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

Abstract

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

21 pages, 1 figure