CG Versus MINRES: An Empirical Comparison

Abstract

For iterative solution of symmetric systems the conjugate gradient method (CG) is commonly used when A is positive definite, while the minimum residual method (MINRES) is typically reserved for indefinite systems. We investigate the sequence of approximate solutions generated by each method and suggest that even if A is positive definite, MINRES may be preferable to CG if iterations are to be terminated early. In particular, we show for MINRES that the solution norms are monotonically increasing when A is positive definite (as was already known for CG), and the solution errors are monotonically decreasing. We also show that the backward errors for the MINRES iterates are monotonically decreasing.

Keywords

• Conjugate gradient method (CG), Conjugate residual method (CR), Iterative method, Krylov subspace method, Linear equations, Minimum residual method (MINRES), Sparse matrix, Trust-region method.