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function [Q,H] = arnoldi(A,q1,m) %ARNOLDI Arnoldi iteration % [Q,H] = ARNOLDI(A,q1,M) carries out M iterations of the % Arnoldi iteration with N-by-N matrix A and starting vector q1 % (which need not have unit 2-norm). For M < N it produces % an N-by-(M+1) matrix Q with orthonormal columns and an % (M+1)-by-M upper Hessenberg matrix H such that % A*Q(:,1:M) = Q(:,1:M)*H(1:M,1:M) + H(M+1,M)*Q(:,M+1)*E_M', % where E_M is the M'th column of the M-by-M identity matrix. n = length(A); if nargin < 3, m = n; end q1 = q1/norm(q1); Q = zeros(n,m); Q(:,1) = q1; H = zeros(min(m+1,m),n); for k=1:m z = A*Q(:,k); for i=1:k H(i,k) = Q(:,i)'*z; z = z - H(i,k)*Q(:,i); end if k < n H(k+1,k) = norm(z); if H(k+1,k) == 0, return, end Q(:,k+1) = z/H(k+1,k); end end