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Unformatted text preview: (0) , then the results of Theorem
9.2.1 imply kM k ! 0 as ! 1 hence, (M) < 1.
Proving that (9.2.2) converges when (M) < 1 is slightly more involved. We'll establish the result when M is diagonalizable. Isaacson and Keller 3], Section 1.1, establish
the result under more general conditions.
If M is diagonalizable, then there is a matrix P such that PMP;1 =
where is a diagonal matrix. Now, M = (P;1 P)(P;1 P) : : : (P;1 P) = P;1 P: 9.2. Basic Iterative Solution Methods 11 If j ij < 1, i = 1 2 : : : N , then
lim kP;1 Pk = 0
!1
and the iteration (9.2.2) converges.
Theorems 9.2.1 and 9.2.2 prescribe convergence conditions. We also want an indication of the convergence rate of the iteration. Many measures are possible and we'll settle
on the following. De nition 9.2.1. The average convergence rate of the iteration (9.2.2) is
R (M) ; ln kM k :
Using (9.2.5) (9.2.7a) ke( ) k kM kke(0) k = e; R ke(0) k: Thus, convergence is fast when R is large or, equivalently, when kMk is small. Additionally, since (M ) kM k and kM k 1 for a converging iteration, R (M) ; ln (M): (9.2.7b) Thus, we may take ; ln (M) as a measure of the convergence rate. Although the spectral
radius is more di cult to compute than a matrix norm, this rate is independent of and
the particular norm.
Many iterative procedures partition A as A=D;L;U
where 2
a11
6
a22
D=6
6
...
4 aNN 3
7
7
7
5 2
0
6 ;a21
L = 6 ..
6
4. 2
0 ;a12
6
0
U=6
6
4 0
... . . .
;aN 1 ;aN 2
0
3
;a1N
;a2N 7
7
. . . ... 7 :
5
0 (9.2.8a)
3
7
7
7
5 (9.2.8b) (9.2.8c) 12 Solution Techniques for Elliptic Problems 9.2.1 Jacobi and GaussSeidel Iteration Three classical iterative methods have the splitting de ned by (9.2.8). With the Jacobi
method, we solve for the diagonal terms of (9.1.1). Thus, using (9.2.8a), we write (9.1.1)
in the form
(D ; L ; U)x = b (9.2.9) and consider the iteration x( +1) = D;1(L + U)x( ) + D;1b (9.2.10a) which has the form of (9.2.2) with
^
bJ = D;1b: MJ = D;1(L + U) (9.2.10b) The scalar form (9.2.10) is x(i +1) = ; N
X a...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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