CG versus minres : an empirical comparison

الملخص EN

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.