Recent zbMATH articles in MSC 15A06https://zbmath.org/atom/cc/15A062024-03-13T18:33:02.981707ZWerkzeugProxy-GMRES: preconditioning via GMRES in polynomial spacehttps://zbmath.org/1528.650202024-03-13T18:33:02.981707Z"Ye, Xin"https://zbmath.org/authors/?q=ai:ye.xin"Xi, Yuanzhe"https://zbmath.org/authors/?q=ai:xi.yuanzhe"Saad, Yousef"https://zbmath.org/authors/?q=ai:saad.yousefSummary: This paper proposes a class of polynomial preconditioners for solving non-Hermitian linear systems of equations. The polynomial is obtained from a least-squares approximation in polynomial space instead of a standard Krylov subspace. The process for building the polynomial relies on an Arnoldi-like procedure in a small dimensional polynomial space and is equivalent to performing GMRES in polynomial space. It is inexpensive and produces the desired polynomial in a numerically stable way. A few improvements to the basic scheme are discussed including the development of a short-term recurrence and the use of compound preconditioners. Numerical experiments, including a test with challenging nonnormal three-dimensional Helmholtz equations and a few publicly available sparse matrices, are provided to illustrate the performance of the proposed preconditioners.