$\mathcal O(n)$ working precision inverses for symmetric tridiagonal Toeplitz matrices with $\mathcal O(1)$ floating point calculations

TL;DR

This paper introduces an efficient linear time algorithm for inverting large symmetric tridiagonal Toeplitz matrices with strict diagonal dominance, using only constant floating point operations, significantly improving computational efficiency.

Contribution

The paper presents a novel $ ext{O}(1)$ floating point operation algorithm for inverting strictly diagonally dominant symmetric tridiagonal Toeplitz matrices in linear time.

Findings

Inversion is banded and sparse for strictly diagonally dominant matrices.

The algorithm achieves $ ext{O}(1)$ complexity per operation.

It enables efficient parallelizable approximate solvers.

Abstract

A well known numerical task is the inversion of large symmetric tridiagonal Toeplitz matrices, i.e., matrices whose entries equal $a$ on the diagonal and $b$ on the extra diagonals ($a, b\in \mathbb R$). The inverses of such matrices are dense and there exist well known explicit formulas by which they can be calculated in $\mathcal O(n^2)$. In this note we present a simplification of the problem that has proven to be rather useful in everyday practice: If $\vert a\vert > 2\vert b\vert$, that is, if the matrix is strictly diagonally dominant, its inverse is a band matrix to working precision and the bandwidth is independent of $n$ for sufficiently large $n$. Employing this observation, we construct a linear time algorithm for an explicit tridiagonal inversion that only uses $\mathcal O(1)$ floating point operations. On the basis of this simplified inversion algorithm we outline the cornerstones for an efficient parallelizable approximative equation solver.