change order of exponent for neg. matrix powers #20071
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According to Higham-Lin [1], numerical accuracy is higher for the calculation `A^-k`, for `A` a matrix and `k` an integer, if it is calculated as `(A^-1)^k` [1] A Schur-Pade algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. & Appl., Vol 32/3 1056-1078 Section 6: "Algorithm 6.1 [A^-k = (A^k)^-1] inverts `A^k`, which is potentially a much more ill conditioned matrix than `A`. Intuitively, Algorithm 6.2 [A^-k = (A^-1)^k] should therefore be preferred. Section 9, Experiment 7: "Algorithms 6.2 ... clearly produce much more accurate results than Algorithm 6.1, as we expected."
andreasnoack
approved these changes
Jan 16, 2017
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For something like a tridiagonal matrix it could be cheaper to calculate |
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I think we should just add a specialized method for |
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Was included in #21184 |
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According to Higham-Lin [1], numerical accuracy is higher for the calculation
A^-k, whenAis a matrix andka positive integer, if it is calculated as(A^-1)^k.[1] A Schur-Pade algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. & Appl., Vol 32/3 1056-1078
Section 6:
"Algorithm 6.1 [
A^-k = (A^k)^-1] invertsA^k, which is potentially a much more ill conditioned matrix thanA. Intuitively, Algorithm 6.2 [A^-k = (A^-1)^k] should therefore be preferred.Section 9, Experiment 7: "Algorithms 6.2 ... clearly produce much more accurate results than Algorithm 6.1, as we expected."