Efficient Computation for Invertibility Sequence of Banded Toeplitz Matrices

TL;DR

This paper presents an efficient method to determine the invertibility of large banded Toeplitz matrices by reducing the problem to smaller matrices, significantly speeding up computations for solving systems and finding inverses.

Contribution

It introduces a novel algorithm that links invertibility of large matrices to small ones, with optimized time and space complexity for practical applications.

Findings

Invertibility of large banded Toeplitz matrices can be reduced to small matrix invertibility.

The proposed algorithm has a time complexity of approximately 2.5k^2 n + kn.

Space complexity is limited to 3k^2, enabling efficient preprocessing.

Abstract

When solving systems of banded Toeplitz equations or calculating their inverses, it is necessary to determine the invertibility of the matrices beforehand. In this paper, we equate the invertibility of an $n$-order banded Toeplitz matrix with bandwidth $2k+1$ to that of a small $k*k$ matrix. By utilizing a specially designed algorithm, we compute the invertibility sequence of a class of banded Toeplitz matrices with a time complexity of $5k^2n/2+kn$ and a space complexity of $3k^2$ where $n$ is the size of the largest matrix. This enables efficient preprocessing when solving equation systems and inverses of banded Toeplitz matrices.