A New Method for Computing $\varphi$-functions and Their Condition Numbers of Large Sparse Matrices

We propose a new method for computing the $\varphi$-functions of large sparse matrices with low rank or fast decaying singular values. The key is to reduce the computation of $\varphi_{\ell}$-functions of a large matrix to $\varphi_{\ell+1}$-functions of some $r$-by-$r$ matrices, where $r$ is the numerical rank of the large matrix in question. Some error analysis on the new method is given. Furthermore, we propose two novel strategies for estimating 2-norm condition numbers of the $\varphi$-functions. Numerical experiments illustrate the numerical behavior of the new algorithms and show the effectiveness of our theoretical results.