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Computing roots of matrix products
Author(s) -
Fassbender H.,
Benner P.
Publication year - 2001
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200108115134
Subject(s) - root (linguistics) , rounding , square root , computation , matrix (chemical analysis) , mathematics , matrix multiplication , product (mathematics) , polynomial , lu decomposition , algorithm , matrix decomposition , computer science , mathematical analysis , eigenvalues and eigenvectors , geometry , physics , philosophy , linguistics , materials science , quantum mechanics , composite material , quantum , operating system
The problem of computing a kth root of a matrix product W = Π k i=1 A i is considered. The explicit computation of W may produce a highly inaccurate result due to rounding errors, such that the computed root W 1/k is far from the actual root W 1/k . An algorithm for computing the square root of W is presented which avoids the explicit computation of W by employing the periodic Schur decomposition and therefore yields better accuracy in the computed root W 1/2 . In principle, the techniques are applicable to k > 2 as well but lead to solving 2 × 2 polynomial matrix equations which are difficult to treat. The case k = 3 is also addressed briefly.