Two Iterative algorithms for the matrix sign function based on the adaptive filtering technology

TL;DR

This paper introduces two efficient iterative algorithms for computing the matrix sign function of large sparse matrices, combining adaptive filtering with Newton-based methods to improve efficiency while maintaining accuracy.

Contribution

The paper presents novel algorithms that integrate adaptive filtering with Newton and Newton-Schultz methods for large sparse matrices, enhancing computational efficiency.

Findings

Algorithms outperform traditional Newton and Newton-Schultz methods in efficiency.

Filtering has minimal impact on iterative accuracy.

Numerical results confirm theoretical error bounds.

Abstract

In this paper, two new efficient algorithms for calculating the sign function of the large-scale sparse matrix are proposed by combining filtering algorithm with Newton method and Newton Schultz method respectively. Through the theoretical analysis of the error diffusion in the iterative process, we designed an adaptive filtering threshold, which can ensure that the filtering has little impact on the iterative process and the calculation result. Numerical experiments are consistent with our theoretical analysis, which shows that the computational efficiency of our method is much better than that of Newton method and Newton Schultz method, and the computational error is of the same order of magnitude as that of the two methods.