How to compute Eigenvalues PHASEII QRAlgorithm How to

How to compute Eigenvalues PHASE-II QR-Algorithm

How to redeuce into Hessenberg

How to deuce into Hessenberg PHASE-I 1 Using Householder Matrices 2 Using Gram-Schmidt (modefied)

Arnoldi Iteration

Arnoldi Iteration Arnoldi iteration: function [Q, H] = Arnoldi(A, k, b); m = length(b); H = zeros(k); Q = zeros(m, k); Q(: , 1) = b/norm(b); for n = 1: k, v=A*Q(: , n); for j=1: n H(j, n)=Q(: , j)'*v; v=v-H(j, n)*Q(: , j); end H(n+1, n)=norm(v); Q(: , n+1)=v/H(n+1, n); end

Arnoldi Iteration A= 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 >> [Q, H = Arnoldi(A, 4, b); Q= 0. 1891 -0. 9666 -0. 1611 0. 7878 0. 1061 0. 0495 -0. 3466 0. 0296 -0. 7862 0. 4727 0. 2315 -0. 5945 >> H H= 12. 7219 2. 0648 0. 0000 2. 0648 10. 1219 1. 9825 0 1. 9825 8. 7845 0 0 1. 5497 0 0. 0636 0. 8359 -0. 6047 0. 1644 -0. 5107 0. 3563 0. 6078 0. 3837 -0. 0000 1. 5497 7. 3716 0. 0000

Arnoldi Iteration A= 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 Q= 0. 1891 -0. 9666 -0. 1611 0. 0636 0. 8359 0. 7878 0. 1061 0. 0495 -0. 6047 0. 1644 -0. 3466 0. 0296 -0. 7862 -0. 5107 0. 3563 0. 4727 0. 2315 -0. 5945 0. 6078 0. 3837 Example: span<b, A*b>= 6 13 25 325 -11 -138 15 206 span<q 1, q 2>= 0. 1891 -0. 9666 0. 7878 0. 1061 -0. 3466 0. 0296 0. 4727 0. 2315

Arnoldi Iteration Example: K 2= 6 25 -11 15 A= 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 Q= 0. 1891 -0. 9666 -0. 1611 0. 0636 0. 8359 0. 7878 0. 1061 0. 0495 -0. 6047 0. 1644 -0. 3466 0. 0296 -0. 7862 -0. 5107 0. 3563 0. 4727 0. 2315 -0. 5945 0. 6078 0. 3837 13 325 -138 206 Q 2= 0. 1891 -0. 9666 0. 7878 0. 1061 -0. 3466 0. 0296 0. 4727 0. 2315 R 2 = 31. 7333 403. 7089 -0. 0000 65. 5221 K 2=Q 2*R 2

Arnoldi Iteration

Arnoldi Iteration A= 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 Q= 0. 1891 -0. 9666 -0. 1611 0. 0636 0. 8359 0. 7878 0. 1061 0. 0495 -0. 6047 0. 1644 -0. 3466 0. 0296 -0. 7862 -0. 5107 0. 3563 0. 4727 0. 2315 -0. 5945 0. 6078 0. 3837

Arnoldi Iteration