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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 TwoDimensional Poisson Equation
Example 1. Consider Poisson's equation on an N
(cf. Example 2.4)
{ Finite di erencing for a 4 4 problem
{ Rowbyrow 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 OddEven Ordering
The structure of the matrix with oddeven 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 pentadiagonal 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.
 Fall '98
 JosephE.Flaherty

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