## Abstract

This is an experimental computational account of projection algorithms for the linear best approximation problem. We focus on the sequential and simultaneous versions of Dykstra's algorithm and the Halpern-Lions-Wittmann-Bauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the Euclidean space. These algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes. We compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms. For the simultaneous versions we present a new component-averaging variant that substantially accelerates their initial behavior for sparse systems.

Original language | English |
---|---|

Pages (from-to) | 111-123 |

Number of pages | 13 |

Journal | Linear Algebra and Its Applications |

Volume | 416 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 2006 |

### Bibliographical note

Funding Information:We thank Yaniv Zaks for preliminary programming, Arik F. Hatwell for his skillful programming and Dan Gordon, from the Department of Computer Science at the University of Haifa, for his remarks on generating test examples. This research is supported by grant No. 2003275 from the United States–Israel Binational Science Foundation (BSF) and by a National Institutes of Health (NIH) grant No. HL70472. Part of this work was done at the Center for Computational Mathematics and Scientific Computation (CCMSC) at the University of Haifa and supported by research grant No. 522/04 from the Israel Science Foundation (ISF).

## Keywords

- Computational acceleration
- Linear best approximation
- Sequential projection algorithms
- Simultaneous projection algorithms

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics