A New Fast Direct Method For Solving Quasi-Toeplitz System of Equations

TL;DR

This paper introduces a new fast direct method specifically designed for solving linear systems with tridiagonal Quasi-Toeplitz matrices, common in differential equations and stochastic processes, outperforming classical methods in speed.

Contribution

The paper proposes a novel direct solution technique optimized for tridiagonal Quasi-Toeplitz matrices, enhancing computational efficiency over traditional LU-based methods.

Findings

Faster solution times for large systems compared to LU and PLU methods.

Effective for discretized differential equations with Neumann boundary conditions.

Applicable to Quasi-Birth-and-Death process matrix equations.

Abstract

The objective of this study is to present a novel, efficient, and fast direct method for solving linear systems of equations whose coefficient matrix is a tridiagonal Quasi-Toeplitz matrix. Such matrices are frequently encountered in the discretization of second-order differential equation problems with Neumann boundary conditions, the discretization of Quasi-Birth-and-Death processes known as QBD matrix equations, and other related applications. The presented method has been demonstrated to produce favorable results in terms of run time for moderately large systems when compared to classical direct methods, such as the LU and PLU methods.