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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 an $n$-by-$n$ large sparse matrix with low rank or with 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 small matrices of size $r$-by-$r$, where $r$ is the numerical rank of the large matrix in question. For storage, the new method only needs to store two $n$-by-$r$ sparse matrices and some $r$-by-$r$ matrices, rather than some $n$-by-$n$ possibly dense matrices. The error analysis on the proposed method is given. Based on the new method, we then propose two novel strategies for estimating the 2-norm condition numbers of the $\varphi$-functions of large matrices. Numerical experiments illustrate the numerical behavior of the new algorithms and show the effectiveness of our theoretical results.