Because gaussseidel computes xi k with newer data

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: istic equation, and this guarantees that An+j (j = 0, 1, 2, . . .) can be expressed as a polynomial in A of at most degree n − 1. Since f (A) is always a polynomial in A, the Cayley– Hamilton theorem insures that f (A) can be expressed as a polynomial in A of at most degree n − 1. Such a polynomial can be determined whenever f (j ) (λi ), j = 0, 1, . . . , ai − 1 exists for each λi ∈ σ (A) , where ai = alg mult (λi ) . The strategy is the same as that in Example 7.9.4 except that ai is used in place of ki . If we can find a polynomial p(z ) = α0 + α1 z + · · · + αn−1 z n−1 such that for each λi ∈ σ (A) , IG R Y P p(λi ) = f (λi ), p (λi ) = f (λi ), ..., p(ai −1) (λi ) = f (ai −1) (λi ), then p(A) = f (A). Why? These equations are an n × n linear system with the αi ’s as the unknowns, and, for the same reason outlined in Example 7.9.4, a solution is always possible. (a) What advantages and disadvantages does this approach have with respect to the approach in Example 7.9.4? (b) Use this method to find a polynomial p(z ) such that p(A) = eA O C for A = 7.9.17. Show that if f ⎛ αβ A=⎝0 α 00 3 −3 −3 2 −2 −2 1 −1 −1 . Compare with Exercise 7.9.15. is a function defined at ⎞ γ β ⎠ = α I + β N + γ N2 , α ⎛ where then f (A) = f (α)I + βf (α)N + γ f (α) + Copyright c 2000 SIAM 0 N = ⎝0 0 1 0 0 ⎞ 0 1⎠, 0 β 2 f (α) 2 N. 2! Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Buy from AMAZON.com 7.9 Functions of Nondiagonalizable Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 615 7.9.18. Composition of Matrix Functions. If h(z ) = f (g (z )), where f and g are functions such that g (A) and f g (A) each exist, then h(A) = f g (A) . However, it’s not legal to prove this simply by saying “replace z by A. ” One way to prove that h(A) = f g (A) is to demonstrate that h(J ) = f g (J ) for a generic Jordan block and then invoke (7.9.3). Do this for a 3 × 3 Jordan block—the generalization to k × k blocks is similar...
View Full Document

This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online