qpms/dipdip-dirty/lrhankel_recspace_dirty.c

292 lines
9.1 KiB
C

/*
* This is a dirty implementation of lrhankel_recpart_fill() that calculates
* the (cylindrical) Hankel transforms of the regularised part of the spherical Hankel
* functions that are to be summed in the reciprocal space.
*
* For now, only the regularisation with κ == 5 && q <= 2 && n <= 5 is implemented
* by writing down the explicit formula for each q,n pair and k>k0 vs k<k0 case,
* only with the help of some macros to make the whole thing shorter.
*
* N.B. the results for very small k/k0 differ significantly (sometimes even in the first
* digit for n >= 3, probably due to catastrophic cancellation (hopefully not due
* to an error in the formula!). On the other hand,
* these numbers are tiny in their absolute value, so their contribution to the
* lattice sum should be negligible.
*/
#define MAXQM 1
#define MAXN 5
#define MAXKAPPA 5
#include "bessels.h"
//#include "mdefs.h"
#include <complex.h>
#include <string.h>
#include <assert.h>
#define SQ(x) ((x)*(x))
#define P4(x) (((x)*(x))*((x)*(x)))
/*
* General form of the κ == 5 transforms. One usually has to put a (-1) factor
* to some part of the zeroth term of one of the cases k < k0 or k > k0 in order
* to stay on the correct branch of complex square root...
*/
#define KAPPA5SUM(form) (\
(form(0, 1)) \
- 5*(form(1, 1)) \
+10*(form(2, 1)) \
-10*(form(3, 1)) \
+ 5*(form(4, 1)) \
- (form(5, 1)) \
)
#define KAPPA5SUMFF(form) (\
(form(0, (-1))) \
- 5*(form(1, 1)) \
+10*(form(2, 1)) \
-10*(form(3, 1)) \
+ 5*(form(4, 1)) \
- (form(5, 1)) \
)
/*
* Prototype for the individual (per q,n) Bessel transform calculating functions.
* a, b, d, e, ash are recurring pre-calculated intermediate results, see the definition
* of lrhankel_recpart_fill() below to see their meaning
*/
#define LRHANKELDEF(fname) complex double fname(const double c, const double k0, const double k, \
const complex double *a, const complex double *b, const complex double *d, \
const complex double *e, const complex double *ash)
typedef complex double (*lrhankelspec)(const double, const double, const double,
const complex double *,
const complex double *,
const complex double *,
const complex double *,
const complex double *);
// complex double fun(double c, double k0, double k, ccd *a, ccd *b, ccd *d, ccd *e)
#define FORMK5Q1N0(i,ff) (ff*e[i])
LRHANKELDEF(fk5q1n0l){
return (KAPPA5SUMFF(FORMK5Q1N0))/k0;
}
LRHANKELDEF(fk5q1n0s){
return (KAPPA5SUM(FORMK5Q1N0))/k0;
}
#undef FORMK5Q1N0
#define FORMK5Q1N1(i,ff) (-ff*d[i])
LRHANKELDEF(fk5q1n1l){
return (KAPPA5SUMFF(FORMK5Q1N1))/(k0*k);
}
LRHANKELDEF(fk5q1n1s){
return (KAPPA5SUM(FORMK5Q1N1))/(k0*k);
}
#undef FORMK5Q1N1
#define FORMK5Q1N2(i,ff) (ff*e[i] - t*a[i] + ff*t*d[i]*a[i])
LRHANKELDEF(fk5q1n2l){
double t = 2/(k*k);
return (KAPPA5SUMFF(FORMK5Q1N2))/k0;
}
LRHANKELDEF(fk5q1n2s){
double t = 2/(k*k);
return (KAPPA5SUM(FORMK5Q1N2))/k0;
}
#undef FORMK5Q1N2
#define FORMK5Q1N3(i,ff) (-ff*d[i] * (kk3 + 4*a[i]*a[i]))
LRHANKELDEF(fk5q1n3l){
double kk3 = 3*k*k;
return (KAPPA5SUMFF(FORMK5Q1N3))/(k0*k*k*k);
}
LRHANKELDEF(fk5q1n3s){
double kk3 = 3*k*k;
return (KAPPA5SUM(FORMK5Q1N3))/(k0*k*k*k);
}
#undef FORMK5Q1N3
#define FORMK5Q1N4(i,ff) (ff*e[i] * (kkkk + kk8*a[i]*a[i] + 8*P4(a[i])))
LRHANKELDEF(fk5q1n4l){
double kk8 = k*k*8, kkkk = P4(k);
return (KAPPA5SUMFF(FORMK5Q1N4))/(k0*kkkk);
}
LRHANKELDEF(fk5q1n4s){
double kk8 = k*k*8, kkkk = P4(k);
return (KAPPA5SUM(FORMK5Q1N4))/(k0*kkkk);
}
#undef FORMK5Q1N4
#define FORMK5Q1N5(i,ff) (d[i]*(kkkk*(-5*ff+b[i])-ff*kk20*a[i]*a[i]-ff*16*P4(a[i])))
LRHANKELDEF(fk5q1n5l){
double kk20 = k*k*20, kkkk = P4(k);
return (KAPPA5SUMFF(FORMK5Q1N5))/(k0*kkkk*k);
}
LRHANKELDEF(fk5q1n5s){
double kk20 = k*k*20, kkkk = P4(k);
return (KAPPA5SUM(FORMK5Q1N5))/(k0*kkkk*k);
}
#undef FORMK5Q1N5
#define FORMK5Q2N0(i,ff) (-ash[i])
LRHANKELDEF(fk5q2n0){
return (KAPPA5SUM(FORMK5Q2N0)) / (k0*k0);
}
const lrhankelspec fk5q2n0s = fk5q2n0, fk5q2n0l = fk5q2n0;
#undef FORMK5Q2N0
#define FORMK5Q2N1(i,ff) (ff*b[i]*a[i])
LRHANKELDEF(fk5q2n1l){
return (KAPPA5SUMFF(FORMK5Q2N1))/(k*k0*k0);
}
LRHANKELDEF(fk5q2n1s){
return (KAPPA5SUM(FORMK5Q2N1))/(k*k0*k0);
}
#undef FORMK5Q2N1
#define FORMK5Q2N2(i,ff) (-ff*b[i]*a[i]*a[i])
LRHANKELDEF(fk5q2n2l){
return (KAPPA5SUMFF(FORMK5Q2N2)) / (k*k*k0*k0);
}
LRHANKELDEF(fk5q2n2s){
return (KAPPA5SUM(FORMK5Q2N2)) / (k*k*k0*k0);
}
#if 0
complex double fk5q3n0l(double c, double k0, double k,
const complex double *a, const complex double *b, const complex double *d, const complex double *e, const complex double *ash) { // FIXME
return ( /* FIXME */
- k*b[0] + a[0] * ash[0]
+ 5 * k*b[1] + a[1] * ash[1]
-10 * k*b[2] + a[2] * ash[2]
+10 * k*b[3] + a[3] * ash[3]
- 5 * k*b[4] + a[4] * ash[4]
+ k*b[5] + a[5] * ash[5]
)/(k0*k0*k0);
}
#endif
#define FORMK5Q2N3(i,ff) (ff*a[i]*b[i]*(kk + 4*a[i]*a[i]))
LRHANKELDEF(fk5q2n3l){
double kk = k*k;
return (KAPPA5SUMFF(FORMK5Q2N3))/(3*k0*k0*kk*k);
}
LRHANKELDEF(fk5q2n3s){
double kk = k*k;
return (KAPPA5SUM(FORMK5Q2N3))/(3*k0*k0*kk*k);
}
#undef FORMK5Q2N3
#define FORMK5Q2N4(i,ff) (-ff*b[i]*a[i]*a[i]*(kk+2*a[i]*a[i]))
LRHANKELDEF(fk5q2n4l){
double kk = k*k;
return (KAPPA5SUMFF(FORMK5Q2N4))/(k0*k0*kk*kk);
}
LRHANKELDEF(fk5q2n4s){
double kk = k*k;
return (KAPPA5SUM(FORMK5Q2N4))/(k0*k0*kk*kk);
}
#undef FORMK5Q2N4
#define FORMK5Q2N5(i,ff) (ff*a[i]*b[i]*(kkkk+12*kk*(a[i]*a[i])+16*P4(a[i])) )
LRHANKELDEF(fk5q2n5l){
double kk = k*k;
double kkkk = kk * kk;
return (
KAPPA5SUMFF(FORMK5Q2N5)
+16*120*P4(c)*c // Stirling S2(5,5) is no longer zero
)/(5*k0*k0*kkkk*k);
}
LRHANKELDEF(fk5q2n5s){
double kk = k*k;
double kkkk = kk * kk;
return (
KAPPA5SUM(FORMK5Q2N5)
+16*120*P4(c)*c // Stirling S2(5,5) is no longer zero
)/(5*k0*k0*kkkk*k);
}
#undef FORMK5Q2N5
static const lrhankelspec transfuns_f[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{fk5q1n0l,fk5q1n1l,fk5q1n2l,fk5q1n3l,fk5q1n4l,fk5q1n5l},{fk5q2n0,fk5q2n1l,fk5q2n2l,fk5q2n3l,fk5q2n4l,fk5q2n5l}}
};
static const lrhankelspec transfuns_n[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
{{fk5q1n0s,fk5q1n1s,fk5q1n2s,fk5q1n3s,fk5q1n4s,fk5q1n5s},{fk5q2n0,fk5q2n1s,fk5q2n2s,fk5q2n3s,fk5q2n4s,fk5q2n5s}}
};
void lrhankel_recpart_fill(complex double *target,
size_t maxp /*max. degree of transformed spherical Hankel fun,
also the max. order of the Hankel transform */,
size_t lrk_cutoff,
complex double *hct,
unsigned kappa, double c, double k0, double k)
{
assert(5 == kappa); // Only kappa == 5 implemented so far
assert(maxp <= 5); // only n <= implemented so far
// assert(lrk_cutoff <= TODO);
const lrhankelspec (*funarr)[MAXQM+1][MAXN+1] = (k>k0) ? transfuns_f : transfuns_n;
memset(target, 0, maxp*(maxp+1)/2*sizeof(complex double));
complex double a[kappa+1], b[kappa+1], d[kappa+1], e[kappa+1], ash[kappa+1];
for (size_t sigma = 0; sigma <= kappa; ++sigma) {
a[sigma] = (sigma * c - I * k0);
b[sigma] = csqrt(1+k*k/(a[sigma]*a[sigma]));
d[sigma] = 1/b[sigma];
e[sigma] = d[sigma] / a[sigma];
ash[sigma] = casinh(a[sigma]/k);
}
for (size_t ql = 0; (ql <= maxp) && (ql < lrk_cutoff); ++ql) // ql is q-1, i.e. corresponds to the hankel term power
for (size_t deltam = 0; deltam <= maxp; ++deltam){
complex double result = funarr[kappa][ql][deltam](c,k0,k,a,b,d,e,ash);
for (size_t p = 0; p <= maxp; ++p)
trindex_cd(target,p)[deltam] += result * hankelcoeffs_get(hct,p)[ql];
}
}
#ifdef TESTING
#include <stdio.h>
int main() {
double k0 = 0.7;
double c = 0.1324;
double kmin = 0.000;
double kmax = 20;
double kstep = 0.001;
size_t kappa = 5;
for (double k = kmin; k <= kmax; k += kstep) {
printf("%f ", k);
complex double a[kappa+1], b[kappa+1], d[kappa+1], e[kappa+1], ash[kappa+1];
for (size_t sigma = 0; sigma <= kappa; ++sigma) {
a[sigma] = (sigma * c - I * k0);
b[sigma] = csqrt(1+k*k/(a[sigma]*a[sigma]));
d[sigma] = 1/b[sigma];
e[sigma] = d[sigma] / a[sigma];
ash[sigma] = casinh(a[sigma]/k);
}
for (size_t qm = 0; qm <= MAXQM; ++qm)
for (size_t n = 0; n <= MAXN; ++n) {
//if (/*!*/((qm==1)&&(n==0))){ // not skip q==2, n=0 for now
// complex double fun(double c, double k0, double k, ccd *a, ccd *b, ccd *d, ccd *e)
complex double result =
(k < k0 ? transfuns_n : transfuns_f)[kappa][qm][n](c,k0,k,a,b,d,e,ash);
printf("%.16e %.16e ", creal(result), cimag(result));
}
printf("\n");
}
return 0;
}
#endif