Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid
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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^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.
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