Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

J. M. Alonso, J. Sastre, J. Ibáñez, E. Defez

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.

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

Abstract

A method for evaluating matrix polynomials have recently been developed that require one fewer matrix product () 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 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 saving while maintaining numerical accuracy comparable to the PS method.
Paper Structure (10 sections, 16 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 16 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Paterson--Stockmeyer matrix polynomial evaluation method
  3. Matrix polynomial evaluation beyond the PS method
  4. Coefficient solutions for $P_8(A)$ evaluation with $3M$
  5. Handling the singular case $c_3=0$
  6. Numerical stability and coefficient selection
  7. General evaluation formula for $P_{m}(A)$ for $m \ge 8$
  8. Numerical implementation and solver selection
  9. Numerical results
  10. Conclusions

Figures (3)

  • Figure 1: Relative error comparison for the evaluation of $\Psi(9,A)$. Plots show sorted 1-norm relative errors in IEEE double precision for matrix dimensions $n=100$ (top), $n=500$ (middle), and $n=1000$ (bottom). The methods compared are the Paterson--Stockmeyer (PS) formula \ref{['PPS']}, Westreich's approach \ref{['Psi9Wes']}89Wes, and the optimized evaluation formula obtained via MatrixPolEval1. Left-hand plots correspond to MCT test matrices, while right-hand plots correspond to EMP matrices.
  • Figure 2: Relative error comparison for the evaluation of $\Psi(13,A)$. Plots show sorted 1-norm relative errors in IEEE double precision for matrix dimensions $n=100$ (top), $n=500$ (middle), and $n=1000$ (bottom). The methods compared are the PS formula \ref{['PPS']}, Westreich's approach \ref{['Psi13Wes']}89Wes, and the optimized evaluation formula obtained via MatrixPolEval1. Left-hand plots correspond to MCT test matrices, while right-hand plots correspond to EMP matrices.
  • Figure 3: Relative error comparison for the evaluation of $\Psi(17,A)$. Plots show sorted 1-norm relative errors in IEEE double precision for matrix dimensions $n=100$ (top), $n=500$ (middle), and $n=1000$ (bottom). The methods compared are the PS formula \ref{['PPS']}, Westreich's approach \ref{['Psi17Wes']}89Wes, and the optimized evaluation formula obtained via MatrixPolEval1. Left-hand plots correspond to MCT test matrices, while right-hand plots correspond to EMP matrices.

Theorems & Definitions (1)

  • proof