Question

code in matlab

1. [2+1+1pt] Power Method and Inverse Iteration. (a) Implement the Power Method. Use your code to find an eigenvector of -2 1 4 A= 1 1 2 4 1 -2 starting with Xo = (1, 2, -1)7 and Xo = (1, 2, 1)7. Report the first 5 iterates for each of the two initial vectors. Then use MATLAB's eig(A) to examine the eigenvalues and eigenvectors of A. Where do the sequences converge to? Why do the limits not seem to be the same? (b) Implement the Inverse Power Method for an arbitrary matrix A E Rnxn, an initial XO E R” and an initial eigenvalue guess 0 E R. (c) Use your code from (c) to calculate all eigenvectors of A. You may pick appropriate values for 0 and the initial vector as you wish (obviously not the eigenvectors them- selves). Always report the first 5 iterates and explain where the sequence converges to and why. ve

Answer 1

clc

clear all

close all

format long

A=[-2,1,4;1,1,1;4,1,-2];

x0=[1;-2;-1];

disp('First 5 iterates for first x0 are');

[x,lam]=powerMethod(A,x0);

x0=[1;2;1];

disp('First 5 iterates for second x0 are');

[x,lam]=powerMethod(A,x0);

disp('Eigen vectors using eig are');

[E,~]=eig(A)

function [x,lam]=powerMethod(A,x)

%Power method for finding largest eigenvalue and eigenvector of A

maxIter = 5; %maximum no;of iterations

tol = 1e-10; %tolerance

converged = 0; %flag for convergence

  

n = length(A);

x = x/norm(x); %random vector for starting

iter = 1;

  

while (iter <=maxIter) %while still not converged and below maximum Iterations

x_old = x;

x = A*x;

x = x/norm(x)

lam = x'*A*x;

converged = (norm(x - x_old) < tol);

iter = iter + 1;

end

end