3.0 API Reference

NFCT/NFST - Nonequispaced fast (co)sine transform

Direct and fast computation of the discrete nonequispaced (co)sine transform. More...


Data Structures
struct	nfct_plan
	Structure for a transform plan. More...
struct	nfst_plan
	Structure for a transform plan. More...
Functions
void	nfct_init_1d (nfct_plan *ths_plan, int N0, int M_total)
	Creates a 1-dimensional transform plan.
void	nfct_init_2d (nfct_plan *ths_plan, int N0, int N1, int M_total)
	Creates a 3-dimensional transform plan.
void	nfct_init_3d (nfct_plan *ths_plan, int N0, int N1, int N2, int M_total)
	Creates a 3-dimensional transform plan.
void	nfct_init (nfct_plan ths_plan, int d, int N, int M_total)
	Creates a d-dimensional transform plan.
void	nfct_init_guru (nfct_plan ths_plan, int d, int N, int M_total, int *n, int m, unsigned nfct_flags, unsigned fftw_flags)
	Creates a d-dimensional transform plan.
void	nfct_precompute_psi (nfct_plan *ths_plan)
	precomputes the values psi if the PRE_PSI is set the application program has to call this routine after setting the nodes this_plan->x
void	nfct_trafo (nfct_plan *ths_plan)
	executes a NFCT (approximate,fast), computes for $j=0,...,M\_total-1$ $f_j^C(x_j) = \sum_{k \in I_0^{N,d}} \hat{f}_k^C * cos(2 \pi k x_j)$
void	ndct_trafo (nfct_plan *ths_plan)
	executes a NDCT (exact,slow), computes for $j=0,...,M\_total-1$ $f_j^C(x_j) = \sum_{k \in I_0^{N,d}} \hat{f}_k^C * cos(2 \pi k x_j)$
void	nfct_adjoint (nfct_plan *ths_plan)
	executes a transposed NFCT (approximate,fast), computes for $k \in I_0^{N,d}$ $h^C(k) = \sum_{j \in I_0^{(M\_total,1)}} f_j^C * cos(2 \pi k x_j)$
void	ndct_adjoint (nfct_plan *ths_plan)
	executes a direct transposed NDCT (exact,slow), computes for $k \in I_0^{N,d}$ $h^C(k) = \sum_{j \in I_0^{(M\_total,1)}} f_j^C * cos(2 \pi k x_j)$
void	nfct_finalize (nfct_plan *ths_plan)
	Destroys a plan.
double	nfct_phi_hut (nfct_plan *ths_plan, int k, int d)
	do some adjustments (N,n) then compute PHI_HUT
double	nfct_phi (nfct_plan *ths_plan, double x, int d)
	do some adjustments (N,n) then compute PHI
int	nfct_fftw_2N (int n)
	returns 2(n-1), fftw related issue
int	nfct_fftw_2N_rev (int n)
	returns 0.5n+1, fftw related issue
void	nfst_init_1d (nfst_plan *ths_plan, int N0, int M_total)
	Creates a 1-dimensional transform plan.
void	nfst_init_2d (nfst_plan *ths_plan, int N0, int N1, int M_total)
	Creates a 3-dimensional transform plan.
void	nfst_init_3d (nfst_plan *ths_plan, int N0, int N1, int N2, int M_total)
	Creates a 3-dimensional transform plan.
void	nfst_init (nfst_plan ths_plan, int d, int N, int M_total)
	Creates a d-dimensional transform plan.
void	nfst_init_m (nfst_plan ths_plan, int d, int N, int M_total, int m)
	Creates a d-dimensional transform plan with pcific m.
void	nfst_init_guru (nfst_plan ths_plan, int d, int N, int M_total, int *n, int m, unsigned nfst_flags, unsigned fftw_flags)
	Creates a d-dimensional transform plan.
void	nfst_precompute_psi (nfst_plan *ths_plan)
	precomputes the values psi if the PRE_PSI is set the application program has to call this routine after setting the nodes this_plan->x
void	nfst_trafo (nfst_plan *ths_plan)
	executes a NFST (approximate,fast), computes for $j=0,...,M\_total-1$ $f_j^S(x_j) = \sum_{k \in I_1^{N,d}} \hat{f}_k^S * sin(2 \pi k x_j)$
void	ndst_trafo (nfst_plan *ths_plan)
	executes a NDST (exact,slow), computes for $j=0,...,M\_total-1$ $f_j^S(x_j) = \sum_{k \in I_1^{N,d}} \hat{f}_k^S * sin(2 \pi k x_j)$
void	nfst_adjoint (nfst_plan *ths_plan)
	executes a transposed NFST (approximate,fast), computes for $k \in I_1^{N,d}$ $h^S(k) = \sum_{j \in I_0^{M\_total,1}} f_j^S * cos(2 \pi k x_j)$
void	ndst_adjoint (nfst_plan *ths_plan)
	executes a direct transposed NDST (exact,slow), computes for $k \in I_1^{N,d}$ $h^S(k) = \sum_{j \in I_0^{M\_total,1}} f_j^S * cos(2 \pi k x_j)$
void	nfst_finalize (nfst_plan *ths_plan)
	Destroys a plan.
void	nfst_full_psi (nfst_plan *ths_plan, double eps)
	more memory usage, a bit faster
double	nfst_phi_hut (nfst_plan *ths_plan, int k, int d)
	do some adjustments (N,n) then compute PHI_HUT
double	nfst_phi (nfst_plan *ths_plan, double x, int d)
	do some adjustments (N,n) then compute PHI
int	nfst_fftw_2N (int n)
	returns 2(n+1), fftw related issue
int	nfst_fftw_2N_rev (int n)
	returns 0.5n-1, fftw related issue

Detailed Description

Direct and fast computation of the discrete nonequispaced (co)sine transform.

Function Documentation

void nfct_init_1d ( nfct_plan * ths_plan,

int N0,

int M_total

)

Creates a 1-dimensional transform plan.

ths_plan The plan for the transform

N0 The bandwidth $N$

M_total The number of nodes $x$

Author:
Steffen Klatt

Definition at line 996 of file nfct.c.
References nfct_init().

void nfct_init_2d ( nfct_plan * ths_plan,

int N0,

int N1,

int M_total

)

Creates a 3-dimensional transform plan.

ths_plan The plan for the transform

N0 The bandwidth of dimension 1

N1 The bandwidth of dimension 2

M_total The number of nodes $x$

Author:
Steffen Klatt

Definition at line 1004 of file nfct.c.
References nfct_init().

void nfct_init_3d ( nfct_plan * ths_plan,

int N0,

int N1,

int N2,

int M_total

)

Creates a 3-dimensional transform plan.

ths_plan The plan for the transform

N0 The bandwidth of dimension 1

N1 The bandwidth of dimension 2

N2 The bandwidth of dimension 3

M_total The number of nodes $x$

Author:
Steffen Klatt

Definition at line 1013 of file nfct.c.
References nfct_init().

void nfct_init ( nfct_plan * ths_plan,

int d,

int * N,

int M_total

)

Creates a d-dimensional transform plan.

ths_plan The plan for the transform

d the dimension

N The bandwidths

M_total The number of nodes $x$

Author:
Steffen Klatt

Definition at line 931 of file nfct.c.
References nfct_plan::d, nfct_plan::fftw_flags, nfct_plan::M_total, nfct_plan::N, nfct_plan::n, nfct_fftw_2N(), nfct_plan::nfct_flags, and nfft_next_power_of_2().
Referenced by nfct_init_1d(), nfct_init_2d(), and nfct_init_3d().

void nfct_init_guru ( nfct_plan * ths,

int d,

int * N,

int M_total,

int * n,

int m,

unsigned nfct_flags,

unsigned fftw_flags

)

Creates a d-dimensional transform plan.
< index over all dimensions
Definition at line 968 of file nfct.c.
References nfct_plan::d, nfct_plan::fftw_flags, nfct_plan::m, nfct_plan::M_total, nfct_plan::N, nfct_plan::n, and nfct_plan::nfct_flags.

void nfct_precompute_psi ( nfct_plan * ths )

precomputes the values psi if the PRE_PSI is set the application program has to call this routine after setting the nodes this_plan->x
< index over all dimensions
< index over all nodes
< index 0<=lj<u+o+1
< depends on x_j
Definition at line 838 of file nfct.c.
References nfct_plan::d, nfct_plan::M_total, and nfct_plan::psi.

void nfct_trafo ( nfct_plan * ths )

executes a NFCT (approximate,fast), computes for $j=0,...,M\_total-1$ $f_j^C(x_j) = \sum_{k \in I_0^{N,d}} \hat{f}_k^C * cos(2 \pi k x_j)$
use ths->my_fftw_r2r_plan
form $\hat g_k = \frac{\hat f_k}{c_k\left(\phi\right)} \text{ for } k \in I_N$
compute by d-variate discrete Fourier transform $g_l = \sum_{k \in I_N} \hat g_k {\rm e}^{-2\pi {\rm i} \frac{kl}{n}} \text{ for } l \in I_n$
set $f_j = \sum_{l \in I_n,m(x_j)} g_l \psi\left(x_j-\frac{l}{n}\right) \text{ for } j=0,\hdots,M-1$
Definition at line 711 of file nfct.c.
References PRE_FULL_PSI.

void ndct_trafo ( nfct_plan * ths_plan )

executes a NDCT (exact,slow), computes for $j=0,...,M\_total-1$ $f_j^C(x_j) = \sum_{k \in I_0^{N,d}} \hat{f}_k^C * cos(2 \pi k x_j)$

ths_plan The plan for the transform

Author:
Steffen Klatt

Definition at line 256 of file nfct.c.

void nfct_adjoint ( nfct_plan * ths )

executes a transposed NFCT (approximate,fast), computes for $k \in I_0^{N,d}$ $h^C(k) = \sum_{j \in I_0^{(M\_total,1)}} f_j^C * cos(2 \pi k x_j)$
use ths->my_fftw_plan
set $g_l = \sum_{j=0}^{M-1} f_j \psi\left(x_j-\frac{l}{n}\right) \text{ for } l \in I_n,m(x_j)$
compute by d-variate discrete cosine transform $\hat g_k = \sum_{l \in I_n} g_l {\rm e}^{-2\pi {\rm i} \frac{kl}{n}} \text{ for } k \in I_N$
form $\hat f_k = \frac{\hat g_k}{c_k\left(\phi\right)} \text{ for } k \in I_N$
Definition at line 765 of file nfct.c.
References nfct_plan::g, nfct_plan::g1, nfct_plan::g2, nfct_plan::g_hat, nfct_plan::my_fftw_r2r_plan, nfct_plan::nfct_flags, PRE_FULL_PSI, nfct_plan::psi_index_f, and nfct_plan::psi_index_g.

void ndct_adjoint ( nfct_plan * ths_plan )

executes a direct transposed NDCT (exact,slow), computes for $k \in I_0^{N,d}$ $h^C(k) = \sum_{j \in I_0^{(M\_total,1)}} f_j^C * cos(2 \pi k x_j)$

ths_plan The plan for the transform

Author:
Steffen Klatt

Definition at line 257 of file nfct.c.
References nfct_fftw_2N().

void nfct_finalize ( nfct_plan * ths )

Destroys a plan.
index over dimensions
Definition at line 1027 of file nfct.c.
References nfct_plan::c_phi_inv, nfct_plan::d, nfct_plan::f, nfct_plan::f_hat, FFT_OUT_OF_PLACE, FFTW_INIT, nfct_plan::g1, nfct_plan::g2, MALLOC_F, MALLOC_F_HAT, MALLOC_X, nfct_plan::my_fftw_r2r_plan, nfct_plan::N, nfct_plan::n, nfct_plan::nfct_flags, PRE_PHI_HUT, PRE_PSI, nfct_plan::psi, nfct_plan::sigma, and nfct_plan::x.

double nfct_phi_hut ( nfct_plan * ths_plan,

int k,

int d

)

do some adjustments (N,n) then compute PHI_HUT

ths_plan the plan for the transform

k index of c_phi

d dimension

Author:
Steffen Klatt

Definition at line 63 of file nfct.c.

double nfct_phi ( nfct_plan * ths_plan,

double x,

int d

)

do some adjustments (N,n) then compute PHI

ths_plan the plan for the transform

x node $x$

d dimension

Author:
Steffen Klatt

Definition at line 72 of file nfct.c.

int nfct_fftw_2N ( int n )

returns 2(n-1), fftw related issue

n i.e. length of dct-1

Author:
Steffen Klatt

Definition at line 81 of file nfct.c.
Referenced by ndct_adjoint(), and nfct_init().

int nfct_fftw_2N_rev ( int n )

returns 0.5n+1, fftw related issue

n i.e. length of dct-1

Author:
Steffen Klatt

Definition at line 86 of file nfct.c.

void nfst_init_1d ( nfst_plan * ths_plan,

int N0,

int M_total

)

Creates a 1-dimensional transform plan.

ths_plan The plan for the transform

N0 The bandwidth $N$

M_total The number of nodes $x$

Author:
Steffen Klatt

Definition at line 979 of file nfst.c.
References nfst_init().

void nfst_init_2d ( nfst_plan * ths_plan,

int N0,

int N1,

int M_total

)

Creates a 3-dimensional transform plan.

ths_plan The plan for the transform

N0 The bandwidth of dimension 1

N1 The bandwidth of dimension 2

M_total The number of nodes $x$

Author:
Steffen Klatt

Definition at line 987 of file nfst.c.
References nfst_init().

void nfst_init_3d ( nfst_plan * ths_plan,

int N0,

int N1,

int N2,

int M_total

)

Creates a 3-dimensional transform plan.

ths_plan The plan for the transform

N0 The bandwidth of dimension 1

N1 The bandwidth of dimension 2

N2 The bandwidth of dimension 3

M_total The number of nodes $x$

Author:
Steffen Klatt

Definition at line 996 of file nfst.c.
References nfst_init().

void nfst_init ( nfst_plan * ths,

int d,

int * N,

int M_total

)

Creates a d-dimensional transform plan.
< index over all dimensions
Definition at line 913 of file nfst.c.
References nfst_plan::d, nfst_plan::fftw_flags, nfst_plan::M_total, nfst_plan::N, nfst_plan::n, nfft_next_power_of_2(), and nfst_plan::nfst_flags.
Referenced by nfst_init_1d(), nfst_init_2d(), and nfst_init_3d().

void nfst_init_m ( nfst_plan * ths_plan,

int d,

int * N,

int M_total,

int m

)

Creates a d-dimensional transform plan with pcific m.
(just for convenience)

ths_plan The plan for the transform

d the dimension

N The bandwidths

M_total The number of nodes $x$

m cut-off parameter

Author:
Steffen Klatt

Definition at line 940 of file nfst.c.
References nfft_next_power_of_2(), nfst_fftw_2N(), and nfst_init_guru().

void nfst_init_guru ( nfst_plan * ths,

int d,

int * N,

int M_total,

int * n,

int m,

unsigned nfst_flags,

unsigned fftw_flags

)

Creates a d-dimensional transform plan.
< index over all dimensions
Definition at line 951 of file nfst.c.
References nfst_plan::d, nfst_plan::fftw_flags, nfst_plan::m, nfst_plan::M_total, nfst_plan::N, nfst_plan::n, and nfst_plan::nfst_flags.
Referenced by nfst_init_m().

void nfst_precompute_psi ( nfst_plan * ths )

precomputes the values psi if the PRE_PSI is set the application program has to call this routine after setting the nodes this_plan->x
< index over all dimensions
< index over all nodes
< index 0<=lj<u+o+1
< depends on x_j
Definition at line 743 of file nfst.c.
References nfst_plan::d, nfst_plan::M_total, nfst_plan::nfst_flags, nfst_full_psi(), nfst_plan::nfst_full_psi_eps, PRE_FULL_PSI, and nfst_plan::psi.

void nfst_trafo ( nfst_plan * ths )

executes a NFST (approximate,fast), computes for $j=0,...,M\_total-1$ $f_j^S(x_j) = \sum_{k \in I_1^{N,d}} \hat{f}_k^S * sin(2 \pi k x_j)$
use ths->my_fftw_r2r_plan
form $\hat g_k = \frac{\hat f_k}{c_k\left(\phi\right)} \text{ for } k \in I_N$
compute by d-variate discrete Fourier transform $g_l = \sum_{k \in I_N} \hat g_k {\rm e}^{-2\pi {\rm i} \frac{kl}{n}} \text{ for } l \in I_n$
set $f_j = \sum_{l \in I_n,m(x_j)} g_l \psi\left(x_j-\frac{l}{n}\right) \text{ for } j=0,\hdots,M-1$
Definition at line 630 of file nfst.c.

void ndst_trafo ( nfst_plan * ths_plan )

executes a NDST (exact,slow), computes for $j=0,...,M\_total-1$ $f_j^S(x_j) = \sum_{k \in I_1^{N,d}} \hat{f}_k^S * sin(2 \pi k x_j)$

ths_plan The plan for the transform

Author:
Steffen Klatt

Definition at line 254 of file nfst.c.

void nfst_adjoint ( nfst_plan * ths )

executes a transposed NFST (approximate,fast), computes for $k \in I_1^{N,d}$ $h^S(k) = \sum_{j \in I_0^{M\_total,1}} f_j^S * cos(2 \pi k x_j)$
use ths->my_fftw_plan
set $g_l = \sum_{j=0}^{M-1} f_j \psi\left(x_j-\frac{l}{n}\right) \text{ for } l \in I_n,m(x_j)$
compute by d-variate discrete cosine transform $\hat g_k = \sum_{l \in I_n} g_l {\rm e}^{-2\pi {\rm i} \frac{kl}{n}} \text{ for } k \in I_N$
form $\hat f_k = \frac{\hat g_k}{c_k\left(\phi\right)} \text{ for } k \in I_N$
Definition at line 675 of file nfst.c.
References nfst_plan::g, nfst_plan::g1, nfst_plan::g2, nfst_plan::g_hat, and nfst_plan::my_fftw_r2r_plan.

void ndst_adjoint ( nfst_plan * ths_plan )

executes a direct transposed NDST (exact,slow), computes for $k \in I_1^{N,d}$ $h^S(k) = \sum_{j \in I_0^{M\_total,1}} f_j^S * cos(2 \pi k x_j)$

ths_plan The plan for the transform

Author:
Steffen Klatt

Definition at line 255 of file nfst.c.
References nfst_fftw_2N().

void nfst_finalize ( nfst_plan * ths_plan )

Destroys a plan.

ths_plan The plan for the transform

Author:
Steffen Klatt

Definition at line 1006 of file nfst.c.
References nfst_plan::c_phi_inv, nfst_plan::d, nfst_plan::f, nfst_plan::f_hat, FFT_OUT_OF_PLACE, FFTW_INIT, nfst_plan::g1, nfst_plan::g2, MALLOC_F, MALLOC_F_HAT, MALLOC_X, nfst_plan::my_fftw_r2r_plan, nfst_plan::N, nfst_plan::n, nfst_plan::nfst_flags, PRE_FULL_PSI, PRE_PHI_HUT, PRE_PSI, nfst_plan::psi, nfst_plan::psi_index_f, nfst_plan::psi_index_g, nfst_plan::sigma, and nfst_plan::x.

void nfst_full_psi ( nfst_plan * ths,

double eps

)

more memory usage, a bit faster
< index over all dimensions
< index over all nodes
< plain index 0<=l_L<lprod
< multi index 0<=lj<u+o+1
< postfix plain index
< 'bandwidth' of matrix B
< depends on x_j
Definition at line 771 of file nfst.c.
References nfst_plan::d, nfst_plan::M_total, nfst_plan::nfst_flags, PRE_PSI, nfst_plan::psi, nfst_plan::psi_index_f, nfst_plan::psi_index_g, and nfst_plan::size_psi.
Referenced by nfst_precompute_psi().

double nfst_phi_hut ( nfst_plan * ths_plan,

int k,

int d

)

do some adjustments (N,n) then compute PHI_HUT

ths_plan the plan for the transform

k index of c_phi

d dimension

Author:
Steffen Klatt

Definition at line 63 of file nfst.c.

double nfst_phi ( nfst_plan * ths_plan,

double x,

int d

)

do some adjustments (N,n) then compute PHI

ths_plan the plan for the transform

x node $x$

d dimension

Author:
Steffen Klatt

Definition at line 72 of file nfst.c.

int nfst_fftw_2N ( int n )

returns 2(n+1), fftw related issue

n i.e. length of dst-1

Author:
Steffen Klatt

Definition at line 81 of file nfst.c.
Referenced by ndst_adjoint(), and nfst_init_m().

int nfst_fftw_2N_rev ( int n )

returns 0.5n-1, fftw related issue

n i.e. length of dct-1

Author:
Steffen Klatt

Definition at line 86 of file nfst.c.

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NFCT/NFST - Nonequispaced fast (co)sine transform

Data Structures

Functions

Detailed Description

Function Documentation