Correction to Partial Fraction Decomposition Coefficients for Chebyshev Rational Approximation on the Negative Real Axis

TL;DR

This paper corrects previously published partial fraction decomposition coefficients for Chebyshev rational approximations on the negative real axis, improving the accuracy of matrix exponential computations in scientific applications.

Contribution

It provides corrected PFD coefficients for orders 14 and 16, addressing errors in earlier published data and enhancing the reliability of Chebyshev rational approximation methods.

Findings

Incorrect coefficients led to 100 times poorer accuracy.

Corrected coefficients improve approximation accuracy.

Discussion of the impact of errors on approximation quality.

Abstract

Chebyshev rational approximation can be a viable method to compute the exponential of matrices with eigenvalues in the vicinity of the negative real axis, and it was recently applied successfully to solving nuclear fuel burnup equations. Determining the partial fraction decomposition (PFD) coefficients of this approximation can be difficult and they have been provided (for approximation orders 10 and 14) by Gallopoulos and Saad in "Efficient solution of parabolic equations by Krylov approximation methods", SIAM J. Sci. Stat. Comput., 13(1992). It was recently discovered that the order 14 coefficients contain errors and result in 100 times poorer accuracy than expected by theory. The purpose of this note is to provide the correct PFD coefficients for approximation orders 14 and 16 and to briefly discuss the approximation accuracy resulting from the erroneous coefficients.