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

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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 Buy from 7.3 Functions of Diagonalizable Matrices 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...
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