On numerical solution of full rank linear systems

TL;DR

This paper extends the augmented block Cimmino method to efficiently solve full rank underdetermined and overdetermined linear systems by augmenting matrices and utilizing projections onto orthogonal subspaces.

Contribution

It generalizes the augmented block Cimmino method to full rank systems, including underdetermined and overdetermined cases, with a detailed analysis of these extensions.

Findings

Method effectively solves full rank underdetermined systems.

Method efficiently handles overdetermined systems through row augmentation.

Analysis confirms the equivalence and stability of the extended method.

Abstract

Matrices can be augmented by adding additional columns such that a partitioning of the matrix in blocks of rows defines mutually orthogonal subspaces. This augmented system can then be solved efficiently by a sum of projections onto these subspaces. The equivalence to the original linear system is ensured by adding additional rows to the matrix in a specific form. The resulting solution method is known as the augmented block Cimmino method. Here this method is extended to full rank underdetermined systems and to overdetermined systems. In the latter case, rows of the matrix, not columns, must be suitably augmented. The article presents an analysis of these methods.