Práce na 1D ewaldovi; du dom.

Former-commit-id: dbecec91884dd005a2001c71d4bbe50de74fc8d0
This commit is contained in:
Marek Nečada 2018-11-22 16:49:37 +02:00
parent 577a4a5a28
commit b90bf2875b
3 changed files with 199 additions and 10 deletions

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@ -242,6 +242,11 @@ theorems-starred
\end_inset \end_inset
\begin_inset FormulaMacro
\newcommand{\expint}{\mathrm{E}}
\end_inset
\end_layout \end_layout
\begin_layout Title \begin_layout Title
@ -628,7 +633,7 @@ reference "eq:W definition"
\end_inset \end_inset
and and legendre
\begin_inset CommandInset ref \begin_inset CommandInset ref
LatexCommand eqref LatexCommand eqref
reference "eq:W sum in reciprocal space" reference "eq:W sum in reciprocal space"
@ -3720,6 +3725,70 @@ reference "eq:Ewald in 3D origin part"
can be used directly without modifications. can be used directly without modifications.
\end_layout \end_layout
\begin_layout Standard
Another possibility is to consider the chain to be aligned along the
\begin_inset Formula $z$
\end_inset
-axis and to apply the formula
\begin_inset CommandInset citation
LatexCommand cite
after "(4.64)"
key "linton_lattice_2010"
\end_inset
instead.
Let us rewrite it again in the spherical-harmonic-normalisation-agnostic
way (N.B.
the relations
\begin_inset CommandInset citation
LatexCommand cite
after "(4.10)"
key "linton_lattice_2010"
\end_inset
\begin_inset Formula $\sigma_{n}^{m}=\left(-1\right)^{m}\hat{\tau}_{n}^{m}$
\end_inset
,
\begin_inset CommandInset citation
LatexCommand cite
after "(A.5)"
key "linton_lattice_2010"
\end_inset
\begin_inset Formula $P_{n}^{m}\left(\pm1\right)=\left(\pm1\right)^{n}\delta_{m0}$
\end_inset
and
\begin_inset CommandInset citation
LatexCommand cite
after "(A.8)"
key "linton_lattice_2010"
\end_inset
\begin_inset Formula $Y_{n}^{m}\left(\theta,\phi\right)=\left(-1\right)^{m}\sqrt{\frac{\left(2n+1\right)\left(n-m\right)!}{4\pi\left(n+m\right)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}$
\end_inset
)
\begin_inset Formula
\begin{eqnarray*}
\sigma_{n}^{m} & = & -\frac{i^{n+1}}{k^{n+1}a}\delta_{m0}\sqrt{\frac{2n+1}{4\pi}}\sum_{\mu\in\ints}\sum_{j=0}^{\left[n/2\right]}\frac{\left(-1\right)^{j}}{j!}\eta^{2j}\expint_{j+1}\left(\frac{k^{2}\gamma^{\mu}}{4\eta^{2}}\right)\frac{n!\tilde{\beta}_{\mu}^{n-2j}}{\left(n-2j\right)!}\\
& = & -\frac{i^{n+1}}{k^{n+1}a}Y_{n}^{m}\left(\hat{\vect z}\sgn\tilde{\beta}_{\mu}\right)\delta_{m0}\left(\sgn\tilde{\beta}_{\mu}\right)^{-n}\sum_{\mu\in\ints}\sum_{j=0}^{\left[n/2\right]}\frac{\left(-1\right)^{j}}{j!}\eta^{2j}\expint_{j+1}\left(\frac{k^{2}\gamma^{\mu}}{4\eta^{2}}\right)\frac{n!\tilde{\beta}_{\mu}^{n-2j}}{\left(n-2j\right)!}
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Standard \begin_layout Standard
\begin_inset Note Note \begin_inset Note Note
status open status open
@ -3914,7 +3983,7 @@ where the spherical Hankel transform
2) 2)
\begin_inset Formula \begin_inset Formula
\[ \[
\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right). \bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).
\] \]
\end_inset \end_inset
@ -3924,7 +3993,7 @@ Using this convention, the inverse spherical Hankel transform is given by
3) 3)
\begin_inset Formula \begin_inset Formula
\[ \[
g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k), g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k),
\] \]
\end_inset \end_inset
@ -3937,7 +4006,7 @@ so it is not unitary.
An unitary convention would look like this: An unitary convention would look like this:
\begin_inset Formula \begin_inset Formula
\begin{equation} \begin{equation}
\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition} \usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition}
\end{equation} \end{equation}
\end_inset \end_inset
@ -3991,8 +4060,8 @@ where the Hankel transform of order
is defined as is defined as
\begin_inset Formula \begin_inset Formula
\begin{eqnarray} \begin{eqnarray}
\pht mg\left(k\right) & = & \int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}\\ \pht mg\left(k\right) & = & \int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}\\
& = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\, g(r)J_{-m}(kr)r & = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\,g(r)J_{-m}(kr)r
\end{eqnarray} \end{eqnarray}
\end_inset \end_inset

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@ -355,8 +355,9 @@ int ewald3_21_xy_sigma_long (
const cart3_t beta, const cart3_t beta,
const cart3_t particle_shift const cart3_t particle_shift
) )
{ {
assert((latdim & LAT_XYONLY) && (latdim & SPACE3D));
assert((latdim & LAT1D) || (latdim & LAT2D));
const qpms_y_t nelem_sc = c->nelem_sc; const qpms_y_t nelem_sc = c->nelem_sc;
const qpms_l_t lMax = c->lMax; const qpms_l_t lMax = c->lMax;
@ -379,6 +380,7 @@ int ewald3_21_xy_sigma_long (
// recycleable values if rbeta_pq stays the same: // recycleable values if rbeta_pq stays the same:
complex double gamma_pq; complex double gamma_pq;
complex double z; complex double z;
double factor1d = 1; // the "additional" factor for the 1D case (then it is not 1)
// space for Gamma_pq[j]'s // space for Gamma_pq[j]'s
qpms_csf_result Gamma_pq[lMax/2+1]; qpms_csf_result Gamma_pq[lMax/2+1];
@ -408,6 +410,7 @@ int ewald3_21_xy_sigma_long (
const bool new_rbeta_pq = (!pgen_generates_shifted_points) || (pgen_retdata.flags & PGEN_NEWR); const bool new_rbeta_pq = (!pgen_generates_shifted_points) || (pgen_retdata.flags & PGEN_NEWR);
if (!new_rbeta_pq) assert(rbeta_pq == rbeta_pq_prev); if (!new_rbeta_pq) assert(rbeta_pq == rbeta_pq_prev);
// R-DEPENDENT BEGIN // R-DEPENDENT BEGIN
if (new_rbeta_pq) { if (new_rbeta_pq) {
gamma_pq = lilgamma(rbeta_pq/k); gamma_pq = lilgamma(rbeta_pq/k);
@ -417,9 +420,10 @@ int ewald3_21_xy_sigma_long (
// we take the other branch, cf. [Linton, p. 642 in the middle]: FIXME instead use the C11 CMPLX macros and fill in -O*I part to z in the line above // we take the other branch, cf. [Linton, p. 642 in the middle]: FIXME instead use the C11 CMPLX macros and fill in -O*I part to z in the line above
if(creal(z) < 0) if(creal(z) < 0)
Gamma_pq[j].val = conj(Gamma_pq[j].val); //FIXME as noted above Gamma_pq[j].val = conj(Gamma_pq[j].val); //FIXME as noted above
if(!(retval==0 ||retval==GSL_EUNDRFLW)) abort(); if(!(retval==0 || retval==GSL_EUNDRFLW)) abort();
} }
// --------------- ZDE JSEM SKONČIL. TODO NEZAPOMEŇ TAKY POŘEŠIT PŘÍPAD 1D VS 2D if (latdim & LAT1D)
factor1d = k * M_SQRT1_2 * .5 * gamma_pq;
} }
// R-DEPENDENT END // R-DEPENDENT END
@ -446,7 +450,7 @@ int ewald3_21_xy_sigma_long (
summand *= Gamma_pq[j].val; // GGG summand *= Gamma_pq[j].val; // GGG
ckahanadd(&jsum, &jsum_c, summand); ckahanadd(&jsum, &jsum_c, summand);
} }
jsum *= phasefac; // PFC jsum *= phasefac * factor1d; // PFC
ckahanadd(target + y, target_c + y, jsum); ckahanadd(target + y, target_c + y, jsum);
if(err) kahanadd(err + y, err_c + y, jsum_err); if(err) kahanadd(err + y, err_c + y, jsum_err);
} }
@ -466,6 +470,115 @@ int ewald3_21_xy_sigma_long (
} }
// 1D chain along the z-axis; not many optimisations here as the same
// shifted beta radius could be recycled only once anyways
int ewald3_1_z_sigma_long (
complex double *target, // must be c->nelem_sc long
double *err,
const qpms_ewald32_constants_t *c,
const double unitcell_volume /* length in this case */,
const LatticeDimensionality latdim,
PGenSph *pgen_K, const bool pgen_generates_shifted_points
/* If false, the behaviour corresponds to the old ewald32_sigma_long_points_and_shift,
* so the function assumes that the generated points correspond to the unshifted reciprocal Bravais lattice,
* and adds beta to the generated points before calculations.
* If true, it assumes that they are already shifted.
*/,
const cart3_t beta,
const cart3_t particle_shift
)
{
assert(LatticeDimensionality_checkflags(latdim, LAT_1D_IN_3D_ZONLY));
assert(beta.x == 0 && beta.y == 0);
assert(particle_shift.x == 0 && particle_shift.y == 0);
const double beta_z = beta.z;
const double particle_shift_z = particle_shift_z;
const qpms_y_t nelem_sc = c->nelem_sc;
const qpms_l_t lMax = c->lMax;
// Manual init of the ewald summation targets
complex double *target_c = calloc(nelem_sc, sizeof(complex double));
memset(target, 0, nelem_sc * sizeof(complex double));
double *err_c = NULL;
if (err) {
err_c = calloc(nelem_sc, sizeof(double));
memset(err, 0, nelem_sc * sizeof(double));
}
PGenSingleReturnData pgen_retdata;
// CHOOSE POINT BEGIN
while ((pgen_retdata = PGenSph_next(pgen_K)).flags & PGEN_NOTDONE) { // BEGIN POINT LOOP
assert(pgen_retdata.flags & PGEN_AT_Z);
double K_z, beta_mu_z;
if (pgen_generates_shifted_points) {
beta_mu_z = ((pgen_retdata.point_sph.theta == 0) ?
pgen_retdata.point_sph.r : -pgen_retdata.point_sph.r); //!!!CHECKSIGN!!!
K_z = beta_mu_z - beta_z;
} else { // as in the old _points_and_shift functions
K_z = ((pgen_retdata.point_sph.theta == 0) ?
pgen_retdata.point_sph.r : -pgen_retdata.point_sph.r); // !!!CHECKSIGN!!!
beta_mu_z = K_z + beta_z;
}
// CHOOSE POINT END
const complex double phasefac = cexp(I * K_z * particle_shift_z); // POINT-DEPENDENT (PFC) // !!!CHECKSIGN!!!
// R-DEPENDENT BEGIN
gamma_pq = lilgamma(rbeta_pq/k);
z = csq(gamma_pq*k/(2*eta)); // Když o tom tak přemýšlím, tak tohle je vlastně vždy reálné
for(qpms_l_t j = 0; j <= lMax/2; ++j) {
int retval = complex_gamma_inc_e(0.5-j, z, Gamma_pq+j);
// we take the other branch, cf. [Linton, p. 642 in the middle]: FIXME instead use the C11 CMPLX macros and fill in -O*I part to z in the line above
if(creal(z) < 0)
Gamma_pq[j].val = conj(Gamma_pq[j].val); //FIXME as noted above
if(!(retval==0 || retval==GSL_EUNDRFLW)) abort();
}
if (latdim & LAT1D)
factor1d = k * M_SQRT1_2 * .5 * gamma_pq;
// R-DEPENDENT END
// TODO optimisations: all the j-dependent powers can be done for each j only once, stored in array
// and just fetched for each n, m pair
for(qpms_l_t n = 0; n <= lMax; ++n)
for(qpms_m_t m = -n; m <= n; ++m) {
if((m+n) % 2 != 0) // odd coefficients are zero.
continue;
const qpms_y_t y = qpms_mn2y_sc(m, n);
const complex double e_imalpha_pq = cexp(I*m*arg_pq);
complex double jsum, jsum_c; ckahaninit(&jsum, &jsum_c);
double jsum_err, jsum_err_c; kahaninit(&jsum_err, &jsum_err_c); // TODO do I really need to kahan sum errors?
assert((n-abs(m))/2 == c->s1_jMaxes[y]);
for(qpms_l_t j = 0; j <= c->s1_jMaxes[y]/*(n-abs(m))/2*/; ++j) { // FIXME </<= ?
complex double summand = pow(rbeta_pq/k, n-2*j)
* e_imalpha_pq * c->legendre0[gsl_sf_legendre_array_index(n,abs(m))] * min1pow_m_neg(m) // This line can actually go outside j-loop
* cpow(gamma_pq, 2*j-1) // * Gamma_pq[j] bellow (GGG) after error computation
* c->s1_constfacs[y][j];
if(err) {
// FIXME include also other errors than Gamma_pq's relative error
kahanadd(&jsum_err, &jsum_err_c, Gamma_pq[j].err * cabs(summand));
}
summand *= Gamma_pq[j].val; // GGG
ckahanadd(&jsum, &jsum_c, summand);
}
jsum *= phasefac * factor1d; // PFC
ckahanadd(target + y, target_c + y, jsum);
if(err) kahanadd(err + y, err_c + y, jsum_err);
}
#ifndef NDEBUG
rbeta_pq_prev = rbeta_pq;
#endif
} // END POINT LOOP
free(err_c);
free(target_c);
for(qpms_y_t y = 0; y < nelem_sc; ++y) // CFC common factor from above
target[y] *= commonfac;
if(err)
for(qpms_y_t y = 0; y < nelem_sc; ++y)
err[y] *= commonfac;
return 0;
}
struct sigma2_integrand_params { struct sigma2_integrand_params {
int n; int n;

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@ -21,10 +21,17 @@ typedef enum LatticeDimensionality {
LAT_2D_IN_3D = 34, LAT_2D_IN_3D = 34,
LAT_3D_IN_3D = 40, LAT_3D_IN_3D = 40,
// special coordinate arrangements (indicating possible optimisations) // special coordinate arrangements (indicating possible optimisations)
LAT_ZONLY = 64,
LAT_XYONLY = 128,
LAT_1D_IN_3D_ZONLY = 97, // LAT1D | SPACE3D | 64 LAT_1D_IN_3D_ZONLY = 97, // LAT1D | SPACE3D | 64
LAT_2D_IN_3D_XYONLY = 162 // LAT2D | SPACE3D | 128 LAT_2D_IN_3D_XYONLY = 162 // LAT2D | SPACE3D | 128
} LatticeDimensionality; } LatticeDimensionality;
inline static bool LatticeDimensionality_checkflags(
LatticeDimensionality a, LatticeDimensionality flags_a_has_to_contain) {
return ((a & flags_a_has_to_contain) == flags_a_has_to_contain);
}
#include <math.h> #include <math.h>
#include <stdbool.h> #include <stdbool.h>
#include <assert.h> #include <assert.h>