For iterative solution of symmetric systems Ax = b, the conjugate gradient method (CG) is commonly used when A is positive definite, while the minimal residual method (MINRES) is typically reserved for indefinite systems. We investigate the sequence of 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. The classic symmetric positive-definite system comes from the full-rank least-squares (LS) problem min ‖Ax − b‖. Specialization of CG and MINRES to the associated normal equation A T Ax = A T b leads to LSQR and LSMR respectively. We include numerical comparisons of these two LS solvers becaus
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