Approximation of matrix operators applied to multiple vectors (Q1010051)

The authors propose a numerical method for the computation of the product of a matrix function with multiple vectors \[ f(\tau A)g_i, \quad g_i = g(t+c_i\tau), \quad i=1,\dots,n, \] where \(g\) is a smooth function, \(A\) is a square real matrix, \(\tau > 0\) is the time step of a numerical integration scheme, and \(c_1,\dots,c_n\) are given quadrature nodes. This method is based on the use of Krylov subspaces and the \(QR\) decomposition. The main idea is to reuse information from previous iterations when computing the \(QR\) decomposition of \([g_1,\dots,g_n]\) and to compute bases of the Krylov subspaces with respect to the columns \(q_i\) of \(Q\) instead of \(g_i\). The authors incorporate also Ritz vectors into the \(QR\) decomposition in order to speed up the computation. A number of numerical examples is presented and a comparative analysis is given.