# Robust Task-Parallel Solution of the Triangular Sylvester Equation

## TL;DR

This paper introduces a robust, task-parallel solver for the triangular Sylvester equation that prevents overflow through dynamic scaling, matching the performance of non-robust solvers while expanding solvable problem scope.

## Contribution

It develops a level-3 BLAS-based task-parallel solver with overflow protection, enhancing robustness without sacrificing performance.

## Key findings

- Achieves robustness comparable to LAPACK's dtrsyl
- Prevents overflow in backward substitution
- Maintains similar performance to non-robust solvers

## Abstract

The Bartels-Stewart algorithm is a standard approach to solving the dense Sylvester equation. It reduces the problem to the solution of the triangular Sylvester equation. The triangular Sylvester equation is solved with a variant of backward substitution. Backward substitution is prone to overflow. Overflow can be avoided by dynamic scaling of the solution matrix. An algorithm which prevents overflow is said to be robust. The standard library LAPACK contains the robust scalar sequential solver dtrsyl. This paper derives a robust, level-3 BLAS-based task-parallel solver. By adding overflow protection, our robust solver closes the gap between problems solvable by LAPACK and problems solvable by existing non-robust task-parallel solvers. We demonstrate that our robust solver achieves a similar performance as non-robust solvers.