343 lines
14 KiB
C
343 lines
14 KiB
C
/*! \file ewald.h
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* \brief Lattice sums of spherical waves.
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*
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* Implementation of two-dimensional lattice sum in three dimensions
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* according to:
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* - [1] C.M. Linton, I. Thompson
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* Journal of Computational Physics 228 (2009) 1815–1829
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* - [2] C.M.Linton
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* SIAM Review Vol 52, No. 4, pp. 630–674
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*
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* N.B.!!! currently, the long-range parts are calculated
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* not according to [1,(4.5)], but rather
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* according to the spherical-harmonic-normalisation-independent
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* formulation in my notes notes/ewald.lyx.
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* Both parts of lattice sums are then calculated with
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* the \f$ P_n^{|m|} e^{im\phi} \f$
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* (N.B. or \f$ P_n^{|m|} e^{imf} (-1)^m \f$ for negative m)
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* substituted in place of \f$ Y_n^m \f$
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* (this is quite a weird normalisation especially
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* for negative \f$ |m| \f$, but it is consistent
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* with the current implementation of the translation coefficients in
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* @ref translations.c;
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* in the long run, it might make more sense to replace it everywhere with normalised
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* Legendre polynomials).
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*/
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#ifndef EWALD_H
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#define EWALD_H
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#include <gsl/gsl_sf_result.h>
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#include <stdlib.h>
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#include <gsl/gsl_sf_legendre.h>
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#include <gsl/gsl_errno.h>
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#include <math.h> // for inlined lilgamma
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#include <complex.h>
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#include "qpms_types.h"
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#include "lattices.h"
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typedef enum {
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QPMS_EWALD_LONG_RANGE = 1,
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QPMS_EWALD_SHORT_RANGE = 2,
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QPMS_EWALD_0TERM = 4,
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QPMS_EWALD_FULL = QPMS_EWALD_LONG_RANGE | QPMS_EWALD_SHORT_RANGE | QPMS_EWALD_0TERM,
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} qpms_ewald_part;
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/// Use this handler to ignore underflows of incomplete gamma.
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gsl_error_handler_t IgnoreUnderflowsGSLErrorHandler;
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/// Object holding the Ewald sum constant factors.
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/**
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* Used internally by qpms_translation_calculator_t.
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* Initialised by qpms_ewald3_constants_init() and freed by qpms_ewald3_constants_free().
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*/
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typedef struct qpms_ewald3_constants_t {
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qpms_l_t lMax;
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qpms_y_t nelem_sc;
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/// The values of maximum \a j's in the long-range part summation, `[(l-|m|/2)]`.
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qpms_l_t *s1_jMaxes;
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/// The constant factors for the long range part of a 2D Ewald sum.
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complex double **s1_constfacs; // indices [y][j] where j is same as in [1, (4.5)]
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/* These are the actual numbers now: (in the EWALD32_CONSTANTS_AGNOSTIC version)
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* for m + n EVEN:
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*
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* s1_constfacs[y(m,n)][j] =
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*
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* -2 * I**(n+1) * sqrt(π) * ((n-m)/2)! * ((n+m)/2)! * (-1)**j
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* -----------------------------------------------------------
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* j! * ((n-m)/2 - j)! * ((n+m)/2 + j)!
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*
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* for m + n ODD:
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*
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* s1_constfacs[y(m,n)][j] = 0
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*/
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complex double *s1_constfacs_base; ///< Internal pointer holding memory for the 2D Ewald sum constant factors.
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// similarly for the 1D z-axis aligned case; now the indices are [n][j] (as m == 0)
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/// The constant factors for the long range part of a 1D Ewald sum along the \a z axis.
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/** If the summation points lie along a different direction, use the formula for
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* 2D sum with additional factor of
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* \f$ \sqrt{pi} \kappa \gamma(\abs{\vect{k}+\vect{K}}/\kappa) \f$.
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*/
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///=============== NEW GENERATION GENERAL 2D-IN-3D, including z != 0 =========================
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// TODO indexing mechanisms
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/// The constant factors for the long range part of a 2D Ewald sum.
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complex double **S1_constfacs; // indices [y][j] where j is same as in [1, (4.5)]
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/* These are the actual numbers now: (in the EWALD32_CONSTANTS_AGNOSTIC version)
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* for m + n EVEN:
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*
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* S1_constfacs[y(m,n)][x(j,s)] =
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*
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* -2 * I**(n+1) * sqrt(π) * ((n-m)/2)! * ((n+m)/2)! * (-1)**j / j \
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* ----------------------------------------------------------- | |
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* j! * ((n - m - s)/2)! * ((n + m - s)/2)! \ 2j - s /
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*
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* for m + n ODD:
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*
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* S1_constfacs[y(m,n)][j] = 0
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*/
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complex double *S1_constfacs_base; ///< Internal pointer holding memory for the 2D Ewald sum constant factors.
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/// The constant factors for the long range part of a 1D Ewald sum along the \a z axis.
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/** If the summation points lie along a different direction, use the formula for
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* 2D sum with additional factor of
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* \f$ \sqrt{pi} \kappa \gamma(\abs{\vect{k}+\vect{K}}/\kappa) \f$.
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*/
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complex double **s1_constfacs_1Dz;
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/* These are the actual numbers now:
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* s1_constfacs_1Dz[n][j] =
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*
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* -I**(n+1) (-1)**j * n!
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* --------------------------
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* j! * 2**(2*j) * (n - 2*j)!
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*/
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complex double *s1_constfacs_1Dz_base; ///<Internal pointer holding memory for the 1D Ewald sum constant factors.
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double *legendre0; /* now with GSL_SF_LEGENDRE_NONE normalisation, because this is what is
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* what the multipliers from translations.c count with.
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*/
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double *legendre_plus1; // needed? TODO; in any case, nonzero only for m=0
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double *legendre_minus1; // needed? TODO; in any case, nonzero only for m=0
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gsl_sf_legendre_t legendre_normconv;
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int legendre_csphase; /* 1 or -1; csphase of the Legendre polynomials saved in legendre0 etc.
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This is because I dont't actually consider this fixed in
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translations.c */
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} qpms_ewald3_constants_t;
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/// Constructor for qpms_ewald3_constants_t.
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qpms_ewald3_constants_t *qpms_ewald3_constants_init(qpms_l_t lMax, int csphase);
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/// Destructor for qpms_ewald3_constants_t.
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void qpms_ewald3_constants_free(qpms_ewald3_constants_t *);
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/// Structure for holding complex-valued result of computation and an error estimate.
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/** Similar to gsl_sf_result, but with complex val. */
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typedef struct qpms_csf_result {
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complex double val; ///< Calculation result.
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double err; ///< Error estimate.
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} qpms_csf_result;
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// [1, (A.9)]
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static inline complex double lilgamma(double t) {
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t = fabs(t);
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if (t >= 1)
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return sqrt(t*t - 1);
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else
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return -I * sqrt(1 - t*t);
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}
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// [1, (A.8)], complex version of lilgamma()
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static inline complex double clilgamma(complex double z) {
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complex double a1 = z - 1, a2 = z + 1;
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// ensure -pi/2 < arg(z + 1) < 3*pi/2
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if (creal(a2) < 0 && cimag(a2) <= 0)
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a2 = -csqrt(a2);
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else
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a2 = csqrt(a2);
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// ensure -3*pi/2 < arg(z - 1) < pi/2
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if (creal(a1) < 0 && cimag(a1) >= 0)
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a1 = -csqrt(a1);
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else
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a1 = csqrt(a1);
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return a1 * a2;
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}
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/// Incomplete Gamma function as a series.
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/** DLMF 8.7.3 (latter expression) for complex second argument.
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*
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* The principal value is calculated. On the negative real axis
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* (where the function has branch cut), the sign of the imaginary
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* part is what matters (even if it is zero). Therefore one
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* can have
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* `cx_gamma_inc_series_e(a, z1) != cx_gamma_inc_series_e(a, z2)`
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* even if `z1 == z2`, because `-0 == 0` according to IEEE 754.
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* The side of the branch cut can be determined using `signbit(creal(z))`.
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*/
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int cx_gamma_inc_series_e(double a, complex double z, qpms_csf_result * result);
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/// Incomplete Gamma function as continued fractions.
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/**
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* The principal value is calculated. On the negative real axis
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* (where the function has branch cut), the sign of the imaginary
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* part is what matters (even if it is zero). Therefore one
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* can have
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* `cx_gamma_inc_CF_e(a, z1) != cx_gamma_inc_CF_e(a, z2)`
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* even if `z1 == z2`, because `-0 == 0` according to IEEE 754.
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* The side of the branch cut can be determined using `signbit(creal(z))`.
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*/
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int cx_gamma_inc_CF_e(double a, complex double z, qpms_csf_result * result);
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/// Incomplete gamma for complex second argument.
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/**
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* If x is (almost) real, it just uses gsl_sf_gamma_inc_e().
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*
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* On the negative real axis
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* (where the function has branch cut), the sign of the imaginary
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* part is what matters (even if it is zero). Therefore one
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* can have
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* `complex_gamma_inc_e(a, z1, m) != complex_gamma_inc_e(a, z2, m)`
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* even if `z1 == z2`, because `-0 == 0` according to IEEE 754.
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* The side of the branch cut can be determined using `signbit(creal(z))`.
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*
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* Another than principal branch can be selected using non-zero \a m
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* argument.
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*/
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int complex_gamma_inc_e(double a, complex double x,
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/// Branch index.
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/** If zero, the principal value is calculated.
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* Other branches might be chosen using non-zero \a m.
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* In such case, the returned value corresponds to \f[
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* \Gamma(a,ze^{2\pi mi})=e^{2\pi mia} \Gamma(a,z)
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* + (1-e^{2\pi mia}) \Gamma(a).
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* \f]
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*
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* If \a a is non-positive integer, the limiting value should
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* be used, but this is not yet implemented!
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*/
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int m,
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qpms_csf_result *result);
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/// Exponential integral for complex second argument.
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/** If x is (almost) positive real, it just uses gsl_sf_expint_En_e(). */
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int complex_expint_n_e(int n, complex double x, qpms_csf_result *result);
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/// Hypergeometric 2F2, used to calculate some errors.
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int hyperg_2F2_series(double a, double b, double c, double d, double x,
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gsl_sf_result *result);
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#if 0
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// The integral from (4.6); maybe should be static and not here.
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int ewald32_sr_integral(double r, double k, double n, double eta, double *result, double *err, gsl_integration_workspace *workspace);
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#endif
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/// The Delta_n factor from [Kambe II], Appendix 3, used in 2D-in-3D long range sum.
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/** \f[ \Delta_n = \int_n^\infty t^{-1/2 - n} \exp(-t + z^2/(4t))\ud t \f]
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*
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* \bug The current choice of method, based purely on the value of \a z, might be
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* unsuitable especially for big values of \a maxn.
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*
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*/
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void ewald3_2_sigma_long_Delta(complex double *target, double *target_err, int maxn, complex double x,
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int xbranch, complex double z);
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/// The Delta_n factor from [Kambe II], Appendix 3, used in 2D-in-3D long range sum.
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/** This function always uses Kambe's (corrected) recurrent formula.
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* For production, use ewald3_2_sigma_long_Delta() instead.
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*/
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void ewald3_2_sigma_long_Delta_recurrent(complex double *target, double *target_err, int maxn, complex double x,
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int xbranch, complex double z, _Bool bigimz);
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/// The Delta_n factor from [Kambe II], Appendix 3, used in 2D-in-3D long range sum.
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/** This function always uses Taylor expansion in \a z.
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* For production, use ewald3_2_sigma_long_Delta() instead.
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*
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* \bug The error estimate seems to be wrong (too small) at least in some cases: try
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* parameters maxn = 40, z = 0.5, x = -3. This might be related to the exponential growth
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* of the error.
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*/
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void ewald3_2_sigma_long_Delta_series(complex double *target, double *target_err, int maxn, complex double x,
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int xbranch, complex double z);
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// General functions acc. to [2], sec. 4.6 – currently valid for 2D and 1D lattices in 3D space
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/// The Ewald sum "self-interaction" term that appears in the lattice sums with zero (direct-space) Bravais lattice displacement.
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int ewald3_sigma0(complex double *result, ///< Pointer to save the result (single complex double).
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double *err, ///< Pointer to save the result error estimate (single double).
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const qpms_ewald3_constants_t *c, ///< Constant factors structure initialised by qpms_ewald3_constants_init().
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double eta, ///< Ewald parameter.
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complex double wavenumber ///< Wavenumber of the background medium.
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);
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/// Short-range part of outgoing scalar spherical wavefunctions' lattice sum \f$ \sigma_{l,m}^\mathrm{S}(\vect k,\vect s)\f$.
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int ewald3_sigma_short(
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complex double *target_sigmasr_y, ///< Target array for \f$ \sigma_{l,m}^\mathrm{S} \f$, must be `c->nelem_sc` long.
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double *target_sigmasr_y_err, ///< Target array for error estimates, must be `c->nelem_sc` long or `NULL`.
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const qpms_ewald3_constants_t *c, ///< Constant factors structure initialised by qpms_ewald3_constants_init().
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double eta, ///< Ewald parameter.
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complex double wavenumber, ///< Wavenumber of the background medium.
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/// Lattice dimensionality.
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/** Ignored apart from asserts and possible optimisations, as the SR formula stays the same. */
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LatticeDimensionality latdim,
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/// Lattice point generator for the direct Bravais lattice.
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/** There is a possibility that the whole PGen is not consumed
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* (this might happen if the summand start to be consistently smaller
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* than the (partial) sums * DBL_EPSILON.
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* In such case, it is the responsibility of the caller to deallocate
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* the generator.
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*/
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PGen *pgen_R,
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/// Indicates whether pgen_R already generates shifted points.
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/** If false, the behaviour corresponds to the old ewald32_sigma_short_points_and_shift(),
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* so the function assumes that the generated points correspond to the unshifted Bravais lattice,
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* and adds particle_shift to the generated points before calculations.
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* If true, it assumes that they are already shifted (if calculating interaction between
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* different particles in the unit cell).
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*/
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bool pgen_generates_shifted_points,
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/// Wave vector \f$\vect k\f$.
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cart3_t k,
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/// Lattice offset \f$\vect s\f$ wrt. the Bravais lattice.
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cart3_t particle_shift
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);
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/// Long-range part of outgoing scalar spherical wavefunctions' lattice sum \f$ \sigma_{l,m}^\mathrm{L}(\vect k,\vect s)\f$.
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int ewald3_sigma_long( // calls ewald3_21_sigma_long or ewald3_3_sigma_long, depending on latdim
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complex double *target_sigmalr_y, ///< Target array for \f$ \sigma_{l,m}^\mathrm{L} \f$, must be `c->nelem_sc` long.
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double *target_sigmalr_y_err, ///< Target array for error estimates, must be `c->nelem_sc` long or `NULL`.
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const qpms_ewald3_constants_t *c, ///< Constant factors structure initialised by qpms_ewald3_constants_init().
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double eta, ///< Ewald parameter.
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complex double wavenumber, ///< Wavenumber of the background medium.
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double unitcell_volume, ///< Volume of the (direct lattice) unit cell (with dimension corresponding to the lattice dimensionality).
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/// Lattice dimensionality.
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LatticeDimensionality latdim,
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/// Lattice point generator for the reciprocal lattice.
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/** There is a possibility that the whole PGen is not consumed
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* (this might happen if the summand start to be consistently smaller
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* than the (partial) sums * DBL_EPSILON.
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* In such case, it is the responsibility of the caller to deallocate
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* the generator.
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*/
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PGen *pgen_K,
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/// Indicates whether pgen_K already 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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bool pgen_generates_shifted_points,
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/// Wave vector \f$\vect k\f$.
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cart3_t k,
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/// Lattice offset \f$\vect s\f$ wrt. the Bravais lattice.
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cart3_t particle_shift
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);
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#endif //EWALD_H
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