Blocked Schur Algorithms for Computing the Matrix Square Root

Deadman, Edvin and Higham, Nicholas J. and Ralha, Rui (2012) Blocked Schur Algorithms for Computing the Matrix Square Root. [MIMS Preprint]

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Abstract

The Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using either standard blocking or recursive blocking the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show significant speedups over the point algorithm, both in the square root phase and in the algorithm as a whole. In parallel implementations, recursive blocking is found to provide better performance than standard blocking when the parallelism comes only from threaded BLAS, but the reverse is true when parallelism is explicitly expressed using OpenMP. The excellent numerical stability of the point algorithm is shown to be preserved by blocking. These results are extended to the real Schur method. Blocking is also shown to be effective for multiplying triangular matrices.

Item Type: MIMS Preprint
Additional Information: To appear in Springer Lecture Notes in Computer Science
Uncontrolled Keywords: matrix function square root Schur recursive
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr Edvin Deadman
Date Deposited: 05 Dec 2012
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/1926

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