diff --git a/dev/dev/index.html b/dev/dev/index.html index b37af6e8..047897aa 100644 --- a/dev/dev/index.html +++ b/dev/dev/index.html @@ -34,4 +34,4 @@

Then we wait for the friendly folks at JuliaPackaging to merge the pull request to Yggdrasil, triggering a new release of the FastTransforms_jll.jl meta package that stores all precompiled binaries. With this release, we update the FastTransforms.jl Project.toml to point to the latest release and register the new version.

Since development of Yggdrasil is quite rapid, a fork may easily become stale. Git permits the developer to forcibly make a master branch on a fork even with upstream master:

git fetch upstream
 git checkout master
 git reset --hard upstream/master
-git push origin master --force
+git push origin master --force diff --git a/dev/generated/annulus.html b/dev/generated/annulus.html index 31f8cd62..cca24425 100644 --- a/dev/generated/annulus.html +++ b/dev/generated/annulus.html @@ -1,7 +1,7 @@ -
+
+
-
+
+
+
+
+
+
Edit on GitHub

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:

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

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

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

Computes a nonuniform fast Fourier transform of type II:

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

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

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

Computes a nonuniform fast Fourier transform of type III:

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

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.

source

Calculates the Gaunt coefficients in 64-bit floating-point arithmetic.

source
FastTransforms.sphevaluate — Function

Pointwise evaluation of real orthonormal spherical harmonic:

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

Internal Methods

Miscellaneous Special Functions

FastTransforms.δ — Function

The Kronecker $\delta$ function:

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

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

source

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.

source

The Lambda function $\Lambda(z,λ₁,λ₂) = \frac{\Gamma(z+\lambda_1)}{Γ(z+\lambda_2)}$ for the ratio of gamma functions.

source
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 $x \in [-e^{-1},\infty)$.

source

Modified Chebyshev Moment-Based Quadrature

FastTransforms.chebyshevlogmoments1 — Function

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

\[ \int_{-1}^{+1} T_n(x) \log\left(\frac{1-x}{2}\right){\rm\,d}x.\]
source
FastTransforms.chebyshevlogmoments2 — Function

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

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

Elliptic

FastTransforms.Elliptic — Module

FastTransforms submodule for the computation of some elliptic integrals and functions.

Complete elliptic integrals of the first and second kinds:

\[K(k) = \int_0^{\frac{\pi}{2}} \frac{{\rm d}\theta}{\sqrt{1-k^2\sin^2\theta}},\quad{\rm and},\]
\[E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2\theta} {\rm\,d}\theta.\]

Jacobian elliptic functions:

\[x = \int_0^{\operatorname{sn}(x,k)} \frac{{\rm d}t}{\sqrt{(1-t^2)(1-k^2t^2)}},\]
\[x = \int_{\operatorname{cn}(x,k)}^1 \frac{{\rm d}t}{\sqrt{(1-t^2)[1-k^2(1-t^2)]}},\]
\[x = \int_{\operatorname{dn}(x,k)}^1 \frac{{\rm d}t}{\sqrt{(1-t^2)(t^2-1+k^2)}},\]

and the remaining nine are defined by:

\[\operatorname{pq}(x,k) = \frac{\operatorname{pr}(x,k)}{\operatorname{qr}(x,k)} = \frac{1}{\operatorname{qp}(x,k)}.\]
source
+Home · FastTransforms.jl
Edit on GitHub

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:

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

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

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

Computes a nonuniform fast Fourier transform of type II:

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

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

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

Computes a nonuniform fast Fourier transform of type III:

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

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.

source

Calculates the Gaunt coefficients in 64-bit floating-point arithmetic.

source
FastTransforms.sphevaluate — Function

Pointwise evaluation of real orthonormal spherical harmonic:

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

Internal Methods

Miscellaneous Special Functions

FastTransforms.δ — Function

The Kronecker $\delta$ function:

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

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

source

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.

source

The Lambda function $\Lambda(z,λ₁,λ₂) = \frac{\Gamma(z+\lambda_1)}{Γ(z+\lambda_2)}$ for the ratio of gamma functions.

source
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 $x \in [-e^{-1},\infty)$.

source

Modified Chebyshev Moment-Based Quadrature

FastTransforms.chebyshevlogmoments1 — Function

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

\[ \int_{-1}^{+1} T_n(x) \log\left(\frac{1-x}{2}\right){\rm\,d}x.\]
source
FastTransforms.chebyshevlogmoments2 — Function

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

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

Elliptic

FastTransforms.Elliptic — Module

FastTransforms submodule for the computation of some elliptic integrals and functions.

Complete elliptic integrals of the first and second kinds:

\[K(k) = \int_0^{\frac{\pi}{2}} \frac{{\rm d}\theta}{\sqrt{1-k^2\sin^2\theta}},\quad{\rm and},\]
\[E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2\theta} {\rm\,d}\theta.\]

Jacobian elliptic functions:

\[x = \int_0^{\operatorname{sn}(x,k)} \frac{{\rm d}t}{\sqrt{(1-t^2)(1-k^2t^2)}},\]
\[x = \int_{\operatorname{cn}(x,k)}^1 \frac{{\rm d}t}{\sqrt{(1-t^2)[1-k^2(1-t^2)]}},\]
\[x = \int_{\operatorname{dn}(x,k)}^1 \frac{{\rm d}t}{\sqrt{(1-t^2)(t^2-1+k^2)}},\]

and the remaining nine are defined by:

\[\operatorname{pq}(x,k) = \frac{\operatorname{pr}(x,k)}{\operatorname{qr}(x,k)} = \frac{1}{\operatorname{qp}(x,k)}.\]
source
diff --git a/dev/search/index.html b/dev/search/index.html index 4a32063c..75795e10 100644 --- a/dev/search/index.html +++ b/dev/search/index.html @@ -1,2 +1,2 @@ -Search · FastTransforms.jl

Loading search...

    +Search · FastTransforms.jl

    Loading search...