First_Version

Lanczos Algorithm

Lanczos.asv

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+function [T] = Lanczos(A,n);
+%Lancozs Algorithm to find Eigenvalue spectra
+%Inputs :A hermitian matrix of the order n
+%       :n order of the matrix
+
+    T = zeros(n);
+
+    v = rand(n,1);     %Random vector 
+    v1 = v/norm(v);    %
+
+    t = A*v1;
+    alpha = v1'*t;
+    t = t - v1*alpha;
+
+    T(1,1) = alpha;
+
+    for j = 2:n,
+
+        beta = norm(t); 
+        v0 = v1; v1 = t/beta;
+
+        t = A*v1 - v0*beta;
+        alpha = v1'*t;
+        t = t - v1*alpha;
+
+        T(j,j-1) = beta; T(j-1,j) = beta; T(j,j) = alpha;
+
+
+    end 
+    T
+    eig(T)
+end

Lanczos.m

@@ -0,0 +1,49 @@
+%Lancozs Algorithm to find Eigenvalue spectra
+%Inputs :A matrix of the order n
+%       :n order of the matrix
+%Author:Jishnu Rajendran
+%As a part of semester project
+%
+
+
+function [T,V,u] = Lanczos(A,n)
+
+    T = zeros(n);       %Initailising the T matrix with all  
+                        %All entries 0
+    V = zeros(n,n);
+   % if abs(norm(u)-1)<0.001
+    %    w1=u;
+    %else 
+    w1 = rand(n,1);      %Generating Random vector of norm 1
+    w1 = w1/norm(w1);    %
+    u=w1;
+
+
+    t = A*w1;
+    alpha = w1'*t;
+    t = t - w1*alpha;
+
+    V(:,1) = w1;
+    T(1,1) = alpha;
+
+    for j = 2:n,
+
+        beta = norm(t); 
+        w0 = w1; w1 = t/beta;
+
+        t = A*w1 - w0*beta;
+        alpha = w1'*t;
+        t = t - w1*alpha;
+
+        T(j,j-1) = beta;
+        T(j-1,j) = beta;
+        T(j,j) = alpha;
+        V(:,j)   = w1;
+        [X,~]=eig(T);
+        if(abs(X(j,n))<0.000001)
+           break
+        end
+
+    end 
+
+end

inp.asv

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+%A=load('input.txt');
+%A=[2 0 1;
+%  0 3 0;
+% 1 0 2;];
+%A = rand(5,5);
+%A=rherm(5)
+ A
+eig(A)
+[n,n]=size(A);
+
+Lanczos(A,n);
+
+
+
+

inp.m

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+sHH=load('sparse_ham_sz_2.txt');
+%H = diag(ones(20, 1),0) +  0.5*diag(ones(19,1), 1)  + 0.5*diag(ones(19,1), -1);
+%H(20,1)=1;H(1,20)=1;
+%D=[2 0 1;   0 3 0;   1 0 2;];
+%u =[0.171799151874028;   0.751483050929023;   0.636991582033681;];
+
+A=spconvert(H);
+%save('A_mat.txt','A');
+issparse(A)
+g=eig(A);
+disp(g);
+[~,n]=size(A);
+[S,D]=eigs(A); %S is the matrix with the eigenvectors as column
+S;
+[T,V,u]=Lanczos(A,n);
+T;
+V;
+k=eig(T);
+disp(k);
+[X,Z]=eig(T); % X is the matrix with the eigenvectors as column
+X;
+B=eig(T)-eig(A);
+disp(B);
+disp([g,k]);
+disp(u);
+

rherm.m

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+function [ A ] = rherm(n)
+% Generate a hermitian matrix with random values.
+% rherm(n) returns a random hermitian matrix of size n*n
+if exist('n')~=1, n=2; end
+A = [zeros(1,0),(2*rand(1,1)-1),(2*rand(1,n-1)-1)+(2*rand(1,n-1)-1)*1i;zeros(n-1,n)];
+for i=1:n-1, A = A + [zeros(n-i,n);zeros(1,n-i),(2*rand(1,1)-1),(2*rand(1,i-1)-1)+(2*rand(1,i-1)-1)*1i;zeros(i-1,n)];end
+for i=1:n-1, for j=1:n-i, A = A + [zeros(n-i,n);zeros(1,j-1),conj(A(j,n-i+1)),zeros(1,n-j);zeros(i-1,n)];end;end
+end