Implementation file for the FPT module. More...

#include "config.h"
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>
#include <math.h>
#include "nfft3.h"
#include "infft.h"

Data Structures
struct	fpt_step_
	Holds data for a single multiplication step in the cascade summation. More...

struct	fpt_data_
	Holds data for a single cascade summation. More...

struct	fpt_set_s_
	Holds data for a set of cascade summations. More...

Macros
#define	K_START_TILDE(x, y) (MAX(MIN(x,y-2),0))
	Minimum degree at top of a cascade.

#define	K_END_TILDE(x, y) MIN(x,y-1)
	Maximum degree at top of a cascade.

#define	FIRST_L(x, y) (LRINT(floor((x)/(double)y)))
	Index of first block of four functions at level.

#define	LAST_L(x, y) (LRINT(ceil(((x)+1)/(double)y))-1)
	Index of last block of four functions at level.

#define	N_TILDE(y) (y-1)

#define	IS_SYMMETRIC(x, y, z) (x >= ((y-1.0)/z))

#define	FPT_BREAK_EVEN 4

#define	ABUVXPWY_SYMMETRIC(NAME, S1, S2)

#define	ABUVXPWY_SYMMETRIC_1(NAME, S1)

#define	ABUVXPWY_SYMMETRIC_2(NAME, S1)

#define	FPT_DO_STEP(NAME, M1_FUNCTION, M2_FUNCTION)

#define	FPT_DO_STEP_TRANSPOSED(NAME, M1_FUNCTION, M2_FUNCTION)

Typedefs
typedef struct fpt_step_	fpt_step
	Holds data for a single multiplication step in the cascade summation.

typedef struct fpt_data_	fpt_data
	Holds data for a single cascade summation.

typedef struct fpt_set_s_	fpt_set_s
	Holds data for a set of cascade summations.

Functions
static void	abuvxpwy (double a, double b, double _Complex u, double _Complex x, double v, double _Complex y, double *w, int n)

static void	abuvxpwy_symmetric1 (double a, double b, double _Complex u, double _Complex x, double v, double _Complex y, double *w, int n)

static void	abuvxpwy_symmetric2 (double a, double b, double _Complex u, double _Complex x, double v, double _Complex y, double *w, int n)

static void	abuvxpwy_symmetric1_1 (double a, double b, double _Complex u, double _Complex x, double v, double _Complex y, int n, double *xx)

static void	abuvxpwy_symmetric1_2 (double a, double b, double _Complex u, double _Complex x, double v, double _Complex y, int n, double *xx)

static void	abuvxpwy_symmetric2_1 (double a, double b, double _Complex u, double _Complex x, double _Complex y, double w, int n, double *xx)

static void	abuvxpwy_symmetric2_2 (double a, double b, double _Complex u, double _Complex x, double _Complex y, double w, int n, double *xx)

static void	auvxpwy (double a, double _Complex u, double _Complex x, double v, double _Complex y, double *w, int n)

static void	auvxpwy_symmetric (double a, double _Complex u, double _Complex x, double v, double _Complex y, double *w, int n)

static void	auvxpwy_symmetric_1 (double a, double _Complex u, double _Complex x, double v, double _Complex y, double w, int n, double xx)

static void	auvxpwy_symmetric_2 (double a, double _Complex u, double _Complex x, double v, double _Complex y, double w, int n, double xx)

static void	fpt_do_step (double _Complex a, double _Complex b, double a11, double a12, double a21, double a22, double g, int tau, fpt_set set)

static void	fpt_do_step_symmetric (double _Complex a, double _Complex b, double a11, double a12, double a21, double a22, double g, int tau, fpt_set set)

static void	fpt_do_step_symmetric_u (double _Complex a, double _Complex b, double a11, double a12, double a21, double a22, double *x, double gam, int tau, fpt_set set)

static void	fpt_do_step_symmetric_l (double _Complex a, double _Complex b, double a11, double a12, double a21, double a22, double *x, double gam, int tau, fpt_set set)

static void	fpt_do_step_t (double _Complex a, double _Complex b, double a11, double a12, double a21, double a22, double g, int tau, fpt_set set)

static void	fpt_do_step_t_symmetric (double _Complex a, double _Complex b, double a11, double a12, double a21, double a22, double g, int tau, fpt_set set)

static void	fpt_do_step_t_symmetric_u (double _Complex a, double _Complex b, double a11, double a12, double *x, double gam, int tau, fpt_set set)

static void	fpt_do_step_t_symmetric_l (double _Complex a, double _Complex b, double a21, double a22, double *x, double gam, int tau, fpt_set set)

static void	eval_clenshaw (const double x, double y, int size, int k, const double alpha, const double beta, const double *gam)

static void	eval_clenshaw2 (const double x, double z, double y, int size1, int size, int k, const double alpha, const double beta, const double gam)

static int	eval_clenshaw_thresh2 (const double x, double z, double y, int size, int k, const double alpha, const double beta, const double gam, const double threshold)

static void	eval_sum_clenshaw_fast (const int N, const int M, const double _Complex a, const double x, double _Complex y, const double alpha, const double beta, const double gam, const double lambda)

static void	eval_sum_clenshaw_transposed (int N, int M, double _Complex a, double x, double _Complex y, double _Complex temp, double alpha, double beta, double *gam, double lambda)
	Clenshaw algorithm. More...

fpt_set	fpt_init (const int M, const int t, const unsigned int flags)

void	fpt_precompute (fpt_set set, const int m, double alpha, double beta, double *gam, int k_start, const double threshold)

void	fpt_trafo_direct (fpt_set set, const int m, const double _Complex x, double _Complex y, const int k_end, const unsigned int flags)

void	fpt_trafo (fpt_set set, const int m, const double _Complex x, double _Complex y, const int k_end, const unsigned int flags)

void	fpt_transposed_direct (fpt_set set, const int m, double _Complex x, double _Complex y, const int k_end, const unsigned int flags)

void	fpt_transposed (fpt_set set, const int m, double _Complex x, double _Complex y, const int k_end, const unsigned int flags)

void	fpt_finalize (fpt_set set)

Detailed Description

Implementation file for the FPT module.

Author: Jens Keiner

Definition in file fpt.c.

Macro Definition Documentation

#define ABUVXPWY_SYMMETRIC	(	NAME,
		S1,
		S2
	)

Value:

static inline void NAME(double a, double b, double _Complex* u, double _Complex* x, \
  double* v, double _Complex* y, double* w, int n) \
{ \
  const int n2 = n>>1; \
  int l; double _Complex *u_ptr = u, *x_ptr = x, *y_ptr = y; \
  double *v_ptr = v, *w_ptr = w; \
  for (l = 0; l < n2; l++) \
    *u_ptr++ = a * (b * (*v_ptr++) * (*x_ptr++) + (*w_ptr++) * (*y_ptr++)); \
  v_ptr--; w_ptr--; \
  for (l = 0; l < n2; l++) \
    *u_ptr++ = a * (b * S1 * (*v_ptr--) * (*x_ptr++) + S2 * (*w_ptr--) * (*y_ptr++)); \
}

Definition at line 136 of file fpt.c.

#define ABUVXPWY_SYMMETRIC_1	(	NAME,
		S1
	)

Value:

static inline void NAME(double a, double b, double _Complex* u, double _Complex* x, \
  double* v, double _Complex* y, int n, double *xx) \
{ \
  const int n2 = n>>1; \
  int l; double _Complex *u_ptr = u, *x_ptr = x, *y_ptr = y; \
  double *v_ptr = v, *xx_ptr = xx; \
  for (l = 0; l < n2; l++, v_ptr++) \
    *u_ptr++ = a * (b * (*v_ptr) * (*x_ptr++) + ((*v_ptr)*(1.0+*xx_ptr++)) * (*y_ptr++)); \
  v_ptr--; \
  for (l = 0; l < n2; l++, v_ptr--) \
    *u_ptr++ = a * (b * S1 * (*v_ptr) * (*x_ptr++) + (S1 * (*v_ptr) * (1.0+*xx_ptr++)) * (*y_ptr++)); \
}

Definition at line 153 of file fpt.c.

#define ABUVXPWY_SYMMETRIC_2	(	NAME,
		S1
	)

Value:

static inline void NAME(double a, double b, double _Complex* u, double _Complex* x, \
  double _Complex* y, double* w, int n, double *xx) \
{ \
  const int n2 = n>>1; \
  int l; double _Complex *u_ptr = u, *x_ptr = x, *y_ptr = y; \
  double *w_ptr = w, *xx_ptr = xx; \
  for (l = 0; l < n2; l++, w_ptr++) \
    *u_ptr++ = a * (b * (*w_ptr/(1.0+*xx_ptr++)) * (*x_ptr++) + (*w_ptr) * (*y_ptr++)); \
  w_ptr--; \
  for (l = 0; l < n2; l++, w_ptr--) \
    *u_ptr++ = a * (b * (S1 * (*w_ptr)/(1.0+*xx_ptr++) ) * (*x_ptr++) + S1 * (*w_ptr) * (*y_ptr++)); \
}

Definition at line 170 of file fpt.c.

#define FPT_DO_STEP_TRANSPOSED	(	NAME,
		M1_FUNCTION,
		M2_FUNCTION
	)

Value:

static inline void NAME(double _Complex  *a, double _Complex *b, double *a11, \
  double *a12, double *a21, double *a22, double g, int tau, fpt_set set) \
{ \ \
  int length = 1<<(tau+1); \ \
  double norm = 1.0/(length<<1); \
  \
  /* Compute function values from Chebyshev-coefficients using a DCT-III. */ \
  fftw_execute_r2r(set->plans_dct3[tau-1],(double*)a,(double*)a); \
  fftw_execute_r2r(set->plans_dct3[tau-1],(double*)b,(double*)b); \
  \
  /* Perform matrix multiplication. */ \
  M1_FUNCTION(norm,g,set->z,a,a11,b,a21,length); \
  M2_FUNCTION(norm,g,b,a,a12,b,a22,length); \
  memcpy(a,set->z,length*sizeof(double _Complex)); \
  \
  /* Compute Chebyshev-coefficients using a DCT-II. */ \
  fftw_execute_r2r(set->plans_dct2[tau-1],(double*)a,(double*)a); \
  fftw_execute_r2r(set->plans_dct2[tau-1],(double*)b,(double*)b); \
}

Definition at line 394 of file fpt.c.

Function Documentation

static void eval_sum_clenshaw_transposed	(	int	N,
		int	M,
		double _Complex *	a,
		double *	x,
		double _Complex *	y,
		double _Complex *	temp,
		double *	alpha,
		double *	beta,
		double *	gam,
		double	lambda
	)

static

Clenshaw algorithm.

Evaluates a sum of real-valued functions $P_k : \mathbb{R} \rightarrow \mathbb{R}$

$f(x) = \sum_{k=0}^N a_k P_k(x) \quad (N \in \mathbb{N}_0)$

obeying a three-term recurrence relation

$P_{k+1}(x) = (alpha_k * x + beta_k)*P_{k}(x) + gamma_k P_{k-1}(x) \quad (alpha_k, beta_k, gamma_k \in \mathbb{R},\; k \ge 0)$

with initial conditions $P_{-1}(x) := 0$ , $P_0(x) := \lambda$ for given double _Complex coefficients $\left(a_k\right)_{k=0}^N \in \mathbb{C}^{N+1}$ at given nodes $\left(x_j\right)_{j=0}^M \in \mathbb{R}^{M+1}$ , $M \in \mathbb{N}_0$ .

Definition at line 712 of file fpt.c.

Data Structures

Macros

Typedefs

Functions

Detailed Description

Macro Definition Documentation

Function Documentation