[dirty dipoles] Hankel transform of the long-range part of Hankel
functions, dimensionful TODO more comments, dimensionless everything Former-commit-id: f05f55c3d444bb633b0fe4877120ea001dd34aa7
This commit is contained in:
Marek Nečada
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60a600a7f3
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f6118d6ed9
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@ -6,27 +6,35 @@
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complex double *hankelcoefftable_init(size_t maxn);
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// general, gives the offset such that result[k] is
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// the coefficient corresponding to the e**(I * x) * x**(-k-1)
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// term of the Hankel function; no boundary checks!
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static inline complex double *
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hankelcoeffs_get(complex double *hankelcoefftable, size_t n){
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return hankelcoefftable + n*(n+1)/2;
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trindex_cd(complex double *arr, size_t n){
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return arr + n*(n+1)/2;
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}
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// general, gives the offset such that result[ql] is
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// the coefficient corresponding to the e**(I * x) * x**(-ql-1)
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// term of the n-th Hankel function; no boundary checks!
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static inline complex double *
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hankelcoeffs_get(complex double *hankelcoefftable, size_t n){
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return trindex_cd(hankelcoefftable, n);
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}
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// general; target_longrange and target_shortrange are of size (maxn+1)
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// if target_longrange is NULL, only the short-range part is calculated
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void hankelparts_fill(complex double *target_longrange, complex double *target_shortrange,
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size_t maxn, size_t longrange_order_cutoff, // x**(-(order+1)-1) terms go completely to short-range part
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complex double *hankelcoefftable,
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unsigned kappa, double c, double x); // x = k0 * r
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unsigned kappa, double vc, double x); // x = k0 * r
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// this declaration is general; however, the implementation
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// is so far only for kappa == ???, maxn == ??? TODO
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void lrhankel_recpart_fill(complex double *target_longrange_kspace,
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size_t maxn, size_t longrange_k_cutoff,
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void lrhankel_recpart_fill(complex double *target_longrange_kspace /*Must be of size maxn*(maxn+1)/2*/,
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size_t maxp, size_t longrange_k_cutoff /* terms e**(I x)/x**(k+1), k>= longrange_k_cutoff go
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completely to the shortrange part
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index with hankelcoeffs_get(target,p)l[delta_m] */,
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complex double *hankelcoefftable,
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unsigned kappa, double c, double k0, double k);
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@ -1,18 +1,38 @@
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#include "bessels.h"
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//#include "mdefs.h"
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#include <complex.h>
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#include <string.h>
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#define SQ(x) ((x)*(x))
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/*
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* This is a dirty implementation of lrhankel_recpart_fill() that calculates
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* the (cylindrical) Hankel transforms of the regularised part of the spherical Hankel
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* functions that are to be summed in the reciprocal space.
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*
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* For now, only the regularisation with κ == 5 && q <= 2 && n <= 5 is implemented
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* by writing down the explicit formula for each q,n pair and k>k0 vs k<k0 case,
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* only with the help of some macros to make the whole thing shorter.
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*
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* N.B. the results for very small k/k0 differ significantly (sometimes even in the first
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* digit for n >= 3, probably due to catastrophic cancellation (hopefully not due
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* to an error in the formula!). On the other hand,
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* these numbers are tiny in their absolute value, so their contribution to the
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* lattice sum should be negligible.
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*/
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#define MAXQM 1
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#define MAXN 5
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#define MAXKAPPA 5
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#define FF (-1)
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#include "bessels.h"
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//#include "mdefs.h"
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#include <complex.h>
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#include <string.h>
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#include <assert.h>
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#define SQ(x) ((x)*(x))
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#define P4(x) (((x)*(x))*((x)*(x)))
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/*
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* General form of the κ == 5 transforms. One usually has to put a (-1) factor
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* to some part of the zeroth term of one of the cases k < k0 or k > k0 in order
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* to stay on the correct branch of complex square root...
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*/
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#define KAPPA5SUM(form) (\
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(form(0, 1)) \
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- 5*(form(1, 1)) \
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@ -31,11 +51,16 @@
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- (form(5, 1)) \
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)
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#define LRHANKELDEF(fname) complex double fname(double c, double k0, double k, \
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/*
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* Prototype for the individual (per q,n) Bessel transform calculating functions.
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* a, b, d, e, ash are recurring pre-calculated intermediate results, see the definition
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* of lrhankel_recpart_fill() below to see their meaning
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*/
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#define LRHANKELDEF(fname) complex double fname(const double c, const double k0, const double k, \
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const complex double *a, const complex double *b, const complex double *d, \
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const complex double *e, const complex double *ash)
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typedef complex double (*lrhankelspec)(double, double, double,
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typedef complex double (*lrhankelspec)(const double, const double, const double,
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const complex double *,
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const complex double *,
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const complex double *,
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@ -185,7 +210,6 @@ LRHANKELDEF(fk5q2n5s){
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#undef FORMK5Q2N5
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static const lrhankelspec transfuns_f[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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@ -195,7 +219,7 @@ static const lrhankelspec transfuns_f[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
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{{fk5q1n0l,fk5q1n1l,fk5q1n2l,fk5q1n3l,fk5q1n4l,fk5q1n5l},{fk5q2n0,fk5q2n1l,fk5q2n2l,fk5q2n3l,fk5q2n4l,fk5q2n5l}}
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};
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static lrhankelspec transfuns_n[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
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static const lrhankelspec transfuns_n[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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@ -203,19 +227,33 @@ static lrhankelspec transfuns_n[MAXKAPPA+1][MAXQM+1][MAXN+1] = {
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{{NULL,NULL,NULL,NULL,NULL,NULL},{NULL,NULL,NULL,NULL,NULL,NULL}},
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{{fk5q1n0s,fk5q1n1s,fk5q1n2s,fk5q1n3s,fk5q1n4s,fk5q1n5s},{fk5q2n0,fk5q2n1s,fk5q2n2s,fk5q2n3s,fk5q2n4s,fk5q2n5s}}
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};
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void lrhankel_recpart_fill(complex double *target,
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size_t maxn, size_t lrk_cutoff,
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size_t maxp /*max. degree of transformed spherical Hankel fun,
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also the max. order of the Hankel transform */,
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size_t lrk_cutoff,
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complex double *hct,
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unsigned kappa, double c, double k0, double k)
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{
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memset(target, 0, (maxn+1)*sizeof(complex double));
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complex double a[kappa+1], b[kappa+1], d[kappa+1], e[kappa+1];
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assert(5 == kappa); // Only kappa == 5 implemented so far
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assert(maxp <= 5); // only n <= implemented so far
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// assert(lrk_cutoff <= TODO);
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const lrhankelspec (*funarr)[MAXQM+1][MAXN+1] = (k>k0) ? transfuns_f : transfuns_n;
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memset(target, 0, maxp*(maxp+1)/2*sizeof(complex double));
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complex double a[kappa+1], b[kappa+1], d[kappa+1], e[kappa+1], ash[kappa+1];
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for (size_t sigma = 0; sigma <= kappa; ++sigma) {
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a[sigma] = (sigma * c - I * k0);
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b[sigma] = csqrt(1+k*k/(a[sigma]*a[sigma]));
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d[sigma] = 1/b[sigma];
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e[sigma] = d[sigma] / a[sigma];
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ash[sigma] = casinh(a[sigma]/k);
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}
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for (size_t ql = 0; (ql <= maxp) && (ql < lrk_cutoff); ++ql) // ql is q-1, i.e. corresponds to the hankel term power
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for (size_t deltam = 0; deltam <= maxp; ++deltam){
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complex double result = funarr[kappa][ql][deltam](c,k0,k,a,b,d,e,ash);
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for (size_t p = 0; p <= maxp; ++p)
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trindex_cd(target,p)[deltam] += result * hankelcoeffs_get(hct,p)[ql];
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}
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}
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#ifdef TESTING
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