RECSY and SCASY Library Software: Recursive Blocked and Parallel Algorithms for Sylvester-Type Matrix Equations with Some Applications

In this contribution, we review state-of-the-art high-performance computing software for solving common standard and generalized continuous-time and discrete-time Sylvester-type matrix equations. The analysis is based on RECSY and SCASY software libraries. Our algorithms and software rely on the standard Schur method. Two ways of introducing blocking for solving matrix equations in reduced (quasi-triangular) form are reviewed. Most common is to perform a fix block partitioning of the matrices involved and rearrange the loop nests of a single-element algorithm so that the computations are performed on submatrices (matrix blocks). Another successful approach is to combine recursion and blocking. We consider parallelization of algorithms for reduced matrix equations at two levels: globally in a distributed memory paradigm, and locally on shared memory or multicore nodes as part of the distributed memory environment. Distributed wave-front algorithms are considered to compute the solution to the reduced triangular systems. Parallelization of recursive blocked algorithms is done in two ways. The simplest way is so-called implicit data parallelization, which is obtained by using SMP-aware implementations of level 3 BLAS. Complementary to this, there is also the possibility of invoking task parallelism. This is done by explicit parallelization of independent tasks in a recursion tree using OpenMP. A brief account of some software issues for the RECSY and SCASY libraries is given. Theoretical results are confirmed by experimental results.