Efficient and accurate computation for the \(\varphi\)-functions arising from exponential integrators

In this paper, we develop efficient and accurate algorithms for evaluating both \(\varphi _l(A)\) and \(\varphi _l(A)b,\) where \(\varphi _l(\cdot )\) is the function related to the exponential defined by \(\varphi _l(z)\equiv \sum \nolimits ^{\infty }_{k=0}\frac{z^k}{(l+k)!}\), A is an \(N\times N\) matrix and b is an N dimensional vector. Such matrix functions play a key role in a class of numerical methods well-known as exponential integrators. The algorithms use the modified scaling and squaring procedure combined with truncated Taylor series. A quasi-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.