A Robust Hermitian and Skew-Hermitian Based Multiplicative Splitting Iterative Method for the Continuous Sylvester Equation

Abstract

For solving the continuous Sylvester equation, a class of Hermitian and skew-Hermitian based multiplicative splitting iteration methods is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations, and it can be equivalently written as two multiplicative splitting matrix equations. When both coefficient matrices in the continuous Sylvester equation are (non-symmetric) positive semi-definite, and at least one of them is positive definite, we can choose Hermitian and skew-Hermitian (HS) splittings of matrices A and B in the first equation, and the splitting of the Jacobi iterations for matrices A and B in the second equation in the multiplicative splitting iteration method. Convergence conditions of this method are studied in depth, and numerical experiments show the efficiency of this method. Moreover, by numerical computation, we show that multiplicative splitting can be used as a splitting preconditioner and induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the continuous Sylvester equation.

Keywords

• Sylvester equation
• matrix equation
• multiplicative splitting
• Hermitian and skew-Hermitian splitting
• iterative methods