Distributed and Shared Memory Block Algorithms for the Triangular Sylvester Equation with $\operatorname{sep}^{ - 1} $ Estimators

Coarse grain message passing and shared memory algorithms for solving the quasi-triangular Sylvester equation are discussed. The basic algorithm is of block type, i.e., rich in matrix-matrix operations. The focus is on computing reliable estimates of the ${\operatorname{sep}}^{ - 1} $ function (a natural condition number for the Sylvester equation and the invariant subspace problem). Estimators based on the Frobenius norm and the 1-norm, respectively, are presented. Accuracy, efficiency, and reliability results are presented. The applicability of the estimators to both the shared memory and distributed memory paradigms are discussed. Some performance results of the parallel block algorithms with condition estimators are also presented. The reliability of both estimators are very good. The Frobenius norm–based estimator is much more efficient in both sequential and parallel settings (on average between four to five times). Further, it is applicable to both the standard and generalized problems.