qpms/qpms/legendre.c

#include "qpms_specfunc.h"
#include "qpms_types.h"
#include <gsl/gsl_sf_legendre.h>
#include <gsl/gsl_math.h>
#include <stdlib.h>
#include "indexing.h"
#include <string.h>

// Legendre functions also for negative m, see DLMF 14.9.3
qpms_errno_t qpms_legendre_deriv_y_fill(double *target, double *target_deriv, double x, qpms_l_t lMax,
    gsl_sf_legendre_t lnorm, double csphase)
{
  size_t n = gsl_sf_legendre_array_n(lMax);
  double *legendre_tmp = malloc(n * sizeof(double));
  double *legendre_deriv_tmp = malloc(n * sizeof(double));
  int gsl_errno = gsl_sf_legendre_deriv_array_e(
      lnorm, (size_t)lMax, x, csphase, legendre_tmp,legendre_deriv_tmp);
  for (qpms_l_t l = 1; l <= lMax; ++l)
    for (qpms_m_t m = 0; m <= l; ++m) {
      qpms_y_t y = qpms_mn2y(m,l);
      size_t i = gsl_sf_legendre_array_index(l,m);
      target[y] = legendre_tmp[i];
      target_deriv[y] = legendre_deriv_tmp[i];
    }
  switch(lnorm) {
    case GSL_SF_LEGENDRE_NONE:
      for (qpms_l_t l = 1; l <= lMax; ++l)
        for (qpms_m_t m = 1; m <= l; ++m) {
          qpms_y_t y = qpms_mn2y(-m,l);
          size_t i = gsl_sf_legendre_array_index(l,m);
          // viz DLMF 14.9.3, čert ví, jak je to s cs fasí.
          double factor = exp(lgamma(l-m+1)-lgamma(l+m+1))*((m%2)?-1:1);
          target[y] = factor * legendre_tmp[i];
          target_deriv[y] = factor * legendre_deriv_tmp[i];
        }
      break;
    case GSL_SF_LEGENDRE_SCHMIDT:
    case GSL_SF_LEGENDRE_SPHARM:
    case GSL_SF_LEGENDRE_FULL:
      for (qpms_l_t l = 1; l <= lMax; ++l)
        for (qpms_m_t m = 1; m <= l; ++m) {
          qpms_y_t y = qpms_mn2y(-m,l);
          size_t i = gsl_sf_legendre_array_index(l,m);
          // viz DLMF 14.9.3, čert ví, jak je to s cs fasí.
          double factor = ((m%2)?-1:1); // this is the difference from the unnormalised case
          target[y] = factor * legendre_tmp[i];
          target_deriv[y] = factor * legendre_deriv_tmp[i];
        }
      break;
    default:
      abort(); //NI
      break;
  }
  free(legendre_tmp);
  free(legendre_deriv_tmp);
  return QPMS_SUCCESS;
}

qpms_errno_t qpms_legendre_deriv_y_get(double **target, double **dtarget, double x, qpms_l_t lMax, gsl_sf_legendre_t lnorm,
    double csphase)
{

  *target = malloc(sizeof(double)*qpms_lMax2nelem(lMax));
  *dtarget = malloc(sizeof(double)*qpms_lMax2nelem(lMax));
  return qpms_legendre_deriv_y_fill(*target, *dtarget, x, lMax, lnorm, csphase);
}

qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, qpms_normalisation_t norm)
{
  const double csphase = qpms_normalisation_t_csphase(norm);
  norm = qpms_normalisation_t_normonly(norm);
  qpms_pitau_t res;
  qpms_y_t nelem = qpms_lMax2nelem(lMax);
  res.pi = malloc(nelem * sizeof(double));
  res.tau = malloc(nelem * sizeof(double));
  double ct = cos(theta), st = sin(theta);
  if (1 == fabs(ct)) { // singular case, use DLMF 14.8.2
    memset(res.pi, 0, nelem*sizeof(double));
    memset(res.tau, 0, nelem*sizeof(double));
    res.leg = calloc(nelem, sizeof(double));
    switch(norm) {
      /* FIXME get rid of multiplicating the five lines */
      case QPMS_NORMALISATION_NONE:
        for (qpms_l_t l = 1; l <= lMax; ++l) {
          res.leg[qpms_mn2y(0, l)] = (l%2)?ct:1.;
          double p = l*(l+1)/2;
          const double n = 0.5;
          int lpar = (l%2)?-1:1;
          res.pi [qpms_mn2y(+1, l)] = -((ct>0) ? -1 : lpar) * p * csphase;
          res.pi [qpms_mn2y(-1, l)] = -((ct>0) ? -1 : lpar) * n * csphase;
          res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * p * csphase;
          res.tau[qpms_mn2y(-1, l)] = -((ct>0) ? +1 : lpar) * n * csphase;
        }
        break;
#ifdef USE_XU_ANTINORMALISATION // broken
      case QPMS_NORMALISATION_XU: // Rather useless except for testing.
        for (qpms_l_t l = 1; l <= lMax; ++l) {
          res.leg[qpms_mn2y(0, l)] = ((l%2)?ct:1.)*sqrt(0.25*M_1_PI *(2*l+1)/(l*(l+1)));
          double p = l*(l+1)/2 /* as in _NONE */ 
            * sqrt(0.25 * M_1_PI * (2*l + 1));
          double n = 0.5 /* as in _NONE */
            * sqrt(0.25 * M_1_PI * (2*l + 1)) / (l * (l+1));
          int lpar = (l%2)?-1:1;
          res.pi [qpms_mn2y(+1, l)] = -((ct>0) ? -1 : lpar) * p * csphase;
          res.pi [qpms_mn2y(-1, l)] = -((ct>0) ? -1 : lpar) * n * csphase;
          res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * p * csphase;
          res.tau[qpms_mn2y(-1, l)] = -((ct>0) ? +1 : lpar) * n * csphase;
        }
        break;
#endif
      case QPMS_NORMALISATION_TAYLOR:
        for (qpms_l_t l = 1; l <= lMax; ++l) {
          res.leg[qpms_mn2y(0, l)] = ((l%2)?ct:1.)*sqrt((2*l+1)*0.25*M_1_PI);
          double fl = 0.25 * sqrt((2*l+1)*l*(l+1)*M_1_PI);
          int lpar = (l%2)?-1:1;
          res.pi [qpms_mn2y(+1, l)] = -((ct>0) ? -1 : lpar) * fl * csphase;
          res.pi [qpms_mn2y(-1, l)] = -((ct>0) ? -1 : lpar) * fl * csphase;
          res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * fl * csphase;
          res.tau[qpms_mn2y(-1, l)] = -((ct>0) ? +1 : lpar) * fl * csphase;
        }
        break;
      case QPMS_NORMALISATION_POWER:
        for (qpms_l_t l = 1; l <= lMax; ++l) {
          res.leg[qpms_mn2y(0, l)] = ((l%2)?ct:1.)*sqrt((2*l+1)/(4*M_PI *l*(l+1)));
          double fl = 0.25 * sqrt((2*l+1)*M_1_PI);
          int lpar = (l%2)?-1:1;
          res.pi [qpms_mn2y(+1, l)] = -((ct>0) ? -1 : lpar) * fl * csphase;
          res.pi [qpms_mn2y(-1, l)] = -((ct>0) ? -1 : lpar) * fl * csphase;
          res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * fl * csphase;
          res.tau[qpms_mn2y(-1, l)] = -((ct>0) ? +1 : lpar) * fl * csphase;
        }
        break;
      default:
        abort();
    }
  }
  else { // cos(theta) in (-1,1), use normal calculation
    double *legder = malloc(sizeof(double)*qpms_lMax2nelem(lMax));
    res.leg = malloc(sizeof(double)*qpms_lMax2nelem(lMax));
    if (qpms_legendre_deriv_y_fill(res.leg, legder, ct, lMax,
          norm == QPMS_NORMALISATION_NONE ? GSL_SF_LEGENDRE_NONE
          : GSL_SF_LEGENDRE_SPHARM, csphase))
      abort();
    if (norm == QPMS_NORMALISATION_POWER)
      /* for None (=non-normalized) and Taylor (=sph. harm. normalized)
       * the correct normalisation is already obtained from gsl_sf_legendre_deriv_array_e().
       * However, Kristensson ("power") normalisation differs from Taylor
       * by 1/sqrt(l*(l+1)) factor.
       */
      for (qpms_l_t l = 1; l <= lMax; ++l) {
        double prefac = 1./sqrt(l*(l+1));
        for (qpms_m_t m = -l; m <= l; ++m) {
          res.leg[qpms_mn2y(m,l)] *= prefac;
          legder[qpms_mn2y(m,l)] *= prefac;
        }
      }
#ifdef USE_XU_ANTINORMALISATION
    else if (norm == QPMS_NORMALISATION_XU)
      /* for Xu (anti-normalized), we start from spharm-normalized Legendre functions
       * Do not use this normalisation except for testing
       */
      // FIXME PROBABLY BROKEN HERE
      for (qpms_l_t l = 1; l <= lMax; ++l) {
        double prefac = (2*l + 1) / sqrt(4*M_PI / (l*(l+1)));
        for (qpms_m_t m = -l; m <= l; ++m) {
          double fac = prefac * exp(lgamma(l+m+1) - lgamma(l-m+1));
          res.leg[qpms_mn2y(m,l)] *= fac;
          legder[qpms_mn2y(m,l)] *= fac;
        }
      }
#endif
    for (qpms_l_t l = 1; l <= lMax; ++l) {
      for (qpms_m_t m = -l; m <= l; ++m) {
        res.pi [qpms_mn2y(m,l)] = m / st * res.leg[qpms_mn2y(m,l)];
        res.tau[qpms_mn2y(m,l)] = - st * legder[qpms_mn2y(m,l)];
      }
    }
    free(legder);
  }
  res.lMax = lMax;
  return res;
}

void qpms_pitau_free(qpms_pitau_t x) {
  free(x.leg);
  free(x.pi);
  free(x.tau);
}