different from that of a horizontal axis turbine.  Preliminary research has shown that VAWTs are capable of being placed closer together and even increasing their overall power production by harnessing the slight wind velocity increases of neighboring turbines.&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;/posts/vawt.png&quot; width=&quot;250px&quot; class=&quot;img-left&quot; /&gt;&lt;/p&gt;

&lt;p&gt;We developed a wake model that computes the wind velocity profile around a VAWT from input parameters such as rotation speed, turbine blade size, and wind speed. To establish a foundation for the wake model, we ran hundreds of computer simulations of a two-dimensional VAWT at many different speed and geometry configurations, as shown by the example at the top of the post. Using the wake behavior obtained from these simulations, we used statistical analysis to simply the complex behavior down to fundamental nondimensional relationships between turbine speed and geometry that influence the wake’s profile both downstream and to the sides of the turbine. This predictive ability is important to take advantage of closely spaced VAWT pairs. Typical horizontal axis turbine wake models predict wake velocities behind a turbine, but to take advantage of the unique behavior of VAWTs, we needed to predict velocities &lt;em&gt;around&lt;/em&gt; the turbine (there is often a speedup around the sides).&lt;/p&gt;

&lt;p&gt;The new wake model runs in about a tenth of second, as compared to the original CFD simulation that takes several hours, and shows reasonable agreement against experimental wake studies.  The VAWT wake model was further combined with the aerodynamic loading of turbine blades to translate wind velocity speeds around the turbine to turbine power production. While this wake model is a preliminary attempt to parametrically define VAWT wakes, it does show promise in its use for wind farm analysis and optimization.&lt;/p&gt;

&lt;p&gt;The journal paper describing the wake model development and access to the wake model code can be found in the links below.&lt;/p&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;Tingey2019&quot;&gt;Tingey, E., and Ning, A., “Development of a Parameterized Reduced-Order Vertical-Axis Wind Turbine Wake Model,” &lt;i&gt;Wind Engineering&lt;/i&gt;, Jun. 2019. doi:10.1177/0309524X19849864&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;Tingey2019-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.1177/0309524X19849864&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=4122&amp;amp;context=facpub&quot;&gt;[PDF]&lt;/a&gt; &lt;a href=&quot;https://github.com/byuflowlab/vawt-wake-model&quot;&gt;[Code]&lt;/a&gt; 

&lt;pre id=&quot;Tingey2019-bibtex&quot; class=&quot;bibtex&quot;&gt;@article{Tingey2019,
  author = {Tingey, Eric and Ning, Andrew},
  doi = {10.1177/0309524X19849864},
  journal = {Wind Engineering},
  month = jun,
  title = {Development of a Parameterized Reduced-Order Vertical-Axis Wind Turbine Wake Model},
  year = {2019}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;
</content>
 </entry>
 
 <entry>
   <title>Polynomial Chaos for Wind Farm Power Prediction</title>
   <link href="http://flow.byu.edu/posts/pc-windfarm"/>
   <updated>2019-05-28T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/pc-windfarm</id>
   <content type="html">&lt;center&gt;
&lt;img src=&quot;/posts/binned.png&quot; alt=&quot;convergence of AEP&quot; width=&quot;300px&quot; /&gt;
&lt;/center&gt;

&lt;p&gt;In a wind farm, it is important to understand how both the wind speed and wind direction varies.  The variability in wind direction is captured in what is called a wind rose (see above).  The longer bars in the wind rose indicate directions from which wind is more likely to come from.  The power produced by the wind farm can change dramatically as the wind direction changes.  This variation in power is often reduced to a single number—the expected value of power (an average).  In the wind energy field, a common metric of interest is the annual energy production (AEP), which is the expected value of power (multiplied by time in a year and availability).&lt;/p&gt;

&lt;p&gt;The wind rose is really a continuous distribution, but we often discretize it into bins like shown above.  This discretization provides a convenient way to compute average power production.  This is done by dividing the uncertain variable (wind direction in this example) into discrete bins, multiplying the power for each direction by its probability, then summing up over all bins.  This approach is the familiar rectangle rule.  It’s a simple approach and is almost always the approach used in wind farm analysis.&lt;/p&gt;

&lt;p&gt;In a collaboration between our lab and the &lt;a href=&quot;http://adl.stanford.edu&quot;&gt;Aerospace Design Laboratory&lt;/a&gt; at Stanford we’ve been exploring an alternative approach based on polynomial chaos.  Polynomial chaos is not a new method.  It is widely known and used in applications of uncertainty quantification, but has not yet been applied to wind farm analysis.  The main idea is that rather than discretize into bins, we can use polynomials to approximate the continuous power response, allowing us to use fewer samples to approximate the AEP.&lt;/p&gt;

&lt;p&gt;Santiago led a recent study where we explored using polynomial chaos, with uncertainty in both wind direction and wind speed, to predict AEP.  We explored using both quadrature approaches and regression approaches and found the latter to be more effective.  On average, polynomial chaos reduced the number of simulations required by a factor of 5 as compared to the rectangle rule. For example, the figure below shows how AEP varies with the number of samples with polynomial chaos based on regression to the left, and the rectangle rule to the right.  We see that polynomial chaos converges with fewer samples and produces less variability in the AEP estimates from starting the quadrature at different points.&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;/posts/uq.png&quot; alt=&quot;convergence of AEP&quot; /&gt;&lt;/p&gt;

&lt;p&gt;Additionally, we applied polynomial chaos to optimizing wind farm layouts.  We found that using polynomial chaos with regression allowed for about 1/3 of the simulations required for the rectangle rule (while producing a slightly better AEP).  One such case is shown below.&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;/posts/pc-opt.png&quot; alt=&quot;optimization using PC&quot; /&gt;&lt;/p&gt;

&lt;p&gt;Further details are available in the open-access paper and data linked below.&lt;/p&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;Padron2019&quot;&gt;Padrón, A. S., Thomas, J., Stanley, A. P. J., Alonso, J. J., and Ning, A., “Polynomial Chaos to Efficiently Compute the Annual Energy Production in Wind Farm Layout Optimization,” &lt;i&gt;Wind Energy Science&lt;/i&gt;, Vol. 4, pp. 211–231, May 2019. doi:10.5194/wes-4-211-2019&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;Padron2019-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.5194/wes-4-211-2019&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://www.wind-energ-sci.net/4/211/2019/wes-4-211-2019.pdf&quot;&gt;[PDF]&lt;/a&gt; &lt;a href=&quot;https://github.com/byuflowlab/windfarm-ouu&quot;&gt;[Code]&lt;/a&gt; 

&lt;pre id=&quot;Padron2019-bibtex&quot; class=&quot;bibtex&quot;&gt;@article{Padron2019,
  author = {Padr\&apos;{o}n, A. Santiago and Thomas, Jared and Stanley, Andrew P. J. and Alonso, Juan J. and Ning, Andrew},
  doi = {10.5194/wes-4-211-2019},
  journal = {Wind Energy Science},
  month = may,
  pages = {211-231},
  title = {Polynomial Chaos to Efficiently Compute the Annual Energy Production in Wind Farm Layout Optimization},
  volume = {4},
  year = {2019}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;
</content>
 </entry>
 
 <entry>
   <title>Making Julia as Fast as C++</title>
   <link href="http://flow.byu.edu/posts/julia-c++"/>
   <updated>2019-04-18T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/juliacpp</id>
   <content type="html">&lt;center&gt;
&lt;img src=&quot;/posts/vortex00.gif&quot; alt=&quot;Vid here&quot; width=&quot;500px&quot; /&gt;
&lt;/center&gt;

&lt;h2 id=&quot;introduction&quot;&gt;Introduction&lt;/h2&gt;

&lt;p&gt;The rumor says that Julia can achieve the &lt;a href=&quot;https://julialang.org/benchmarks/&quot;&gt;same computing
performance&lt;/a&gt; as any other compiled language
like C++ and FORTRAN. After coding in Julia for the past two years I have
definitely fell in love with its pythonic syntax, multiple dispatch, and MATLAB-like
handiness in linear algebra, while being able to use compilation features
like explicit type declaration for bug-preventive programming. In summary,
Julia’s philosophy brings all the flexibility of an interpreted language,
meanwhile its Just-In-Time (JIT) compilation makes it a defacto
compiled language.&lt;/p&gt;

&lt;p&gt;Julia’s high level syntax makes the language easygoing for programmers from any
background, however, achieving high performance is sort of
an art. In this post I summarize some of the things I’ve learned while crafting
my Julia codes for high-performance computing. I will attempt to show the
process of code optimization through a real-world computing application from
aerodynamics: the &lt;a href=&quot;https://scholarsarchive.byu.edu/facpub/2116/&quot;&gt;vortex particle
method&lt;/a&gt;\(^{[1,\,2]}\).&lt;/p&gt;

&lt;h2 id=&quot;problem-definition&quot;&gt;Problem Definition&lt;/h2&gt;

&lt;p&gt;In the &lt;a href=&quot;https://scholarsarchive.byu.edu/facpub/2116/&quot;&gt;vortex particle method&lt;/a&gt; we
are interested in calculating the velocity \(\mathbf{u}\) and velocity Jacobian
\(\mathbf{J}\) that a field of \(N\) vortex particles induces at an arbitrary
position \(\mathbf{x}\). This is calculated as&lt;/p&gt;

\[{\bf u}\left( {\bf x} \right) = \sum\limits_p^N g_\sigma\left(
{\bf x}-{\bf x}_p \right)
                                                {\bf K}\left( {\bf x}-{\bf x}_p
\right)   \times
                                    \boldsymbol\Gamma_p\]

\[\frac{\partial {\bf u}}{\partial x_j}\left( {\bf x} \right)
        = \sum\limits_p^N \left[
            \left(
                \frac{1}{\sigma }
                 \frac{\Delta x_j}{\Vert \Delta \mathbf{x} \Vert}
                 \frac{\partial g}{\partial r}
                 \left(
                             \frac{\Vert \Delta\mathbf{x} \Vert}{\sigma}
                 \right) -
                 3 g_\sigma\left( \Delta{\bf x} \right)
                 \frac{\Delta x_j}{\Vert \Delta\mathbf{x} \Vert^2}
            \right)
            {\bf K}\left( \Delta\mathbf{x} \right)  \times \boldsymbol\Gamma_p -
            \frac{g_\sigma\left( \Delta{\bf x} \right) }{4\pi \Vert \Delta{\bf
x} \Vert^3}
            \delta_{ij} \times \boldsymbol\Gamma_p
        \right]
,\]

&lt;p&gt;with \({\bf K}\) the singular Newtonian kernel \({\bf K}\left( {\bf x}\right)=-\frac{ {\bf x} }{4\pi \Vert{\bf x}\Vert^3}\), \(g_\sigma\) a regularizing
function of smoothing radius \(\sigma\), and \(\mathbf{x}_p\) and
\(\boldsymbol{\Gamma}_p\) the position and vectorial strength of the \(p\)-th
particle, respectively. Furthermore, the governing equations of the method
require evaluating the velocity \(\mathbf{u}\) and Jacobian \(\mathbf{J}\) that the
collection of particles induces on itself, leading to the well-known &lt;a href=&quot;https://en.wikipedia.org/wiki/N-body_problem&quot;&gt;\(N\)-body
problem&lt;/a&gt; of computational
complexity \(\mathcal{O}(N^2)\).&lt;/p&gt;

&lt;p&gt;Choosing Winckelmans’ regularizing kernel\(^{[1]}\)&lt;/p&gt;

\[g(r) = r^3 \frac{r^2 + 5/2}{\left( r^2 + 1 \right)^{5/2}}\]

\[\frac{\partial g}{\partial r} (r) = \frac{15}{2}
            \frac{r^2}{\left( r^2 + 1 \right)^{7/2}}
,\]

&lt;p&gt;the above equations are implemented in C++ as follows:&lt;/p&gt;

&lt;div class=&quot;language-cpp highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&lt;span class=&quot;c1&quot;&gt;// Particle-to-particle interactions&lt;/span&gt;
&lt;span class=&quot;kt&quot;&gt;void&lt;/span&gt; &lt;span class=&quot;nf&quot;&gt;P2P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Particle&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;const&lt;/span&gt; &lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;numParticles&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;

  &lt;span class=&quot;n&quot;&gt;real_t&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;vec3&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;

  &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numParticles&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;++&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numParticles&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;++&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;

      &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
      &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
      &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
      &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]);&lt;/span&gt;

      &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;

          &lt;span class=&quot;c1&quot;&gt;// Evaluate g_σ and ∂gσ∂r&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;pow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;1.0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;);&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;7.5&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;1.0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;);&lt;/span&gt;

          &lt;span class=&quot;c1&quot;&gt;// u(x) = ∑g_σ(x−xp) K(x−xp) × Γp&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;

          &lt;span class=&quot;c1&quot;&gt;// ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp]&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;3.0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;));&lt;/span&gt;
          &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;++&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;

          &lt;span class=&quot;c1&quot;&gt;// Adds the Kronecker delta term&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;);&lt;/span&gt;
          &lt;span class=&quot;c1&quot;&gt;// k=1&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;c1&quot;&gt;// k=2&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;c1&quot;&gt;// k=3&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
      &lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;

    &lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;
  &lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;

&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;The figure at the top of the page is a simulation of a 6x6x6 box of particles with vorticity
initially concentrated at its center, diffusing as the simulation progresses due
to viscous effects:
&lt;!-- &lt;center&gt;
&lt;img src=&quot;vortex00.gif&quot; alt=&quot;Vid here&quot; width=&quot;500px&quot;&gt;
&lt;/center&gt; --&gt;&lt;/p&gt;

&lt;h2 id=&quot;c-benchmark&quot;&gt;C++ Benchmark&lt;/h2&gt;

&lt;p&gt;Here I have &lt;a href=&quot;https://github.com/EdoAlvarezR/post-juliavscpp&quot;&gt;coded a benchmark&lt;/a&gt;
of the C++ code shown above, evaluating &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;P2P&lt;/code&gt; on a box of 6x6x6=216 particles,
and made it callable in this notebook through the
&lt;a href=&quot;https://github.com/JuliaInterop/CxxWrap.jl&quot;&gt;CxxWrap&lt;/a&gt; Julia package:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [1]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;cm&quot;&gt;#=
    This cell loads auxiliary benchmarking functions available
    here: https://github.com/EdoAlvarezR/post-juliavscpp
=#&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;using&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;LinearAlgebra&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Load C++ code wrapped as module CxxVortexTest&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;include&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s&quot;&gt;&quot;code/vortextest.jl&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Load benchmarking tools&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;include&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s&quot;&gt;&quot;code/benchmarking.jl&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;&lt;strong&gt;In [2]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;ntests&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1000&lt;/span&gt;               &lt;span class=&quot;c&quot;&gt;# Tests to run&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;6&lt;/span&gt;                       &lt;span class=&quot;c&quot;&gt;# Particles per side&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;lambda&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;1.5&lt;/span&gt;                &lt;span class=&quot;c&quot;&gt;# Core overlap&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;verbose&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;true&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Run C++ benchmark&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;cpptime&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;CxxVortexTest&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;benchmarkP2P_wrap&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ntests&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;kt&quot;&gt;Float32&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lambda&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;verbose&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Store benchmark result&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;benchtime&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s&quot;&gt;&quot;C++&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cpptime&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;Samples:	1000
min time:	3.99555 ms
ave time:	4.77778 ms
max time:	6.73414 ms
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;This was ran in my Dell Latitude 5580 laptop (Intel® Core™ i7-7820HQ CPU @
2.90GHz × 8 ) in only one process, and we see that &lt;strong&gt;the C++ kernel, best-case
scenario, is evaluated in ~4 ms&lt;/strong&gt;. Let’s move on and code this up in Julia.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;NOTE:&lt;/strong&gt; The C++ code was compiled with the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;-O3&lt;/code&gt; flag for code optimization.&lt;/p&gt;

&lt;h2 id=&quot;optimizing-julia&quot;&gt;Optimizing Julia&lt;/h2&gt;

&lt;h3 id=&quot;julia-baseline-pythonic-programming&quot;&gt;Julia Baseline: Pythonic Programming&lt;/h3&gt;

&lt;p&gt;Tempted by the Python-like syntax available in Julia, our first inclination is
to make the code as general, simple, and easy to understand as possible. Here is
the most general implementation where no types are specified:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [3]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
This is a particle struct made up of ambiguous (unspecified) types
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;struct&lt;/span&gt;&lt;span class=&quot;nc&quot;&gt; ParticleAmbiguous&lt;/span&gt;

    &lt;span class=&quot;c&quot;&gt;# User inputs&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;                               &lt;span class=&quot;c&quot;&gt;# Position&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;                           &lt;span class=&quot;c&quot;&gt;# Vectorial circulation&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;                           &lt;span class=&quot;c&quot;&gt;# Smoothing radius&lt;/span&gt;

    &lt;span class=&quot;c&quot;&gt;# Properties&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;                               &lt;span class=&quot;c&quot;&gt;# Velocity at particle&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;                               &lt;span class=&quot;c&quot;&gt;# Jacobian at particle J[i,j]=dUi/dxj&lt;/span&gt;

    &lt;span class=&quot;n&quot;&gt;ParticleAmbiguous&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                      &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;new&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Empty initializer&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Type&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ParticleAmbiguous&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;})&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ParticleAmbiguous&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;0.0&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;




&lt;span class=&quot;c&quot;&gt;# Winckelmans regularizing kernel&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;g_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;7.5&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mf&quot;&gt;3.5&lt;/span&gt;

&lt;span class=&quot;kd&quot;&gt;const&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;pi&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;




&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
    Pythonic programming approach
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;
        &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;    

            &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;norm&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# K × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cross&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# U = ∑g_σ(x-xp) * K(x-xp) × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
                &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;
                                      &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;
                                      &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cross&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;([&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;],&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;&lt;strong&gt;In [4]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;c&quot;&gt;# Create a 6x6x6 box of ambiguous particles&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;particles_amb&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;generate_particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ParticleAmbiguous&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;lambda&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Run benchmark&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles_amb&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;compare&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;s&quot;&gt;&quot;C++&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;reverse&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;nb&quot;&gt;false&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@benchmark&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;C++                  is 58.48 times faster than         P2P_pythonic (3.996ms vs 233.679ms)

BenchmarkTools.Trial:
  memory estimate:  217.57 MiB
  allocs estimate:  4087152
  --------------
  minimum time:     233.679 ms (5.04% GC)
  median time:      238.482 ms (4.99% GC)
  mean time:        251.540 ms (9.28% GC)
  maximum time:     295.517 ms (19.89% GC)
  --------------
  samples:          4
  evals/sample:     1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Here we see that in our pythonic attempt we’ve got code that is &lt;strong&gt;pretty neat
and concise, but ~58x slower than the C++ implementation&lt;/strong&gt;.&lt;/p&gt;

&lt;h3 id=&quot;fix-1-always-work-with-concrete-types&quot;&gt;Fix #1: Always work with concrete types&lt;/h3&gt;

&lt;p&gt;The problem with the pythonic approach is that variable types are never
declared, and &lt;strong&gt;without foreknowledge of the types to be handled, Julia can’t
optimize the function during compilation&lt;/strong&gt;. In order to help us catch ambiguous
(abstract) types in our code, the Julia &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Base&lt;/code&gt; package provides the macro
&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@code_warntype&lt;/code&gt;, which prints the lowered and type-inferred AST used during
compilation highlighting any abstract types encountered.&lt;/p&gt;

&lt;p&gt;The output is pretty lengthy, so here I have copied and pasted only a snippet:&lt;/p&gt;

&lt;div class=&quot;language-julia highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&lt;span class=&quot;nd&quot;&gt;@code_warntype&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;n&quot;&gt;Body&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Nothing&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;│╻╷╷&lt;/span&gt;  &lt;span class=&quot;n&quot;&gt;iterate34&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;──&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;   &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arraylen&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Int64&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;││╻╷&lt;/span&gt;   &lt;span class=&quot;n&quot;&gt;iterate&lt;/span&gt;   &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;   &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sle_int&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Bool&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;│││╻&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;   &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;   &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bitcast&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;UInt64&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;UInt64&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;││││╻&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;   &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;4&lt;/span&gt;   &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ult_int&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;mh&quot;&gt;0x0000000000000000&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Bool&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;││││╻&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;   &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;5&lt;/span&gt;   &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;and_int&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Bool&lt;/span&gt;
        &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
        &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
        &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;       &lt;span class=&quot;mi&quot;&gt;11&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;┄&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;33&lt;/span&gt;  &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;φ&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;c&quot;&gt;#10 =&amp;gt; %28, #34 =&amp;gt; %149)::ParticleAmbiguous&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;       &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;34&lt;/span&gt;  &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;φ&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;c&quot;&gt;#10 =&amp;gt; %29, #34 =&amp;gt; %150)::Int64&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;│╻&lt;/span&gt;    &lt;span class=&quot;n&quot;&gt;getproperty37&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;35&lt;/span&gt;  &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;getfield&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;16&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Any&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;││&lt;/span&gt;      &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;36&lt;/span&gt;  &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;getfield&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;33&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Any&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;       &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;37&lt;/span&gt;  &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;35&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;36&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Any&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;mi&quot;&gt;38&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;│&lt;/span&gt;    &lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;38&lt;/span&gt;  &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Main&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;norm&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;37&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Any&lt;/span&gt;
        &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
        &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
        &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Understanding this lowered AST syntax is sort of an art, but you’ll soon learn
that &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@code_warntype&lt;/code&gt; is your best friend when optimizing code. As we scroll
down the AST we see that code encounters types &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Any&lt;/code&gt; in the properties of our
&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;ParticleAmbiguous&lt;/code&gt; type, which immediately should raise a red flag to us (&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Any&lt;/code&gt;
is an abstract type). In fact, when running &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@code_warntype&lt;/code&gt; the output
automatically highlights those &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;::Any&lt;/code&gt; asserts in red.&lt;/p&gt;

&lt;p&gt;We can take care of those abstract types by defining the properties of the
&lt;a href=&quot;https://docs.julialang.org/en/v1/manual/performance-
tips/index.html#Type-declarations-1&quot;&gt;struct parametrically&lt;/a&gt;:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [5]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
This is a particle struct with property types
explicitely/parametrically defined.
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;struct&lt;/span&gt;&lt;span class=&quot;nc&quot;&gt; Particle&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt;

  &lt;span class=&quot;c&quot;&gt;# User inputs&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Array&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt;                &lt;span class=&quot;c&quot;&gt;# Position&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Array&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt;            &lt;span class=&quot;c&quot;&gt;# Vectorial circulation&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;                      &lt;span class=&quot;c&quot;&gt;# Smoothing radius&lt;/span&gt;

  &lt;span class=&quot;c&quot;&gt;# Properties&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Array&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt;                &lt;span class=&quot;c&quot;&gt;# Velocity at particle&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Array&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt;                &lt;span class=&quot;c&quot;&gt;# Jacobian at particle J[i,j]=dUi/dxj&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Another initializer alias&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;Particle&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;where&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Particle&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;))&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Empty initializer&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Type&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Particle&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}})&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;where&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Particle&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt;
                                                      &lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt;
                                                      &lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;))&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;No modifications further are needed in our &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;P2P_pythonic&lt;/code&gt; function since Julia’s
multiple dispatch and JIT automatically compiles a version of the function
specialized for our new &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Particle{T}&lt;/code&gt; type on the fly. Still, we will define an
alias to help us compare benchmarks:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [6]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;P2P_concretetypes&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;&lt;strong&gt;In [7]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;c&quot;&gt;# Create a 6x6x6 box of concrete Float64 particles&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;generate_particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Particle&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Float64&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;},&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;lambda&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Run benchmark&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;compare&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_concretetypes&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@benchmark&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_concretetypes&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;P2P_concretetypes    is  3.06 times faster than         P2P_pythonic (76.488ms vs 233.679ms)

BenchmarkTools.Trial:
  memory estimate:  189.93 MiB
  allocs estimate:  2275776
  --------------
  minimum time:     76.488 ms (13.63% GC)
  median time:      77.860 ms (13.79% GC)
  mean time:        84.567 ms (17.89% GC)
  maximum time:     145.918 ms (42.64% GC)
  --------------
  samples:          12
  evals/sample:     1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Voilà! By specifying concrete types in our &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;Particle&lt;/code&gt; struct now we have gained
a 3x speed up (we should run &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@code_warntype&lt;/code&gt; again to make sure we got rid of
all abstract types, but I’ll omit it for brevity).&lt;/p&gt;

&lt;p&gt;Let’s new see how we compare to C++:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [8]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;printcomparison&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_concretetypes&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;s&quot;&gt;&quot;C++&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;false&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;C++                  is 19.14 times faster than    P2P_concretetypes (3.996ms vs 76.488ms)
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Working with concrete types greatly sped up the computation; however, the C++
version is still ~20x faster than Julia. Let’s see what else can we optimize.&lt;/p&gt;

&lt;h3 id=&quot;fix-2-avoid-list-comprehensions&quot;&gt;Fix #2: Avoid List Comprehensions&lt;/h3&gt;

&lt;p&gt;The wonders of list comprehension may tempt you to write some line-efficient
code; however, loosely used may lead to a very inefficient
computation. Take for example this list-comprehension sum:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [9]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;sum_list&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;([&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;])&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@btime&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sum_list&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;  80.975 ns (1 allocation: 896 bytes)
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Here is the version of the same function unrolled without the list
comprehension, which does the job 60 times faster:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [10]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; sum_unrolled&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;out&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;
        &lt;span class=&quot;n&quot;&gt;out&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;return&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;out&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@btime&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sum_unrolled&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;  1.374 ns (0 allocations: 0 bytes)
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;In our P2P function we have a Kronecker delta cross product that we were
calculating in just one line as a list comprehension as&lt;/p&gt;

&lt;div class=&quot;language-julia highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;    &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
        &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;
                          &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;
                          &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cross&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;([&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;],&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;The alternative is to expand it into a few lines as&lt;/p&gt;

&lt;div class=&quot;language-julia highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;
    &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
        &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;
                          &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

    &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = −∑gσ/(4πr^3) δij×Γp&lt;/span&gt;
    &lt;span class=&quot;c&quot;&gt;# Adds the Kronecker delta term&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
    &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
    &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
    &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;The problem with list comprehension operations is that it has to allocate memory
to build the generated array. Just resist the temptation of using list
comprehensions to save yourself a few lines, and simply unroll it. As seen below
we get a 1.5x speed up by unrolling this line:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [11]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
    Unrolling the list comprehension
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P_nocomprehension&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;
        &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;    

            &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;norm&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# K × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cross&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# U = ∑g_σ(x-xp) * K(x-xp) × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
                &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;
                                      &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) += −∑gσ/(4πr^3) δij×Γp&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# Adds the Kronecker delta term&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;&lt;strong&gt;In [12]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;compare&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_nocomprehension&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_concretetypes&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@benchmark&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_nocomprehension&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;P2P_nocomprehension  is  1.52 times faster than    P2P_concretetypes (50.196ms vs 76.488ms)

BenchmarkTools.Trial:
  memory estimate:  126.16 MiB
  allocs estimate:  1300536
  --------------
  minimum time:     50.196 ms (11.28% GC)
  median time:      51.929 ms (11.65% GC)
  mean time:        55.319 ms (15.41% GC)
  maximum time:     107.039 ms (50.23% GC)
  --------------
  samples:          19
  evals/sample:     1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;h3 id=&quot;fix-3-reduce-allocation&quot;&gt;Fix #3: Reduce Allocation&lt;/h3&gt;

&lt;p&gt;Next, we notice that the benchmarking test is allotting an unusual amount of
memory (126MiB) and allocation operations (1.3M). I am suspicious that this is
an issue with Julia allowing arrays of dynamic sizes. The first step to solve
this is to &lt;strong&gt;do away with creating any internal variables of type arrays&lt;/strong&gt;. In
the code bellow, notice that I had replaced the array variables &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dX&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;crss&lt;/code&gt;
with float variables &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dX1, dX2, dX3&lt;/code&gt;, and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;crss1, crss2, crss3&lt;/code&gt;, which leads to
having to fully unroll the inner for-loop:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [13]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
    Reducing memory allocation
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P_noallocation&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;
        &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;    

            &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;norm&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# K × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cross&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# U = ∑g_σ(x-xp) * K(x-xp) × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) += ∑p[(Δxj∂gσ∂r/(σr) − 3Δxjgσ/r^2) K(Δx)×Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# Adds the Kronecker delta term&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;&lt;strong&gt;In [14]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;compare&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_noallocation&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_nocomprehension&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@benchmark&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_noallocation&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;P2P_noallocation     is  3.68 times faster than  P2P_nocomprehension (13.627ms vs 50.196ms)

BenchmarkTools.Trial:
  memory estimate:  21.99 MiB
  allocs estimate:  325296
  --------------
  minimum time:     13.627 ms (7.41% GC)
  median time:      15.615 ms (8.40% GC)
  mean time:        16.582 ms (12.83% GC)
  maximum time:     59.011 ms (66.91% GC)
  --------------
  samples:          61
  evals/sample:     1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Here we have reduced the memory allocated from 126MiB to 22MiB, leading to a
3.5x speed up. Let’s see what else can we do to decrease memory allocation.&lt;/p&gt;

&lt;h3 id=&quot;fix-4-no-linear-algebra&quot;&gt;Fix #4: No Linear Algebra&lt;/h3&gt;

&lt;p&gt;The next thing to consider is that doing &lt;strong&gt;linear algebra operations
using the Julia base library (i.e., &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dot(X,X)&lt;/code&gt;,
&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cross(X,X)&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;norm(X,X)&lt;/code&gt;) is more expensive that explicitely unfolding the
operation into lines of code&lt;/strong&gt;. I am suspicious that this is a memory allocation
problem since these functions need to allocate internal arrays to store the
computation prior to outputting the result. Here is the code without any
functional linear algebra functions from the base library (notice that I no longer use &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;norm()&lt;/code&gt; nor
&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cross()&lt;/code&gt;):&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [15]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
    No linear algebra functions
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P_nolinalg&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;
        &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;    

            &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# K × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# U = ∑g_σ(x-xp) * K(x-xp) × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) += ∑p[(Δxj∂gσ∂r/(σr) − 3Δxjgσ/r^2) K(Δx)×Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# Adds the Kronecker delta term&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;&lt;strong&gt;In [16]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;compare&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_nolinalg&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_noallocation&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@benchmark&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_nolinalg&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;P2P_nolinalg         is  2.10 times faster than     P2P_noallocation (6.487ms vs 13.627ms)

BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     6.487 ms (0.00% GC)
  median time:      7.140 ms (0.00% GC)
  mean time:        7.198 ms (0.00% GC)
  maximum time:     9.172 ms (0.00% GC)
  --------------
  samples:          139
  evals/sample:     1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;By doing away with linear algebra functions we are now &lt;strong&gt;not allocating any
memory, reaching an extra 2x speed up&lt;/strong&gt;.&lt;/p&gt;

&lt;h3 id=&quot;fix-5-fine-tuning&quot;&gt;Fix #5: Fine Tuning&lt;/h3&gt;

&lt;p&gt;Notice that by now we are achieving benchmarks in the same order of magnitude
than C++ (6.5ms in Julia vs 4.0ms in C++):&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [17]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;printcomparison&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_nolinalg&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;s&quot;&gt;&quot;C++&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;false&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;C++                  is  1.62 times faster than         P2P_nolinalg (3.996ms vs 6.487ms)
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;What we did prior to this point were
general principles that apply to any code that attempts to get high performance.
What it follows now is fine tuning of our code in ways that only apply to the
specific computation that we are performing.&lt;/p&gt;

&lt;p&gt;For instance, recall that our P2P function receives any user-defined
regularizing kernel that our function calls through this lines:&lt;/p&gt;

&lt;div class=&quot;language-julia highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sources&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;targets&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Function&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Function&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;targets&lt;/span&gt;
        &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sources&lt;/span&gt;
            &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
            &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
            &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
            &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
            &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
            &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
            &lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;
        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;For the case of Winckelmans’ kernel, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;g&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dgdr&lt;/code&gt; look like this:&lt;/p&gt;
&lt;div class=&quot;language-julia highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&lt;span class=&quot;n&quot;&gt;g_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;7.5&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mf&quot;&gt;3.5&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;We notice that each of these function calculate a power operation independently,
&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;(r^2 + 1)^2.5&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;(r^2 + 1)^3.5&lt;/code&gt;. I have observed that &lt;strong&gt;any sort of non-integer
power operation takes Julia more than any basic arithmetic operation or
even space allocation&lt;/strong&gt;. Thus, we can save computation by merging this two functions
and reusing the square root calculation as&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [45]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; g_dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
    &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;

    &lt;span class=&quot;c&quot;&gt;# Returns g and dgdr&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;return&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;7.5&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;))&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
    Reuses sqrt, reducing to only one power calculation
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P_reusesqrt&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;
        &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;    

            &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# K × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# U = ∑g_σ(x-xp) * K(x-xp) × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) += ∑p[(Δxj∂gσ∂r/(σr) − 3Δxjgσ/r^2) K(Δx)×Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# Adds the Kronecker delta term&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;&lt;strong&gt;In [19]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;compare&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_reusesqrt&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_nolinalg&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@benchmark&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_reusesqrt&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;P2P_reusesqrt        is  1.60 times faster than         P2P_nolinalg (4.050ms vs 6.487ms)

BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     4.050 ms (0.00% GC)
  median time:      4.479 ms (0.00% GC)
  mean time:        4.493 ms (0.00% GC)
  maximum time:     5.361 ms (0.00% GC)
  --------------
  samples:          223
  evals/sample:     1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Hence, by &lt;strong&gt;simply avoiding one extra power calculation we have now gained a
1.6x speed up.&lt;/strong&gt;&lt;/p&gt;

&lt;h3 id=&quot;final-fix-inbounds-simd-fastmath&quot;&gt;Final Fix: &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@inbounds&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@simd&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@fastmath&lt;/code&gt;&lt;/h3&gt;

&lt;p&gt;This is straight from the Julia official documentation:&lt;/p&gt;

&lt;p&gt;&lt;a href=&quot;https://docs.julialang.org/en/v1/devdocs/boundscheck/index.html&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@inbounds&lt;/code&gt;&lt;/a&gt;&lt;/p&gt;
&lt;blockquote&gt;
  &lt;p&gt;Like many modern programming languages, Julia uses bounds checking to ensure
program safety when accessing arrays. In tight inner loops or other performance
critical situations, you may wish to skip these bounds checks to improve runtime
performance. For instance, in order to emit vectorized (SIMD) instructions, your
loop body cannot contain branches, and thus cannot contain bounds checks.
Consequently, Julia includes an @inbounds(…) macro to tell the compiler to
skip such bounds checks within the given block. User-defined array types can use
the @boundscheck(…) macro to achieve context-sensitive code selection.&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;&lt;a href=&quot;https://docs.julialang.org/en/v1/manual/performance-tips/index.html
#man-performance-annotations-1&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@simd&lt;/code&gt;&lt;/a&gt;&lt;/p&gt;
&lt;blockquote&gt;
  &lt;p&gt;Write @simd in front of for loops to promise that the iterations are
independent and may be reordered. Note that in many cases, Julia can
automatically vectorize code without the @simd macro; it is only beneficial in
cases where such a transformation would otherwise be illegal, including cases
like allowing floating-point re-associativity and ignoring dependent memory
accesses (@simd ivdep). Again, be very careful when asserting @simd as
erroneously annotating a loop with dependent iterations may result in unexpected
results. In particular, note that setindex! on some AbstractArray subtypes is
inherently dependent upon iteration order. This feature is experimental and
could change or disappear in future versions of Julia.&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;&lt;a href=&quot;https://docs.julialang.org/en/v1/manual/performance-
tips/index.html#man-performance-annotations-1&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@fastmath&lt;/code&gt;&lt;/a&gt;&lt;/p&gt;
&lt;blockquote&gt;
  &lt;p&gt;Use @fastmath to allow floating point optimizations that are correct for real
numbers, but lead to differences for IEEE numbers. Be careful when doing this,
as this may change numerical results. This corresponds to the -ffast-math option
of clang.&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;I won’t dive into details, but what I have realized is that you can get a speed
up from the three macros listed above only when you are working with &lt;a href=&quot;https://docs.julialang.org/en/v1/devdocs/isbitsunionarrays/&quot;&gt;data
structures that pass the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;isbits&lt;/code&gt;
test&lt;/a&gt;. In practice,
that means that our structs can’t contain any arrays, which lead us to
redefining the struct as follows:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [30]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;k&quot;&gt;struct&lt;/span&gt;&lt;span class=&quot;nc&quot;&gt; ParticleNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt;

  &lt;span class=&quot;c&quot;&gt;# User inputs&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;X1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;X2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;X3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Empty initializer&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;Base&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Type&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ParticleNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}})&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;where&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ParticleNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Another initializer alias&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;ParticleNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;T&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;}(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;where&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;T&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ParticleNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;With that twist, now we implement &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@simd&lt;/code&gt; in the internal for loop, while both
&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@fastmath&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@inbounds&lt;/code&gt; wrap all floating point operations and output
allocation:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [50]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
    Implementation of @simd, @fastmath, and @inbounds
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;enumerate&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
        &lt;span class=&quot;nd&quot;&gt;@simd&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;

            &lt;span class=&quot;nd&quot;&gt;@fastmath&lt;/span&gt; &lt;span class=&quot;nd&quot;&gt;@inbounds&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;begin&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# K × Γp&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# U = ∑g_σ(x-xp) * K(x-xp) × Γp&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) += ∑p[(Δxj∂gσ∂r/(σr) − 3Δxjgσ/r^2) K(Δx)×Γp&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# Adds the Kronecker delta term&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt;

                &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
            &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;&lt;strong&gt;In [51]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;c&quot;&gt;# Create a 6x6x6 box of particles of our new ParticleNoArr struct&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;particlesNoArr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;generate_particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ParticleNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Float64&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;},&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;lambda&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Pre-allocate outputs in separate arrays&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Float64&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particlesNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;zeros&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;Float64&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particlesNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Run benchmark&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particlesNoArr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_dgdr_wnk&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;compare&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_reusesqrt&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;nd&quot;&gt;@benchmark&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;args&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;...&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;);&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;P2P_FINAL            is  1.14 times faster than        P2P_reusesqrt (3.552ms vs 4.050ms)

BenchmarkTools.Trial:
  memory estimate:  0 bytes
  allocs estimate:  0
  --------------
  minimum time:     3.552 ms (0.00% GC)
  median time:      4.090 ms (0.00% GC)
  mean time:        4.056 ms (0.00% GC)
  maximum time:     4.761 ms (0.00% GC)
  --------------
  samples:          247
  evals/sample:     1
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Comparing to C++, &lt;strong&gt;our Julia code is now as fast —and a little
faster—than C++:&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [52]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;printcomparison&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;s&quot;&gt;&quot;C++&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;true&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;P2P_FINAL            is  1.12 times faster than                  C++ (3.552ms vs 3.996ms)
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;However, our Julia code is using clang’s fastmath compilation mode. A more fair
comparison is done by also compiling C++ with the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;-ffast-math&lt;/code&gt; flag in addition
to &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;-O3&lt;/code&gt;, which I have coded, compiled, and wrapped in this
&lt;a href=&quot;https://github.com/JuliaInterop/CxxWrap.jl&quot;&gt;CxxWrap&lt;/a&gt;:&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;In [34]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;c&quot;&gt;# Run C++ benchmark with -ffast-math flag&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;cpptime&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;CxxVortexTestFASTMATH&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;benchmarkP2P_wrap&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ntests&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;kt&quot;&gt;Float32&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lambda&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;verbose&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

&lt;span class=&quot;c&quot;&gt;# Store benchmark result&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;benchtime&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;s&quot;&gt;&quot;C++ -ffast-math&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cpptime&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;Samples:	1000
min time:	1.40114 ms
ave time:	1.72617 ms
max time:	2.71737 ms
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;&lt;strong&gt;In [53]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;printcomparison&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;s&quot;&gt;&quot;C++ -ffast-math&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;false&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;C++ -ffast-math      is  2.54 times faster than            P2P_FINAL (1.401ms vs 3.552ms)
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;And once again, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;-ffast-math&lt;/code&gt; places C++ at a modest 2.5x over the optimized
Julia code; but be not discouraged, we are talking about a dynamic-like language
that is &lt;strong&gt;only 0.4x slower than C++&lt;/strong&gt;! =]&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;NOTE:&lt;/strong&gt; In a subsequent test I commented out the two power operations (&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;r =
sqrt(...)&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;gsgm, dgsgmdr = g_dgdr(r/Pj.sigma)&lt;/code&gt;) and the resulting wall-
clock time went down from 6.35ms to 0.496ms, meaning that at this stage the
overhead is on non-algebraic operations and that the function could further be
optimized only by finding a more efficient way of calculation powers in Julia.&lt;/p&gt;

&lt;h2 id=&quot;julia-as-fast-as-c&quot;&gt;Julia as Fast as C++&lt;/h2&gt;

&lt;p&gt;Finally, let’s take a second to compare where we started and where we ended. Our
initial Julia function was very compact and understandable, however it was more
than 50x slower than its C++ counterpart:&lt;/p&gt;

&lt;div class=&quot;language-julia highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;
&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
    Pythonic programming approach
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;
        &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;

            &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;norm&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# K × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cross&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;),&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# U = ∑g_σ(x-xp) * K(x-xp) × Γp&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt;

                &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
                &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;
                                      &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;
                                      &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;cross&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;([&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;],&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

            &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;&lt;strong&gt;In [48]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;printcomparison&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;s&quot;&gt;&quot;C++&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;false&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;C++                  is 58.48 times faster than         P2P_pythonic (3.996ms vs 233.679ms)
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;After all the optimization we end up with about twice the amount of lines, but
65x faster that the original version and only 2.5x slower than C++ with
&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;-ffastmath&lt;/code&gt;:&lt;/p&gt;

&lt;div class=&quot;language-julia highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&lt;span class=&quot;s&quot;&gt;&quot;&quot;&quot;
    Optimized Julia code
&quot;&quot;&quot;&lt;/span&gt;
&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt; P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;enumerate&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
        &lt;span class=&quot;nd&quot;&gt;@simd&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;particles&lt;/span&gt;

            &lt;span class=&quot;nd&quot;&gt;@fastmath&lt;/span&gt; &lt;span class=&quot;nd&quot;&gt;@inbounds&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;begin&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X1&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X2&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pi&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X3&lt;/span&gt;
                &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# g_σ and ∂gσ∂r&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_dgdr&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# K × Γp&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt; &lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# U = ∑g_σ(x-xp) * K(x-xp) × Γp&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp ]&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# ∂u∂xj(x) += ∑p[(Δxj∂gσ∂r/(σr) − 3Δxjgσ/r^2) K(Δx)×Γp&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgsgmdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX1&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX2&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;crss3&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX3&lt;/span&gt;

                    &lt;span class=&quot;c&quot;&gt;# Adds the Kronecker delta term&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;gsgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=1&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt;
                    &lt;span class=&quot;c&quot;&gt;# j=3&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma2&lt;/span&gt;
                    &lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;Pj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma1&lt;/span&gt;

                &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
            &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

        &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;

&lt;span class=&quot;k&quot;&gt;end&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;&lt;strong&gt;In [54]:&lt;/strong&gt;&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-julia&quot; data-lang=&quot;julia&quot;&gt;&lt;span class=&quot;n&quot;&gt;printcomparison&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P2P_pythonic&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;true&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;printcomparison&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;s&quot;&gt;&quot;C++&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;true&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;n&quot;&gt;printcomparison&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P2P_FINAL&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;s&quot;&gt;&quot;C++ -ffast-math&quot;&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;nb&quot;&gt;false&lt;/span&gt;&lt;span class=&quot;x&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;P2P_FINAL            is 65.79 times faster than         P2P_pythonic (3.552ms vs 233.679ms)
P2P_FINAL            is  1.12 times faster than                  C++ (3.552ms vs 3.996ms)
C++ -ffast-math      is  2.54 times faster than            P2P_FINAL (1.401ms vs 3.552ms)
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Oddly enough, through the optimization process we naturally morphed the code
into something that looks very similar to the C++ version:&lt;/p&gt;

&lt;div class=&quot;language-cpp highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&lt;span class=&quot;c1&quot;&gt;// Particle-to-particle interactions&lt;/span&gt;
&lt;span class=&quot;kt&quot;&gt;void&lt;/span&gt; &lt;span class=&quot;nf&quot;&gt;P2P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Particle&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;k&quot;&gt;const&lt;/span&gt; &lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;numParticles&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;

  &lt;span class=&quot;n&quot;&gt;real_t&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
  &lt;span class=&quot;n&quot;&gt;vec3&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;

  &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numParticles&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;++&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;
    &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numParticles&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;++&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;

      &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
      &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
      &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
      &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]);&lt;/span&gt;

      &lt;span class=&quot;k&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;

          &lt;span class=&quot;c1&quot;&gt;// Evaluate g_σ and ∂gσ∂r&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;pow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;1.0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;);&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;2.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;7.5&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ros&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;1.0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;);&lt;/span&gt;

          &lt;span class=&quot;c1&quot;&gt;// u(x) = ∑g_σ(x−xp) K(x−xp) × Γp&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;U&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt;

          &lt;span class=&quot;c1&quot;&gt;// ∂u∂xj(x) = ∑[ ∂gσ∂xj(x−xp) * K(x−xp)×Γp + gσ(x−xp) * ∂K∂xj(x−xp)×Γp]&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dgdr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sigma&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;mf&quot;&gt;3.0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;));&lt;/span&gt;
          &lt;span class=&quot;k&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kt&quot;&gt;int&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;;&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;++&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
            &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;

          &lt;span class=&quot;c1&quot;&gt;// Adds the Kronecker delta term&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;const4&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;g_sgm&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;/&lt;/span&gt; &lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;);&lt;/span&gt;
          &lt;span class=&quot;c1&quot;&gt;// k=1&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;c1&quot;&gt;// k=2&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;c1&quot;&gt;// k=3&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;-=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
          &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;J&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;2&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;mi&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;aux&lt;/span&gt; &lt;span class=&quot;o&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;].&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Gamma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mi&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;];&lt;/span&gt;
      &lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;

    &lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;
  &lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;

&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;In conclusion, if you are trying to get a piece of code fit for high-performance
computing, my recommendation is to &lt;strong&gt;forget about elegant and line-efficient
coding (“pythonic” some would say), and code in C++-esque style from the
beginning&lt;/strong&gt;.&lt;/p&gt;

&lt;h2 id=&quot;conclusions&quot;&gt;Conclusions&lt;/h2&gt;

&lt;ul&gt;
  &lt;li&gt;
    &lt;p&gt;Without foreknowledge of the types to be handled, Julia can’t optimize the
code during compilation. Multiple dispatch and JIT will compile a well-defined
version of every function based on the arguments that is given on the fly, but
this is not the case for structs properties. &lt;strong&gt;Always define your structs with
its &lt;a href=&quot;https://docs.julialang.org/en/v1/manual/performance-tips/index.html#Type-
declarations-1&quot;&gt;properties explicitely specified as concrete
types&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;
  &lt;/li&gt;
  &lt;li&gt;
    &lt;p&gt;&lt;a href=&quot;https://docs.julialang.org/en/v1/manual/performance-
tips/index.html#man-code-warntype-1&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@code_warntype&lt;/code&gt;&lt;/a&gt; is your best friend when trying to catch
abstract types in your code.&lt;/p&gt;
  &lt;/li&gt;
  &lt;li&gt;
    &lt;p&gt;&lt;strong&gt;Avoid memory allocation:&lt;/strong&gt; If your function is allocating an insane amount
of memory, that’s an indication that you are saving yourself some lines of code
while sacrificing computation efficiency (&lt;em&gt;e.g.,&lt;/em&gt; inline list-comprehension
operations or array algebra instead of unrolling the operations
into explicit code). I recommend using
&lt;a href=&quot;https://docs.julialang.org/en/v1/manual/performance-tips/index.html
#Measure-performance-with-[@time](@ref)-and-pay-attention-to-memory-
allocation-1&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@time&lt;/code&gt;&lt;/a&gt; or &lt;a href=&quot;https://github.com/JuliaCI/BenchmarkTools.jl/blo
b/master/doc/manual.md&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@benchmark&lt;/code&gt;&lt;/a&gt; to check memory allocation.&lt;/p&gt;
  &lt;/li&gt;
  &lt;li&gt;
    &lt;p&gt;&lt;strong&gt;Avoid storing intermediate calculations in internal arrays, just use Float
variables&lt;/strong&gt;. For some reason Julia spends a lot of time trying to allocate space
for internal arrays.&lt;/p&gt;
  &lt;/li&gt;
  &lt;li&gt;
    &lt;p&gt;Any &lt;strong&gt;non-integer power operations (&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;^&lt;/code&gt; or &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sqrt&lt;/code&gt;) are very expensive&lt;/strong&gt; in
Julia. If possible, reduce these operations by storing and reusing previous
calculations.&lt;/p&gt;
  &lt;/li&gt;
  &lt;li&gt;
    &lt;p&gt;At all cost, &lt;strong&gt;avoid using functions from the LinearAlgebra package&lt;/strong&gt;
(like &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;dot&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;cross&lt;/code&gt;, &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;norm&lt;/code&gt;) as they will require allocating memory for
internal arrays.&lt;/p&gt;
  &lt;/li&gt;
  &lt;li&gt;
    &lt;p&gt;For some reason,
&lt;a href=&quot;https://docs.julialang.org/en/v1/base/base/#Base.SimdLoop.@simd&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@simd&lt;/code&gt;&lt;/a&gt;,
&lt;a href=&quot;https://docs.julialang.org/en/v1/base/math/#Base.FastMath.@fastmath&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@fastmath&lt;/code&gt;&lt;/a&gt;,
and &lt;a href=&quot;https://docs.julialang.org/en/v1/base/base/#Base.@inbounds&quot;&gt;&lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;@inbounds&lt;/code&gt;&lt;/a&gt;
speed up your code only when working with &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;isbits&lt;/code&gt; types.&lt;/p&gt;
  &lt;/li&gt;
  &lt;li&gt;
    &lt;p&gt;&lt;strong&gt;Forget about elegant and line-efficient coding (“pythonic” some would say),
and code in C++-esque style&lt;/strong&gt; in parts of the code where performance is
critical.&lt;/p&gt;
  &lt;/li&gt;
&lt;/ul&gt;

&lt;h3 id=&quot;notes&quot;&gt;Notes&lt;/h3&gt;

&lt;ul&gt;
  &lt;li&gt;This benchmark was done in a Dell Latitude 5580 laptop (Intel® Core™ i7-7820HQ
CPU @ 2.90GHz × 8 ) in only one process.&lt;/li&gt;
  &lt;li&gt;Julia v1.0.3 was used.&lt;/li&gt;
  &lt;li&gt;The official &lt;a href=&quot;https://docs.julialang.org/en/v1/manual
/performance-tips/index.html&quot;&gt;Julia documentation&lt;/a&gt; also provide very useful tips for performance.&lt;/li&gt;
  &lt;li&gt;The code used in this post is available here:
&lt;a href=&quot;https://github.com/EdoAlvarezR/post-juliavscpp&quot;&gt;https://github.com/EdoAlvarezR/post-juliavscpp&lt;/a&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;h3 id=&quot;references&quot;&gt;References&lt;/h3&gt;

&lt;ol&gt;
  &lt;li&gt;Winckelmans, G. S., &amp;amp; Leonard, A. (1993). Contributions to Vortex Particle
Methods for the Computation of Three-Dimensional Incompressible Unsteady Flows.
Journal of Computational Physics, 109(2), 247–273.
https://doi.org/10.1006/jcph.1993.1216&lt;/li&gt;
  &lt;li&gt;Alvarez, E. J., &amp;amp; Ning, A. (2018). Development of a Vortex Particle Code for
the Modeling of Wake Interaction in Distributed Propulsion. In 2018 Applied
Aerodynamics Conference (pp. 1–22). American Institute of Aeronautics and
Astronautics. https://doi.org/10.2514/6.2018-3646 . &lt;a href=&quot;https://scholarsarchive.byu.edu/facpub/2116/&quot;&gt;PDF
Available&lt;/a&gt;&lt;/li&gt;
&lt;/ol&gt;
</content>
 </entry>
 
 <entry>
   <title>Coupled Wind Turbine Design and Layout Optimization</title>
   <link href="http://flow.byu.edu/posts/coupled-turbine-farm"/>
   <updated>2019-02-13T00:00:00-07:00</updated>
   <id>http://flow.byu.edu/posts/coupled-turbine.farm</id>
   <content type="html">&lt;p&gt;&lt;img src=&quot;https://byu.box.com/shared/static/prvuhcsgo0qdnwm4zbqugzxhl0wcxols.gif&quot; alt=&quot;animation&quot; /&gt;&lt;/p&gt;

&lt;p&gt;As wind turbines extract momentum from the air they leave behind wakes of slower moving air. Downstream turbines interact with these wakes decreasing their power production.  Wake interference is unavoidable (a wind farm has many wind turbines, and wind frequently changes direction), but we would like to minimize it.  Over the past several years, our lab and other researchers, have conducted a variety of studies to try to minimize the power loss from wake effects.  One approach is to use layout optimization (repositioning turbines to reduce wake interference).  Another approach is with turbine control optimization (dynamically reorienting the way the turbines face to leave more momentum behind in the wakes).&lt;/p&gt;

&lt;p&gt;More recently, we have been collaborating with Katherine Dykes of NREL to explore a third approach: changing the wind farm layout and the wind turbine design simultaneously.  This differs from the current sequential design approach.  We started by optimizing turbine heights while also optimizing positions (Stanley, Ning, Dykes, doi:10.1002/we.2310).  Effectively, this added a third dimension where turbines can avoid wakes by using vertical separation.  Changing the turbine heights has multiple consequences that must be considered.  Turbines that shrink in size operate closer to the ground where wind speeds are reduced and thus produce less power.  Taller turbines can take advantage of higher wind speeds, but require more mass both because of the increased height but also because taller towers require extra stiffness to resist bending loads.  A system-level study combining multiple disciplines was necessary to understand whether the increased power could significantly offset the increased capital costs for the towers.&lt;/p&gt;

&lt;p&gt;Following that initial study, and with continued NREL support, we added more degrees of freedom to the wind turbine design and allowed the turbines to move.  In the recent paper below, we found that the cost of energy in a wind farm can be significantly decreased by optimizing the wind turbine design at the same time as the wind farm layout. Furthermore, we found that if we didn’t force all of the turbines to be exactly the same we could reduce the cost to produce energy even further. In many cases, it is better for some turbines to be tall with large rotors, and for some turbines to be short with smaller rotors. For wind farms with closely spaced turbines, a wind farm with different turbine designs can have a &lt;strong&gt;cost of energy that is 7-10% lower&lt;/strong&gt; than the same wind farm optimized with all the same turbines.&lt;/p&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;Stanley2018a&quot;&gt;Stanley, A. P. J., and Ning, A., “Coupled Wind Turbine Design and Layout Optimization with Non-Homogeneous Wind Turbines,” &lt;i&gt;Wind Energy Science&lt;/i&gt;, Vol. 4, No. 1, pp. 99–114, Jan. 2019. doi:10.5194/wes-4-99-2019&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;Stanley2018a-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.5194/wes-4-99-2019&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://www.wind-energ-sci.net/4/99/2019/wes-4-99-2019.pdf&quot;&gt;[PDF]&lt;/a&gt; &lt;a href=&quot;https://github.com/byuflowlab/stanley2018-turbine-design&quot;&gt;[Code]&lt;/a&gt; 

&lt;pre id=&quot;Stanley2018a-bibtex&quot; class=&quot;bibtex&quot;&gt;@article{Stanley2018a,
  author = {Stanley, Andrew P.J. and Ning, Andrew},
  doi = {10.5194/wes-4-99-2019},
  journal = {Wind Energy Science},
  month = jan,
  number = {1},
  pages = {99-114},
  title = {Coupled Wind Turbine Design and Layout Optimization with Non-Homogeneous Wind Turbines},
  volume = {4},
  year = {2019}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;

&lt;p&gt;The animation above shows an accelerated example of one optimization from our paper. In this example, you can see reductions in the cost of energy as the turbine locations change (top left) and the turbine designs change (top middle). While not apparent from the visual, in addition to tower height and rotor diameter, we also change the tower diameter and thickness distributions, turbine rating, and blade chord and twist distributions. The bottom left frame shows the wind direction probabilities that were used for this example, and the bottom middle frame shows the power that the wind farm produces for each incoming wind direction. Near the end of the optimization, there is increased power production from all of the wind directions.&lt;/p&gt;
</content>
 </entry>
 
 <entry>
   <title>Trajectory Optimization for High Altitude Long Endurance Aircraft</title>
   <link href="http://flow.byu.edu/posts/hale-trajectory"/>
   <updated>2018-12-18T00:00:00-07:00</updated>
   <id>http://flow.byu.edu/posts/traj-martin</id>
   <content type="html">&lt;p&gt;&lt;img src=&quot;/posts/solar.png&quot; alt=&quot;NASA Helios&quot; /&gt;&lt;/p&gt;

&lt;p&gt;Our lab has been studying solar powered high altitude long endurance aircraft (HALE) for the past couple of years.  HALE aircraft are intended to fly persistently, or at least for very long periods of time, with various applications including telecommunications (replacing more expensive satellites), surveillance, and weather monitoring.  Persistent flight is challenging and requires high altitudes where wind speeds are reduced, large wingspans and lightweight structures (both of which reduce induced drag, the dominant source of drag at low speeds), and efficient solar capture.  A well known example from a decade ago was the &lt;a href=&quot;https://www.nasa.gov/centers/dryden/history/pastprojects/Helios/index.html&quot;&gt;NASA Helios&lt;/a&gt;.  More recent commercial examples include &lt;a href=&quot;https://www.airbus.com/defence/uav/zephyr.html&quot;&gt;Airbus’ Zephyr&lt;/a&gt;, &lt;a href=&quot;https://www.facebook.com/notes/mark-zuckerberg/the-technology-behind-aquila/10153916136506634/&quot;&gt;Facebook’s Aquila&lt;/a&gt;, and &lt;a href=&quot;https://www.aurora.aero/odysseus-high-altitude-pseudo-satellite-haps/&quot;&gt;Aurora’s Odysseus&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;To remain aerodynamically efficient, aircraft have a minimal amount of forward projected area, in other words most of the area available for solar panels faces upward.  That works well when the sun is straight overhead, but during the winter solstice the sun is low in the sky (and is never straight overhead) reducing the amount of incident solar energy.  Furthermore, the days are shorter in the winter, which further reduces the amount of solar energy available.  Thus, winter solstice ends up being a limiting case for the design and operation of these aircraft.&lt;/p&gt;

&lt;p&gt;In a collaboration between our lab and the &lt;a href=&quot;https://apm.byu.edu/prism/index.php&quot;&gt;PRISM Group&lt;/a&gt; we began exploring how much additional energy could be captured, for a given aircraft, just by changing how the aircraft is flown. To explore this question we solved for optimal trajectories, using nonlinear model predictive control, with models for aircraft dynamics, aerodynamics, solar capture, atmosphere, and propulsion.  Trajectories were modeled for a full day with an eight-second time step resolution.  This was a much finer resolution than existing studies, which ended up being important to uncover some interesting behavior.  One major constraint in the problem was that the aircraft needed to stay within a fixed radius, as would be necessary for a telecommunications application.  We hypothesized that the aircraft would try to bank to orient the panels more towards the sun, at least for part of the trajectory, but this did not happen.&lt;/p&gt;

&lt;p&gt;The optimal trajectories during the day looked like the bean shape shown below, with the major axis oriented towards the sun (the yellow line in the middle of the figure points towards the sun).  The outer circle is the radius constraint that the airplane must stay in.  When flying away from the sun, points 4 to 7 in the figure, the aircraft increased its angle of attack, which results in slower flight speeds and increased incidence towards the sun.  In other words, it increased its energy capture and increased the time during increased energy capture.  On the return trip the angle of attack was lowered to quickly return back to the start.  As the sun moved through the day, the bean shape would precess so as to remain oriented towards the sun.  By flying these optimal paths the total energy capture at the end of the day was found to increase by about 8%, over conventional optimal circular paths, without any change in the aircraft (just changes in the way it is flown).&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;fabiform.png&quot; alt=&quot;optimal trajectory&quot; width=&quot;400px&quot; /&gt;&lt;/p&gt;

&lt;p&gt;Other important details were explored including the effect of storing potential energy through altitude changes, orbit changes throughout the day, and seasonal changes.  The journal paper was recently published and the postprint can be accessed here:&lt;/p&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;Martin2018&quot;&gt;Martin, R. A., Gates, N. S., Ning, A., and Hedengren, J. D., “Dynamic Optimization of High-Altitude Solar Aircraft Trajectories Under Station-Keeping Constraints,” &lt;i&gt;Journal of Guidance, Control, and Dynamics&lt;/i&gt;, Vol. 42, No. 3, pp. 538–552, Mar. 2019. doi:10.2514/1.G003737&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;Martin2018-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.2514/1.G003737&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=3890&amp;amp;context=facpub&quot;&gt;[PDF]&lt;/a&gt; &lt;a href=&quot;https://github.com/BYU-PRISM/hale-trajectory&quot;&gt;[Code]&lt;/a&gt; 

&lt;pre id=&quot;Martin2018-bibtex&quot; class=&quot;bibtex&quot;&gt;@article{Martin2018,
  author = {Martin, R. Abraham and Gates, Nathaniel S. and Ning, Andrew and Hedengren, John D.},
  doi = {10.2514/1.G003737},
  journal = {Journal of Guidance, Control, and Dynamics},
  month = mar,
  number = {3},
  pages = {538-552},
  title = {Dynamic Optimization of High-Altitude Solar Aircraft Trajectories Under Station-Keeping Constraints},
  volume = {42},
  year = {2019}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;

&lt;p&gt;In related work, we did a study on optimal HALE vehicle design (which is still ongoing):&lt;/p&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;McDonnell2018&quot;&gt;McDonnell, T., Mehr, J., and Ning, A., “Multidisciplinary Design Optimization of Flexible Solar-Regenerative High-Altitude Long-Endurance Aircraft,” &lt;i&gt;AIAA Structures, Structural Dynamics, and Materials Conference&lt;/i&gt;, Kissimmee, FL, Jan. 2018. doi:10.2514/6.2018-0107&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;McDonnell2018-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.2514/6.2018-0107&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=3086&amp;amp;context=facpub&quot;&gt;[PDF]&lt;/a&gt;  

&lt;pre id=&quot;McDonnell2018-bibtex&quot; class=&quot;bibtex&quot;&gt;@inproceedings{McDonnell2018,
  address = {Kissimmee, FL},
  author = {McDonnell, Taylor and Mehr, Judd and Ning, Andrew},
  booktitle = {{AIAA} Structures, Structural Dynamics, and Materials Conference},
  doi = {10.2514/6.2018-0107},
  month = jan,
  title = {Multidisciplinary Design Optimization of Flexible Solar-Regenerative High-Altitude Long-Endurance Aircraft},
  year = {2018}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;

&lt;p&gt;More recently we combined our efforts on trajectory optimization with some aspects of aircraft design.  As the trajectory changes, the optimal operating conditions for the propulsion system change as well. By coupling these design decisions we can increase end of day energy capture by over 10% as compared to optimizing these systems separately.  These latter findings are still preliminary as this was a conference paper, but will be discussed in more detail once they become journal ready.&lt;/p&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;Gates2019&quot;&gt;Gates, N. S., Moore, K. R., Ning, A., and Hedengren, J. D., “Combined Trajectory, Propulsion, and Battery Mass Optimization for Solar-Regenerative High-Altitude Long Endurance Unmanned Aircraft,” &lt;i&gt;AIAA Scitech 2019 Forum&lt;/i&gt;, San Diego, CA, Jan. 2019. doi:10.2514/6.2019-1221&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;Gates2019-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.2514/6.2019-1221&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=3932&amp;amp;context=facpub&quot;&gt;[PDF]&lt;/a&gt;  

&lt;pre id=&quot;Gates2019-bibtex&quot; class=&quot;bibtex&quot;&gt;@inproceedings{Gates2019,
  address = {San Diego, CA},
  author = {Gates, Nathaniel S. and Moore, Kevin R. and Ning, Andrew and Hedengren, John D.},
  booktitle = {AIAA Scitech 2019 Forum},
  doi = {10.2514/6.2019-1221},
  month = jan,
  title = {Combined Trajectory, Propulsion, and Battery Mass Optimization for Solar-Regenerative High-Altitude Long Endurance Unmanned Aircraft},
  year = {2019}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;

&lt;p&gt;P.S. For fun, here is a &lt;a href=&quot;http://prismweb.groups.et.byu.net/360/&quot;&gt;3D visualization&lt;/a&gt; Abe put together based on an Aquila-like configuration at winter solstice.&lt;/p&gt;
</content>
 </entry>
 
 <entry>
   <title>Multirotor - Vortex Particle Method</title>
   <link href="http://flow.byu.edu/posts/multirotor-vpm"/>
   <updated>2018-07-12T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/multirotor-vpm</id>
   <content type="html">&lt;iframe width=&quot;560&quot; height=&quot;315&quot; src=&quot;https://www.youtube.com/embed/CP4jI7lLags&quot; frameborder=&quot;0&quot; allow=&quot;accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture&quot; allowfullscreen=&quot;&quot;&gt;&lt;/iframe&gt;

&lt;p&gt;Counter-rotating APC 10x7 propellers ramping up from 0 to 9400 RPM with a constant freestream (advance ratio of 0.35 at 9400 RPM) modeled with a viscous vortex particle method. From the following paper:&lt;/p&gt;

&lt;ol class=&quot;bibliography nonum&quot; reversed=&quot;&quot;&gt;&lt;li&gt;&lt;span id=&quot;Alvarez2018&quot;&gt;Alvarez, E. J., and Ning, A., “Development of a Vortex Particle Code for the Modeling of Wake Interaction in Distributed Propulsion,” &lt;i&gt;AIAA Applied Aerodynamics Conference&lt;/i&gt;, Atlanta, GA, Jun. 2018. doi:10.2514/6.2018-3646&lt;/span&gt;
&lt;a href=&quot;#nowhere&quot; onclick=&quot;toggle_visibility(&apos;Alvarez2018-bibtex&apos;);&quot;&gt;[BibTeX]&lt;/a&gt; &lt;a href=&quot;http://dx.doi.org/10.2514/6.2018-3646&quot;&gt;[DOI]&lt;/a&gt; &lt;a href=&quot;https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=3148&amp;amp;context=facpub&quot;&gt;[PDF]&lt;/a&gt;  

&lt;pre id=&quot;Alvarez2018-bibtex&quot; class=&quot;bibtex&quot;&gt;@inproceedings{Alvarez2018,
  address = {Atlanta, GA},
  author = {Alvarez, Eduardo J. and Ning, Andrew},
  booktitle = {{AIAA} Applied Aerodynamics Conference},
  doi = {10.2514/6.2018-3646},
  month = jun,
  title = {Development of a Vortex Particle Code for the Modeling of Wake Interaction in Distributed Propulsion},
  year = {2018}
}
&lt;/pre&gt;
&lt;/li&gt;&lt;/ol&gt;
</content>
 </entry>
 
 <entry>
   <title>Julia for the Win</title>
   <link href="http://flow.byu.edu/posts/julia-ftw"/>
   <updated>2016-05-16T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/julia-ftw</id>
   <content type="html">&lt;p&gt;In updating a paper to prepare for journal submission I needed to revisit the accompanying Julia code.  I chose Julia at the time because this was a mostly self-contained project and I wanted to give Julia a trial run on something of moderate complexity (see &lt;a href=&quot;http://flow.byu.edu/posts/julia-first-imp&quot;&gt;first impressions&lt;/a&gt;).  I cleaned up the code, added some capabilities, and really tried to improve performance.  I read all the primary documentation on Julia, including the sections on performance, updated to 0.4.x, explicitly declared all types, and profiled quite a bit.  This made some difference, but my code was still about an order of magnitude slower than a Python/Fortran version.&lt;/p&gt;

&lt;p&gt;The original version was mostly Fortran so I wasn’t necessarily seeking a speed improvement.  For the benefits of working in one language I would be ok with a 20-30% slowdown.  However, an order of magnitude slow down was a lot and it really bogged things down when creating figures and data for the paper.  I had about given up on performance, but two separate encounters with Julia-using colleagues made me revisit the topic once again.  I sent my code to a colleague who had been using Julia for some time.  His PhD student pointed out some suggestions, which I tried, but the impact was negligible.&lt;/p&gt;

&lt;p&gt;Profiling suggested that a lot of time was spent in conversion between PyCall and scipy.  I tried using the lower-level calls in PyCall to be more explicit in the types, but it didn’t help.  I was using PyCall because I needed an N-dimensional root solver and nothing was available in native Julia.  Roots.jl was available, but it only solves 1-dimensional root finding problems.  PyCall allowed me to call Python’s scipy.optimize.root, which was quite convenient.&lt;/p&gt;

&lt;p&gt;When I first wrote this code a year or so ago I directly wrapped the Fortran code hybrd from minpack (using Julia 0.2.x and 0.3.x), which is what is used in Scipy’s root finder.  I later discovered PyCall and that was way more convenient with no real change in computation time.  However, I noted that a lot of improvements were made to ccall and cfunction in 0.4.x so I thought I’d try the direct route again.  I reverted to my direct wrapper of hybrd (with a few minor updates to comply with 0.4.x), to eliminate the Python middleman.  Performance problem solved!   After that change I found that the Julia code was actually 3X faster than the Python/Fortran version!&lt;/p&gt;

&lt;p&gt;The other problem I had was that plotting and just running Julia in general was painfully slow.  I was using a Python-like workflow where I would just run the full script each time (using Sublime Text or the terminal).  I knew that Julia had a REPL, but I couldn’t stand working in the browser as opposed to a text editor.  Sometimes with plotting I would use the REPL, but it was a real pain switching back and forth.  My colleague informed me about &lt;a href=&quot;https://github.com/JunoLab/atom-ink&quot;&gt;Ink&lt;/a&gt; and the &lt;a href=&quot;https://github.com/JunoLab/atom-julia-client&quot;&gt;Julia Client in Atom&lt;/a&gt; and it changed everything.  With this tool I could avoid the recompilation of packages and code that before was occurring on every run.&lt;/p&gt;

&lt;p&gt;I’m now very happy with using Julia and for one of our new major research projects we are going all in.  Everything isn’t perfect.  Some libraries are nonexistent (although wrapping Fortran/C libraries is pretty easy), and the Atom Julia Client is still a bit rough around the edges and could use a debugger.  But overall, working within one language that is both performant and is very easy to work with is a big win for us.&lt;/p&gt;

&lt;p&gt;&lt;span style=&quot;font-size: 80%;&quot;&gt;
P.S. If interested in the mentioned code, I have open-sourced it and it is available on &lt;a href=&quot;https://github.com/byuflowlab/vawt-ac&quot;&gt;GitHub&lt;/a&gt;.  It computes aerodynamic loading for multiple vertical axis wind turbines using an extended version of actuator cylinder theory.  A preprint of the accompanying theory is available on &lt;a href=&quot;/publications&quot;&gt;this website&lt;/a&gt; under “Actuator Cylinder Theory for Multiple Vertical Axis Wind Turbines”.
&lt;/span&gt;&lt;/p&gt;
</content>
 </entry>
 
 <entry>
   <title>Julia First Impressions</title>
   <link href="http://flow.byu.edu/posts/julia-first-imp"/>
   <updated>2015-08-26T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/julia-first-imp</id>
   <content type="html">&lt;p&gt;I’m primarily a Python user, and my primary use is scientific computing.  Over the last few years, I’ve followed Julia’s development and looked through documentation and various benchmarks multiple times.  I was intrigued by the potential.  However, it’s one thing to read about a programming language, and another to use it for yourself.  I asked a couple of my undergraduate students to use it for an exploratory project on aircraft design.  These students had some programming experience, and seemed to be able to complete the tasks just fine.  However, those problems were relatively simple and I needed to take Julia for a test drive myself on a larger problem to evaluate whether or not it was something worth switching to, for at least some of our lab projects.&lt;/p&gt;

&lt;p&gt;I had been working on a new method in vertical axis wind turbine simulation, and thought developing the code for this paper would give me a good chance to try Julia.  I’ll describe my impressions in working with Julia on a real project. This was not a large code, but not small either.  It was around 500 lines of code (not including some of the extra modules I needed to develop that are discussed later), and included reading data from a file, integration, forming matrices, multidimensional root finding, and multiple loops,  Once the journal paper is accepted somewhere, I’ll open source the code. This was done over a month ago.  I updated my version of Julia once or twice during development.  The last version I used was 0.3.10.&lt;/p&gt;

&lt;h1 id=&quot;tldr&quot;&gt;TL;DR&lt;/h1&gt;

&lt;p&gt;Julia was easy to use, and I liked the syntax a lot.  I didn’t think it was quite ready for our usage.  I needed to write/wrap many libraries that are just available in Python or Matlab.  Using the plotting modules was a real pain.  Performance was not particularly impressive, but I spent no effort in optimizing for speed.  Despite the limitations, the future of Julia looks very promising.&lt;/p&gt;

&lt;h1 id=&quot;the-good&quot;&gt;The Good&lt;/h1&gt;

&lt;p&gt;Julia’s syntax is my favorite of any programming language I’ve used.  It’s concise, simple, well-thought-out, and an improvement over both Matlab and Python in my opinion (although there is not a whole lot to complain about in those languages either).  Everything was pretty easy, even calling external Fortran code and reading files.&lt;/p&gt;

&lt;p&gt;Syntax was probably closest to Matlab.  One thing that drives me crazy with Matlab is the one function per file requirement (except for nested functions), which of course isn’t an issue with pretty much any other language.&lt;/p&gt;

&lt;p&gt;Julia uses 1-based indexing.  Although it’s a source of debate, and I’ve read some very good arguments for 0-based, I still prefer 1-based.  Python, C, and Java use 0-based and Matlab and Fortran use 1-based (although you could really use any base in Fortran if you wanted to).  I recognize that in many disciplines, algorithms are 0-based, but not so in my fields (generally).  I find 1-based much more intuitive.  I also like that Julia uses the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;end&lt;/code&gt; keyword for the end of the array, I never cared much for the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;-1&lt;/code&gt; syntax in Python.  Array slicing is similar to Matlab’s, both of which I find superior to that of Python’s (NumPy).&lt;/p&gt;

&lt;p&gt;I thought that I liked Python’s indentation-based grouping, but after using Julia, and going back and forth, I find having an explicit end statement makes code easier to read.&lt;/p&gt;

&lt;p&gt;All of these points are relatively minor though.  What is different and exciting, although I didn’t explore it much yet, is the ability to explicitly define types (or not) and for the compiler to take advantage of types.  The ability to optimize performance within a single language is very appealing.  For the purposes of this paper, optimizing performance wasn’t necessary so I spent no time on it.&lt;/p&gt;

&lt;h1 id=&quot;the-not-so-good&quot;&gt;The Not So Good&lt;/h1&gt;

&lt;p&gt;Development time was hindered a fair amount just because a lot of the routines I’m used having in numpy/scipy or Matlab, weren’t around in Julia.  The main detour for this project, was that there was no n-dimensional root solver.  I found &lt;a href=&quot;https://github.com/JuliaLang/Roots.jl&quot;&gt;Roots&lt;/a&gt;, but unfortunately the only available implementations were for 1D problems.  Fortunately, a good solver exists in Fortran.  I grabbed the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;hybrd&lt;/code&gt; routines from minpack (which is also the default method that scipy.optimize.root uses).&lt;/p&gt;

&lt;p&gt;The &lt;a href=&quot;http://julia.readthedocs.org/en/latest/manual/calling-c-and-fortran-code/&quot;&gt;documentation&lt;/a&gt; for calling C and Fortran code wasn’t that clear, in my opinion, but it was adequate enough that with a little trial and error I was able to make things work.  After I had done it once, I thought that calling Fortran or C from Julia was even easier than doing so in Python (which is already pretty easy).&lt;/p&gt;

&lt;p&gt;While wrapping this external code wasn’t terribly difficult, it did take some time to get the external code and build a fortran shared library, create a proper wrapper to hybrd, wrap a callback function in a generic way, and do a little testing.  In the end it took about 100 lines of code and a couple hours, instead of a one-liner using a built-in function in Matlab or Python.&lt;/p&gt;

&lt;p&gt;I developed a couple of other small functions as well that I normally wouldn’t need to, but weren’t available in any core Julia package (or at least I couldn’t find them).  The added functions were a couple of linear interpolation methods, and a couple trapezoidal integration methods (I did use the built-in quadgk in another part of the code, but also needed a routine that didn’t require a functional form for integration).  These were simple tasks, but did distract from the main development.&lt;/p&gt;

&lt;p&gt;Performance was not impressive.  I had developed a previous version of the code in Python, with the computationally intensive parts written in Fortran and  called as a shared library using f2py.  The Julia code and the Python/Fortran code had different capabilities, but I setup a problem that utilized common capabilities so that the algorithms used in the test were effectively identical.  I did not do any profiling, or make any effort to optimize for speed in any of the languages, but I didn’t try to make things slow either.  To run one simulation (a computation of turbine power) took 0.25 seconds in Python/Fortran and 2.5 seconds in Julia—an order of magnitude difference. I’m not sure why my implementation in Julia was so slow.  I’m not really counting this as a negative against Julia yet because I haven’t explored performance improvements at all.  Once I open-source the code, and module caching in Julia is available, I’ll be interested in exploring this in more detail (To be clear, I did not leave in any modules that I didn’t need, like plotting, in the timing analysis).&lt;/p&gt;

&lt;h1 id=&quot;the-bad&quot;&gt;The Bad&lt;/h1&gt;

&lt;p&gt;Plotting data was horrendous.  Actually, it wasn’t the plotting that was bad.  I mainly used PyPlot, which of course I was right at home with since it’s just an interface to Python’s matplotlib.pyplot.  The real problem was how Julia imports packages.  Apparently, Julia compiles each package every time the code is executed so just putting in &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;using PyPlot&lt;/code&gt; would add 10 seconds or more to the run time.  This was ridiculously painful when I needed to iterate multiple times when trying to debug something.  I didn’t even bother trying to create figures from Julia.  I’d dump the data and then tweak the plots to my liking in Python.  However, for debugging that process didn’t work because the data kept changing so I was forced to do it in Julia.  I found myself avoiding plots and just printing results to the screen if possible, just because it was so painfully slow.&lt;/p&gt;

&lt;p&gt;This is not much of a problem if you use the Julia REPL.  For example, you can use Juno or JuliaBox and then you can execute portions of the code and not have to reload modules.  However, I can develop much faster in my own text editor of choice (was Sublime Text, recently switched to Atom).  There are a whole host of keyboard shortcuts, snippets, and the like that I am used to using.  For a while, I started inputting a bunch of these in Juno.  That mostly worked, but I found Juno a bit buggy at times with plotting and it was cumbersome to use a different editor just for Julia.&lt;/p&gt;

&lt;p&gt;It looks like this won’t be a problem for long.  Following some of the issues on GitHub, you can see that a lot of discussion and work has gone on to allow static compilation and module caching, and it looks like it will be introduced in 0.4.&lt;/p&gt;

&lt;h1 id=&quot;summary&quot;&gt;Summary&lt;/h1&gt;

&lt;p&gt;Julia is a pleasure to use.  The promise of developing code with speed comparable to C or Fortran, but with the ease of use of Matlab or Python, all within &lt;em&gt;one language&lt;/em&gt; is incredibly appealing.  In my experience, it’s not there yet.  Lots of important scientific libraries are not yet directly available, and module importing is horrendously slow.  I look forward to continued developments, and anticipate that I probably will move most of our work to Julia in the future, but that day is not today.&lt;/p&gt;
</content>
 </entry>
 
 <entry>
   <title>Scientific Programming Languages</title>
   <link href="http://flow.byu.edu/posts/sci-prog-lang"/>
   <updated>2015-08-11T00:00:00-06:00</updated>
   <id>http://flow.byu.edu/posts/sci-prog-lang</id>
   <content type="html">&lt;p&gt;I’ve used a number of scientific programming languages over the past 16 years: C++, C, Matlab, Java, Fortran, Python, and Julia, and I wouldn’t name any one as the “best” (I’ve also used Objective-C, JavaScript, and PHP quite a bit, but not for scientific computing).  Usually I try to pick the right tool for the job, not necessarily just the tool I happen to know best (as they say: if all you have is a hammer, everything looks like a nail).  However, my usage has evolved over the years from Matlab-centric, to Python-centric, and I’m contemplating a move to Julia-centric.  Before explaining why, let’s discuss some of the reasons why I might choose one language over the others.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Edit (May 2019)&lt;/strong&gt;: We’ve gone all in with Julia shortly after this post was written (see bottom of post)&lt;/p&gt;

&lt;h1 id=&quot;matlab&quot;&gt;Matlab&lt;/h1&gt;

&lt;p&gt;Matlab is widely used in university settings.  For students it is very affordable, and it is very easy to use.  Matlab is oriented towards scientific computing and it comes pre-packaged with a built-in IDE, debugger, and a large collection of built-in methods and toolboxes.  I did much of my graduate work and PhD Dissertation using Matlab. Most students like it a lot.&lt;/p&gt;

&lt;p&gt;It’s great as a student, particularly an undergraduate student, but as you move to larger problems and/or move out of a university setting its weaknesses become more apparent.  Matlab is quite expensive outside of universities, and it runs very slowly.  The first concern could be addressed with Octave.  Octave is an open-source software designed to mimic Matlab, but it runs even slower and is far less capable in terms of available packages (it’s been years since I’ve used Octave so maybe the gap isn’t so large anymore, but Matlab hasn’t been idle either and has made quite a few performance and capability improvements including a JIT).&lt;/p&gt;

&lt;p&gt;Because of the speed and parallelization issues, a typical workflow for me was  to prototype a code in Matlab, and then if needed rewrite the entire code in either C, C++, or Fortran.  Obviously, rewriting is a pain, but if needed the speedup was usually worth it (usually 1-3 orders of magnitude).  Mex files can be used here, but I found them more painful than helpful with large multi-language projects.&lt;/p&gt;

&lt;p&gt;I haven’t used Matlab for research work in several years (with one exception noted below).  My graduate students rarely use it either.  I do, however, often use it in undergraduate classes because it is widely available and easy for students to use.  I also have most of my undergraduate research assistants use Matlab.  It’s perfectly suited for their problems where high performance is not necessary, and familiarity and ease of use is much more important.  That’s not necessarily a deliberate choice, I’d be happy to use Python with them as well, but Matlab is what they already know from other classes.  For the level of time they have available and the complexity of their problems, it’s just not worth trying to teach them something else.&lt;/p&gt;

&lt;p&gt;There is one exception for my usage, which is that I sometimes still use &lt;em&gt;fmincon&lt;/em&gt; from the Optimization Toolbox (which usually involves some sort of terrible hack to make the wrapping work).  I’ve used a lot of optimization packages for constrained nonconvex problems, and fmincon is still one of the most robust on the types of problems I solve.  These days I primarily use SNOPT through Python, but still find &lt;em&gt;fmincon&lt;/em&gt; useful from time to time.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Update (10/1/2015)&lt;/strong&gt;: With Mathworks relatively new &lt;a href=&quot;http://www.mathworks.com/help/matlab/matlab-engine-for-python.html&quot;&gt;Matlab Engine for Python&lt;/a&gt;, connected to fmincon from Python was relatively easy.  I posted an example &lt;a href=&quot;https://bitbucket.org/mdolab/pyoptsparse/pull-requests/13/an-example-bridge-to-fmincon/diff&quot;&gt;here&lt;/a&gt;, to hopefully be integrated with pyopt-sparse later.  Having access to fmincon is great, and makes me once again interested in keeping a Matlab license around.&lt;/p&gt;

&lt;h1 id=&quot;c-c-and-fortran&quot;&gt;C, C++, and Fortran&lt;/h1&gt;

&lt;p&gt;If the end result is a re-write in a compiled language, why not just start there to begin with?  Good question.  I have done that on occasion. The problem is that while these languages have great run times, development time is usually much slower (even when accounting for the re-write), and re-writing may not be necessary.  Development time is usually a much bigger bottleneck as compared to run time, at least for my use cases.&lt;/p&gt;

&lt;p&gt;In languages like Matlab, debugging and inspecting variables, plotting, making small changes and retesting, and using existing functions is just much faster.  All these things can be done in C, C++, and Fortran, but it just takes more work and time to repeatedly compile, integrate existing libraries or functions yourself, debug and plot results, etc.  It is always more important for your algorithms to be correct than to be fast.  Using a language like Matlab allows for rapid development, with more testing and inspecting for a give time allotment.  Once you are confident in a code’s correctness, then you can start thinking about speeding it up, &lt;em&gt;if necessary&lt;/em&gt;, usually by rewriting in one of these compiled languages.  The key point is that rewriting often isn’t necessary.  For example, if my vortex lattice code was running in a tenth of a second in Matlab, that was already good enough for the number of cases I needed to run.  Code can always be pushed off to other computers or clusters to run overnight and on the weekends easily.&lt;/p&gt;

&lt;p&gt;Of all of the languages I am discussing in the post I would say C++ is the most difficult to work with.  Despite its complexity, there are still times when C++ is a good choice.  It’s object-oriented and it’s fast (if done well).  If that is the cross-section you need, like in a CFD code, then it may be the best way to go.  XCode is actually a nice IDE for C++ development.  It has a great static analyzer that eases some of the pain of working with it.&lt;/p&gt;

&lt;p&gt;For people with limited Fortran experience it often has a bad reputation, and in my opinion undeservedly so.  It is true that Fortran 77 and older are pretty horrible to work with. GOTO statements, implicit typing, fixed layout, and so on just made life difficult.  Lots of awesome numerical packages were developed in Fortran 77 (or older), and so the only exposure many people have had to Fortran is looking at the syntax of some of these old codes.&lt;/p&gt;

&lt;p&gt;However, modern Fortran (90 and up) is actually quite nice to work with.  A lot of improvements were made including free-form input, array notation (like Matlab’s), modules, dynamic memory allocation, etc.  Recent versions have even added object-oriented (OO) features.  Fortran is designed for scientific programming (unlike C which is more general), and the syntax is actually easy to use and similar to Matlab’s.  It does requiring specifying all of the types at the beginning of a function or subroutine, which slows down development, especially when your interface to the functions is still evolving quite a bit.&lt;/p&gt;

&lt;p&gt;As an aside: one thing that drives me crazy in Fortran is how it handles double precision constants.  If you define a variable like this:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;real(8) :: x
x = 1.0
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;x will actually hold a single precision number and not double even though its type is double (compiler-dependent).  Instead you need to specify the constant as a double as well: &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;x = 1.0d0&lt;/code&gt;.  In practice I actually use something like below for portability:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;integer, parameter :: dp = selected_real_kind(15, 307)
real(dp) :: x
x = 1.0_dp
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Littering &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;_dp&lt;/code&gt; or &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;d0&lt;/code&gt; all over the place is a bit of a pain.  I didn’t know this early on, and it was a big gotcha.  I was working on an optimization application where I needed exact gradients, and the difference between single and double precision in some places was causing my gradients to be very close but just barely off to the point where it caused some numerical issues.  Took me a while to figure this out.&lt;/p&gt;

&lt;p&gt;If I needed speed, but didn’t need OO, how would I decide between Fortran and C?  Usually just by what packages I needed, and which language I thought would be easier to do the integration in.  I still use both Fortran and C quite a bit, but almost always only in connection with Python as I’ll discuss below.  I don’t use C++ very much anymore, unless I want something that can stand alone without Python.  Otherwise, the combination of Python with C I find to be much easier to develop in and just as fast to run (more on this later).&lt;/p&gt;

&lt;h1 id=&quot;java&quot;&gt;Java&lt;/h1&gt;

&lt;p&gt;I used Java quite a bit during graduate school.  Most folks don’t think of Java as a scientific langauge.  My advisor liked Java.  He was ahead of his time in developing interactive modules for teaching.  His course notes (e.g., &lt;a href=&quot;http://adg.stanford.edu/aa241&quot;&gt;http://adg.stanford.edu/aa241&lt;/a&gt;) contained Java applets that allowed you to run interactive examples on wing design, etc., which were super helpful.  Recently, there has been a lot of interest in the scientific community to do similar things with Jupyter notebooks.&lt;/p&gt;

&lt;p&gt;From some CS courses I took, I had developed a pretty deep familiarity with Java.  I also liked object-oriented programming and Java was way easier than C++ for development.  None of the other languages I used (Matlab, C, Fortran) accomodated OOP.  I used Java for several aircraft analysis tools and it worked very nicely. As a bonus you could load Java *.jar files in Matlab and call the functions from Matlab for plotting and other visualizations.&lt;/p&gt;

&lt;p&gt;As a graduate student I also worked on the side for a startup, Complete Solar Solution.  I developed an analysis tool, HelioQuote, for them in Java.  This was a comprehensive tool that would automatically pull in electric rates for an address, download a Google map of their roof, perform an optimization analysis, and layout solar panels on their house that you could then drag and drop around if desired.&lt;/p&gt;

&lt;p&gt;Java was great because it was a full featured programming language, was easy to use, had pretty good GUI support if needed, and actually had surprisingly good performance through an aggressive JIT compiler.  The main problem with Java was numerical support was weak.  There were very few scientific libraries available through Java.  I wrote a bunch of &lt;a href=&quot;https://github.com/andrewning/jMath&quot;&gt;my own methods and interfaces&lt;/a&gt; to do basic, but frequently used stuff like integration, root finding, linear solves, etc.  I used that library quite a bit, but it was very limited compared to what’s available in Matlab or Python.&lt;/p&gt;

&lt;p&gt;Java was pretty nice to work with, but I haven’t used it in years.  After diving into Python I found it superior in every way.  It is an even easier language to work with and has great scientific support.  Its only disadvantage is performance, but this is remedied through its easy connections to C/Fortran.&lt;/p&gt;

&lt;h1 id=&quot;python&quot;&gt;Python&lt;/h1&gt;

&lt;p&gt;I dabbled in Python a bit during graduate school, but really only for random fun side projects (like this &lt;a href=&quot;https://code.google.com/p/movie-content-editor/&quot;&gt;movie filter project&lt;/a&gt; we made as a proof of concept).  As a postdoc I started using Python for everything.  Python is pretty great.  It is a very easy language to use.  It has simplistic syntax like Matlab, but unlike Matlab it allows for objected oriented programming (I know Matlab has some OO features, but they are pretty weak), functional programming, or procedural programming.&lt;/p&gt;

&lt;p&gt;Python is free. This may have saved us some money (not really we already had Matlab licenses at the lab), but the bigger benefit is that it allowed others to use our code.  We had some users/collaborators who did not have Matlab and so developing in Matlab limited our ability to collaborate.  Matlab does have a runtime you can distribute to allow users to run your code, but it doesn’t allow them to develop as well.&lt;/p&gt;

&lt;p&gt;It’s also open-source.  In scientific computing, I’ve needed to dive into the details of certain algorithms many times.  In Python I can do this, in Matlab I can rarely do this.&lt;/p&gt;

&lt;p&gt;One of the main benefits of Python, for me, was performance.  This seems odd to say because Python is not fast.  It’s an interpreted langauge with speeds similar to Matlab’s.  However, Python allows you to wrap C/Fortran code pretty easily.  Now, instead of rewriting my entire code in a compiled language, I would profile, find the bottleneck, and rewrite just that portion in Fortran or C.  This allowed me to approach effectively the same speeds I would get in pure Fortran or C, but with an easy-to-use, rapid development environment in Python for scripting, plotting, debugging, etc.  It also allowed me to easily take advantage of existing C/C++/Fortran libraries.  A wide variety of compiled numerical libraries are already available in numpy, scipy, etc., but for specialized tools (like finite element codes, aircraft panel codes, etc.) I was able to wrap these in Python.  Being able to directly pass variables, instead of trying to read/write input files, made using and integrating these codes much easier—especially for optimization applications.&lt;/p&gt;

&lt;p&gt;This is perhaps the primary use of Python in scientific computing.  It is often referred to as a “glue code”.  Meaning, that most of the work is not actually being done in Python, but Python serves as the glue that links codes together.  I’ve linked a blade element code in Fortran 95, a beam finite element code in C++, a cost model in C, and dozens of other components in pure Python, all with an optimizer written in Fortran 77.  All the interfacing is done in Python making data passing and scripting very easy.&lt;/p&gt;

&lt;p&gt;Python has completely replaced Java and C++ for me and almost completely replaced Matlab as well.  The combination of Python with either Fortran or C gives me the benefits of a fast compiled language with a lightweight, interpreted, easy-to-use interface.  What’s the downside?  The main downside is that I have to work with multiple languages still.  There are some inefficiencies (in terms of development) in switching between languages.&lt;/p&gt;

&lt;p&gt;I should say of course that your mileage may vary.  I don’t work in controls, but as far as I’m aware there is nothing as capable as Simulink and I don’t think the support for controls is as strong in Python as compared to Matlab.  But that’s not my area, just my impression.  For the types of things I do, the only thing I’ve missed are optimization algorithms.  The Optimization Toolbox in Matlab is pretty capable and robust.  Conversely, the optimization tools built into scipy are not very good in my opinion.  Fortunately, this is remedied through the use of &lt;a href=&quot;http://www.pyopt.org&quot;&gt;pyOpt&lt;/a&gt; and now &lt;a href=&quot;https://bitbucket.org/mdolab/pyoptsparse&quot;&gt;pyOptSparse&lt;/a&gt;.&lt;/p&gt;

&lt;h1 id=&quot;julia&quot;&gt;Julia&lt;/h1&gt;

&lt;p&gt;Julia appears to be the holy-grail of scientific programming languages.  An interpreted, easy-to-use language, &lt;em&gt;and&lt;/em&gt; with speeds comparable to those of C.  In addition, its designed for mathematical computing, including parallelism and cloud computing, and if needed it makes it even easier to call C or Fortran code (no wrappers needed).&lt;/p&gt;

&lt;p&gt;I’ve been testing it with my students and with one of my own projects.  I’ll provide more details in a subsequent post, but in short I’ve concluded that it’s promising but still too early to fully switch.  The surrounding ecosystem is lacking, and so I needed to develop or wrap a lot of methods myself that are just available in Python.  There were also a few other minor issues that slowed down development, and in the end even the run time was disappointing (though I made no effort to optimize and I suspect there are some small changes I could make that would make a significant impact on run time).&lt;/p&gt;

&lt;p&gt;Julia is at a point where I would say it is fully functional for any of our projects, but in terms of development time (and even run time) it wasn’t yet superior to our current workflow in Python with C/Fortran.  I’m definitely rooting for it to get there, and with all the ongoing work by the Julia team I imagine it won’t be long.  Working in multiple languages has been frustrating at times, and I’d love to go back to spending most of my time in a single language.&lt;/p&gt;

&lt;p&gt;I should mention that Python has several JITs that look very promising, notably Numba and Pyston.  However, these are not quite there yet either for the full stack of scipy tools.  Additionally, I often rely on automatic differentiation and I’m quite sure how that would work with a JIT.  Perhaps these will soon reach a point of full usability across the scientific stack to where one could do high-performance development fully in Python.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Edit (May 2019)&lt;/strong&gt;: In our lab we completely switched over to Julia about three years ago (at least for for all new development).  We’ve been very impressed with the performance and happy with the ease of one language.  There were a few growing pains, but once we started using &lt;a href=&quot;https://github.com/timholy/Revise.jl&quot;&gt;Revise.jl&lt;/a&gt;, and text editors with Julia support, the new workflow became more natural.  We also had to learn (or un-learn) styles for better performance.&lt;/p&gt;

&lt;p&gt;I haven’t adopted Julia in teaching yet but would like to.  In many of my classes programming is a tool and not the point of the class so it’s hard to justify taking time out to teach Julia when they already know something else.  In my graduate optimization class it would be a natural fit except that the optimization solver support (for general nonlinear optimization) isn’t quite to the point where I’d be comfortable with a switch.  In our research this isn’t an issue as we use commercial optimizers that we’ve licensed and wrapped in Julia.&lt;/p&gt;
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