Projection methods for large-scale T-Sylvester equations

The matrix Sylvester equation for congruence, or T-Sylvester equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning T-Sylvester equations is rather well understood and there are stable and efficient numerical algorithms which solve these equations for smallto medium-sized matrices. However, developing numerical algorithms for solving large-scale T-Sylvester equations still remains an open problem. In this paper, we present several projection algorithms based on different Krylov spaces for solving this problem when the right-hand side of the T-Sylvester equation is a low-rank matrix. The new algorithms have been extensively tested, and the reported numerical results show that they work very well in practice, offering a clear guidance on which algorithm is the most convenient in each situation.