Block Gauss and Anti-Gauss Quadrature with Application to Networks

Abstract

Abstract. Approximations of matrix-valued functions of the form W T f (A)W , where A ∈ R m×m is symmetric, W ∈ R m×k , with m large and k m, has orthonormal columns, and f is a function, can be computed by applying a few steps of the symmetric block Lanczos method to A with initial block-vector W ∈ R m×k . Golub and Meurant have shown that the approximants obtained in this manner may be considered block Gauss quadrature rules associated with a matrix-valued measure. This paper generalizes anti-Gauss quadrature rules, introduced by Laurie for real-valued measures, to matrix-valued measures, and shows that under suitable conditions pairs of block Gauss and block anti-Gauss rules provide upper and lower bounds for the entries of the desired matrix-valued function. Extensions to matrix-valued functions of the form W T f (A)V , where A ∈ R m×m may be nonsymmetric, and the matrices V, W ∈ R m×k satisfy V T W = I k are also discussed. Approximations of the latter functions are computed by applying a few steps of the nonsymmetric block Lanczos method to A with initial block-vectors V and W . We describe applications to the evaluation of functions of a symmetric or nonsymmetric adjacency matrix for a network. Numerical examples illustrate that a combination of block Gauss and anti-Gauss quadrature rules typically provides upper and lower bounds for such problems. We introduce some new quantities that describe properties of nodes in directed or undirected networks, and demonstrate how these and other quantities can be computed inexpensively with the quadrature rules of the present paper.

Network Analysis via Partial Spectral Factorization and Gauss Quadrature

“…Other quantities of the form (1.3) used to characterize nodes of a graph are described in [5,11,13,16]. For instance, the f -starting convenience of node i, n e…”

“…This is the sum of the communicabilities from node i to all other nodes, scaled so that the average of the quantity over all nodes is one. This quantity is defined in [16], where also the f -ending convenience of node i is introduced. The latter also can be computed by evaluating expressions of the form (1.3).…”

“…A brief review of this technique is provided in section 4. An application of pairs of block Gauss-type quadrature rules to simultaneously determine approximate upper and lower bounds for several entries of f (A) is described in [16].…”

“…The choice u = e i and w = e j allows the computation of upper and lower bounds for the f -communicability between the nodes i and j. Computations with these and other choices of vectors u = w are illustrated in [4,16].…”

“…The symmetry of the adjacency matrix for this kind of graph leads to some simplifications of the computations. However, there are many networks that give rise to directed graphs; see [5,8,10,11,14,15,16,27] for discussions and illustrations. Hybrid methods also can be developed for directed graphs.…”