CG Versus MINRES: An Empirical Comparison

For iterative solution of symmetric systems Ax =b, 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 x k 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 ║x k║ are monotonically increasing when A is positive definite (as was already known for CG), and the solution errors ║x * - x k║ are monotonically decreasing. We also show that the backward errors for the MINRES iterates x k are monotonically decreasing.