  The computation of approximating e^tA B, where A is a large sparse matrix and
B is a rectangular matrix, serves as a crucial element in numerous scientific
and engineering calculations. A powerful way to consider this problem is to use
Krylov subspace methods. The purpose of this work is to approximate the matrix
exponential and some Cauchy-Stieltjes functions on a block vectors B of R^n*p
using a rational block Lanczos algorithm. We also derive some error estimates
and error bound for the convergence of the rational approximation and finally
numerical results attest to the computational efficiency of the proposed
method.
