Two sided Matrix Algorithms

Two sided Matrix Algorithms - Recursive Blocked Algorithms...

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Recursive Blocked Algorithms for Solving Triangular Systems—Part II: Two-Sided and Generalized Sylvester and Lyapunov Matrix Equations ISAK JONSSON and BO KA b GSTR ¨ OM Ume˚a University We continue our study of high-performance algorithms for solving triangular matrix equations. They appear naturally in different condition estimation problems for matrix equations and var- ious eigenspace computations, and as reduced systems in standard algorithms. Building on our successful recursive approach applied to one-sided matrix equations (Part I), we now present novel recursive blocked algorithms for two-sided matrix equations, which include matrix product terms such as AX B T . Examples are the discrete-time standard and generalized Sylvester and Lyapunov equations. The means for achieving high performance is the recursive variable blocking, which has the potential of matching the memory hierarchies of today’s high-performance computing systems, and level-3 computations which mainly are performed as GEMM operations. Different implemen- tation issues are discussed, including the design of efficient new algorithms for two-sided matrix products. We present uniprocessor and SMP parallel performance results of recursive blocked al- gorithms and routines in the state-of-the-art SLICOT library. Although our recursive algorithms with optimized kernels for the two-sided matrix equations perform more operations, the perfor- mance improvements are remarkable, including 10-fold speedups or more, compared to standard algorithms. Categories and Subject Descriptors: F.2.1 [ Analysis of Algorithms and Problem Complexity ]: Numerical Algorithms and Problems— computations on matrices ; G.1.3 [ Numerical Analysis ]: Numerical Linear Algebra— conditioning, linear systems ; G.4 [ Mathematical Software ]: Algo- rithm design and analysis, efficiency, parallel and vector implementations, portability, reliability and robustness General Terms: Algorithms, Performance Additional Key Words and Phrases: Matrix equations, standard discrete-time Sylvester and Lyapunov, generalized Sylvester and Lyapunov, recursion, automatic blocking, superscalar, GEMM- based, level-3 BLAS, SMP parallelization, LAPACK, SLICOT This research was conducted using the resources of the High Performance Computing Center North (HPC2N) and PDC-Parallelldatorcentrum at KTH, Stockholm. Financial support was provided by the Swedish Research Council under grants TFR 98-604 and VR 621-2001-3284. Authors’ address: Department of Computing Science and HPC2N, Ume˚a University, SE-901 87 Ume˚a, Sweden; email: { isak,bokg } @cs.umu.se. Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee.
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