Práce na 1D ewaldovi; du dom.
Former-commit-id: dbecec91884dd005a2001c71d4bbe50de74fc8d0
This commit is contained in:
Marek Nečada
parent
577a4a5a28
commit
b90bf2875b
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@ -242,6 +242,11 @@ theorems-starred
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\expint}{\mathrm{E}}
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\end_inset
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\end_layout
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\begin_layout Title
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@ -628,7 +633,7 @@ reference "eq:W definition"
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\end_inset
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and
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and legendre
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W sum in reciprocal space"
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@ -3720,6 +3725,70 @@ reference "eq:Ewald in 3D origin part"
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can be used directly without modifications.
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\end_layout
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\begin_layout Standard
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Another possibility is to consider the chain to be aligned along the
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\begin_inset Formula $z$
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\end_inset
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-axis and to apply the formula
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\begin_inset CommandInset citation
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LatexCommand cite
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after "(4.64)"
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key "linton_lattice_2010"
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\end_inset
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instead.
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Let us rewrite it again in the spherical-harmonic-normalisation-agnostic
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way (N.B.
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the relations
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\begin_inset CommandInset citation
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LatexCommand cite
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after "(4.10)"
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key "linton_lattice_2010"
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\end_inset
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\begin_inset Formula $\sigma_{n}^{m}=\left(-1\right)^{m}\hat{\tau}_{n}^{m}$
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\end_inset
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,
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\begin_inset CommandInset citation
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LatexCommand cite
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after "(A.5)"
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key "linton_lattice_2010"
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\end_inset
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\begin_inset Formula $P_{n}^{m}\left(\pm1\right)=\left(\pm1\right)^{n}\delta_{m0}$
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\end_inset
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and
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\begin_inset CommandInset citation
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LatexCommand cite
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after "(A.8)"
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key "linton_lattice_2010"
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\end_inset
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\begin_inset Formula $Y_{n}^{m}\left(\theta,\phi\right)=\left(-1\right)^{m}\sqrt{\frac{\left(2n+1\right)\left(n-m\right)!}{4\pi\left(n+m\right)!}}P_{n}^{m}\left(\cos\theta\right)e^{im\phi}$
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\end_inset
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)
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\begin_inset Formula
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\begin{eqnarray*}
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\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)!}\\
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& = & -\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)!}
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Note Note
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status open
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@ -3914,7 +3983,7 @@ where the spherical Hankel transform
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2)
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\begin_inset Formula
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\[
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\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).
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\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).
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\]
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\end_inset
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@ -3924,7 +3993,7 @@ Using this convention, the inverse spherical Hankel transform is given by
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3)
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\begin_inset Formula
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\[
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g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k),
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g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k),
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\]
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\end_inset
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@ -3937,7 +4006,7 @@ so it is not unitary.
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An unitary convention would look like this:
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\begin_inset Formula
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\begin{equation}
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\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}
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\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}
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\end{equation}
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\end_inset
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@ -3991,8 +4060,8 @@ where the Hankel transform of order
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is defined as
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\begin_inset Formula
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\begin{eqnarray}
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\pht mg\left(k\right) & = & \int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}\\
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& = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\, g(r)J_{-m}(kr)r
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\pht mg\left(k\right) & = & \int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}\\
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& = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\,g(r)J_{-m}(kr)r
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\end{eqnarray}
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\end_inset
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121
qpms/ewald.c
121
qpms/ewald.c
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@ -355,8 +355,9 @@ int ewald3_21_xy_sigma_long (
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const cart3_t beta,
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const cart3_t particle_shift
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)
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{
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assert((latdim & LAT_XYONLY) && (latdim & SPACE3D));
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assert((latdim & LAT1D) || (latdim & LAT2D));
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const qpms_y_t nelem_sc = c->nelem_sc;
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const qpms_l_t lMax = c->lMax;
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@ -379,6 +380,7 @@ int ewald3_21_xy_sigma_long (
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// recycleable values if rbeta_pq stays the same:
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complex double gamma_pq;
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complex double z;
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double factor1d = 1; // the "additional" factor for the 1D case (then it is not 1)
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// space for Gamma_pq[j]'s
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qpms_csf_result Gamma_pq[lMax/2+1];
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@ -408,6 +410,7 @@ int ewald3_21_xy_sigma_long (
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const bool new_rbeta_pq = (!pgen_generates_shifted_points) || (pgen_retdata.flags & PGEN_NEWR);
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if (!new_rbeta_pq) assert(rbeta_pq == rbeta_pq_prev);
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// R-DEPENDENT BEGIN
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if (new_rbeta_pq) {
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gamma_pq = lilgamma(rbeta_pq/k);
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@ -417,9 +420,10 @@ int ewald3_21_xy_sigma_long (
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// 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
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if(creal(z) < 0)
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Gamma_pq[j].val = conj(Gamma_pq[j].val); //FIXME as noted above
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if(!(retval==0 ||retval==GSL_EUNDRFLW)) abort();
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if(!(retval==0 || retval==GSL_EUNDRFLW)) abort();
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}
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// --------------- ZDE JSEM SKONČIL. TODO NEZAPOMEŇ TAKY POŘEŠIT PŘÍPAD 1D VS 2D
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if (latdim & LAT1D)
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factor1d = k * M_SQRT1_2 * .5 * gamma_pq;
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}
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// R-DEPENDENT END
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@ -446,7 +450,7 @@ int ewald3_21_xy_sigma_long (
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summand *= Gamma_pq[j].val; // GGG
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ckahanadd(&jsum, &jsum_c, summand);
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}
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jsum *= phasefac; // PFC
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jsum *= phasefac * factor1d; // PFC
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ckahanadd(target + y, target_c + y, jsum);
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if(err) kahanadd(err + y, err_c + y, jsum_err);
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}
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@ -466,6 +470,115 @@ int ewald3_21_xy_sigma_long (
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}
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// 1D chain along the z-axis; not many optimisations here as the same
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// shifted beta radius could be recycled only once anyways
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int ewald3_1_z_sigma_long (
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complex double *target, // must be c->nelem_sc long
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double *err,
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const qpms_ewald32_constants_t *c,
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const double unitcell_volume /* length in this case */,
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const LatticeDimensionality latdim,
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PGenSph *pgen_K, const bool pgen_generates_shifted_points
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/* If false, the behaviour corresponds to the old ewald32_sigma_long_points_and_shift,
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* so the function assumes that the generated points correspond to the unshifted reciprocal Bravais lattice,
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* and adds beta to the generated points before calculations.
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* If true, it assumes that they are already shifted.
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*/,
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const cart3_t beta,
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const cart3_t particle_shift
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)
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{
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assert(LatticeDimensionality_checkflags(latdim, LAT_1D_IN_3D_ZONLY));
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assert(beta.x == 0 && beta.y == 0);
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assert(particle_shift.x == 0 && particle_shift.y == 0);
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const double beta_z = beta.z;
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const double particle_shift_z = particle_shift_z;
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const qpms_y_t nelem_sc = c->nelem_sc;
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const qpms_l_t lMax = c->lMax;
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// Manual init of the ewald summation targets
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complex double *target_c = calloc(nelem_sc, sizeof(complex double));
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memset(target, 0, nelem_sc * sizeof(complex double));
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double *err_c = NULL;
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if (err) {
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err_c = calloc(nelem_sc, sizeof(double));
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memset(err, 0, nelem_sc * sizeof(double));
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}
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PGenSingleReturnData pgen_retdata;
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// CHOOSE POINT BEGIN
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while ((pgen_retdata = PGenSph_next(pgen_K)).flags & PGEN_NOTDONE) { // BEGIN POINT LOOP
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assert(pgen_retdata.flags & PGEN_AT_Z);
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double K_z, beta_mu_z;
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if (pgen_generates_shifted_points) {
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beta_mu_z = ((pgen_retdata.point_sph.theta == 0) ?
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pgen_retdata.point_sph.r : -pgen_retdata.point_sph.r); //!!!CHECKSIGN!!!
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K_z = beta_mu_z - beta_z;
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} else { // as in the old _points_and_shift functions
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K_z = ((pgen_retdata.point_sph.theta == 0) ?
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pgen_retdata.point_sph.r : -pgen_retdata.point_sph.r); // !!!CHECKSIGN!!!
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beta_mu_z = K_z + beta_z;
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}
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// CHOOSE POINT END
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const complex double phasefac = cexp(I * K_z * particle_shift_z); // POINT-DEPENDENT (PFC) // !!!CHECKSIGN!!!
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// R-DEPENDENT BEGIN
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gamma_pq = lilgamma(rbeta_pq/k);
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z = csq(gamma_pq*k/(2*eta)); // Když o tom tak přemýšlím, tak tohle je vlastně vždy reálné
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for(qpms_l_t j = 0; j <= lMax/2; ++j) {
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int retval = complex_gamma_inc_e(0.5-j, z, Gamma_pq+j);
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// 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
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if(creal(z) < 0)
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Gamma_pq[j].val = conj(Gamma_pq[j].val); //FIXME as noted above
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if(!(retval==0 || retval==GSL_EUNDRFLW)) abort();
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}
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if (latdim & LAT1D)
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factor1d = k * M_SQRT1_2 * .5 * gamma_pq;
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// R-DEPENDENT END
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// TODO optimisations: all the j-dependent powers can be done for each j only once, stored in array
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// and just fetched for each n, m pair
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for(qpms_l_t n = 0; n <= lMax; ++n)
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for(qpms_m_t m = -n; m <= n; ++m) {
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if((m+n) % 2 != 0) // odd coefficients are zero.
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continue;
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const qpms_y_t y = qpms_mn2y_sc(m, n);
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const complex double e_imalpha_pq = cexp(I*m*arg_pq);
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complex double jsum, jsum_c; ckahaninit(&jsum, &jsum_c);
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double jsum_err, jsum_err_c; kahaninit(&jsum_err, &jsum_err_c); // TODO do I really need to kahan sum errors?
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assert((n-abs(m))/2 == c->s1_jMaxes[y]);
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for(qpms_l_t j = 0; j <= c->s1_jMaxes[y]/*(n-abs(m))/2*/; ++j) { // FIXME </<= ?
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complex double summand = pow(rbeta_pq/k, n-2*j)
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* 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
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* cpow(gamma_pq, 2*j-1) // * Gamma_pq[j] bellow (GGG) after error computation
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* c->s1_constfacs[y][j];
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if(err) {
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// FIXME include also other errors than Gamma_pq's relative error
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kahanadd(&jsum_err, &jsum_err_c, Gamma_pq[j].err * cabs(summand));
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}
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summand *= Gamma_pq[j].val; // GGG
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ckahanadd(&jsum, &jsum_c, summand);
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}
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jsum *= phasefac * factor1d; // PFC
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ckahanadd(target + y, target_c + y, jsum);
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if(err) kahanadd(err + y, err_c + y, jsum_err);
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}
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#ifndef NDEBUG
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rbeta_pq_prev = rbeta_pq;
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#endif
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} // END POINT LOOP
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free(err_c);
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free(target_c);
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for(qpms_y_t y = 0; y < nelem_sc; ++y) // CFC common factor from above
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target[y] *= commonfac;
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if(err)
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for(qpms_y_t y = 0; y < nelem_sc; ++y)
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err[y] *= commonfac;
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return 0;
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}
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struct sigma2_integrand_params {
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int n;
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@ -21,10 +21,17 @@ typedef enum LatticeDimensionality {
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LAT_2D_IN_3D = 34,
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LAT_3D_IN_3D = 40,
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// special coordinate arrangements (indicating possible optimisations)
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LAT_ZONLY = 64,
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LAT_XYONLY = 128,
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LAT_1D_IN_3D_ZONLY = 97, // LAT1D | SPACE3D | 64
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LAT_2D_IN_3D_XYONLY = 162 // LAT2D | SPACE3D | 128
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} LatticeDimensionality;
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inline static bool LatticeDimensionality_checkflags(
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LatticeDimensionality a, LatticeDimensionality flags_a_has_to_contain) {
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return ((a & flags_a_has_to_contain) == flags_a_has_to_contain);
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}
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#include <math.h>
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#include <stdbool.h>
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#include <assert.h>
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