Computing f(A)b via Least Squares Polynomial Approximations
|Title||Computing f(A)b via Least Squares Polynomial Approximations|
|Publication Type||Journal Article|
|Year of Publication||2009|
|Authors||Chen, J, Anitescu, M, Saad, Y|
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 effectiveness of the proposed technique.