Page 392 - Applied Numerical Methods Using MATLAB

JACOBI METHOD 381
Moreover, each of the disks contains at least one eigenvalue of the
matrix A.
The power method introduced in Section 8.3.1 is cast into the routine
“eig_power()”. The MATLAB program “nm831.m” uses it to perform the power
method, the inverse power method and the shifted inverse power method for
finding the eigenvalues of a matrix and compares the results with that of the
MATLAB built-in routine “eig()” for cross-check.

function [lambda,v] = eig_power(A,x,EPS,MaxIter)
% The power method to find the largest eigenvalue (lambda) and
% the corresponding eigenvector (v) of a matrix A.
if nargin < 4, MaxIter = 100; end % maximum number of iterations
if nargin < 3, EPS = 1e-8; end % difference between successive values
N = size(A,2);
if nargin < 2, x = [1:N]; end % the initial vector
x = x(:);
lambda = 0;
for k = 1:MaxIter
x1 = x; lambda1 = lambda;
x = A*x/norm(x,inf); %Eq.(8.3.4)
[xm,m] = max(abs(x));
lambda = x(m); % the component with largest magnitude(absolute value)
if norm(x1 - x) < EPS & abs(lambda1-lambda) < EPS, break; end
end
if k == MaxIter, disp(’Warning: you may have to increase MaxIter’); end
v = x/norm(x);
%nm831
%Apply the power method to find the largest/smallest/medium eigenvalue
clear
A = [2 0 1;0 -2 0;1 0 2];
x = [1 2 3]’; %x = [1 1 1]’; % with different initial vector
EPS = 1e-8; MaxIter = 100;
%the largest eigenvalue and its corresponding eigenvector
[lambda_max,v] = eig_power(A,x,EPS,MaxIter)
%the smallest eigenvalue and its corresponding eigenvector
[lambda,v] = eig_power(A^ - 1,x,EPS,MaxIter);
lambda_min = 1/lambda, v %Eq.(8.3.6)
%eigenvalue nearest to a number and its corresponding eigenvector
s = -3; AsI = (A - s*eye(size(A)))^ - 1;
[lambda,v] = eig_power(AsI,x,EPS,MaxIter);
lambda = 1/lambda+s %Eq.(8.3.8)
fprintf(’Eigenvalue closest to %4.2f = %8.4f\nwith eigenvector’,s,lambda)
v
[V,LAMBDA] = eig(A) %modal matrix composed of eigenvectors

8.4 JACOBI METHOD
This method finds us all the eigenvalues of a real symmetric matrix. Its idea is
based on the following theorem.