Journal of Computational and Applied Mathematics
Two polynomial methods of calculating functions of symmetric matrices
USSR Computational Mathematics and Mathematical Physics
Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
Efficient solution of parabolic equations by Krylov approximation methods
SIAM Journal on Scientific and Statistical Computing
Some large-scale matrix computation problems
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
SIAM Journal on Scientific Computing
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions
SIAM Journal on Matrix Analysis and Applications
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
Multigrid
A polynomial method based on Fejèr points for the computation of functions of unsymmetric matrices
Applied Numerical Mathematics
Preconditioning Lanczos Approximations to the Matrix Exponential
SIAM Journal on Scientific Computing
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Filtered Conjugate Residual-type Algorithms with Applications
SIAM Journal on Matrix Analysis and Applications
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Stopping Criteria for Rational Matrix Functions of Hermitian and Symmetric Matrices
SIAM Journal on Scientific Computing
Computing $A^\alpha, \log(A)$, and Related Matrix Functions by Contour Integrals
SIAM Journal on Numerical Analysis
Scalable Parallel 3d FFTs for Electronic Structure Codes
High Performance Computing for Computational Science - VECPAR 2008
Matrices, Moments and Quadrature with Applications
Matrices, Moments and Quadrature with Applications
The university of Florida sparse matrix collection
ACM Transactions on Mathematical Software (TOMS)
A Matrix-free Approach for Solving the Parametric Gaussian Process Maximum Likelihood Problem
SIAM Journal on Scientific Computing
Computation of matrix functions with deflated restarting
Journal of Computational and Applied Mathematics
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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.