Matrix Functions

1. Introduction. 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. This chapter is organized as follows: In section 2 we introduce the concept of a matrix function f (A) in detail, essentially following [37] and [27]. Section 3 gives an assessment of various general computational approaches for either obtaining the whole matrix f (A) or its action f (A)v on a vector v. Sections 4 and 5 then give much more details for two specific functions, the exponential and the sign functions, which, as we will show, are particularly important in many areas like control theory, simulation of physical systems and other application fields involving the solution of certain ordinary or partial differential equations. The applicability of matrix functions in general, and of the exponential and the sign functions in particular, is vast. However, we will limit our discussion to characterizations and to application problems that are mostly related to Model Order Reduction.