jacobian.py

import sympy as sp
from sympy import pprint

import numpy as np

# jacobian matricies for multivariate functions

# 2d example
def a1():
    x1, x2 = sp.symbols('x1 x2')
    f1 = 5*x1*x2
    f2 = x1**2 * x2**2 + x1 + 2*x2
    f = sp.Matrix([f1, f2])
    X = sp.Matrix([x1, x2])
    Df = f.jacobian(X)
    Df0 = Df.subs([(x1, 1), (x2, 2)])
    pprint(Df)
    pprint(Df0)
# a1()

# 3d example
def a2():
    x1, x2, x3 = sp.symbols('x1 x2 x3')

    # assemble vectors f and x
    f1 = sp.log(x1**2+x2**2)+x3**2
    f2 = sp.exp(x2**2+x3**2)+x1**2
    f3 = 1/(x3**2+x1**2)+x2**2
    f = sp.Matrix([f1, f2, f3])
    X = sp.Matrix([x1, x2, x3])

    Df = f.jacobian(X)

    # compute for actual values
    Df0 = Df.subs([(x1, 1), (x2, 2), (x3, 3)])
    pprint(Df)
    pprint(Df0)
# a2()


# example linearization of a multivariable function via tangent plane
def a3():
    x1, x2, x3 = sp.symbols('x1 x2 x3')

    # define vectors
    f1 = x1+x2**2-x3**2-13
    f2 = sp.log(x2/4)+sp.exp(x3/2-1)
    f3 = (x2-3)**2-x3**3+7
    f = sp.Matrix([f1,f2,f3])
    X = sp.Matrix([x1,x2,x3])

    # Df(x) substitutes variables i.e 3*x1^2 inside partial derivative formuli with value
    Df = f.jacobian(X)

    # lambdify for use with numpy values
    func = sp.lambdify([(x1,x2,x3)], f, "numpy")
    jac = sp.lambdify([(x1,x2,x3)], Df, "numpy")


    # evaluate f(v0) 
    v0 = np.array([1.5,3,2.5])

    # evaluate jacobian at point v0 for slopes in all dimensions
    DF = jac(v0).flatten()

    # now choose v1 in vicinity of v0
    offset = np.array([0.1,0.1,0.1])
    v1 = v0+offset
    
    # evaluating f(v0) and following the slope along the linearization 
    # should yeald a result close to actually evaluating f(v1) while still in v0's vicinity.
    pprint(func(v1))

    # tangent plane equation for linearization
    # around neighbourhood of point v 
    def g(v_next,v): return (func(v).flatten()+jac(v)@(v_next-v)).flatten()
    pprint(g(v1,v0))
a3()