A Systematic Framework for the Stable Evaluation of Matrix Polynomials with Product Savings - Group of High Performance Scientific Computing (HiPerSC)

Evaluating high-degree matrix polynomials is a common task in matrix function approximation. A recently proposed evaluation scheme reduces the computational cost by one matrix multiplication (1M) compared with the Paterson–Stockmeyer (PS) method, offering a direct reduction in execution time for large matrices. The main difficulty, however, lies in determining suitable coefficients for these optimized formulas. Since they are obtained by solving nonlinear polynomial systems, several possible solutions may arise, with substantially different numerical stability properties. Selecting an unsuitable solution can lead to significant accuracy loss in floating-point arithmetic.

J. M. Alonso, J. Sastre, J. Ibáñez, and E. Defez address this issue by developing a systematic framework and its MATLAB implementation, MatrixPolEval1, for computing and validating stable coefficient sets for matrix polynomials of degrees $m \in {8, 10, 12}$ and beyond. The framework automates coefficient selection using variable precision arithmetic (VPA) and assesses numerical stability through coefficient reconstruction errors, providing a reliability indicator for the expected accuracy. It also incorporates alternative evaluation structures to handle cases in which the original formulation fails to provide a robust solution, including degenerate configurations that require separate treatment.

Numerical experiments involving matrix polynomials arising in applications such as the matrix exponential and geometric series demonstrate that the proposed framework retains the one-multiplication advantage over the PS method while achieving comparable numerical accuracy. These results provide a practical basis for applying multiplication-efficient evaluation schemes without compromising numerical reliability.

Publication Details

Title: A Systematic Framework for Stable and Cost-Efficient Matrix Polynomial Evaluation
Authors: J. M. Alonso, J. Sastre, J. Ibáñez, E. Defez
Preprint: arXiv:2603.23143 (march 24 of2026)
Enlace: https://arxiv.org/abs/2603.23143
Preprint (ResearchGate): https://www.researchgate.net/publication/403112848

Main Contributions

A systematic framework for identifying stable coefficient sets in evaluation schemes that save one matrix multiplication.
MATLAB implementation (MatrixPolEval1), featuring variable-precision verification and a reliability indicator.
Numerical validation for polynomials of degrees 8, 10, 12, and higher, including applications to the matrix exponential and geometric series.

Why This Matters

Saving one matrix multiplication can directly reduce execution time in large-scale problems. The proposed framework facilitates the reliable use of these optimized schemes without compromising numerical stability.