akk where at the j th step of the process the vector

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Unformatted text preview: Schmidt works fine to produce m(x) in exact arithmetic, things are not so nice in floating-point arithmetic. For example, if A has a dominant eigenvalue, then, as explained in the power method (Example 7.3.7, p. 533), Ak asymptotically approaches the dominant spectral projector G1 , so, as k grows, Ak becomes increasingly close to span I, A, . . . , Ak−1 . Consequently, finding the first Ak that is truly in span I, A, . . . , Ak−1 is an ill-conditioned problem, and Gram–Schmidt may not work well in floatingpoint arithmetic—the modified Gram–Schmidt algorithm (p. 316), or a version of Householder reduction (p. 341), or Arnoldi’s method (p. 653) works better. Fortunately, explicit knowledge of the minimum polynomial often is not needed in applied work. T H IG R The relationship between the characteristic polynomial c(x) and the minimum polynomial m(x) for A is now transparent. Since Y P c(x) = (x − λ1 )a1 (x − λ2 )a2 · · · (x − λs )as , where aj = alg mult (λj ), m(x) = (x − λ1 )k1 (x − λ2 )k2 · · · (x − λs )ks , where kj = index (λj ), and O C it’s clear that m(x) divides c(x). Furthermore, m(x) = c(x) if and only if alg mult (λj ) = index (λj ) for each λj ∈ σ (A) . Matrices for which m(x) = c(x) are said to be nonderogatory matrices, and they are precisely the ones for which geo mult (λj ) = 1 for each eigenvalue λj because m(x) = c(x) ⇐⇒ alg mult (λj ) = index (λj ) for each j ⇐⇒ there is only one Jordan block for each λj ⇐⇒ there is only one independent eigenvector for each λj ⇐⇒ geo mult (λj ) = 1 for each λj . In addition to dividing the characteristic polynomial c(x), the minimum polynomial m(x) divides all other annihilating polynomials p(x) for A because deg[m(x)] ≤ deg[p(x)] insures the existence of polynomials q (x) and r(x) (quotient and remainder) such that p(x) = m(x)q (x) + r(x), Copyright c 2000 SIAM where deg[r(x)] < deg[m(x)]. Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.11 Minimum Polynomials and Krylov Methods http://www.amazon.com/exec...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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