# Geometrical inverse matrix approximation for least-squares problems and acceleration strategies

## TL;DR

This paper extends a geometrical inverse approximation method for least-squares problems, integrating acceleration strategies like matrix extrapolation and heuristic schemes, demonstrated through large-scale numerical experiments.

## Contribution

It adapts the MinCos iterative scheme to least-squares problems and combines it with various acceleration techniques for improved convergence.

## Key findings

- Acceleration strategies enhance convergence speed.
- Numerical experiments validate effectiveness on large-scale problems.
- Proposed methods outperform traditional approaches in tests.

## Abstract

We extend the geometrical inverse approximation approach for solving linear least-squares problems. For that we focus on the minimization of $1-\cos(X(A^TA),I)$, where $A$ is a given rectangular coefficient matrix and $X$ is the approximate inverse. In particular, we adapt the recently published simplified gradient-type iterative scheme MinCos to the least-squares scenario. In addition, we combine the generated convergent sequence of matrices with well-known acceleration strategies based on recently developed matrix extrapolation methods, and also with some deterministic and heuristic acceleration schemes which are based on affecting, in a convenient way, the steplength at each iteration. A set of numerical experiments, including large-scale problems, are presented to illustrate the performance of the different accelerations strategies.