QPMS
/
qpms
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

Pi, tau auxillary functions. still bugs in the poles. 

Former-commit-id: 99d550f5ffa03d001f89e41a07e70652105bdc36
This commit is contained in:
Marek Nečada 2018-01-28 09:18:06 +00:00
parent 3312bc61e2
commit 92e9622b64
6 changed files with 337 additions and 39 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

177
notes/Scattering and Shit.lyx
View File

@ -240,6 +240,117 @@ Orthonormality:
Pi and tau
\end_layout
\begin_layout Subsection
Xu
\begin_inset CommandInset label
LatexCommand label
name "sub:Xu pitau"
\end_inset
\end_layout
\begin_layout Standard
As in (37)
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{eqnarray*}
\pi_{mn}\left(\cos\theta\right) & = & \frac{m}{\sin\theta}P_{n}^{m}\left(\cos\theta\right)\\
\tau_{mn}\left(\cos\theta\right) & = & \frac{\ud}{\ud\theta}P_{n}^{m}\left(\cos\theta\right)=-\left(\sin\theta\right)\frac{\ud P_{n}^{m}\left(\cos\theta\right)}{\ud\left(\cos\theta\right)}
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard
The expressions
\begin_inset Formula $\left(\sin\theta\right)^{-1}$
\end_inset
and
\begin_inset Formula $\frac{\ud P_{n}^{m}\left(\cos\theta\right)}{\ud\left(\cos\theta\right)}$
\end_inset
are singular for
\begin_inset Formula $\cos\theta=\pm1$
\end_inset
, the limits
\begin_inset Formula $\tau_{mn}\left(\pm1\right),\pi_{mn}\left(\pm1\right)$
\end_inset
however exist.
Labeling
\begin_inset Formula $x\equiv\cos\theta$
\end_inset
,
\begin_inset Formula $\sqrt{\left(1+x\right)\left(1-x\right)}=\sqrt{1-x^{2}}\equiv\sin\theta$
\end_inset
and using the asymptotic expression (DLMF 14.8.2) we obtain that the limits
are nonzero only for
\begin_inset Formula $m=\pm1$
\end_inset
and
\begin_inset Formula
\begin{eqnarray*}
\pi_{1\nu}(1-) & = & -\frac{\nu\left(\nu+1\right)\left(\nu+2\right)}{2}\\
\tau_{1\nu}(1-) & = & \frac{\nu\left(\nu+1\right)\left(\nu+2\right)}{2}
\end{eqnarray*}
\end_inset
and using the parity property
\begin_inset Formula $P_{n}^{m}\left(-x\right)=\left(-1\right)^{m+n}P_{n}^{m}\left(x\right)$
\end_inset
\begin_inset Formula
\begin{eqnarray*}
\pi_{1\nu}(-1+) & = & \left(-1\right)^{\nu}\frac{\nu\left(\nu+1\right)\left(\nu+2\right)}{2}\\
\tau_{1\nu}(-1+) & = & \left(-1\right)^{\nu}\frac{\nu\left(\nu+1\right)\left(\nu+2\right)}{2}
\end{eqnarray*}
\end_inset
For
\begin_inset Formula $m=1$
\end_inset
, we simply use the relation
\begin_inset Formula $P_{n}^{-m}=\left(CS\right)^{m}P_{n}^{m}\frac{\left(n-m\right)!}{\left(n+m\right)!}$
\end_inset
to get
\begin_inset Formula
\begin{eqnarray*}
\pi_{-1\nu}(1-) & = & -CS\frac{\nu+2}{2}\\
\tau_{-1\nu}(1-) & = & CS\frac{\nu+2}{2}\\
\pi_{-1\nu}(-1+) & = & CS\left(-1\right)^{\nu}\frac{\nu+2}{2}\\
\tau_{-1\nu}(-1+) & = & CS\left(-1\right)^{\nu}\frac{\nu+2}{2}
\end{eqnarray*}
\end_inset
where
\begin_inset Formula $CS$
\end_inset
is
\begin_inset Formula $-1$
\end_inset
if the Condon-Shortley phase is employed on the level of Legendre polynomials,
1 otherwise.
\end_layout
\begin_layout Subsection
Taylor
\end_layout
@ -256,6 +367,51 @@ Taylor
\end_layout
\begin_layout Standard
The limiting expressions are obtained simply by multiplying the expressions
from sec.
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Xu pitau"
\end_inset
by the normalisation factor,
\begin_inset Formula
\begin{eqnarray*}
\tilde{\pi}_{1\nu}(1-) & = & -\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}\\
\tilde{\tau}_{1\nu}(1-) & = & \sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}\\
\tilde{\pi}_{1\nu}(-1+) & = & \left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}\\
\tilde{\tau}_{1\nu}(-1+) & = & \left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}
\end{eqnarray*}
\end_inset
\begin_inset Formula
\begin{eqnarray*}
\tilde{\pi}_{-1\nu}(1-) & = & -CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}\\
\tilde{\tau}_{-1\nu}(1-) & = & CS\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}\\
\tilde{\pi}_{-1\nu}(-1+) & = & CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}\\
\tilde{\tau}_{-1\nu}(-1+) & = & CS\left(-1\right)^{\nu}\sqrt{\frac{2\nu+1}{4\pi}}\frac{\sqrt{\nu\left(\nu+1\right)}\left(\nu+2\right)}{2}
\end{eqnarray*}
\end_inset
i.e.
the expressions for
\begin_inset Formula $m=-1$
\end_inset
are the same as for
\begin_inset Formula $m=1$
\end_inset
except for the sign if Condon-Shortley phase is used on the Legendre polynomial
level.
\end_layout
\begin_layout Section
Vector spherical harmonics (?)
\end_layout
@ -550,6 +706,27 @@ Definition [T](2.40);
\end_inset
\end_layout
\begin_layout Subsection
Xu
\end_layout
\begin_layout Standard
are the electric and magnetic waves, respectively:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{eqnarray*}
\vect N_{mn}^{(j)} & = & \frac{n(n+1)}{kr}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}z_{n}^{j}\left(kr\right)\hat{\vect r}\\
& & +\left[\tau_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}+i\pi_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}\frac{1}{kr}\frac{\ud\left(kr\, z_{n}^{j}\left(kr\right)\right)}{\ud(kr)}\\
\vect M_{mn}^{(j)} & = & \left[i\pi_{mn}\left(\cos\theta\right)\hat{\vect{\theta}}-\tau_{mn}\left(\cos\theta\right)\hat{\vect{\phi}}\right]e^{im\phi}z_{n}^{j}\left(kr\right)
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Subsection

3
qpms/indexing.h
View File

@ -2,9 +2,10 @@
#define QPMS_INDEXING_H
#include "qpms_types.h"
#include <math.h>
static inline qpms_y_t qpms_mn2y(qpms_m_t m, qpms_l_t n) {
return (qpms_y_t) n * (n + 1) + m - 1;
return n * (n + 1) + m - 1;
}
static inline qpms_lm_t qpms_y2n(qpms_y_t y) {

22
qpms/qpms_types.h
View File

@ -7,29 +7,25 @@
// integer index types
typedef int qpms_lm_t;
typedef unsigned int qpms_l_t;
typedef int qpms_l_t; // can't be unsigned because of the behaviour under - operator
typedef qpms_lm_t qpms_m_t;
typedef size_t qpms_y_t;
typedef enum {
QPMS_SUCCESS = 0;
QPMS_ERROR = 1;
QPMS_SUCCESS = 0,
QPMS_ERROR = 1
} qpms_errno_t;
// Normalisations
typedef enum {
// As in TODO
QPMS_NORMALIZATION_XU = 3, // NI!
QPMS_NORMALISATION_XU = 3, // NI!
// As in http://www.eit.lth.se/fileadmin/eit/courses/eit080f/Literature/book.pdf, power-normalised
QPMS_NORMALIZATION_KRISTENSSON = 2, // NI!
QPMS_NORMALIZATION_POWER = QPMS_NORMALIZATION_KRISTENSSON, // NI!
QPMS_NORMALIZATION_TAYLOR = 1,
QPMS_NORMALIZATION_UNDEF = 0
} qpms_normalization_t;
typedef enum {
QPMS_SUCCESS = 0;
} qpms_errno_t;
QPMS_NORMALISATION_KRISTENSSON = 2, // NI!
QPMS_NORMALISATION_POWER = QPMS_NORMALISATION_KRISTENSSON, // NI!
QPMS_NORMALISATION_TAYLOR = 1,
QPMS_NORMALISATION_UNDEF = 0
} qpms_normalisation_t;
typedef enum {
QPMS_BESSEL_REGULAR = 1, // regular function j

25
qpms/test_vswf.c Normal file
View File

@ -0,0 +1,25 @@
#include "vswf.h"
#include "indexing.h"
#include <stdio.h>
#include <gsl/gsl_math.h>
const double dtheta = 0.01 * M_PI;
const qpms_l_t lMax = 3;
int main() {
qpms_y_t nelem = qpms_lMax2nelem(lMax);
for (double theta = 0.; theta <= M_PI; theta += dtheta) {
printf("%.5e ", theta);
for(qpms_normalisation_t norm = 1; norm <= 3; ++norm) {//fujka :D
qpms_pitau_t pt = qpms_pitau_get(theta, lMax, norm);
for (qpms_y_t y = 0; y < nelem; ++y)
printf("%.5e %.5e %.5e ", pt.leg[y], pt.pi[y], pt.tau[y]);
qpms_pitau_free(pt);
}
printf("\n");
}
}

121
qpms/vswf.c
View File

@ -1,28 +1,37 @@
#include "vswf.h"
#include "indexing.h"
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_sf_legendre.h>
#include <stdlib.h>
#include <string.h>
#ifndef CSPHASE
#define CSPHASE (1.) // FIXME this should be later determined by qpms_normalization_t
#endif
// Legendre functions also for negative m, see DLMF 14.9.3
int qpms_errno_t qpms_legendre_deriv_y_fill(double *target, double *target_deriv, double x, qpms_l_t lMax,
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_legenre_array_n(lMax);
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_tmp_deriv);
for (qpms_l_t l = 0; l <= lMax; ++l)
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_legenre_array_index(l,m);
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_LEGEDRE_NONE:
for (qpms_l_t l = 0; l <= lMax; ++l)
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_legenre_array_index(l,m);
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(n+m+1))*((m%2)?-1:1);
target[y] = factor * legendre_tmp[i];
@ -32,10 +41,10 @@ int qpms_errno_t qpms_legendre_deriv_y_fill(double *target, double *target_deriv
case GSL_SF_LEGENDRE_SCHMIDT:
case GSL_SF_LEGENDRE_SPHARM:
case GSL_SF_LEGENDRE_FULL:
for (qpms_l_t l = 0; l <= lMax; ++l)
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_legenre_array_index(l,m);
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];
@ -51,13 +60,101 @@ int qpms_errno_t qpms_legendre_deriv_y_fill(double *target, double *target_deriv
return QPMS_SUCCESS;
}
int qpms_legendre_deriv_y_get(double **target, double **dtarget, double x, qpms_l_t lMax, gsl_sf_legendre_t lnorm,
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(ar, x, lMax, lnorm, csphase);
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)
{
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) {
case QPMS_NORMALISATION_XU:
for (qpms_l_t l = 1; l <= lMax; ++l) {
res.leg[qpms_mn2y(0, l)] = (l%2)?ct:1.;
double p = l*(l+1)*(l+2)/2;
double n = (l+2)/2.;
int lpar = (l%2)?-1:1;
res.pi [qpms_mn2y(+1, l)] = ((ct>0) ? -1 : lpar) * p;
res.pi [qpms_mn2y(-1, l)] = ((ct>0) ? -1 : lpar) * n * CSPHASE;
res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * p;
res.tau[qpms_mn2y(-1, l)] = ((ct>0) ? +1 : lpar) * n * CSPHASE;
}
break;
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);
int lpar = (l%2)?-1:1;
double fl = 0.25 * (l+2) * sqrt((2*l+1)*l*(l+1)*M_1_PI);
res.pi [qpms_mn2y(+1, l)] = ((ct>0) ? -1 : lpar) * fl;
res.pi [qpms_mn2y(-1, l)] = ((ct>0) ? -1 : lpar) * fl * CSPHASE;
res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * fl;
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)));
int lpar = (l%2)?-1:1;
double fl = 0.25 * (l+2) * sqrt((2*l+1)*M_1_PI);
res.pi [qpms_mn2y(+1, l)] = ((ct>0) ? -1 : lpar) * fl;
res.pi [qpms_mn2y(-1, l)] = ((ct>0) ? -1 : lpar) * fl * CSPHASE;
res.tau[qpms_mn2y(+1, l)] = ((ct>0) ? +1 : lpar) * fl;
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_XU ? GSL_SF_LEGENDRE_NONE
: GSL_SF_LEGENDRE_SPHARM, CSPHASE))
abort();
if (norm == QPMS_NORMALISATION_POWER)
/* for Xu (=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;
}
}
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);
}

28
qpms/vswf.h
View File

@ -5,10 +5,10 @@
// Electric wave N; NI
csphvec_t qpms_vswf_single_el(int m, int n, sph_t kdlj,
qpms_bessel_t btyp, qpms_normalization_t norm);
qpms_bessel_t btyp, qpms_normalisation_t norm);
// Magnetic wave M; NI
csphvec_t qpms_vswf_single_mg(int m, int n, sph_t kdlj,
qpms_bessel_t btyp, qpms_normalization_t norm);
qpms_bessel_t btyp, qpms_normalisation_t norm);
// Set of electric and magnetic VSWF in spherical coordinate basis
typedef struct {
@ -26,29 +26,31 @@ typedef struct {
* ( gsl/specfunc/legendre_source.c and 7.24.2 of gsl docs
*/
int qpms_legendre_deriv_y_get(double **result, double **result_deriv, double x, qpms_l_t lMax,
gsle_sf_legendre_t lnorm, double csphase);
int qpms_errno_t qpms_legendre_deriv_y_fill(double *where, double *where_deriv, double x,
qpms_errno_t qpms_legendre_deriv_y_get(double **result, double **result_deriv, double x, qpms_l_t lMax,
gsl_sf_legendre_t lnorm, double csphase); // free() result and result_deriv yourself!
qpms_errno_t qpms_legendre_deriv_y_fill(double *where, double *where_deriv, double x,
qpms_l_t lMax, gsl_sf_legendre_t lnorm, double csphase);
qpms_vswfset_sph_t *qpms_vswfset_make(qpms_l_t lMax, sph_t kdlj,
qpms_bessel_t btyp, qpms_normalization_t norm);//NI
void qpms_vswfst_sph_pfree(qpms_vswfset_t *);//NI
qpms_bessel_t btyp, qpms_normalisation_t norm);//NI
void qpms_vswfset_sph_pfree(qpms_vswfset_sph_t *);//NI
double *qpms_legendre_y_get(double x, qpms_l_t lMax, qpms_normalisation_t norm);//NI
double *qpms_legendre0d_y_get(qpms_l_t lMax, qpms_normalization_t norm); //NI
double *qpms_legendre_plus1d_y_get(qpms_l_t lMax, qpms_normalization_t norm); //NI
double *qpms_legendre_minus1d_y_get(qpms_l_t lMax, qpms_normalization_t norm); //NI
double *qpms_legendre0d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
double *qpms_legendre_plus1d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
double *qpms_legendre_minus1d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
// array of pi, tau auxillary function (see [1,(37)])
// array of Legendre and pi, tau auxillary functions (see [1,(37)])
// This should handle correct evaluation for theta -> 0 and theta -> pi
typedef struct {
//qpms_normalization_t norm;
//qpms_normalisation_t norm;
qpms_l_t lMax;
//qpms_y_t nelem;
double *pi, *tau;
double *leg, *pi, *tau;
} qpms_pitau_t;
qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, qpms_normalisation_t norm);
void qpms_pitau_free(qpms_pitau_t);//NI
void qpms_pitau_pfree(qpms_pitau_t*);//NI