A Matrix Hyperbolic Cosine Algorithm and Applications

In this paper, we generalize Spencer's hyperbolic cosine algorithm to the matrix-valued setting. We apply the proposed algorithm to several problems by analyzing its computational efficiency under two special cases of matrices; one in which the matrices have a group structure and an other in which they have rank-one. As an application of the former case, we present a deterministic algorithm that, given the multiplication table of a finite group of size $n$, it constructs an expanding Cayley graph of logarithmic degree in near-optimal O(n^2 log^3 n) time. For the latter case, we present a fast deterministic algorithm for spectral sparsification of positive semi-definite matrices, which implies an improved deterministic algorithm for spectral graph sparsification of dense graphs. In addition, we give an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrix sparsification. As a consequence, we obtain improved element-wise sparsification algorithms for diagonally dominant-like matrices.Comment: 16 pages, simplified proof and corrected acknowledging of prior work in (current) Section

A New Algorithm for Computing the Actions of Trigonometric and Hyperbolic Matrix Functions

A new algorithm is derived for computing the actions $f(tA)B$ and $f(tA^{1/2})B$, where $f$ is cosine, sinc, sine, hyperbolic cosine, hyperbolic sinc, or hyperbolic sine function. $A$ is an $n\times n$ matrix and $B$ is $n\times n_0$ with $n_0 \ll n$. $A^{1/2}$ denotes any matrix square root of $A$ and it is never required to be computed. The algorithm offers six independent output options given $t$, $A$, $B$, and a tolerance. For each option, actions of a pair of trigonometric or hyperbolic matrix functions are simultaneously computed. The algorithm scales the matrix $A$ down by a positive integer $s$, approximates $f(s^{-1}tA)B$ by a truncated Taylor series, and finally uses the recurrences of the Chebyshev polynomials of the first and second kind to recover $f(tA)B$. The selection of the scaling parameter and the degree of Taylor polynomial are based on a forward error analysis and a sequence of the form $\|A^k\|^{1/k}$ in such a way the overall computational cost of the algorithm is optimized. Shifting is used where applicable as a preprocessing step to reduce the scaling parameter. The algorithm works for any matrix $A$ and its computational cost is dominated by the formation of products of $A$ with $n\times n_0$ matrices that could take advantage of the implementation of level-3 BLAS. Our numerical experiments show that the new algorithm behaves in a forward stable fashion and in most problems outperforms the existing algorithms in terms of CPU time, computational cost, and accuracy.Comment: 4 figures, 16 page