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An efficient algorithm for computing real power of a matrix and a related matrix function. (English) Zbl 0637.65036
The algorithm proposed to compute A r and (A \(r-1)(A-1)^{-1}\), for a square matrix A and a real r, uses the binary expansion of r and has the logarithmic computational complexity with respect to r. The main idea is to reduce it by cumulating computations; the same idea has been applied previously for r integer and not just for matrices [cf. D. E. Knuth, The art of computer programming, II (1969; Zbl 0191.180)].
Reviewer: A.de Castro
MSC:
65F30 Other matrix algorithms (MSC2010)
68Q25 Analysis of algorithms and problem complexity
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
Software:
ALGOL 60
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References:
[1] F. R. Gantmacher: Theory of matrices. (in Russian). Moscow 1966. · Zbl 0136.00410
[2] B. Randell L. J. Russel: Algol 60 Implementation. Academic Press 1964. Russian translation: Mir 1967. · Zbl 0149.37603
[3] D. E. Knuth: The art of computer programming, vol 2. Addison-Wesley 1969. Russian translation: Mir 1977. · Zbl 0191.18001
[4] J. Ježek: Computation of matrix exponential, square root and logarithm. (in Czech). Knižnica algoritmov, diel III, symposium Algoritmy, SVTS Bratislava 1975.
[5] J.Ježek: General matrix power and sum of matrix powers. (in Czech). Knižnica algoritmov, diel IX, symposium Algoritmy, SVTS Bratislava 1987.
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