FastTransforms.jl Documentation
Introduction
FastTransforms.jl
allows the user to conveniently work with orthogonal polynomials with degrees well into the millions.
This package provides a Julia wrapper for the C library of the same name. Additionally, all three types of nonuniform fast Fourier transforms available, as well as the Padua transform.
Fast orthogonal polynomial transforms
For this documentation, please see the documentation for FastTransforms. Most transforms have separate forward and inverse plans. In some instances, however, the inverse is in the sense of least-squares, and therefore only the forward transform is planned.
Nonuniform fast Fourier transforms
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FastTransforms.nufft1
— Function
Computes a nonuniform fast Fourier transform of type I:
Computes a 2D nonuniform fast Fourier transform of type I-I:
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FastTransforms.nufft2
— Function
Computes a nonuniform fast Fourier transform of type II:
Computes a 2D nonuniform fast Fourier transform of type II-II:
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FastTransforms.inufft1
— Function
Computes an inverse nonuniform fast Fourier transform of type I.
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FastTransforms.inufft2
— Function
Computes an inverse nonuniform fast Fourier transform of type II.
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FastTransforms.paduatransform
— Function
Padua Transform maps from interpolant values at the Padua points to the 2D Chebyshev coefficients.
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FastTransforms.ipaduatransform
— Function
Inverse Padua Transform maps the 2D Chebyshev coefficients to the values of the interpolation polynomial at the Padua points.
Other Exported Methods
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FastTransforms.gaunt
— Function
Calculates the Gaunt coefficients, defined by:
or defined by:
This is a Julia implementation of the stable recurrence described in:
Y.-l. Xu, Fast evaluation of Gaunt coefficients: recursive approach, J. Comp. Appl. Math., 85:53—65, 1997.
Calculates the Gaunt coefficients in 64-bit floating-point arithmetic.
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FastTransforms.paduapoints
— Function
Returns coordinates of the Padua points.
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FastTransforms.sphevaluate
— Function
Pointwise evaluation of real orthonormal spherical harmonic:
Internal Methods
Miscellaneous Special Functions
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FastTransforms.half
— Function
Compute a typed 0.5.
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FastTransforms.two
— Function
Compute a typed 2.
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FastTransforms.Λ
— Function
The Lambda function for the ratio of gamma functions.
For 64-bit floating-point arithmetic, the Lambda function uses the asymptotic series for in Appendix B of
I.Bogaert and B. Michiels and J. Fostier, 𝒪(1) computation of Legendre polynomials and Gauss—Legendre nodes and weights for parallel computing, SIAM J. Sci. Comput., 34:C83—C101, 2012.
The Lambda function for the ratio of gamma functions.
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FastTransforms.lambertw
— Function
The principal branch of the Lambert-W function, defined by , computed using Halley’s method for .
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FastTransforms.pochhammer
— Function
Pochhammer symbol for the rising factorial.
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FastTransforms.stirlingseries
— Function
Stirling’s asymptotic series for .
Modified Chebyshev Moment-Based Quadrature
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FastTransforms.clenshawcurtisnodes
— Function
Compute nodes of the Clenshaw—Curtis quadrature rule.
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FastTransforms.clenshawcurtisweights
— Function
Compute weights of the Clenshaw—Curtis quadrature rule with modified Chebyshev moments of the first kind .
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FastTransforms.fejernodes1
— Function
Compute nodes of Fejer’s first quadrature rule.
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FastTransforms.fejerweights1
— Function
Compute weights of Fejer’s first quadrature rule with modified Chebyshev moments of the first kind .
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FastTransforms.fejernodes2
— Function
Compute nodes of Fejer’s second quadrature rule.
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FastTransforms.fejerweights2
— Function
Compute weights of Fejer’s second quadrature rule with modified Chebyshev moments of the second kind .
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FastTransforms.chebyshevjacobimoments1
— Function
Modified Chebyshev moments of the first kind with respect to the Jacobi weight:
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FastTransforms.chebyshevlogmoments1
— Function
Modified Chebyshev moments of the first kind with respect to the logarithmic weight:
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FastTransforms.chebyshevjacobimoments2
— Function
Modified Chebyshev moments of the second kind with respect to the Jacobi weight:
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FastTransforms.chebyshevlogmoments2
— Function
Modified Chebyshev moments of the second kind with respect to the logarithmic weight:
Elliptic
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FastTransforms.Elliptic
— Module
FastTransforms
submodule for the computation of some elliptic integrals and functions.
Complete elliptic integrals of the first and second kinds:
Jacobian elliptic functions:
and the remaining nine are defined by: