Computing f(A)b via Least Squares Polynomial Approximations

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 $f(t)=\sqrt{t}$ 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 effectiveness of the proposed technique.