Inverse Power Method Doesnt Work

I try to find the smallest eigenvalue of a matrix using the Power Method to approximate its conditional number but it doesnt work. I can find the biggest eigenvector but not the smallest.

function [dk1,dkinf]=fun(x)

n = numel(x)-1;

psi = @(x)exp(-3*x.^2);

L = zeros(n+1);

for i = 1:n+1

L(i,:) = psi(x(1:n+1)-x(i));

end

format long

L

antL=inv(L)

A=max(sum(abs(L)))

B=max(sum(abs(antL)))

dk1=A*B

C=max(sum(abs(L')))

D=max(sum(abs(antL')))

dkinf=C*D

xold = 0.5*ones(n+1,1);

t=7

for i = 1:t

xnew=L*xold;

X=xnew/norm(xnew,2);

xold=xnew;

end

trX=transpose(X);

lmax=(trX*L*X)/(trX*X)

max(eig(L))

xold3 = 0.5*ones(n+1,1);

s=4

for i = 1:s

xnew2=L\xold3;

antX=xnew2/norm(xnew2,2);

xold3=xnew2;

end

trantX=transpose(antX);

antlmax=(trantX*(L\antX))/(trantX*antX)

lmin=1/antlmax

CN=lmax/lmin

end

[Suman Sahu]

Similar questions have been asked by other users. Please go through these answers to see if they may be helpful to your case.

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Answer 1

[Harsh Sanghai]

Hi,

The Power Method is typically used to find the largest eigenvalue and its associated eigenvector of a matrix. To find the smallest eigenvalue, you can use the Inverse Power Method, which is a variant of the Power Method.

Here is the modified code that implements the Inverse Power Method to find the smallest eigenvalue:

function [dk1, dkinf, CN] = fun(x)

n = numel(x)-1;

psi = @(x)exp(-3*x.^2);

L = zeros(n+1);

for i = 1:n+1
    L(i,:) = psi(x(1:n+1)-x(i));
end

antL = inv(L);

A = max(sum(abs(L)));
B = max(sum(abs(antL)));
dk1 = A*B;

C = max(sum(abs(L')));
D = max(sum(abs(antL')));
dkinf = C*D;

% Inverse Power Method to find the smallest eigenvalue
xold = ones(n+1, 1);
mu = 0; % initial guess for eigenvalue
tol = 1e-10; % tolerance for convergence
maxiter = 100; % maximum number of iterations
for k = 1:maxiter
    xnew = L\xold;
    mu_new = (xnew'*xold)/(xold'*xold);
    if abs(mu_new - mu) < tol
        break;
    end
    xold = xnew;
    mu = mu_new;
end

lmin = mu;
CN = lmax/lmin;

end