Introduction to Krylov subspace methods - IMAGe

et r n = f − Au n ,Let r n = f − Au n ,n+1 n −1 n stationary iterative methodu n+1 = u n + M −1 r n : stationary iterative methodn en u 0 0Given:u n n → u u ?0nvergencevergence :: eu n n = → u n u −?uGiven u 0Matrix splittingsConvergence : e n = u n − uu n+1 = u n + M −1 r n : stationary iterativeu n → u ?Convergence Error: : e n = u n − uu n+1 n+1 = u n + M −1 −1 (f (f − Au Au n n ))e n+1 n+1 = e n + M −1 −1 (f (f − Au Au n ))u n+1 = u n + Me n+1 = e n + Mu n+1 = u n + M −1 (f − Au n )= e n + M −1 −1 A(u − u n ))e n+1 = e n + M −1 (f − Au n )= e n − M −1 Ae n= e n + M −1 A(u − u n )= e n + M= e n − M∴ e n+1 = (I − M −1 = A)e e n − n M −1 Ae n∴∴ e n+1 = (I − M −1 A)e n5e n+1 = (I − M