This module implements nonuniform fast spherical Fourier transforms. More...

Data Structures
struct	nfsft_wisdom
	Wisdom structure. More...

struct	nfsft_plan
	data structure for an NFSFT (nonequispaced fast spherical Fourier transform) plan with double precision More...

Macros
#define	NFSFT_NORMALIZED (1U << 0)

#define	NFSFT_USE_NDFT (1U << 1)

#define	NFSFT_USE_DPT (1U << 2)

#define	NFSFT_MALLOC_X (1U << 3)

#define	NFSFT_MALLOC_F_HAT (1U << 5)

#define	NFSFT_MALLOC_F (1U << 6)

#define	NFSFT_PRESERVE_F_HAT (1U << 7)

#define	NFSFT_PRESERVE_X (1U << 8)

#define	NFSFT_PRESERVE_F (1U << 9)

#define	NFSFT_DESTROY_F_HAT (1U << 10)

#define	NFSFT_DESTROY_X (1U << 11)

#define	NFSFT_DESTROY_F (1U << 12)

#define	NFSFT_NO_DIRECT_ALGORITHM (1U << 13)

#define	NFSFT_NO_FAST_ALGORITHM (1U << 14)

#define	NFSFT_ZERO_F_HAT (1U << 16)

#define	NFSFT_INDEX(k, n, plan) ((2(plan)->N+2)((plan)->N-n+1)+(plan)->N+k+1)

#define	NFSFT_F_HAT_SIZE(N) ((2N+2)(2*N+2))

#define	BWEXP_MAX 10

#define	BW_MAX 1024

#define	ROW(k) (k*(wisdom.N_MAX+2))

#define	ROWK(k) (k*(wisdom.N_MAX+2)+k)

#define	NFSFT_DEFAULT_NFFT_CUTOFF 6
	The default NFFT cutoff parameter. More...

#define	NFSFT_DEFAULT_THRESHOLD 1000
	The default threshold for the FPT. More...

#define	NFSFT_BREAK_EVEN 5
	The break-even bandwidth $N \in \mathbb{N}_0$ . More...

Enumerations
enum	bool { false = 0, true = 1 }

Functions
void	alpha_al_row (R *alpha, const int N, const int n)

void	beta_al_row (R *beta, const int N, const int n)

void	gamma_al_row (R *gamma, const int N, const int n)

void	alpha_al_all (R *alpha, const int N)
	Compute three-term-recurrence coefficients $\alpha_{k-1}^n$ of associated Legendre functions for $k,n = 0,1,\ldots,N$ . More...

void	beta_al_all (R *beta, const int N)
	Compute three-term-recurrence coefficients $\beta_{k-1}^n$ of associated Legendre functions for $k,n = 0,1,\ldots,N$ . More...

void	gamma_al_all (R *gamma, const int N)
	Compute three-term-recurrence coefficients $\gamma_{k-1}^n$ of associated Legendre functions for $k,n = 0,1,\ldots,N$ . More...

void	eval_al (R x, R y, const int size, const int k, R alpha, R beta, R *gamma)
	Evaluates an associated Legendre polynomials using the Clenshaw-algorithm. More...

int	eval_al_thresh (R x, R y, const int size, const int k, R alpha, R beta, R *gamma, R threshold)
	Evaluates an associated Legendre polynomials using the Clenshaw-algorithm if it no exceeds a given threshold. More...

static void	c2e (nfsft_plan *plan)
	Converts coefficients $\left(b_k^n\right)_{k=0}^M$ with $M \in \mathbb{N}_0$ , $-M \le n \le M$ from a linear combination of Chebyshev polynomials $f(\cos\vartheta) = \sum_{k=0}^{2\lfloor\frac{M}{2}\rfloor} a_k (\sin\vartheta)^{n\;\mathrm{mod}\;2} T_k(\cos\vartheta)$ to coefficients $\left(c_k^n\right)_{k=0}^M$ matching the representation by complex exponentials $f(\cos\vartheta) = \sum_{k=-M}^{M} c_k \mathrm{e}^{\mathrm{i}k\vartheta}$ for each order $n=-M,\ldots,M$ . More...

static void	c2e_transposed (nfsft_plan *plan)
	Transposed version of the function c2e. More...

void	nfsft_init (nfsft_plan *plan, int N, int M)

void	nfsft_init_advanced (nfsft_plan *plan, int N, int M, unsigned int flags)

void	nfsft_init_guru (nfsft_plan *plan, int N, int M, unsigned int flags, unsigned int nfft_flags, int nfft_cutoff)

void	nfsft_precompute (int N, double kappa, unsigned int nfsft_flags, unsigned int fpt_flags)

void	nfsft_forget (void)

void	nfsft_finalize (nfsft_plan *plan)

void	nfsft_trafo_direct (nfsft_plan *plan)

void	nfsft_adjoint_direct (nfsft_plan *plan)

void	nfsft_trafo (nfsft_plan *plan)

void	nfsft_adjoint (nfsft_plan *plan)

void	nfsft_precompute_x (nfsft_plan *plan)

Variables
static struct nfsft_wisdom	wisdom = {false,0U,-1,-1,0,0,0,0,0}
	The global wisdom structure for precomputed data. More...

Detailed Description

This module implements nonuniform fast spherical Fourier transforms.

In the following, we abbreviate the term "nonuniform fast spherical Fourier transform" by NFSFT.

Preliminaries

This section summarises basic definitions and properties related to spherical Fourier transforms.

Spherical Coordinates

Every point in $\mathbb{R}^3$ can be described in spherical coordinates by a vector $(r,\vartheta,\varphi)^{\mathrm{T}}$ with the radius $r \in \mathbb{R}^{+}$ and two angles $\vartheta \in [0,\pi]$ , $\varphi \in [-\pi,\pi)$ . We denote by $\mathbb{S}^2$ the two-dimensional unit sphere embedded into $\mathbb{R}^3$ , i.e.

$\mathbb{S}^2 := \left\{\mathbf{x} \in \mathbb{R}^{3}:\; \|\mathbf{x}\|_2=1\right\}$

and identify a point from $\mathbb{S}^2$ with the corresponding vector $(\vartheta,\varphi)^{\mathrm{T}}$ . The spherical coordinate system is illustrated in the following figure:

For consistency with the other modules and the conventions used there, we also use swapped scaled spherical coordinates $x_1 := \frac{\varphi}{2\pi}$ , $x_2 := \frac{\vartheta}{2\pi}$ and identify a point from $\mathbb{S}^2$ with the vector $\mathbf{x} := \left(x_1,x_2\right) \in [-\frac{1}{2}, \frac{1}{2}) \times [0,\frac{1}{2}]$ .

Legendre Polynomials

The Legendre polynomials $P_k : [-1,1] \rightarrow \mathbb{R}$, $k \in \mathbb{N}_{0}$ as classical orthogonal polynomials are given by their corresponding Rodrigues formula

$P_k(t) := \frac{1}{2^k k!} \frac{\text{d}^k}{\text{d} t^k} \left(t^2-1\right)^k.$

The corresponding three-term recurrence relation is

$(k+1)P_{k+1}(t) = (2k+1) x P_{k}(t) - k P_{k-1}(t) \quad (k \in \mathbb{N}_0).$

With

$\left< f,g \right>_{\text{L}^2\left([-1,1]\right)} := \int_{-1}^{1} f(t) g(t) \text{d} t$

being the usual $\text{L}^2\left([-1,1]\right)$ inner product, the Legendre polynomials obey the orthogonality condition

$\left< P_k,P_l \right>_{\text{L}^2\left([-1,1]\right)} = \frac{2}{2k+1} \delta_{k,l}.$

Remarks: The normalisation constant $c_k := \sqrt{\frac{2k+1}{2}}$ renders the scaled Legendre polynomials orthonormal with respect to the induced $\text{L}^2\left([-1,1]\right)$ norm
$\|f\|_{\text{L}^2\left([-1,1]\right)} := \left(<f,f>_{\text{L}^2\left([-1,1]\right)}\right)^{1/2} = \left(\int_{-1}^{1} |f(t)|^2 \; \text{d} t\right)^{1/2}.$

Associated Legendre Functions

The associated Legendre functions $P_k^n : [-1,1] \rightarrow \mathbb{R}$ , $n \in \mathbb{N}_0$ , $k \ge n$ are defined by

$P_k^n(t) := \left(\frac{(k-n)!}{(k+n)!}\right)^{1/2} \left(1-t^2\right)^{n/2} \frac{\text{d}^n}{\text{d} t^n} P_k(t).$

For $n = 0$ , they coincide with the Legendre polynomials, i.e. $P_k^0 = P_k$ . The associated Legendre functions obey the three-term recurrence relation

$P_{k+1}^n(t) = v_{k}^n t P_k^n(t) + w_{k}^n P_{k-1}^n(t) \quad (k \ge n),$

with $P_{n-1}^n(t) := 0$ , $P_{n}^n(t) := \frac{\sqrt{(2n)!}}{2^n n!} \left(1-t^2\right)^{n/2}$ , and

$v_{k}^n := \frac{2k+1}{((k-n+1)(k+n+1))^{1/2}}\; ,\qquad w_{k}^n := - \frac{((k-n)(k+n))^{1/2}}{((k-n+1)(k+n+1))^{1/2}}.$

For fixed $n$ , the set $\left\{P_k^n:\: k \ge n\right\}$ forms a complete set of orthogonal functions in $\text{L}^2\left([-1,1]\right)$ with

$\left< P_k^n,P_l^n \right>_{\text{L}^2\left([-1,1]\right)} = \frac{2}{2k+1} \delta_{k,l} \quad (0 \le n \le k,l).$

Remarks: The normalisation constant $c_k = \sqrt{\frac{2k+1}{2}}$ renders the scaled associated Legendre functions orthonormal with respect to the induced $\text{L}^2\left([-1,1]\right)$ norm
$\|f\|_{\text{L}^2\left([-1,1]\right)} := \left(<f,f>_{\text{L}^2\left([-1,1]\right)}\right)^{1/2} = \left(\int_{-1}^{1} |f(t)|^2 \; \text{d} t\right)^{1/2}.$

Spherical Harmonics

The standard orthogonal basis of spherical harmonics for $\text{L}^2 \left(\mathbb{S}^2\right)$ with yet unnormalised basis functions $\tilde{Y}_k^n : \mathbb{S}^2 \rightarrow \mathbb{C}$ is given by

$\tilde{Y}_k^n(\vartheta,\varphi) := P_k^{|n|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}$

with the usual $\text{L}^2\left(\mathbb{S}^2\right)$ inner product

$\left< f,g \right>_{\mathrm{L}^2\left(\mathbb{S}^2\right)} := \int_{\mathbb{S}^2} f(\vartheta,\varphi) \overline{g(\vartheta,\varphi)} \: \mathrm{d} \mathbf{\xi} := \int_{-\pi}^{\pi} \int_{0}^{\pi} f(\vartheta,\varphi) \overline{g(\vartheta,\varphi)} \sin \vartheta \; \mathrm{d} \vartheta \; \mathrm{d} \varphi.$

The normalisation constant $c_k^n := \sqrt{\frac{2k+1}{4\pi}}$ renders the scaled basis functions

$Y_k^n(\vartheta,\varphi) := c_k^n P_k^{|n|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}$

orthonormal with respect to the induced $\text{L}^2\left(\mathbb{S}^2 \right)$ norm

$\|f\|_{\text{L}^2\left(\mathbb{S}^2\right)} = \left(<f,f>_{\text{L}^2\left(\mathbb{S}^2\right)}\right)^{1/2} = \left(\int_{-\pi}^{\pi} \int_{0}^{\pi} |f(\vartheta,\varphi)|^2 \sin \vartheta \; \mathrm{d} \vartheta \; \mathrm{d} \varphi\right)^{1/2}.$

A function $f \in \mathrm{L}^2\left(\mathbb{S}^2\right)$ has the orthogonal expansion

$f = \sum_{k=0}^{\infty} \sum_{n=-k}^{k} \hat{f}(k,n) Y_k^n,$

where the coefficients $\hat{f}(k,n) := \left< f, Y_k^{n} \right>_{\mathrm{L}^2\left(\mathbb{S}^2\right)}$ are the spherical Fourier coefficients and the equivalence is understood in the $\mathrm{L}^2$ -sense.

Nonuniform Fast Spherical Fourier Transforms

This section describes the input and output relation of the spherical Fourier transform algorithms and the layout of the corresponding plan structure.

Nonuniform Discrete Spherical Fourier Transform

The nonuniform discrete spherical Fourier transform (NDSFT) is defined as follows:

$\begin{array}{rcl} \text{\textbf{Input}} & : & \text{coefficients } \hat{f}(k,n) \in \mathbb{C} \text{ for } k=0,\ldots,N,\;n=-k, \ldots,k,\; N \in \mathbb{N}_0,\\[1ex] & & \text{arbitrary nodes } \mathbf{x}(m) \in [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}] \text{ for } m=0,\ldots,M-1, M \in \mathbb{N}. \\[1ex] \text{\textbf{Task}} & : & \text{evaluate } f(m) := f\left( \mathbf{x}(m)\right) = \sum_{k=0}^N \sum_{n=-k}^k \hat{f}_k^n Y_k^n\left(\mathbf{x}(m)\right) \text{ for } m=0,\ldots,M-1. \\[1ex] \text{\textbf{Output}} & : & \text{coefficients } f(m) \in \mathbb{C} \text{ for } m=0,\ldots,M-1.\\ \end{array}$

Adjoint Nonuniform Discrete Spherical Fourier Transform

The adjoint nonuniform discrete spherical Fourier transform (adjoint NDSFT) is defined as follows:

$\begin{array}{rcl} \text{\textbf{Input}} & : & \text{coefficients } f(m) \in \mathbb{C} \text{ for } m=0,\ldots,M-1, M \in \mathbb{N},\\ & & \text{arbitrary nodes } \mathbf{x}(m) \in [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}] \text{ for } m=0,\ldots,M-1, N \in \mathbb{N}_0.\\[1ex] \text{\textbf{Task}} & : & \text{evaluate } \hat{f}(k,n) := \sum_{m=0}^{M-1} f(m) \overline{Y_k^n\left(\mathbf{x}(m)\right)}cd Do \text{ for } k=0,\ldots,N,\;n=-k,\ldots,k.\\[1ex] \text{\textbf{Output}} & : & \text{coefficients } \hat{f}(k,n) \in \mathbb{C} \text{ for } k=0,\ldots,N,\;n=-k,\ldots,k.\\[1ex] \end{array}$

Data Layout

This section describes the public layout of the nfsft_plan structure which contains all data for the computation of the aforementioned spherical Fourier transforms. The structure contains private (no read or write allowed), public read-only (only read access permitted), and public read-write (read and write access allowed) members. In the following, we indicate read and write access by read and write. The public members are structured as follows:

N_total (read) The total number of components in f_hat. If the bandwidth is $N \in \mathbb{N}_0$ , the total number of components in f_hat is N_total .
M_total (read) the total number of samples
f_hat (read-write) The flattened array of spherical Fourier coefficents. The array has length such that valid indices $i \in \mathbb{N}_0$ for array access f_hat [ ] are $i=0,1,\ldots,(2N+2)^2-1$ . However, only read and write access to indices corresponding to spherical Fourier coefficients $\hat{f}(k,n)$ is defined. The index corresponding to the spherical Fourier coefficient $\hat{f}(k,n)$ with $0 \le k \le M$ , $-k \le n \le k$ is . For convenience, the helper macro NFSFT_INDEX(k,n) provides the necessary index calculations such that one can write f_hat[ NFSFT_INDEX( )] = ... to access the component corresponding to $\hat{f}(k,n)$ . The data layout is due to implementation details.
f (read-write) the array of coefficients for $m=0,\ldots,M-1$ such that f[ ] =
N (read) the bandwidth $N \in \mathbb{N}_0$
x the array of nodes $\mathbf{x}(m) \in [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}]$ for $m = 0, \ldots,M-1$ such that f[ ] = and f[ ] =

Good to know...

When using the routines of this module you should bear in mind the following:

The bandwidth $N_{\text{max}}$ up to which precomputation is performed is always chosen as the next power of two with respect to the specified maximum bandwidth.
By default, the NDSFT transforms (see nfsft_direct_trafo, nfsft_trafo) are allowed to destroy the input f_hat while the input x is preserved. On the contrary, the adjoint NDSFT transforms (see nfsft_direct_adjoint, nfsft_adjoint) do not destroy the input f and x by default. The desired behaviour can be assured by using the NFSFT_PRESERVE_F_HAT, NFSFT_PRESERVE_X, NFSFT_PRESERVE_F and NFSFT_DESTROY_F_HAT, NFSFT_DESTROY_X, NFSFT_DESTROY_F flags.

Macro Definition Documentation

#define NFSFT_NORMALIZED (1U << 0)

By default, all computations are performed with respect to the unnormalized basis functions

$\tilde{Y}_k^n(\vartheta,\varphi) = P_k^{|n|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}.$

If this flag is set, all computations are carried out using the $L_2$ - normalized basis functions

$Y_k^n(\vartheta,\varphi) = \sqrt{\frac{2k+1}{4\pi}} P_k^{|n|}(\cos\vartheta) \mathrm{e}^{\mathrm{i} n \varphi}.$

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 575 of file nfft3.h.

Referenced by main(), nfsft_adjoint(), and nfsft_trafo().

#define NFSFT_USE_NDFT (1U << 1)

If this flag is set, the fast NFSFT algorithms (see nfsft_trafo, nfsft_adjoint) will use internally the exact but usually slower direct NDFT algorithm in favor of fast but approximative NFFT algorithm.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 576 of file nfft3.h.

Referenced by main(), nfsft_adjoint(), and nfsft_trafo().

#define NFSFT_USE_DPT (1U << 2)

If this flag is set, the fast NFSFT algorithms (see nfsft_trafo, nfsft_adjoint) will use internally the usually slower direct DPT algorithm in favor of the fast FPT algorithm.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Warning: This feature is not implemented yet!

Definition at line 577 of file nfft3.h.

Referenced by main(), nfsft_adjoint(), and nfsft_trafo().

#define NFSFT_MALLOC_X (1U << 3)

If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array x for you. Otherwise, you have to assure by yourself that x points to an array of proper size before excuting a transform and you are responsible for freeing the corresponding memory before program termination.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 578 of file nfft3.h.

Referenced by main(), nfsft_finalize(), and nfsft_init().

#define NFSFT_MALLOC_F_HAT (1U << 5)

If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array f_hat for you. Otherwise, you have to assure by yourself that f_hat points to an array of proper size before excuting a transform and you are responsible for freeing the corresponding memory before program termination.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 579 of file nfft3.h.

Referenced by main(), nfsft_finalize(), and nfsft_init().

#define NFSFT_MALLOC_F (1U << 6)

If this flag is set, the init methods (see nfsft_init , nfsft_init_advanced , and nfsft_init_guru) will allocate memory and the method nfsft_finalize will free the array f for you. Otherwise, you have to assure by yourself that f points to an array of proper size before excuting a transform and you are responsible for freeing the corresponding memory before program termination.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 580 of file nfft3.h.

Referenced by main(), nfsft_finalize(), and nfsft_init().

#define NFSFT_PRESERVE_F_HAT (1U << 7)

If this flag is set, it is guaranteed that during an execution of nfsft_direct_trafo or nfsft_trafo the content of f_hat remains unchanged.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 581 of file nfft3.h.

Referenced by nfsft_finalize(), and nfsft_trafo().

#define NFSFT_PRESERVE_X (1U << 8)

If this flag is set, it is guaranteed that during an execution of nfsft_direct_trafo, nfsft_trafo or nfsft_direct_adjoint, nfsft_adjoint the content of x remains unchanged.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 582 of file nfft3.h.

#define NFSFT_PRESERVE_F (1U << 9)

If this flag is set, it is guaranteed that during an execution of nfsft_direct_adjoint or nfsft_adjoint the content of f remains unchanged.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 583 of file nfft3.h.

#define NFSFT_DESTROY_F_HAT (1U << 10)

If this flag is set, it is explicitely allowed that during an execution of nfsft_direct_trafo or nfsft_trafo the content of f_hat may be changed.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 584 of file nfft3.h.

#define NFSFT_DESTROY_X (1U << 11)

If this flag is set, it is explicitely allowed that during an execution of nfsft_direct_trafo, nfsft_trafo or nfsft_direct_adjoint, nfsft_adjoint the content of x may be changed.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 585 of file nfft3.h.

#define NFSFT_DESTROY_F (1U << 12)

If this flag is set, it is explicitely allowed that during an execution of nfsft_direct_adjoint or nfsft_adjoint the content of f may be changed.

See Also: nfsft_init; nfsft_init_advanced; nfsft_init_guru

Author: Jens Keiner

Definition at line 586 of file nfft3.h.

#define NFSFT_NO_DIRECT_ALGORITHM (1U << 13)

If this flag is set, the transforms nfsft_direct_trafo and nfsft_direct_adjoint do not work. Setting this flag saves some memory for precomputed data.

See Also: nfsft_precompute; nfsft_direct_trafo; nfsft_direct_adjoint

Author: Jens Keiner

Definition at line 589 of file nfft3.h.

Referenced by nfsft_forget(), and nfsft_precompute().

#define NFSFT_NO_FAST_ALGORITHM (1U << 14)

If this flag is set, the transforms nfsft_trafo and nfsft_adjoint do not work. Setting this flag saves memory for precomputed data.

See Also: nfsft_precompute; nfsft_trafo; nfsft_adjoint

Author: Jens Keiner

Definition at line 590 of file nfft3.h.

Referenced by main(), nfsft_adjoint(), nfsft_forget(), nfsft_precompute(), and nfsft_trafo().

#define NFSFT_ZERO_F_HAT (1U << 16)

If this flag is set, the transforms nfsft_adjoint and nfsft_direct_adjoint set all unused entries in f_hat not corresponding to spherical Fourier coefficients to zero.

Author: Jens Keiner

Definition at line 591 of file nfft3.h.

Referenced by main(), and nfsft_adjoint().

#define NFSFT_INDEX	(	k,
		n,
		plan
	)	((2(plan)->N+2)((plan)->N-n+1)+(plan)->N+k+1)

This helper macro expands to the index $i$ corresponding to the spherical Fourier coefficient $f_hat(k,n)$ for $0 \le k \le N$ , $-k \le n \le k$ with

$(N+2)(N-n+1)+N+k+1$

Definition at line 594 of file nfft3.h.

Referenced by c2e(), c2e_transposed(), main(), nfsft_adjoint(), and nfsft_trafo().

#define NFSFT_F_HAT_SIZE ( N ) ((2*N+2)*(2*N+2))

This helper macro expands to the logical size of a spherical Fourier coefficients array for a bandwidth N.

Definition at line 595 of file nfft3.h.

Referenced by main().

#define NFSFT_DEFAULT_NFFT_CUTOFF 6

The default NFFT cutoff parameter.

Author: Jens Keiner

Definition at line 62 of file nfsft.c.

Referenced by nfsft_init_advanced().

#define NFSFT_DEFAULT_THRESHOLD 1000

The default threshold for the FPT.

Author: Jens Keiner

Definition at line 69 of file nfsft.c.

#define NFSFT_BREAK_EVEN 5

The break-even bandwidth $N \in \mathbb{N}_0$ .

Author: Jens Keiner

Definition at line 76 of file nfsft.c.

Referenced by nfsft_adjoint(), nfsft_forget(), nfsft_precompute(), and nfsft_trafo().

Function Documentation

void alpha_al_all	(	R *	alpha,
		const int	N
	)

inline

Compute three-term-recurrence coefficients $\alpha_{k-1}^n$ of associated Legendre functions for $k,n = 0,1,\ldots,N$ .

alpha A pointer to an array of doubles of size where the coefficients will be stored such that alpha[n+(N+1)+k] = $\alpha_{k-1}^n$ .
N The upper bound .

Definition at line 89 of file legendre.c.

Referenced by nfsft_precompute().

void beta_al_all	(	R *	beta,
		const int	N
	)

inline

Compute three-term-recurrence coefficients $\beta_{k-1}^n$ of associated Legendre functions for $k,n = 0,1,\ldots,N$ .

beta A pointer to an array of doubles of size where the coefficients will be stored such that beta[n+(N+1)+k] = $\beta_{k-1}^n$ .
N The upper bound .

Definition at line 98 of file legendre.c.

Referenced by nfsft_precompute().

void gamma_al_all	(	R *	gamma,
		const int	N
	)

inline

Compute three-term-recurrence coefficients $\gamma_{k-1}^n$ of associated Legendre functions for $k,n = 0,1,\ldots,N$ .

beta A pointer to an array of doubles of size where the coefficients will be stored such that gamma[n+(N+1)+k] = $\gamma_{k-1}^n$ .
N The upper bound .

Definition at line 107 of file legendre.c.

Referenced by nfsft_precompute().

void eval_al	(	R *	x,
		R *	y,
		const int	size,
		const int	k,
		R *	alpha,
		R *	beta,
		R *	gamma
	)

Evaluates an associated Legendre polynomials $P_k^n(x,c)$ using the Clenshaw-algorithm.

x A pointer to an array of nodes where the function is to be evaluated
y A pointer to an array where the function values are returned
size The length of x and y
k The index
alpha A pointer to an array containing the recurrence coefficients $\alpha_c^n,\ldots,\alpha_{c+k}^n$
beta A pointer to an array containing the recurrence coefficients $\beta_c^n,\ldots,\beta_{c+k}^n$
gamma A pointer to an array containing the recurrence coefficients $\gamma_c^n,\ldots,\gamma_{c+k}^n$

Definition at line 116 of file legendre.c.

int eval_al_thresh	(	R *	x,
		R *	y,
		const int	size,
		const int	k,
		R *	alpha,
		R *	beta,
		R *	gamma,
		R	threshold
	)

Evaluates an associated Legendre polynomials $P_k^n(x,c)$ using the Clenshaw-algorithm if it no exceeds a given threshold.

x A pointer to an array of nodes where the function is to be evaluated
y A pointer to an array where the function values are returned
size The length of x and y
k The index
alpha A pointer to an array containing the recurrence coefficients $\alpha_c^n,\ldots,\alpha_{c+k}^n$
beta A pointer to an array containing the recurrence coefficients $\beta_c^n,\ldots,\beta_{c+k}^n$
gamma A pointer to an array containing the recurrence coefficients $\gamma_c^n,\ldots,\gamma_{c+k}^n$
threshold The threshold

Definition at line 161 of file legendre.c.

static void c2e ( nfsft_plan * plan )

inlinestatic

Converts coefficients $\left(b_k^n\right)_{k=0}^M$ with $M \in \mathbb{N}_0$ , $-M \le n \le M$ from a linear combination of Chebyshev polynomials

$f(\cos\vartheta) = \sum_{k=0}^{2\lfloor\frac{M}{2}\rfloor} a_k (\sin\vartheta)^{n\;\mathrm{mod}\;2} T_k(\cos\vartheta)$

to coefficients $\left(c_k^n\right)_{k=0}^M$ matching the representation by complex exponentials

$f(\cos\vartheta) = \sum_{k=-M}^{M} c_k \mathrm{e}^{\mathrm{i}k\vartheta}$

for each order $n=-M,\ldots,M$ .

plan The nfsft_plan containing the coefficients $\left(b_k^n\right)_{k=0,\ldots,M;n=-M,\ldots,M}$

Remarks: The transformation is computed in place.

Author: Jens Keiner

Definition at line 108 of file nfsft.c.

References nfsft_plan::f_hat_intern, nfsft_plan::N, and NFSFT_INDEX.

Referenced by nfsft_trafo(), and nfsoft_trafo().

static void c2e_transposed ( nfsft_plan * plan )

inlinestatic

Transposed version of the function c2e.

plan The nfsft_plan containing the coefficients $\left(c_k^n\right)_{k=-M,\ldots,M;n=-M,\ldots,M}$

Remarks: The transformation is computed in place.

Author: Jens Keiner

Definition at line 186 of file nfsft.c.

References nfsft_plan::f_hat, nfsft_plan::N, and NFSFT_INDEX.

Referenced by nfsft_adjoint().

void nfsft_init	(	nfsft_plan *	plan,
		int	N,
		int	M
	)

Creates a transform plan.

plan a pointer to a nfsft_plan structure
N the bandwidth $N \in \mathbb{N}_0$
M the number of nodes $M \in \mathbb{N}$

Author: Jens Keiner

Definition at line 253 of file nfsft.c.

References nfsft_init_advanced(), NFSFT_MALLOC_F, NFSFT_MALLOC_F_HAT, and NFSFT_MALLOC_X.

void nfsft_init_advanced	(	nfsft_plan *	plan,
		int	N,
		int	M,
		unsigned int	flags
	)

Creates a transform plan.

plan a pointer to a
```
nfsft_plan 
```
structure
N the bandwidth $N \in \mathbb{N}_0$
M the number of nodes $M \in \mathbb{N}$
nfsft_flags the NFSFT flags

Author: Jens Keiner

Definition at line 260 of file nfsft.c.

References FFT_OUT_OF_PLACE, FFTW_INIT, NFSFT_DEFAULT_NFFT_CUTOFF, PRE_PHI_HUT, and PRE_PSI.

Referenced by nfsft_init().

void nfsft_precompute	(	int	N,
		double	kappa,
		unsigned int	nfsft_flags,
		unsigned int	fpt_flags
	)

Performes precomputation up to the next power of two with respect to a given bandwidth $N \in \mathbb{N}_2$ . The threshold parameter $\kappa \in \mathbb{R}^{+}$ determines the number of stabilization steps computed in the discrete polynomial transform and thereby its accuracy.