Lanczos Iteration function q TlanczosA b k mlengthb

Lanczos Iteration function [q, T]=lanczos(A, b, k) m=length(b); q=zeros(m, k+1); alpha=zeros(k, 1); beta=zeros(k-1, 1); T=zeros(k, k); beta 0=0; q 0=zeros(m, 1); q(: , 1)=b/norm(b); v=A*q(: , 1); alpha(1)=q(: , 1)'*v; v=v-beta 0*q 0 -alpha(1)*q(: , 1); beta(1)=norm(v); q(: , 2)=v/beta(1); for n=2: k v=A*q(: , n); alpha(n)=q(: , n)'*v; v=v-beta(n-1)*q(: , n-1)alpha(n)*q(: , n); beta(n)=norm(v); q(: , n+1)=v/beta(n); end length(alpha) length(beta) T=diag(alpha)+diag(beta(1: k 1), 1)+diag(beta(1: k-1), -1); T(k+1, k)=beta(k); Lanczos Iterations

Lanczos Iteration Q= 0. 3421 0. 4211 0. 3421 -0. 3539 0. 2519 0. 4408 -0. 1795 -0. 6872 -0. 3539 -0. 0862 -0. 5003 -0. 0898 0. 4388 0. 4819 0. 2831 -0. 1017 -0. 7405 -0. 1096 0. 1862 -0. 8354 0. 3002 0. 4488 -0. 2220 0. 4819 0. 2831 -0. 1017 -0. 0224 -0. 2761 -0. 3539 -0. 0862 -0. 5003 -0. 0898 0. 4388 -0. 3539 0. 2519 0. 4408 0. 4488 -0. 0719 -0. 0190 -0. 0749 -0. 2269 0. 4208 -0. 2192 -0. 0749 -0. 8437 0. 4558 0. 0622 -0. 0694 -0. 1141 -0. 8601 0. 0622 0. 1639 H= 14. 5208 0. 8010 0 0. 8010 8. 5564 2. 6233 0 2. 6233 12. 2010 0 0 2. 7747 0 0 0 0 2. 7747 0 7. 7218 0. 0000 10. 3535 0 3. 1405 0 0 0 3. 1405 0 9. 5836 2. 6879 11. 0555 0 2. 6024 A= 10 0 3 0 0 0 3 0 10 0 3 0 10 0 0 3 0 10 b= 13 13 16 16 16 13 13

Minres EXAMPLE: A= 10 0 3 0 0 0 3 0 10 0 3 0 10 0 0 3 0 10 b= 13 13 16 16 16 13 13 Case: n=1 H 1= 14. 5208 0. 8010 y= 2. 6090 Least Square Q 1 = 0. 3421 0. 4211 0. 3421 x 1 = 0. 8926 1. 0985 0. 8926

Minres EXAMPLE: A= 10 0 3 0 0 0 3 0 10 0 3 0 10 0 0 3 0 10 b= 13 13 16 16 16 13 13 Case: n=2 Hi = 14. 5208 0. 8010 8. 5564 0 2. 6233 y= 2. 6287 -0. 2248 Least Square Q 2 = 0. 3421 0. 4211 0. 3421 -0. 3539 0. 4819 0. 1862 0. 4819 -0. 3539 x 2= 0. 9788 0. 9985 1. 0649 0. 9985 0. 9788

Minres x= 0. 8926 1. 0985 0. 8926 0. 9788 0. 9985 1. 0649 0. 9985 0. 9788 1. 0061 0. 9881 0. 9970 1. 0148 0. 9970 0. 9881 1. 0061 1. 0000 1. 0000 0. 0728 0. 0000 relres =res/res 0 1. 0000 0. 2949

Minres