Redefine qpms_pitau_get to always use the "power" normalisation.
Former-commit-id: 20cd5494f7e3faab4660dde15a00f948fd7391ef
This commit is contained in:
Marek Nečada
parent
d3ef942dee
commit
616095890f
138
qpms/legendre.c
138
qpms/legendre.c
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@ -5,14 +5,16 @@
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#include <stdlib.h>
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#include "indexing.h"
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#include <string.h>
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#include "qpms_error.h"
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// Legendre functions also for negative m, see DLMF 14.9.3
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qpms_errno_t qpms_legendre_deriv_y_fill(double *target, double *target_deriv, double x, qpms_l_t lMax,
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gsl_sf_legendre_t lnorm, double csphase)
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{
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size_t n = gsl_sf_legendre_array_n(lMax);
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double *legendre_tmp = malloc(n * sizeof(double));
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double *legendre_deriv_tmp = malloc(n * sizeof(double));
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const size_t n = gsl_sf_legendre_array_n(lMax);
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double *legendre_tmp, *legendre_deriv_tmp;
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QPMS_CRASHING_MALLOC(legendre_tmp, n * sizeof(double));
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QPMS_CRASHING_MALLOC(legendre_deriv_tmp, n * sizeof(double));
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int gsl_errno = gsl_sf_legendre_deriv_array_e(
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lnorm, (size_t)lMax, x, csphase, legendre_tmp,legendre_deriv_tmp);
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for (qpms_l_t l = 1; l <= lMax; ++l)
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@ -22,104 +24,46 @@ qpms_errno_t qpms_legendre_deriv_y_fill(double *target, double *target_deriv, do
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target[y] = legendre_tmp[i];
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target_deriv[y] = legendre_deriv_tmp[i];
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}
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switch(lnorm) {
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case GSL_SF_LEGENDRE_NONE:
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// Fill negative m's.
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for (qpms_l_t l = 1; l <= lMax; ++l)
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for (qpms_m_t m = 1; m <= l; ++m) {
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qpms_y_t y = qpms_mn2y(-m,l);
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size_t i = gsl_sf_legendre_array_index(l,m);
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// viz DLMF 14.9.3, čert ví, jak je to s cs fasí.
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double factor = exp(lgamma(l-m+1)-lgamma(l+m+1))*((m%2)?-1:1);
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// cf. DLMF 14.9.3, but we're normalised.
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double factor = ((m%2)?-1:1);
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target[y] = factor * legendre_tmp[i];
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target_deriv[y] = factor * legendre_deriv_tmp[i];
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}
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break;
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case GSL_SF_LEGENDRE_SCHMIDT:
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case GSL_SF_LEGENDRE_SPHARM:
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case GSL_SF_LEGENDRE_FULL:
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for (qpms_l_t l = 1; l <= lMax; ++l)
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for (qpms_m_t m = 1; m <= l; ++m) {
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qpms_y_t y = qpms_mn2y(-m,l);
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size_t i = gsl_sf_legendre_array_index(l,m);
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// viz DLMF 14.9.3, čert ví, jak je to s cs fasí.
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double factor = ((m%2)?-1:1); // this is the difference from the unnormalised case
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target[y] = factor * legendre_tmp[i];
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target_deriv[y] = factor * legendre_deriv_tmp[i];
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}
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break;
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default:
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abort(); //NI
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break;
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}
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free(legendre_tmp);
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free(legendre_deriv_tmp);
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return QPMS_SUCCESS;
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return gsl_errno;
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}
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qpms_errno_t qpms_legendre_deriv_y_get(double **target, double **dtarget, double x, qpms_l_t lMax, gsl_sf_legendre_t lnorm,
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double csphase)
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{
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*target = malloc(sizeof(double)*qpms_lMax2nelem(lMax));
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*dtarget = malloc(sizeof(double)*qpms_lMax2nelem(lMax));
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const qpms_y_t nelem = qpms_lMax2nelem(lMax);
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QPMS_CRASHING_MALLOC(target, nelem * sizeof(double));
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QPMS_CRASHING_MALLOC(dtarget, nelem * sizeof(double));
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return qpms_legendre_deriv_y_fill(*target, *dtarget, x, lMax, lnorm, csphase);
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}
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qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, qpms_normalisation_t norm)
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qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, const double csphase)
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{
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const double csphase = qpms_normalisation_t_csphase(norm);
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norm = qpms_normalisation_t_normonly(norm);
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QPMS_ENSURE(fabs(csphase) == 1, "The csphase argument must be either 1 or -1, not %g.", csphase);
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qpms_pitau_t res;
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qpms_y_t nelem = qpms_lMax2nelem(lMax);
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res.pi = malloc(nelem * sizeof(double));
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res.tau = malloc(nelem * sizeof(double));
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const qpms_y_t nelem = qpms_lMax2nelem(lMax);
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QPMS_CRASHING_MALLOC(res.leg, nelem * sizeof(double));
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QPMS_CRASHING_MALLOC(res.pi, nelem * sizeof(double));
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QPMS_CRASHING_MALLOC(res.tau, nelem * sizeof(double));
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double ct = cos(theta), st = sin(theta);
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if (1 == fabs(ct)) { // singular case, use DLMF 14.8.2
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memset(res.pi, 0, nelem*sizeof(double));
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memset(res.tau, 0, nelem*sizeof(double));
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res.leg = calloc(nelem, sizeof(double));
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switch(norm) {
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/* FIXME get rid of multiplicating the five lines */
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case QPMS_NORMALISATION_NONE:
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for (qpms_l_t l = 1; l <= lMax; ++l) {
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res.leg[qpms_mn2y(0, l)] = (l%2)?ct:1.;
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double p = l*(l+1)/2;
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const double n = 0.5;
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int lpar = (l%2)?-1:1;
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res.pi [qpms_mn2y(+1, l)] = -((ct>0) ? -1 : lpar) * p * csphase;
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res.pi [qpms_mn2y(-1, l)] = -((ct>0) ? -1 : lpar) * n * csphase;
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res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * p * csphase;
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res.tau[qpms_mn2y(-1, l)] = -((ct>0) ? +1 : lpar) * n * csphase;
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}
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break;
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#ifdef USE_XU_ANTINORMALISATION // broken
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case QPMS_NORMALISATION_XU: // Rather useless except for testing.
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for (qpms_l_t l = 1; l <= lMax; ++l) {
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res.leg[qpms_mn2y(0, l)] = ((l%2)?ct:1.)*sqrt(0.25*M_1_PI *(2*l+1)/(l*(l+1)));
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double p = l*(l+1)/2 /* as in _NONE */
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* sqrt(0.25 * M_1_PI * (2*l + 1));
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double n = 0.5 /* as in _NONE */
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* sqrt(0.25 * M_1_PI * (2*l + 1)) / (l * (l+1));
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int lpar = (l%2)?-1:1;
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res.pi [qpms_mn2y(+1, l)] = -((ct>0) ? -1 : lpar) * p * csphase;
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res.pi [qpms_mn2y(-1, l)] = -((ct>0) ? -1 : lpar) * n * csphase;
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res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * p * csphase;
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res.tau[qpms_mn2y(-1, l)] = -((ct>0) ? +1 : lpar) * n * csphase;
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}
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break;
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#endif
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case QPMS_NORMALISATION_TAYLOR:
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for (qpms_l_t l = 1; l <= lMax; ++l) {
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res.leg[qpms_mn2y(0, l)] = ((l%2)?ct:1.)*sqrt((2*l+1)*0.25*M_1_PI);
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double fl = 0.25 * sqrt((2*l+1)*l*(l+1)*M_1_PI);
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int lpar = (l%2)?-1:1;
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res.pi [qpms_mn2y(+1, l)] = -((ct>0) ? -1 : lpar) * fl * csphase;
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res.pi [qpms_mn2y(-1, l)] = -((ct>0) ? -1 : lpar) * fl * csphase;
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res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * fl * csphase;
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res.tau[qpms_mn2y(-1, l)] = -((ct>0) ? +1 : lpar) * fl * csphase;
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}
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break;
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case QPMS_NORMALISATION_POWER:
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memset(res.pi, 0, nelem * sizeof(double));
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memset(res.tau, 0, nelem * sizeof(double));
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memset(res.leg, 0, nelem * sizeof(double));
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for (qpms_l_t l = 1; l <= lMax; ++l) {
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res.leg[qpms_mn2y(0, l)] = ((l%2)?ct:1.)*sqrt((2*l+1)/(4*M_PI *l*(l+1)));
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double fl = 0.25 * sqrt((2*l+1)*M_1_PI);
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@ -129,24 +73,13 @@ qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, qpms_normalisation_t no
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res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * fl * csphase;
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res.tau[qpms_mn2y(-1, l)] = -((ct>0) ? +1 : lpar) * fl * csphase;
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}
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break;
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default:
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abort();
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}
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}
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else { // cos(theta) in (-1,1), use normal calculation
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double *legder = malloc(sizeof(double)*qpms_lMax2nelem(lMax));
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res.leg = malloc(sizeof(double)*qpms_lMax2nelem(lMax));
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if (qpms_legendre_deriv_y_fill(res.leg, legder, ct, lMax,
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norm == QPMS_NORMALISATION_NONE ? GSL_SF_LEGENDRE_NONE
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: GSL_SF_LEGENDRE_SPHARM, csphase))
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abort();
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if (norm == QPMS_NORMALISATION_POWER)
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/* for None (=non-normalized) and Taylor (=sph. harm. normalized)
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* the correct normalisation is already obtained from gsl_sf_legendre_deriv_array_e().
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* However, Kristensson ("power") normalisation differs from Taylor
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* by 1/sqrt(l*(l+1)) factor.
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*/
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double *legder;
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QPMS_CRASHING_MALLOC(legder, nelem * sizeof(double));
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QPMS_ENSURE_SUCCESS(qpms_legendre_deriv_y_fill(res.leg, legder, ct, lMax,
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GSL_SF_LEGENDRE_SPHARM, csphase));
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// Multiply by the "power normalisation" factor
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for (qpms_l_t l = 1; l <= lMax; ++l) {
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double prefac = 1./sqrt(l*(l+1));
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for (qpms_m_t m = -l; m <= l; ++m) {
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@ -154,21 +87,6 @@ qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, qpms_normalisation_t no
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legder[qpms_mn2y(m,l)] *= prefac;
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}
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}
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#ifdef USE_XU_ANTINORMALISATION
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else if (norm == QPMS_NORMALISATION_XU)
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/* for Xu (anti-normalized), we start from spharm-normalized Legendre functions
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* Do not use this normalisation except for testing
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*/
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// FIXME PROBABLY BROKEN HERE
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for (qpms_l_t l = 1; l <= lMax; ++l) {
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double prefac = (2*l + 1) / sqrt(4*M_PI / (l*(l+1)));
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for (qpms_m_t m = -l; m <= l; ++m) {
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double fac = prefac * exp(lgamma(l+m+1) - lgamma(l-m+1));
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res.leg[qpms_mn2y(m,l)] *= fac;
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legder[qpms_mn2y(m,l)] *= fac;
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}
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}
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#endif
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for (qpms_l_t l = 1; l <= lMax; ++l) {
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for (qpms_m_t m = -l; m <= l; ++m) {
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res.pi [qpms_mn2y(m,l)] = m / st * res.leg[qpms_mn2y(m,l)];
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@ -52,17 +52,40 @@ double *qpms_legendre_minus1d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); /
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// array of Legendre and pi, tau auxillary functions (see [1,(37)])
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// This should handle correct evaluation for theta -> 0 and theta -> pi
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/// Array of Legendre and and auxillary \f$\pi_{lm}, \tau_{lm} \f$ functions.
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/**
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* The leg, pi, tau arrays are indexed using the standard qpms_mn2y() VSWF indexing.
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*/
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typedef struct {
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//qpms_normalisation_t norm;
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qpms_l_t lMax;
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//qpms_y_t nelem;
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double *leg, *pi, *tau;
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} qpms_pitau_t;
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qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, qpms_normalisation_t norm);
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void qpms_pitau_free(qpms_pitau_t);//NI
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void qpms_pitau_pfree(qpms_pitau_t*);//NI
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/// Returns an array of normalised Legendre and and auxillary \f$\pi_{lm}, \tau_{lm} \f$ functions.
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/**
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* The normalised Legendre function here is defined as
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* \f[
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* \Fer[norm.]{l}{m} = \csphase^{-1}
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* \sqrt{\frac{1}{l(l+1)}\frac{(l-m)!(2l+1)}{4\pi(l+m)!}},
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* \f] i.e. obtained using `gsl_sf_legendre_array_e()` with
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* `norm = GSL_SF_LEGENDRE_SPHARM` and multiplied by \f$ \sqrt{l(l+1)} \f$.
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*
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* The auxillary functions are defined as
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* \f[
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* \pi_{lm}(\cos \theta) = \frac{m}{\sin \theta} \Fer[norm.]{l}{m}(\cos\theta),\\
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* \tau_{lm}(\cos \theta) = \frac{\ud}{\ud \theta} \Fer[norm.]{l}{m}(\cos\theta)
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* \f]
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* with appropriate limit expression used if \f$ \abs{\cos\theta} = 1 \f$.
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*
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* When done, don't forget to deallocate the memory using qpms_pitau_free().
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*
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*/
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qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, double csphase);
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/// Frees the dynamically allocated arrays from qpms_pitau_t.
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void qpms_pitau_free(qpms_pitau_t);
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//void qpms_pitau_pfree(qpms_pitau_t*);//NI
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// Associated Legendre polynomial at zero argument (DLMF 14.5.1) DEPRECATED?
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double qpms_legendre0(int m, int n);
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