A Numerical Study for MrR Algorithm for Linear Equations with Symmetric Matrices

We treat with Krylov subspace methods for efficiently solving linear equations. AZMJ variant of Orthomin(2) [1, 2] (abbreviated as AZMJ) has been proposed for solving the linear equations. In this paper, we redesign an alternative AZMJ variant, i.e., propose an alternative minimum residual method for symmetric matrices using the coupled twoterm recurrences formulated by Rutishauser. The recurrence coefficients are determined by imposing the A-orthogonality on the residuals as well as the Conjugate Residual (CR) method. The proposed variant is referred to as MrR. It is mathematically equivalent to CR and AZMJ, but the implementations are different; the recurrence formulas contain alternative expressions for the auxiliary vector and the recurrence coefficients. Moreover, we derive a preconditioned MrR algorithm. By numerical experiments on the linear equations with real symmetric matrices, we demonstrate that the residual norms of MrR converge faster than those of CG and AZMJ, and the preconditioned MrR algorithm is effective.