Global SCD algorithm for real positive definite linear systems with multiple right-hand sides

Abstract

In the present paper, we present the left conjugate direction (LCD) method for linear systems with multiple right-hand sides. We define the method by using left and right conjugate direction matrices and Petrov–Galerkin condition. The method has no breakdown for real positive definite systems. The method reduces to the usual CG-type method when A is symmetric positive definite. We also show how to apply these methods for solving the Lyapunov matrix equation. Finally a numerical experiment is given to illustrate the result.