Recursive blocked algorithms for solving triangular systems—Part II

Abstract

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 various 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 implementation 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 algorithms 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 performance improvements are remarkable, including 10-fold speedups or more, compared to standard algorithms.

Knowledge-Based Automatic Generation of Partitioned Matrix Expressions

“…The approach has been successfully applied to all the operations included in the BLAS [2] and RECSY [3,4] libraries and to many included in the LAPACK [5] library. In general, the methodology applies to any operation that can be expressed in a "divide and conquer" fashion.…”

“…3, while in Sect. 4 we describe how to use partitionings to obtain PMEs. We draw conclusion in Sect.…”

Retracing the residual curve of a Lyapunov equation solver

“…They all require O(n 3 ) arithmetic operations and O(n 2 ) words of storage, but efficient parallel algorithms have been implemented [15,16,8,7].…”

Shifted Kronecker Product Systems

“…A reshaped special case of this problem is the discrete-time Sylvester equation A (1) XA (2) T − X = B. As with many matrix equations of this variety, the first step is to convert A (1) and A (2) to triangular form via the Schur decompoistion. The resulting system can then be solved via a back-substitution process.…”

“…The resulting system can then be solved via a back-substitution process. Jonsson and Kågström [2] have developed block recursive methods for these kinds of problems and they are very effective in high-performance computing environments. The method we present is also recursive and can be regarded as a generalization of their technique.…”