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Recursive blocked algorithms for solving triangular systems—Part I: one-sided and coupled Sylvester-type matrix equations

Recursive blocked algorithms for solving triangular systems—Part I: one-sided and coupled... Triangular matrix equations appear naturally in estimating the condition numbers of matrix equations and different eigenspace computations, including block-diagonalization of matrices and matrix pairs and computation of functions of matrices. To solve a triangular matrix equation is also a major step in the classical Bartels--Stewart method for solving the standard continuous-time Sylvester equation ( AX − XB = C ). We present novel recursive blocked algorithms for solving one-sided triangular matrix equations, including the continuous-time Sylvester and Lyapunov equations, and a generalized coupled Sylvester equation. The main parts of the computations are performed as level-3 general matrix multiply and add (GEMM) operations. In contrast to explicit standard blocking techniques, our recursive approach leads to an automatic variable blocking that has the potential of matching the memory hierarchies of today's HPC systems. Different implementation issues are discussed, including when to terminate the recursion, the design of new optimized superscalar kernels for solving leaf-node triangular matrix equations efficiently, and how parallelism is utilized in our implementations. Uniprocessor and SMP parallel performance results of our recursive blocked algorithms and corresponding routines in the state-of-the-art libraries LAPACK and SLICOT are presented. The performance improvements of our recursive algorithms are remarkable, including 10-fold speedups compared to standard algorithms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Mathematical Software (TOMS) Association for Computing Machinery

Recursive blocked algorithms for solving triangular systems—Part I: one-sided and coupled Sylvester-type matrix equations

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Publisher
Association for Computing Machinery
Copyright
Copyright © 2002 by ACM Inc.
ISSN
0098-3500
DOI
10.1145/592843.592845
Publisher site
See Article on Publisher Site

Abstract

Triangular matrix equations appear naturally in estimating the condition numbers of matrix equations and different eigenspace computations, including block-diagonalization of matrices and matrix pairs and computation of functions of matrices. To solve a triangular matrix equation is also a major step in the classical Bartels--Stewart method for solving the standard continuous-time Sylvester equation ( AX − XB = C ). We present novel recursive blocked algorithms for solving one-sided triangular matrix equations, including the continuous-time Sylvester and Lyapunov equations, and a generalized coupled Sylvester equation. The main parts of the computations are performed as level-3 general matrix multiply and add (GEMM) operations. In contrast to explicit standard blocking techniques, our recursive approach leads to an automatic variable blocking that has the potential of matching the memory hierarchies of today's HPC systems. Different implementation issues are discussed, including when to terminate the recursion, the design of new optimized superscalar kernels for solving leaf-node triangular matrix equations efficiently, and how parallelism is utilized in our implementations. Uniprocessor and SMP parallel performance results of our recursive blocked algorithms and corresponding routines in the state-of-the-art libraries LAPACK and SLICOT are presented. The performance improvements of our recursive algorithms are remarkable, including 10-fold speedups compared to standard algorithms.

Journal

ACM Transactions on Mathematical Software (TOMS) – Association for Computing Machinery

Published: Dec 1, 2002

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