Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

We explore two classes of exponential integrators in this letter to design nonlinear Fourier transform (NFT) algorithms with a desired accuracy-complexity trade-off and a convergence order of \(4\) on an equispaced grid. The integrating factor based method in the class of Runge-Kutta methods yield a...

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Bibliographic Details
Published in:arXiv.org 2019-03
Main Author: Vaibhav, Vishal
Format: Article
Language:English
Subjects:
Algorithms
Commutators
Complexity
Fourier transforms
Integrators
Runge-Kutta method
Series expansion
Tradeoffs
Online Access:Get full text
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Summary:We explore two classes of exponential integrators in this letter to design nonlinear Fourier transform (NFT) algorithms with a desired accuracy-complexity trade-off and a convergence order of \(4\) on an equispaced grid. The integrating factor based method in the class of Runge-Kutta methods yield algorithms with complexity \(O(N\log^2N)\) (where \(N\) is the number of samples of the signal) which have superior accuracy-complexity trade-off than any of the fast methods known currently. The integrators based on Magnus series expansion, namely, standard and commutator-free Magnus methods yield algorithms of complexity \(O(N^2)\) that have superior error behavior even for moderately small step-sizes and higher signal strengths.
ISSN:2331-8422
DOI:10.48550/arxiv.1903.11702