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Toward efficient polynomial preconditioning for GMRES

Journal Article · · Numerical Linear Algebra with Applications
DOI:https://doi.org/10.1002/nla.2427· OSTI ID:1838187
 [1];  [2]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Baylor Univ., Waco, TX (United States)
+ Show Author Affiliations

Here, we present a polynomial preconditioner for solving large systems of linear equations. The polynomial is derived from the minimum residual polynomial (the GMRES polynomial) and is more straightforward to compute and implement than many previous polynomial preconditioners. Our current implementation of this polynomial using its roots is naturally more stable than previous methods of computing the same polynomial. We implement further stability control using added roots, and this allows for high degree polynomials. We discuss the effectiveness and challenges of root-adding and give an additional check for stability. In this article, we study the polynomial preconditioner applied to GMRES; however it could be used with any Krylov solver. This polynomial preconditioning algorithm can dramatically improve convergence for some problems, especially for difficult problems, and can reduce dot products by an even greater margin.

Research Organization:
Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
Grant/Contract Number:
NA0003525
OSTI ID:
1838187
Report Number(s):
SAND2021-16131J; 702364
Journal Information:
Numerical Linear Algebra with Applications, Journal Name: Numerical Linear Algebra with Applications Journal Issue: 4 Vol. 29; ISSN 1070-5325
Publisher:
WileyCopyright Statement
Country of Publication:
United States
Language:
English

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Related Subjects

97 MATHEMATICS AND COMPUTING
GMRES
linear equations
polynomial preconditioning