Arnoldi received his undergraduate degree in

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Unformatted text preview: is to determine the first matrix Ak for which {I, A, A2 , . . . , Ak } is linearly dependent. In other words, if k is the smallest positive integer such that k−1 Ak = j =0 αj Aj , then the minimum polynomial for A is T H IG k−1 m(x) = xk − R Y αj xj . j =0 The Gram–Schmidt orthogonalization procedure (p. 309) with the standard inner product A B = trace (A∗ B) (p. 286) is the perfect theoretical tool for determining k and the αj ’s.√ Gram–Schmidt applied to {I, A, A2 , . . .} begins by setting U0 = I/ I F = I/ n, and it proceeds by sequentially computing P Aj − O C Uj = Aj − j −1 i=0 j −1 i=0 Ui Aj Ui Ui Aj Ui for j = 1, 2, . . . (7.11.2) F k−1 until Ak − i=0 Ui Ak Ui = 0. The first such k is the smallest positive integer such that Ak ∈ span {U0 , U1 , . . . , Uk−1 } = span I, A, . . . , Ak−1 . The k−1 j coefficients αj such that Ak = are easily determined from the j =0 αj A upper-triangular matrix R in the QR factorization produced by the Gram– Schmidt process. To see how, extend the notation in the discussion on p. 311 in an obvious way to write (7.11.2) in matrix form as ⎛ ⎞ r0 k ν0 r01 · · · r0k−1 ⎜ 0 ν1 · · · r1k−1 r1 k ⎟ ⎜. . .⎟ .. .. ⎜. k . . ⎟ , (7.11.3) I | A | · · · | A = U0 | U1 | · · · | Uk ⎜ . . . . .⎟ ⎜ ⎟ ⎝0 0 νk−1 rk−1k ⎠ 0 Copyright c 2000 SIAM 0 ··· 0 0 Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 644 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 where ν0 = I F ⎛ If we set R = ⎝ = √ ··· .. . ν0 j −1 i=0 n , νj = A j − ⎞ r0k−1 .⎠ . . νk−1 ⎛ and c = ⎝ Ui Aj Ui ⎞ r 0k . . . F , and rij = Ui Aj . ⎠, then (7.11.3) implies that r k −1 k Ak = U0 | · · · |Uk−1 c = I| · · · |Ak−1 R−1 c, so R−1 c = k−1 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu ⎞ ⎛ α0 ⎝.⎠ . . αk−1 contains the coefficients such that Ak = j =0 αj Aj , and thus the coefficients in the minimum polynomial are determined. D E Caution! While Gram–...
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