The fast Gauss transform with complex parameters
type: Item
instance of: scholarly article
title: The fast Gauss transform with complex parameters
zbMATH Open document ID: 1066.65164
author: Fredrik Andersson
author: Gregory Beylkin
published in: Journal of Computational Physics
publication date: 2005-02-09T00:00:00Z
review text: For given coefficients \(q_k \in {\mathbb C}\) and source knots \(s_k \in [-1/2,\,1/2]\) \((k=1,\dots,N)\), the authors evaluate approximately the sum \[ G(x)=\sum_{k=1}^N q_k \, \exp(-\alpha | x-s_k| ^2) \] at the target knots \(x_j \in [-1/2,\, 1/2]\) \((j=1,\dots, M)\), where \(\alpha = a - \text{ i}b\) \((a>0,\, b\in {\mathbb R})\) is a complex parameter. Such sums appear in numerical computation of integrals \[ Q(x)=\int q(s)\, \exp(-\alpha | x-s| ^2)\, \text{ d}s. \] Using a fast algorithm of discrete Fourier transform for nonequally spaced knots, this paper presents a fast Gauss transform with \({\mathcal O}(N\,\log N + M)\) flops. Remark of the reviewer: The general fast summation algorithm developed by \textit{D. Potts, G. Steidl} and \textit{A. Nieslony} [Numer. Math. 98, 329--351 (2004; Zbl 1056.65146)] can be specified for the Gaussian kernel with complex parameter \(\alpha\) to obtain a fast Gauss transform with \({\mathcal O}(N+M)\) flops.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65T50
Mathematics Subject Classification ID: 65R10
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 2132612
zbMATH Keywords: fast Gauss transform
zbMATH Keywords: fast algorithm
zbMATH Keywords: chirped Gaussian
zbMATH Keywords: nonequally spaced fast Fourier transform
zbMATH Keywords: discrete Fourier transform
MaRDI profile type: Publication
full work available at URL: j.jcp.2004.07.020
OpenAlex ID: W2137541069
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DOI: 10.1016/J.JCP.2004.07.020
description: scientific article; zbMATH DE number 2132612