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

We propose a new method for computing the $φ$-functions of large sparse matrices with low rank or fast decaying singular values. The key is to reduce the computation of $φ_{\ell}$-functions of a large matrix to $φ_{\ell+1}$-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 $φ$-functions. Numerical experiments illustrate the numerical behavior of the new algorithms and show the effectiveness of our theoretical results.

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