 
Summary: Preprint ANL/MCSP16931109
COMPUTING f(A)b VIA LEAST SQUARES POLYNOMIAL
APPROXIMATIONS
JIE CHEN, MIHAI ANITESCU, AND YOUSEF SAAD
Abstract. Given a certain function f, various methods have been proposed in the past for
addressing the important problem of computing the matrixvector product f(A)b without explicitly
computing the matrix f(A). Such methods were typically developed for a specific function f, a
common case being that of the exponential. This paper discusses a procedure based on least squares
polynomials that can, in principle, be applied to any (continuous) function f. The idea is to start
by approximating the function by a spline of a desired accuracy. Then, a particular definition of the
function inner product is invoked that facilitates the computation of the least squares polynomial to
this spline function. Since the function is approximated by a polynomial, the matrix A is referenced
only through a matrixvector multiplication. In addition, the choice of the inner product makes it
possible to avoid numerical integration. As an important application, we consider the case when
f(t) =
t and A is a sparse, symmetric positivedefinite matrix, which arises in sampling from
a Gaussian process distribution. The covariance matrix of the distribution is defined by using a
covariance function that has a compact support, at a very large number of sites that are on a regular
or irregular grid. We derive error bounds and show extensive numerical results to illustrate the
