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

FastTransforms.nufft1 — Function

Computes a nonuniform fast Fourier transform of type I:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} \frac{j}{N} \omega_k},\quad{\rm for}\quad 0 \le j \le N-1.\]

Computes a 2D nonuniform fast Fourier transform of type I-I:

\[F_{i,j} = \sum_{k=0}^{M-1}\sum_{\ell=0}^{N-1} C_{k,\ell} e^{-2\pi{\rm i} (\frac{i}{M} \omega_k + \frac{j}{N} \pi_{\ell})},\quad{\rm for}\quad 0 \le i \le M-1,\quad 0 \le j \le N-1.\]
FastTransforms.nufft2 — Function

Computes a nonuniform fast Fourier transform of type II:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} x_j k},\quad{\rm for}\quad 0 \le j \le N-1.\]

Computes a 2D nonuniform fast Fourier transform of type II-II:

\[F_{i,j} = \sum_{k=0}^{M-1}\sum_{\ell=0}^{N-1} C_{k,\ell} e^{-2\pi{\rm i} (x_i k + y_j \ell)},\quad{\rm for}\quad 0 \le i \le M-1,\quad 0 \le j \le N-1.\]
FastTransforms.nufft3 — Function

Computes a nonuniform fast Fourier transform of type III:

\[f_j = \sum_{k=0}^{N-1} c_k e^{-2\pi{\rm i} x_j \omega_k},\quad{\rm for}\quad 0 \le j \le N-1.\]
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

FastTransforms.gaunt — Function

Calculates the Gaunt coefficients, defined by:

\[a(m,n,\mu,\nu,q) = \frac{2(n+\nu-2q)+1}{2} \frac{(n+\nu-2q-m-\mu)!}{(n+\nu-2q+m+\mu)!} \int_{-1}^{+1} P_n^m(x) P_\nu^\mu(x) P_{n+\nu-2q}^{m+\mu}(x) {\rm\,d}x.\]

or defined by:

\[P_n^m(x) P_\nu^\mu(x) = \sum_{q=0}^{q_{\rm max}} a(m,n,\mu,\nu,q) P_{n+\nu-2q}^{m+\mu}(x)\]

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.

FastTransforms.sphevaluate — Function

Pointwise evaluation of real orthonormal spherical harmonic:

\[Y_\ell^m(\theta,\varphi) = (-1)^{|m|}\sqrt{(\ell+\frac{1}{2})\frac{(\ell-|m|)!}{(\ell+|m|)!}} P_\ell^{|m|}(\cos\theta) \sqrt{\frac{2-\delta_{m,0}}{2\pi}} \left\{\begin{array}{ccc} \cos m\varphi & {\rm for} & m \ge 0,\\ \sin(-m\varphi) & {\rm for} & m < 0.\end{array}\right.\]

Internal Methods

Miscellaneous Special Functions

FastTransforms.δ — Function

The Kronecker $\delta$ function:

\[\delta_{k,j} = \left\{\begin{array}{ccc} 1 & {\rm for} & k = j,\\ 0 & {\rm for} & k \ne j.\end{array}\right.\]
FastTransforms.Λ — Function

The Lambda function $\Lambda(z) = \frac{\Gamma(z+\frac{1}{2})}{\Gamma(z+1)}$ for the ratio of gamma functions.

For 64-bit floating-point arithmetic, the Lambda function uses the asymptotic series for $\tau$ 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 $\Lambda(z,λ₁,λ₂) = \frac{\Gamma(z+\lambda_1)}{Γ(z+\lambda_2)}$ for the ratio of gamma functions.

FastTransforms.lambertw — Function

The principal branch of the Lambert-W function, defined by $x = W_0(x) e^{W_0(x)}$, computed using Halley's method for $x \in [-e^{-1},\infty)$.

Modified Chebyshev Moment-Based Quadrature

FastTransforms.fejerweights1 — Function

Compute weights of Fejer's first quadrature rule with modified Chebyshev moments of the first kind $\mu$.

FastTransforms.fejerweights2 — Function

Compute weights of Fejer's second quadrature rule with modified Chebyshev moments of the second kind $\mu$.

FastTransforms.chebyshevjacobimoments1 — Function

Modified Chebyshev moments of the first kind with respect to the Jacobi weight:

\[ \int_{-1}^{+1} T_n(x) (1-x)^\alpha(1+x)^\beta{\rm\,d}x.\]
FastTransforms.chebyshevlogmoments1 — Function

Modified Chebyshev moments of the first kind with respect to the logarithmic weight:

\[ \int_{-1}^{+1} T_n(x) \log\left(\frac{1-x}{2}\right){\rm\,d}x.\]
FastTransforms.chebyshevjacobimoments2 — Function

Modified Chebyshev moments of the second kind with respect to the Jacobi weight:

\[ \int_{-1}^{+1} U_n(x) (1-x)^\alpha(1+x)^\beta{\rm\,d}x.\]
FastTransforms.chebyshevlogmoments2 — Function

Modified Chebyshev moments of the second kind with respect to the logarithmic weight:

\[ \int_{-1}^{+1} U_n(x) \log\left(\frac{1-x}{2}\right){\rm\,d}x.\]