Approximation of row sums Sample Clauses

Approximation of row sums of matrix functions To compute Xxxx centrality or total communicability (which is introduced in Chapter 2), all that is needed is the row sums of the matrix resolvent or exponential: (I −αA)−11 or eβA1. For digraphs, AT may be used in place of A. In recent years, efficient algorithms have been developed for computing the action of a matrix function on a vector, that is, for computing the vector f (A)b for a given matrix A (usually large and sparse), vector b, and function f . A particularly important case is that of the matrix exponential, since this provides a solution method for initial value problems for first-order systems of linear ordinary differential equations. These algorithms, based on variants of the Lanczos, Arnoldi or other Krylov subspace method, access the matrix A only in the form of (sparse) matrix-vector products and have O(n) storage cost for a sparse n × n matrix A [65, Chapter 13]. When b = 1, the vector with all its entries equal to 1, the ith entry of the resulting vector f (A)1 contains the ith row sum of f (A): Σn
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