A Systematic Framework for Stable and Cost-Efficient Matrix Polynomial Evaluation

TL;DR

This paper introduces a systematic framework and MATLAB tool for stable, cost-efficient evaluation of matrix polynomials, achieving fewer matrix multiplications without sacrificing numerical stability.

Contribution

It provides a new framework and implementation for obtaining stable coefficients for optimized matrix polynomial evaluation formulas, ensuring numerical stability and efficiency.

Findings

Achieves 1M matrix product savings compared to Paterson--Stockmeyer method.

Maintains numerical accuracy comparable to traditional methods.

Provides a MATLAB tool with stability validation and reliability indicators.

Abstract

A method for evaluating matrix polynomials have recently been developed that require one fewer matrix product ($1M$) than the Paterson--Stockmeyer (PS) method. Since the computational cost for large-scale matrices is asymptotically determined by the number of matrix products, this reduction directly affects the total execution time. However, the coefficients in these optimized formulas emerge as solutions to systems of nonlinear polynomial equations, resulting in multiple potential solution sets. An inappropriate selection of these coefficients can lead to numerical instability in floating-point arithmetic. This paper presents a systematic framework and a MATLAB implementation, MatrixPolEval1, used to obtain and validate stable coefficient sets for matrix polynomials of degrees $m \in \{8, 10, 12\}$ and above. The framework introduces structural variants to maintain stability even when the original configuration fails to yield a robust solution. The provided tool identifies stable coefficient sets using variable precision arithmetic (VPA) and provides a reliability indicator for expected accuracy. Numerical experiments on polynomials arising in applications, including the matrix exponential and geometric series, show that the framework achieves the $1M$ saving while maintaining numerical accuracy comparable to the PS method.