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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
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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
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Please report violations to email@example.com 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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