Introduction to Krylov subspace methods - IMAGe
Introduction to Krylov subspace methods - IMAGe
Introduction to Krylov subspace methods - IMAGe
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GMRES5 KRYLOV SUBSPACE METHODS 36r 0 = b − Ax 0 , β = ‖r 0 ‖ 2 and v 1 = r 0 /βfor j = 1, · · · , m doendw j = Av jfor i = 1, · · · , j doendh ij = (w j , v i )w j = w j − h ij v ih j+1,j = ‖w j ‖ 2 , if h j+1,j = 0, m = j exitv j+1 = w j /h j+1.jV = [v 1 , · · · , v m ] , H m = {h ij } 1≤i≤m+1, 1≤j≤mMinimizer y m of ‖βẽ 1 − H m y‖ 2x m = x 0 + V m y m⎫⎪ ⎬⎪ ⎭MGSIf P is an orthogonal projec<strong>to</strong>r, then P x and (I − P )x are orthogonal,Saad Schultz (1986)(P x, (I − P )x) = (P x, x) − (P x, P x) = (P x, x) − (x, P H P x)= (P x, x) − (x, P 2 x) = (P x, x) − (x, P x) = 0,37