 # Mathematics Questions

## Transcribed Text

Problem 1 Numerical computation of eigerrvalues Write " Mutlab program performing " simple QR-iteration on " matrix A: QR-ileration: Giwen Ao. uv iderate i 0 repent Factor A, Q.R. (the QR-decomposition) R,Q, i-i+1 until convergenos. You can une " Muilab gr function to achieve QR-decomposition. Your program whould be able to compute sigenvalues up to any amall arror e and also avoid infinite loogs. The output thould abow successive iterations of the matrions A.. Teat your on the following matrices: 1. Creade - matrix D - diag(1,2,3,4,5). Create 5a5 matrix S with rundom entries using rand function from Matlab. Compute the matrix A - SDS 2. Repont the proom with D - ding(1,2,3.4.5.6.7.8.9.10) and " randorn 10x10 matrix S. Problem 2 Extracting eigenvectors Suppore A - U*TU is . given Schur decomposition and is - fas ia " simple eigenvalue. Write an O(m') routine to compute left and right eigenvectora with N. Problem 3 Quadratic eigenvalue problem In the of dumped vibrations, the quadratic sigenvalue problem frequeritly playa a role: where K is typioally symmertric and positive definite, and B - typically positive 1. Show how to commert the quadratic eigenvalue problem into . sigernalue problem by adding the equation D - Au. 2. Produce a MATLAB furaction to produce all 2re aolutions to the quadratic sigenvalue problem. You may call est - " subroutine. 1

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Problem 1.
We apply the QR algorithm to determine the eigenvalues of the matrix A via the algorithm explained in the problem. This produces a sequence of matrices, and we check if the diagonal elements converge by comparing each with the diagonal elements of the previous matrix.
If the absolute value of the difference in diagonal elements is small enough (given by a user prescribed tolerance) we say that the algorithm has converged.

Sample runs:
D=diagonal(1,2,3,4,5)
A=S D S^{-1}, with S a random matrix as produced by rand
Tol= 1.e-7
75 iterations till convergence
D=diagonal(1,2,3,4,5,6,7,8,9,10)
A=S D S^{-1}, with S a random matrix as produced by rand
Tol= 1.e-7
140 iterations till convergence.
Matlab code:
function [ev,iter]=qr_diagonalize(A,tol)
%INPUT:
%   A: n by n matrix
%   tol: tolerance for convergence
%OUTPUT:
%   eigenvalues - of equation A obtain by QR

n=size(A,1);
if size(A,2) ~= n
error('A must be square')
end

if nargin < 2
tol=eps;
end
done=0;
iter=0;
while ~done
iter=iter+1;
[Q,R]=qr(A);...
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