Question

SOMEONE PLEASE HELP ME WITH THIS QUESTION. THUMBS UP WITH BE GIVEN IF DONE CORRECTLY AND CLEARLY. Use MATLAB for both part a and b!!!! THANKS!!!!

2-1 0 Pj [MATLAB] Consider the matrix A =-1 2-1 0-1 2 a) Compute the largest eigenvalue using a power iteration (y random) b) Compute the eigenvalue with smallest modulus using an inverse power iteration (y random)

Answer 1

Q a) Power Iterative

*********************power_iterative.m************************

A=[2 -1 0;-1 2 -1;0 -1 2];

y0=ones(3,1);
err=inf;
tol=1e-6;
iters=0;
max_iters=10000;
e=inf;
while(iters<max_iters)
e0=e;
y_new=A*y0;

e=max(y_new);

y_new=y_new./max(y_new);
y0=y_new;
  
err=e-e0;
if abs(err)<tol
break;
end
iters=iters+1;
end

fprintf('The dominant eigen value is %f\n',e);

Q b

*****************************inversePowerIterative.m************************

A=[2 -1 0;-1 2 -1;0 -1 2];

%A Matrix should not be singular

B=inv(A);
y0=ones(3,1);
err=inf;
tol=1e-6;
iters=0;
max_iters=10000;
e=inf;
while(iters<max_iters)
e0=e;
y_new=B*y0;

e=max(y_new);

y_new=y_new./max(y_new);
y0=y_new;
  
err=e-e0;
if abs(err)<tol
break;
end
iters=iters+1;
end
%e is the dominant eigen value in inverse matrix
% and 1/e is the least dominant eigen value in the original matrix
e=1/e;
fprintf('The least dominant eigen value is %f\n',e);

>PowerIterativeMethod The dominant eigen value is 3.414213 InversePowerIterative The least dominant eigen value is 0.585786