The Akhiezer iteration and inverse-free solvers for Sylvester matrix equations

Two inverse-free iterative methods are developed for solving Sylvester matrix equations when the spectra of the coefficient matrices are on, or near, known disjoint subintervals of the real axis. Both methods use the recently-introduced Akhiezer iteration: one to address an equivalent problem of approximating the matrix sign function applied to a block matrix and the other to directly approximate the inverse of the Sylvester operator. In each case this results in provable and computable geometric rates of convergence. When the right-hand side matrix is low rank, both methods require only low-rank matrix-matrix products. Relative to existing approaches, the methods presented here can be more efficient and require less storage when the coefficient matrices are dense or otherwise costly to invert. Applications include solving partial differential equations and computing Fr\'echet derivatives.