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Register a new userComputing the Reciprocal of a $phi$-function by Rational Approximation
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Added by
Paola Boito
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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In this paper we introduce a family of rational approximations of the reciprocal of a $phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation and properties of this family of approximations applied to scalar and matrix arguments are presented. Moreover, we show that the matrix functions computed by these approximations exhibit decaying properties comparable to the best existing theoretical bounds. Numerical examples highlight the benefits of the proposed rational approximations w.r.t.~the classical Taylor polynomials and other rational functions.
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Let $A$ be a square complex matrix; $z_1$, ..., $z_{N}inmathbb C$ be arbitrary (possibly repetitive) points of interpolation; $f$ be an analytic function defined on a neighborhood of the convex hull of the union of the spectrum $sigma(A)$ of the matrix $A$ and the points $z_1$, ..., $z_{N}$; and the rational function $r=frac uv$ (with the degree of the numerator $u$ less than $N$) interpolates $f$ at these points (counted according to their multiplicities). Under these assumptions estimates of the kind $$ biglVert f(A)-r(A)bigrVertle max_{tin[0,1];muintext{convex hull}{z_1,z_{2},dots,z_{N}}}bigglVertOmega(A)[v(A)]^{-1} frac{bigl(vfbigr)^{{(N)}} bigl((1-t)mumathbf1+tAbigr)}{N!}biggrVert, $$ where $Omega(z)=prod_{k=1}^N(z-z_k)$, are proposed. As an example illustrating the accuracy of such estimates, an approximation of the impulse response of a dynamic system obtained using the reduced-order Arnoldi method is considered, the actual accuracy of the approximation is compared with the estimate based on this paper.
We present a new fast algorithm for computing the Boys function using nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are competitive with previously developed algorithms for the same purpose.
Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincare inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincar e inequality, where the measurement function is of subsampled type, with a small but non-zero lengthscale that will be made precise. Our analysis identifies this inequality as a basic tool for function recovery problems. We discuss and demonstrate the optimality of the inequality concerning the subsampled lengthscale, connecting it to existing results in the literature. In application to function approximation problems, the approximation accuracy using different basis functions and under different regularity assumptions is established by using the subsampled Poincare inequality. We observe that the error bound blows up as the subsampled lengthscale approaches zero, due to the fact that the underlying function is not regular enough to have well-defined pointwise values. A weighted version of the Poincar e inequality is proposed to address this problem; its optimality is also discussed.
We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(mathbf{A}) mathbf{b}$ when $mathbf{A}$ is a Hermitian matrix and $mathbf{b}$ is a given mathbftor. Assuming that $f : mathbb{C} rightarrow mathbb{C}$ is piecewise analytic, we give a framework, based on the Cauchy integral formula, which can be used to derive {em a priori} and emph{a posteriori} error bounds for Lanczos-FA in terms of the error of Lanczos used to solve linear systems. Unlike many error bounds for Lanczos-FA, these bounds account for fine-grained properties of the spectrum of $mathbf{A}$, such as clustered or isolated eigenvalues. Our results are derived assuming exact arithmetic, but we show that they are easily extended to finite precision computations using existing theory about the Lanczos algorithm in finite precision. We also provide generalized bounds for the Lanczos method used to approximate quadratic forms $mathbf{b}^textsf{H} f(mathbf{A}) mathbf{b}$, and demonstrate the effectiveness of our bounds with numerical experiments.
The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this problem; here blo
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