# The inverse power method k is now at work to see how

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ears to make the connection. Inverse iteration was not introduced until 1944 by the German mathematician Helmut Wielandt (1910–2001). Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.3 Functions of Diagonalizable Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 535 method to B produces an eigenpair (λ − α)−1 , x for B from which the eigenpair (λ, x) for A is determined. That is, if x0 ∈ R (B − λI), and if / It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] yn = Bxn = (A − αI)−1 xn , νn = m(yn ), xn+1 = yn νn for n = 0, 1, 2, . . . , then (νn , xn ) → (λ − α)−1 , x , an eigenpair for B, so (7.3.18) guarantees that − (νn 1 + α, xn ) → (λ, x), an eigenpair for A. Rather than using matrix inversion to compute yn = (A − αI)−1 xn , it’s more eﬃcient to solve the linear system (A − αI)yn = xn for yn . Because this is a system in which the coeﬃcient matrix remains the same from step to step, the eﬃciency is further enhanced by computing an LU factorization of (A − αI) at the outset so that at each step only one forward solve and one back solve (as described on pp. 146 and 153) are needed to determine yn . Advantages. Striking results are often obtained (particularly in the case of symmetric matrices) with only one or two iterations, even when x0 is nearly in R (B − λI) = R (A − λI). For α close to λ, computing an accurate ﬂoating-point solution of (A − αI)yn = xn is diﬃcult because A − αI is nearly singular, and this almost surely guarantees that (A−αI)yn = xn is an ill-conditioned system. But only the direction of the solution is important, and the direction of a computed solution is usually reasonable in spite of conditioning problems. Finally, the algorithm can be adapted to compute approximations of eigenvectors associated with complex eigenvalues. Disadvantages. Only one eigenpair at a time is computed, and an approximate eigenvalue must be known in advance...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online