Fast numerical computation on the harmonic Arnoldi algorithm for large-scale φ functions

In the problem of complex networks, it is often necessary to calculate matrix exponential function which is a special case of $\varphi$ functions arising from exponential integrators. However, the computational overhead of matrix exponential function is very high if the matrix of networks is large scale. For generality, in this paper, we consider the fast computation of large-scale $\varphi$ functions. In the previous work (G. Wu, L. Zhang, T. Xu, Adv. Comput. Math., 42(2016) 505–541.), we have given a thick-restated harmonic Arnoldi (TRHA) method for approximating $\varphi$ functions. However, in this algorithm, one needs to solve initial value problems frequently in the practical computation. When the time variable t approaches to zero, it is not easy to solve these initial value problems and is time-consuming. To overcome this problem, in the current work, we propose the exponential quadrature rule (EQR) method for solving the initial value problems. The major benefit of our EQR method is that we can avoid the time t approaching to zero and then reduce the computational time. Numerical examples demonstrate the effectiveness of our new method for approximating large-scale $\varphi$ functions.