## Abstract

This paper introduces a conjugate gradients (CG) acceleration of the coordinate descent algorithm (CD) for linear systems. It is shown that the Kaczmarz algorithm (KACZ) can simulate CD exactly, so CD can be accelerated by CG similarly to the CG acceleration of KACZ (Björck and Elfving in BIT 19:145–163, 1979). Experimental results were carried out on large sets of problems of reconstructing bandlimited functions from random sampling. The randomness causes extreme variance between different instances of these problems, thus causing extreme variance in the advantage of CGCD over CD. The reduction of the number of iterations by CGCD varies from about 50–90% and beyond. The implementation of CGCD is simple. CGCD can also be used for the parallel solution of linear systems derived from partial differential equations, and for the efficient solution of multiple right-hand-side problems and matrix inversion.

Original language | English |
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Article number | 86 |

Journal | Journal of Scientific Computing |

Volume | 96 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2023 |

### Bibliographical note

Funding Information:The author is grateful to the reviewers for their very useful comments.

Publisher Copyright:

© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- CD
- CGCD
- CGMN
- Conjugate gradients acceleration
- Coordinate descent
- Gauss–Seidel
- Kaczmarz algorithm
- Linear systems
- Matrix inversion
- Multiple right-hand-sides
- Parallelism

## ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Numerical Analysis
- Engineering (all)
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics