Find smallest Eigenvalue and the corresponding eigenvector.
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I need to write a program which computes the largest and the smallest (in terms of absolute value) eigenvalues using power method. I can find the largest one using the power method. But I have no idea how to find the smallest one using the power method.
How can I modify the power method so that it computes the smallest eigenvalue?
my power method algorithm :
2. Define matrix X
3. Calculate Y = AX
4. Find the largest element in the magnitude of matrix Y and assign it to K.
5. Calculate fresh value X = (1/K) * Y
6. If [Kn – K(n-1)] > delta, go to step 3.
Below is the coding :
function [ v d] = power_method( A )
% for finding the largest eigen value by power method
disp ( ' Enter the matrix whose eigen value is to be found')
% Calling matrix A
A = input ( ' Enter matrix A : \n')
% check for matrix A
% it should be a square matrix
[na , ma ] = size (A);
if na ~= ma
disp('ERROR:Matrix A should be a square matrix')
% initial guess for X..?
% default guess is [ 1 1 .... 1]'
disp('Suppose X is an eigen vector corresponding to largest eigen value of matrix A')
r = input ( 'Any guess for initial value of X? (y/n): ','s');
% asking for initial guess
X0 = input('Please enter initial guess for X :\n')
% check for initial guess
[nx, mx] = size(X0);
if nx ~= na || mx ~= 1
disp( 'ERROR: please check your input')
X0 = ones(na,1);
%allowed error in final answer
t = input ( 'Enter the error allowed in final answer: ');
tol = t*ones(na,1);
% initialing k and X
X( : , 1 ) = X0;
%initial error assumption
% loop starts
while sum(abs(err) >= tol) ~= 0
X( : ,k+ 1 ) = A*X( : ,k); %POWER METHOD formula
% normalizing the obtained vector
[ v i ] = max(abs(A*X( : ,k+ 1 )));
E = X( : ,k+ 1 );
e = E( i,1);
X(:,k+1) = X(:,k+1)/e;
err = X( :,k+1) - X( :, k);% finding error
k = k + 1;
%display of final result
fprintf (' The largest eigen value obtained after %d itarations is %7.7f \n', k, e)
disp('and the corresponding eigen vector is ')
X( : ,k)
John D'Errico on 9 Jun 2022
Edited: John D'Errico on 9 Jun 2022
Just compute the matrix
Ainv = inv(A);
Now use your code on the matrix Ainv. This works because the smallest eigenvalue is now the largest.
Will this faiil for a singular matrix A? Of course. But then you cannot use the power method there anyway.
So there is absolutely no need to modify your code. Just use it on a different matrix.