| | |
Summary: Preprint ANL/MCS-P1693-1109
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 matrix-vector 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 matrix-vector 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 positive-definite 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
|
