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# Class Note for MATH 6370 with Professor Caboussat at UH

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15
Department of Mathematics Fall 2007 MATH 63707 Section 10583 A Caboussat Theorem 1 Let A E Rn be symmetiic positive de nite then the GaussSeidel method converges Let D dizzga117 227 l l l 7ann be the diagonal part of A IfA and 2D7A are symmetric positive de nite then the Jacobi method converges Proof We remind A D 7 E 7 F GaussSeidel D 7 E is a lower triangular matrix lf aii 07 for all i7 then D 7 E is regular lndeed aii eZTAei gt 0 since A is positive definite7 and the GaussSeidel method is wellde ned We know that if pI 7 B lA pD 7 e 1F lt 17 then the method converges Let A E C and go 6 C go 0 such that D7E 1Fgo Ago Fgo AD7Ego Thus Ago D 7 E 7 Fgo l 7 A D 7 Egol We distinguish two cases i A 1 In this case Ago 07 which is a contradiction ii A f l is therefore required Then we have some 7 17 MIND 7 EM Even if D 7 E is real since A is real7 both sides are complex numbers The complex conjugate of this relation reads 05490 17 VWWD ET80 1 NWWD FM We multiply the rst relation by l 7 Aquot and the second by l 7 A and sum 90As0 17 AsoD 7 EM 17 A WM 7 17 soD 7 FM 17 A 2 7A 7AgtsoAso 7 17 MQWOD 7 E7Fgtso 7 1 7l2s0DAs0 Therefore 2 7 A 7 Now 7 17 Wm Agtso 2 7 7 WWW 17 7 ll2WD AW 17 Mom 7 17 Wow Since A is positive definite7 goquotAgo gt 07 ll 7 AV gt 0 since A f l and goquotDgo gt 0 since D is positive de nite diagonal matrix with all elements aii gt 0 Therefore 17w2gt0 ewa All the eigenvalue of D 7 E 1F are small than one in modulusl Therefore the method convergesl Jacobi The Jacobi method is wellde ned for the same reasons than for the GaussSeidel method We show that pD 1E F lt 1 to ensure convergence Let A E C and go 6 C g0 0 such that D 1EFso A90 EFso ADso Therefore Asa DiEiFso1Dso 90Aso 1 7 A90Dso Since g0quotAg0 gt 0 and g0Dg0 gt 0 are both real numbers7 1 7 A gt 0 is real and A lt 1 Moreover 902D 7 AW 1 AWDso Since g0quot2D 7 Ag0 gt 0 is real7 1 A gt 0 is real and A gt 71 Conclusion is that A lt 1 and the method converges D

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