Evaluating Parallel Algorithms for Solving Sylvester-Type Matrix Equations: Direct Transformation-Based Versus Iterative Matrix-Sign-Function-Based Methods

Recent ScaLAPACK-style implementations of the Bartels-Stewart method and the iterative matrix-sign-function-based method for solving continuous-time Sylvester matrix equations are evaluated with respect to generality of use, execution time and accuracy of computed results. The test problems include well-conditioned as well as ill-conditioned Sylvester equations. A method is considered more general if it can effectively solve a larger set of problems. Ill-conditioning is measured with respect to the separation of the two matrices in the Sylvester operator. Experiments carried out on two different distributed memory machines show that the parallel explicitly blocked Bartels-Stewart algorithm can solve more general problems and delivers far more accuracy for ill-conditioned problems. It is also up to four times faster for large enough problems on the most balanced parallel platform (IBM SP), while the parallel iterative algorithm is almost always the fastest of the two on the less balanced platform (HPC2N Linux Super Cluster).