Efficient and accurate computation to the $\varphi$-function and its action on a vector

TL;DR

This paper introduces efficient algorithms for computing the $oldsymbol{ extphi}$-function of matrices and its action on vectors, crucial for exponential integrators, using scaling, modified squaring, and Taylor series with backward error analysis.

Contribution

It presents novel algorithms combining scaling, modified squaring, and Taylor series for accurate and efficient $oldsymbol{ extphi}$-function computation, with backward error analysis for optimal parameter selection.

Findings

Algorithms outperform existing methods in accuracy.

Computational cost is significantly reduced.

Numerical tests confirm efficiency and precision.

Abstract

In this paper, we develop efficient and accurate algorithms for evaluating $\varphi(A)$ and $\varphi(A)b$, where $A$ is an $N\times N$ matrix, $b$ is an $N$ dimensional vector and $\varphi$ is the function defined by $\varphi(x)\equiv\sum\limits^{\infty}_{k=0}\frac{z^k}{(1+k)!}$. Such matrix function (the so-called $\varphi$-function) plays a key role in a class of numerical methods well-known as exponential integrators. The algorithms use the scaling and modified squaring procedure combined with truncated Taylor series. The backward error analysis is presented to find the optimal value of the scaling and the degree of the Taylor approximation. Some useful techniques are employed for reducing the computational cost. Numerical comparisons with state-of-the-art algorithms show that the algorithms perform well in both accuracy and efficiency.