Proxy-gmres: Preconditioning via gmres in polynomial space


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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.

Original languageEnglish (US)
Pages (from-to)1248-1267
Number of pages20
JournalSIAM Journal on Matrix Analysis and Applications
Issue number3
StatePublished - 2021

Bibliographical note

Funding Information:
\ast Received by the editors June 2, 2020; accepted for publication (in revised form) by L. Giraud May 27, 2021; published electronically August 5, 2021. Funding: The work of the authors was supported by the National Science Foundation grants DMS-1521573, DMS-1912048, and OAC-2003720. \dagger Hewlett Packard Enterprise, Bloomington, MN 55425 USA ( \ddagger Department of Mathematics, Emory University, Atlanta, GA 30322 USA ( \S Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 USA (

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.


  • Helmholtz equation
  • Orthogonal polynomial
  • Polynomial iteration
  • Polynomial preconditioning
  • Shortterm recurrence


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