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On Lanczos-type methods for Wilson fermions

by M. H. Gutknecht

(Report number 2000-03)

Abstract
Numerical simulations of lattice gauge theories with fermions rely heavily on the iterative solution of huge sparse linear systems of equations. Due to short recurrences, which mean small memory requirement, Lanczos-type methods (including suitable versions of the conjugate gradient method when applicable) are best suited for this type of problem. The Wilson formulation of the lattice Dirac operator leads to a matrix with special symmetry properties that makes the application of the classical biconjugate gradient (\BICG) particularly attractive, but other methods, for example \BICGSTAB\ and \BICGSTAB2 have also been widely used. We discuss some of the pros and cons of these methods. In particular, we review the specific simplification of \BICG, clarify some details, and discuss general results on the roundoff behavior.

Keywords: system of linear equations, iterative method, biconjugate gradient method, Lanczos-type method, simplified Lanczos method, roundoff errors, finite precision arithmetic, Wilson fermions, Dirac operator, lattice QCD, quantum chromodynamics

@Techreport{G00_262,
  author = {M. H. Gutknecht},
  title = {On Lanczos-type methods for Wilson fermions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2000-03},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2000/2000-03.pdf },
  year = {2000}
}

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