Single Projection Kaczmarz Extended Algorithms

TL;DR

This paper analyzes deterministic control strategies for the extended Kaczmarz algorithm, demonstrating convergence to least squares solutions in large, inconsistent systems through almost-cyclic or maximal-residual updates.

Contribution

It extends convergence results of the Kaczmarz algorithm to deterministic control strategies, including almost-cyclic and maximal-residual choices.

Findings

Convergence to least squares solutions is proven for deterministic update strategies.

Almost-cyclic and maximal-residual controls ensure asymptotic convergence.

The method is effective for large, inconsistent systems.

Abstract

To find the least squares solution of a very large and inconsistent system of equations, one can employ the extended Kaczmarz algorithm. This method simultaneously removes the error term, such that a consistent system is asymptotically obtained, and applies Kaczmarz iterations for the current approximation of this system. For random corrections of the right hand side and Kaczmarz updates selected at random, convergence to the least squares solution has been shown. We consider the deterministic control strategies, and show convergence to a least squares solution when row and column updates are chosen according to the almost-cyclic or maximal-residual choice.