wigner.c
/*
* Copyright (c) 2002, 2017 Jens Keiner, Stefan Kunis, Daniel Potts
*
* This program is free software; you can redistribute it and/or modify it under
* the terms of the GNU General Public License as published by the Free Software
* Foundation; either version 2 of the License, or (at your option) any later
* version.
*
* This program is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
* FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
* details.
*
* You should have received a copy of the GNU General Public License along with
* this program; if not, write to the Free Software Foundation, Inc., 51
* Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
#include <math.h>
#include <stdio.h>
#include "infft.h"
#include "wigner.h"
#include "infft.h"
double SO3_alpha(const int m1, const int m2, const int j)
{
const int M = MAX(ABS(m1),ABS(m2)), mini = MIN(ABS(m1),ABS(m2));
if (j < 0)
return K(0.0);
else if (j == 0)
{
if (m1 == 0 && m2 == 0)
return K(1.0);
if (m1 == m2)
return K(0.5);
else
return IF((m1+m2)%2,K(0.0),K(-0.5));
}
else if (j < M - mini)
return IF(j%2,K(0.5),K(-0.5));
else if (j < M)
return K(0.5) * SIGNF((R)m1)*SIGNF((R)m2);
else
return
SQRT(((R)(j+1))/((R)(j+1-m1)))
* SQRT(((R)(2*j+1))/((R)(j+1+m1)))
* SQRT(((R)(j+1))/((R)(j+1-m2)))
* SQRT(((R)(2*j+1))/((R)(j+1+m2)));
}
double SO3_beta(const int m1, const int m2, const int j)
{
if (j < 0)
return K(0.0);
else if (j < MAX(ABS(m1),ABS(m2)))
return K(0.5);
else if (m1 == 0 || m2 == 0)
return K(0.0);
else
{
const R m1a = FABS((R)m1), m2a = FABS((R)m2);
return -COPYSIGN(
((SQRT(m1a)*SQRT(m2a))/((R)j))
* SQRT(m1a/((R)(j+1-m1)))
* SQRT(((R)(2*j+1))/((R)(j+1+m1)))
* SQRT(m2a/((R)(j+1-m2)))
* SQRT(((R)(2*j+1))/((R)(j+1+m2))),
SIGNF((R)m1)*SIGNF((R)m2));
}
}
double SO3_gamma(const int m1, const int m2, const int j)
{
if (MAX(ABS(m1),ABS(m2)) < j)
return -(((R)(j+1))/((R)j)) * SQRT((((R)(j-m1))/((R)(j+1-m1)))
*(((R)(j+m1))/((R)(j+1+m1)))*(((R)(j-m2))/((R)(j+1-m2)))
*(((R)(j+m2))/((R)(j+1+m2))));
else if (j == -1)
return IF(m1 > m2 || !((m1 + m2) % 2), K(1.0), K(-1.0))
* nfft_lambda2((R)ABS(m2 - m1),(R)ABS(m2 + m1));
else
return K(0.0);
}
/*compute the coefficients for all degrees*/
inline void SO3_alpha_row(double *alpha, int N, int k, int m)
{
int j;
double *alpha_act = alpha;
for (j = -1; j <= N; j++)
*alpha_act++ = SO3_alpha(k, m, j);
}
inline void SO3_beta_row(double *beta, int N, int k, int m)
{
int j;
double *beta_act = beta;
for (j = -1; j <= N; j++)
*beta_act++ = SO3_beta(k, m, j);
}
inline void SO3_gamma_row(double *gamma, int N, int k, int m)
{
int j;
double *gamma_act = gamma;
for (j = -1; j <= N; j++)
*gamma_act++ = SO3_gamma(k, m, j);
}
/*compute for all degrees l and orders k*/
inline void SO3_alpha_matrix(double *alpha, int N, int m)
{
int i, j;
double *alpha_act = alpha;
for (i = -N; i <= N; i++)
{
for (j = -1; j <= N; j++)
{
*alpha_act = SO3_alpha(i, m, j);
alpha_act++;
}
}
}
inline void SO3_beta_matrix(double *alpha, int N, int m)
{
int i, j;
double *alpha_act = alpha;
for (i = -N; i <= N; i++)
{
for (j = -1; j <= N; j++)
{
*alpha_act = SO3_beta(i, m, j);
alpha_act++;
}
}
}
inline void SO3_gamma_matrix(double *alpha, int N, int m)
{
int i, j;
double *alpha_act = alpha;
for (i = -N; i <= N; i++)
{
for (j = -1; j <= N; j++)
{
*alpha_act = SO3_gamma(i, m, j);
alpha_act++;
}
}
}
/*compute all 3termrecurrence coeffs*/
inline void SO3_alpha_all(double *alpha, int N)
{
int q;
int i, j, m;
double *alpha_act = alpha;
q = 0;
for (m = -N; m <= N; m++)
{
for (i = -N; i <= N; i++)
{
for (j = -1; j <= N; j++)
{
*alpha_act = SO3_alpha(i, m, j);
fprintf(stdout, "alpha_all_%d^[%d,%d]=%f\n", j, i, m,
SO3_alpha(i, m, j));
alpha_act++;
q = q + 1;
}
}
}
}
inline void SO3_beta_all(double *alpha, int N)
{
int i, j, m;
double *alpha_act = alpha;
for (m = -N; m <= N; m++)
{
for (i = -N; i <= N; i++)
{
for (j = -1; j <= N; j++)
{
*alpha_act = SO3_beta(i, m, j);
alpha_act++;
}
}
}
}
inline void SO3_gamma_all(double *alpha, int N)
{
int i, j, m;
double *alpha_act = alpha;
for (m = -N; m <= N; m++)
{
for (i = -N; i <= N; i++)
{
for (j = -1; j <= N; j++)
{
*alpha_act = SO3_gamma(i, m, j);
alpha_act++;
}
}
}
}
inline void eval_wigner(double *x, double *y, int size, int k, double *alpha,
double *beta, double *gamma)
{
/* Evaluate the wigner function d_{k,nleg} (l,x) for the vector
* of knots x[0], ..., x[size-1] by the Clenshaw algorithm
*/
int i, j;
double a, b, x_val_act, a_old;
double *x_act, *y_act;
double *alpha_act, *beta_act, *gamma_act;
/* Traverse all nodes. */
x_act = x;
y_act = y;
for (i = 0; i < size; i++)
{
a = 1.0;
b = 0.0;
x_val_act = *x_act;
if (k == 0)
{
*y_act = 1.0;
}
else
{
alpha_act = &(alpha[k]);
beta_act = &(beta[k]);
gamma_act = &(gamma[k]);
for (j = k; j > 1; j--)
{
a_old = a;
a = b + a_old * ((*alpha_act) * x_val_act + (*beta_act));
b = a_old * (*gamma_act);
alpha_act--;
beta_act--;
gamma_act--;
}
*y_act = (a * ((*alpha_act) * x_val_act + (*beta_act)) + b);
}
x_act++;
y_act++;
}
}
inline int eval_wigner_thresh(double *x, double *y, int size, int k,
double *alpha, double *beta, double *gamma, double threshold)
{
int i, j;
double a, b, x_val_act, a_old;
double *x_act, *y_act;
double *alpha_act, *beta_act, *gamma_act;
/* Traverse all nodes. */
x_act = x;
y_act = y;
for (i = 0; i < size; i++)
{
a = 1.0;
b = 0.0;
x_val_act = *x_act;
if (k == 0)
{
*y_act = 1.0;
}
else
{
alpha_act = &(alpha[k]);
beta_act = &(beta[k]);
gamma_act = &(gamma[k]);
for (j = k; j > 1; j--)
{
a_old = a;
a = b + a_old * ((*alpha_act) * x_val_act + (*beta_act));
b = a_old * (*gamma_act);
alpha_act--;
beta_act--;
gamma_act--;
}
*y_act = (a * ((*alpha_act) * x_val_act + (*beta_act)) + b);
if (fabs(*y_act) > threshold)
{
return 1;
}
}
x_act++;
y_act++;
}
return 0;
}
/************************************************************************/
/* L2 normed wigner little d, WHERE THE DEGREE OF THE FUNCTION IS EQUAL
TO ONE OF ITS ORDERS. This is the function to use when starting the
three-term recurrence at orders (m1,m2)
Note that, by definition, since I am starting the recurrence with this
function, that the degree j of the function is equal to max(abs(m1), abs(m2) ).
*/
double wigner_start(int m1, int m2, double theta)
{
int i, l, delta;
int cosPower, sinPower;
int absM1, absM2;
double dl, dm1, dm2, normFactor, sinSign;
double dCP, dSP;
double max;
double min;
max = (double) (ABS(m1) > ABS(m2) ? ABS(m1) : ABS(m2));
min = (double) (ABS(m1) < ABS(m2) ? ABS(m1) : ABS(m2));
l = max;
delta = l - min;
absM1 = ABS(m1);
absM2 = ABS(m2);
dl = (double) l;
dm1 = (double) m1;
dm2 = (double) m2;
sinSign = 1.;
normFactor = 1.;
for (i = 0; i < delta; i++)
normFactor *= SQRT((2. * dl - ((double) i)) / (((double) i) + 1.));
/* need to adjust to make the L2-norm equal to 1 */
normFactor *= SQRT((2. * dl + 1.) / 2.);
if (l == absM1)
if (m1 >= 0)
{
cosPower = l + m2;
sinPower = l - m2;
if ((l - m2) % 2)
sinSign = -1.;
}
else
{
cosPower = l - m2;
sinPower = l + m2;
}
else if (m2 >= 0)
{
cosPower = l + m1;
sinPower = l - m1;
}
else
{
cosPower = l - m1;
sinPower = l + m1;
if ((l + m1) % 2)
sinSign = -1.;
}
dCP = (double) cosPower;
dSP = (double) sinPower;
return normFactor * sinSign * POW(sin(theta / 2), dSP) * POW(cos(theta / 2),
dCP);
}
wigner_start
double wigner_start(int n1, int n2, double theta)
A method used for debugging, gives the values to start the "old" three-term recurrence generates WHE...
Definition: wigner.c:317
SO3_gamma_row
void SO3_gamma_row(double *gamma, int N, int m, int n)
Compute three-term-recurrence coefficients of Wigner-d functions for all degrees ...
Definition: wigner.c:104
SO3_gamma_matrix
void SO3_gamma_matrix(double *gamma, int N, int n)
Compute three-term-recurrence coefficients of Wigner-d functions for all order and degrees ...
Definition: wigner.c:142
SO3_alpha_all
void SO3_alpha_all(double *alpha, int N)
Compute three-term-recurrence coefficients of Wigner-d functions for all and .
Definition: wigner.c:158
SO3_beta_row
void SO3_beta_row(double *beta, int N, int m, int n)
Compute three-term-recurrence coefficients of Wigner-d functions for all degrees ...
Definition: wigner.c:96
SO3_alpha_matrix
void SO3_alpha_matrix(double *alpha, int N, int n)
Compute three-term-recurrence coefficients of Wigner-d functions for all order and degrees ...
Definition: wigner.c:114
SO3_gamma_all
void SO3_gamma_all(double *gamma, int N)
Compute three-term-recurrence coefficients of Wigner-d functions for all and .
Definition: wigner.c:198
SO3_alpha
double SO3_alpha(int k, int m, int l)
Computes three-term recurrence coefficients of Wigner-d functions.
Definition: wigner.c:25
SO3_beta
double SO3_beta(int k, int m, int l)
Computes three-term recurrence coefficients of Wigner-d functions.
Definition: wigner.c:52
SO3_gamma
double SO3_gamma(int k, int m, int l)
Computes three-term recurrence coefficients of Wigner-d functions.
Definition: wigner.c:73
SO3_alpha_row
void SO3_alpha_row(double *alpha, int N, int m, int n)
Compute three-term-recurrence coefficients of Wigner-d functions for all degrees ...
Definition: wigner.c:88
SO3_beta_matrix
void SO3_beta_matrix(double *beta, int N, int n)
Compute three-term-recurrence coefficients of Wigner-d functions for all order and degrees ...
Definition: wigner.c:128
SO3_beta_all
void SO3_beta_all(double *beta, int N)
Compute three-term-recurrence coefficients of Wigner-d functions for all and .
Definition: wigner.c:181
eval_wigner_thresh
int eval_wigner_thresh(double *x, double *y, int size, int l, double *alpha, double *beta, double *gamma, double threshold)
Evaluates Wigner-d functions using the Clenshaw-algorithm if it not exceeds a given threshold...
Definition: wigner.c:260
wigner.h
Header file for functions related to Wigner-d/D functions.
eval_wigner
void eval_wigner(double *x, double *y, int size, int l, double *alpha, double *beta, double *gamma)
Evaluates Wigner-d functions using the Clenshaw-algorithm.
Definition: wigner.c:215