Linearly scaling direct method for accurately inverting sparse banded matrices

TL;DR

This paper introduces a new linear-time algorithm for accurately inverting sparse banded matrices, improving computational efficiency and extending applicability to matrices with additional non-zero entries, with demonstrated advantages over traditional methods.

Contribution

The paper presents a novel O(n) algorithm for inverting banded matrices, including extensions for matrices with extra non-zero entries and full inverse computation, with implementations and parallelization options.

Findings

The new algorithm is faster and more accurate than Gaussian elimination.

It effectively handles matrices with small off-band non-zero entries.

Performance is validated on large random and Poisson equation matrices.

Abstract

In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main diagonal, need often to be inverted in order to solve the associated linear system of equations. In this work, we introduce a new O(n) algorithm for solving such a system, being n X n the size of the matrix. We produce the analytical recursive expressions that allow to directly obtain the solution, as well as the pseudocode for its computer implementation. Moreover, we review the different options for possibly parallelizing the method, we describe the extension to deal with matrices that are banded plus a small number of non-zero entries outside the band, and we use the same ideas to produce a method for obtaining the full inverse matrix. Finally, we show that the New Algorithm is competitive, both in accuracy and in numerical efficiency, when compared to a standard method based in Gaussian elimination. We do this using sets of large random banded matrices, as well as the ones that appear when one tries to solve the 1D Poisson equation by finite differences.