M in A res : An Iterative Solver for Symmetric Linear Systems
Alexis Montoison, Dominique Orban, Michael A. Saunders · SIAM Journal on Matrix Analysis and Applications · 2025
Abstract. We introduce an iterative solver named MinAres for symmetric linear systems [Formula: see text], where [Formula: see text] is possibly singular. MinAres is based on the symmetric Lanczos process, like Minres and Minres-qlp, but it minimizes [Formula: see text] in each Krylov subspace rather than [Formula: see text], where [Formula: see text] is the current residual vector. When [Formula: see text] is symmetric, MinAres minimizes the same quantity [Formula: see text] as Lsmr, but in more relevant Krylov subspaces, and it requires only one matrix-vector product [Formula: see text] per iteration, whereas Lsmr would need two. Our numerical experiments with Minres-qlp and Lsmr show that MinAres is a pertinent alternative on consistent symmetric systems and the most suitable Krylov method for inconsistent symmetric systems. We derive properties of MinAres from an equivalent solver named CAr that is to MinAres as Cr is to Minres, is not based on the Lanczos process, and minimizes [Formula: see text] in the same Krylov subspace as MinAres. We establish that MinAres and CAr generate monotonic [Formula: see text], [Formula: see text], and [Formula: see text] when [Formula: see text] is positive definite.