2018/12/11 by Vishal Vaibhav, Vaibhav, Vishal
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Computational Physics (physics.comp-ph) #FOS: Mathematics #FOS: Physical sciences #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1812.04701
openalex publication_date 2018/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main objective of this series of papers is to explore the entire\nlandscape of numerical methods for fast nonlinear Fourier transformation (NFT)\nwithin the class of integrators known as the exponential integrators. In this\npaper, we explore the theoretical aspects of exponential Runge-Kutta (RK) and\nlinear multistep (LM) methods, in particular, the stability and convergence of\nthese methods via the transfer matrix formulation. The analysis carried out in\nthe paper shows that while the exponential LM methods are naturally amenable to\nFFT-based fast polynomial arithmetic, the RK methods require equispaced nodes\nto achieve that. Therefore, each these family of methods is capable of yielding\na family of fast NFT algorithms such that the scattering coefficients can be\ncomputed with a complexity of mathscrO(N\log2N) and a rate of convergence\ngiven by mathscrO(N-p) where N is the number of samples of the signal\nand p is order of the underlying discretization scheme. Further, while RK\nmethods can accommodate vanishing as well as periodic boundary conditions, the\nLM methods can only handle the former type of boundary conditions without\nrequiring a starting procedure. The ideas presented in this paper extend\nnaturally to the family of integrators known as general linear methods which\nwill be explored in a forthcoming paper.\n