A shift and invert reorthogonalization Arnoldi algorithm for solving the chemical master equation

The shift and invert Arnoldi (SIA) method is a numerical algorithm for approximating the product of Toeplitz matrix exponential with a vector. In this paper, we extend the SIA method to chemical master equation (CME) and propose a SIA algorithm based on the strategy of reorthogonalization (SIRA). We establish a theoretical error of the resulting approximation of SIRA algorithm. Numerical experiments show that the SIRA algorithm is more efficient than the Krylov FSP algorithm in terms of finite models, and the error estimate can be used to determine whether this result obtained by SIRA algorithm is acceptable or not.