đ¤ AI Summary
This work proposes a systematic framework to enhance the computational efficiency and numerical stability of evaluating high-degree matrix polynomials. Specifically, for polynomial degrees eight and higher, the method generates and validates stable coefficient sets that reduce the number of required matrix multiplications by one compared to the classical PatersonâStockmeyer scheme. To address instability issues in the original formulation, the authors introduce structural variants and design a reliability metric to assess the expected numerical accuracy of candidate coefficient sets. Nonlinear polynomial systems are solved using variable-precision arithmetic (VPA), and an in-house tool, MatrixPolEval1, enables efficient screening and validation. Applied to matrix exponentials and geometric series, the approach achieves a saving of one matrix multiplication while maintaining comparable numerical accuracy.
đ 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.