Introduction to Krylov subspace methods - IMAGe

(2) Ax = b(2) Ax = brve the structure of the matrix (not change it)bMatrix splittingsIterative methodIterative methodSuppose A = MIterative Suppose − N (⇒ A = method M −1 −A N = (⇒I −M −1 −1 A N, = IM− −1Mtive methodSolve: Au = f Suppose:A = M − N (⇒ M −1 A = I − M(M − N)u = fose A = M − N (⇒ M −1 A = I − M −1 N, (M M −1 −N N)u = I= −fM −1 A)n = f − Au n ,(M − N)u = fDefine: Let r n = f − Au n ,Mu n+1 = Nu n + fMu n+1 = (M NuMu n −+ n+1 N)u f = Nu fn + fu n+1 = M −1 Nu n + M −1 fu n+1 = M −1 Mu uNu n+1 n+1 n = + M Nu −1 −1n Nu f+nu n+1 = (I −uM n+1 n+1 −1 = A)u (IM n − −1 + M N−u n+1 = u n u+ M n+1 n+1 −1 =(f u(I − n +Au −M n −)u n+1 = (I − M −1 A)u n + M −1 fu n+1 = u n + M −1 (f − Au n )Let r n = f − Au n ,u n+1 = u n + MStationary iterative Let method: r n = f − Au n ,u n+1 = u n u n+1+ M −1 =r n u n + M −1 r n : statio: stationary iteratiu n+1 = u n + M −1 r n : stationary iterative method4