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A general framework for recursions for Krylov space solvers

by M. H. Gutknecht

(Report number 2005-09)

Abstract
Krylov space methods for solving linear systems of equations come in many flavors and use various types of recursions to generate iterates $x_n$ (approximate solutions of $Ax = b$), corresponding residuals $r_n := b - A x_n$, and direction vectors (or search directions) $v_n := (x_{n+1} - x_n) / \omega_n$. Starting from a general definition for a Krylov space solver we give necessary and sufficient conditions for the existence of various types of recursions, and we recall the relations that exist between the matrix representations of these recursions. Much of this is more or less well known, but there are also some new, perhaps even surprising aspects. In particular, we introduce what we call the general inconsistent OrthoRes algorithm, which in contrast to the other recursions is also applicable in situations where for some $n$ the iterate $\bfx_n$ is not defined due to a so-called pivot breakdown.

Keywords:

@Techreport{G05_348,
  author = {M. H. Gutknecht},
  title = {A general framework for recursions for Krylov space solvers },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2005-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2005/2005-09.pdf },
  year = {2005}
}

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