On MrR (mister R) method for solving linear equations with symmetric matrices

Abstract

Krylov subspace methods, such as the Conjugate Gradient (CG) and Conjugate Residual (CR) methods, are treated for efficiently solving a linear system of equations with symmetric matrices. AZMJ variant of Orthomin(2) (abbreviated as AZMJ) [1] has recently been proposed for solving the linear equations. In this paper, an alternative AZMJ variant is redesigned, i.e., an alternative minimum residual method for symmetric matrices is proposed by using the coupled two-term recurrences formulated by Rutishauser. The recurrence coefficients are determined by imposing the A-orthogonality on the residuals as well as CR. Our proposed variant is referred to as MrR. It is mathematically equivalent to CR and AZMJ, but the implementations are different; the recurrence formulae contain alternative expressions for the auxiliary vectors and the recurrence coefficients. Through numerical experiments on the linear equations with real symmetric matrices, it is demonstrated that the residual norms of MrR converge faster than those of CG and AZMJ.