Matrix Functions

Abstract

In this chapter, we give an overview on methods to compute functions of a (usually square) matrix $A$ with particular emphasis on the matrix exponential and the matrix sign function.We will distinguish between methods which indeed compute the entire matrix function, i.e. they compute a matrix, and those which compute the action of the matrix function on a vector. The latter task is particularly important in the case where we have to deal with a very large (and possibly sparse) matrix $A$ or in situations, where $A$ is not available as a matrix but just as a function which returns $Ax$ for any input vector $x$. Computing the action of a matrix function on a vector is a typical model reduction problem, since the resulting techniques usually rely on approximations from small-dimensional subspaces.