On an n n grid example 24 p q banded factorization

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Unformatted text preview: anded factorization has approximately 2N 4 FLOPs Block factorization has n N 2 and q N for approximately 14N 5=3 FLOPs Some reductions in the block operation count are possible by accounting for the sparsity of Bi and Ci 3 Two-Dimensional Poisson Equation Example 1. Consider Poisson's equation on an N (cf. Example 2.4) { Finite di erencing for a 4 4 problem { Row-by-row numbering with a single index 1 23 1 8 4 2 9 5 6 XX XX 2 3 grid 7 8 X X XX X X XXX X 6 X XX X XX X 7 XXX X 8 XX X 9 Write equations at the odd points rst { Order the odd unknowns rst 1 1 7 8 9 5 2 X X 4 4 X X XXX X 8 8 XXXX X 2 6 46 X 7 3 2 XX 9 1 9 X 3 6 7 X 5 4 35 X XX X X XX XX 9 X X 4 5 1 56 XXX 3 7 4 N X XXX X X Odd-Even Ordering The structure of the matrix with odd-even ordering is V A = D21 D12 V To solve { Factor D1 V1 V2 D2 D1 V1 V2 D2 x1 x2 = = I0 L2 I b1 b2 U1 B1 0 U2 { Expand the factorization U1 = D1 B1 = V1 L2 = V2U;1 U2 = D2 ; L2V1 1 { Forward and backward substitution y1 = b1 y2 = b2 ; V2U;1y1 1 U2x2 = y2 D1x1 = y1 ; V1x2 { U2 is penta-diagonal 5 Domain Decomposition \Dissection," \substructuring," and \domain decomposition" { Partition A into three pieces so that The unknowns x1 are only connected to x3 The unknowns x2 are only connected to x3 { Occurs \naturally" in nite di erence and nite element applications x1 x2 x3 { The structure of the system is 2 32 3 2 3 A11 0 A13 x1 b1 Ax = 4 0 A22 A23 5 4 x2 5 = 4 b2 5 (5) AT AT A33 x3 b3 13 23 Suppose A is symmetric { Factor A...
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This document was uploaded on 03/16/2014 for the course CSCI 6800 at Rensselaer Polytechnic Institute.

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