NDLI: Computing $f(A)b$ via Least Squares Polynomial Approximations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Saad, Yousef Anitescu, Mihai Chen, Jie
Copyright Year	2011
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 $f(t)=\sqrt{t}$ 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.
Starting Page	195
Ending Page	222
Page Count	28
File Format	PDF
ISSN	10648275
DOI	10.1137/090778250
e-ISSN	10957197
Issue Number	1
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-02-01
Access Restriction	Subscribed
Subject Keyword	matrix function least squares polynomials Gaussian process Other matrix algorithms sampling Sparse matrices Matrix norms, conditioning, scaling
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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