Abstract
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 language | English (US) |
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Pages (from-to) | 1248-1267 |
Number of pages | 20 |
Journal | SIAM Journal on Matrix Analysis and Applications |
Volume | 42 |
Issue number | 3 |
DOIs | |
State | Published - 2021 |
Bibliographical note
Publisher Copyright:© 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.
Keywords
- Helmholtz equation
- Orthogonal polynomial
- Polynomial iteration
- Polynomial preconditioning
- Shortterm recurrence