From Sylvester's determinant identity to Cramer's rule

TL;DR

This paper introduces a novel method for solving large linear systems using Sylvester's determinant identity, enabling parallel computation and confirmed by numerical experiments.

Contribution

It presents a new approach combining Sylvester's determinant identity with Cramer's rule, facilitating parallel processing for large linear equations.

Findings

Method enables parallel computation of large linear systems.

Numerical experiments confirm theoretical advantages.

New scheme improves efficiency over traditional methods.

Abstract

The object of this paper is to introduce a new and fascinating method of solving large linear equations, based on Cramer's rule or Gaussian elimination but employing Sylvester's determinant identity in its computation process. In addition, a scheme suitable for parallel computing is presented for this kind of generalized Chi\`{o}'s determinant condensation processes, which makes this new method have a property of natural parallelism. Finally, some numerical experiments also confirm our theoretical analysis.