Implementation of the Δ_n factor as a series; error estimates.
The error estimate for the recurrence approach is buggy. Former-commit-id: f5183eaacf6a592461f07f72e04d346a42f9fca6
This commit is contained in:
Marek Nečada
parent
c056099d5a
commit
c50f40a747
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@ -3,17 +3,28 @@ from libc.stdlib cimport malloc, free, calloc
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import numpy as np
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cdef extern from "ewald.h":
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void ewald3_2_sigma_long_Delta(qpms_csf_result *target, int maxn, cdouble x, cdouble z)
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void ewald3_2_sigma_long_Delta(cdouble *target, double *err, int maxn, cdouble x, cdouble z)
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void ewald3_2_sigma_long_Delta_series(cdouble *target, double *err, int maxn, cdouble x, cdouble z)
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void ewald3_2_sigma_long_Delta_recurrent(cdouble *target, double *err, int maxn, cdouble x, cdouble z)
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int complex_gamma_inc_e(double a, cdouble x, int m, qpms_csf_result *result)
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def e32_Delta(int maxn, cdouble x, cdouble z):
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cdef qpms_csf_result *target = <qpms_csf_result *>malloc((maxn+1)*sizeof(qpms_csf_result))
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cdef np.ndarray[cdouble, ndim=1] target_np = np.empty((maxn+1,), dtype=complex, order='C')
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ewald3_2_sigma_long_Delta(target, maxn, x, z)
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cdef int i
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for i in range(maxn+1):
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target_np[i] = target[i].val
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free(target)
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def e32_Delta(int maxn, cdouble x, cdouble z, get_err=True, method='auto'):
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cdef np.ndarray[double, ndim=1] err_np
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cdef double[::1] err_view
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cdef np.ndarray[np.complex_t, ndim=1] target_np = np.empty((maxn+1,), dtype=complex, order='C')
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cdef cdouble[::1] target_view = target_np
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if get_err:
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err_np = np.empty((maxn+1,), order='C')
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err_view = err_np
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if method == 'recurrent':
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ewald3_2_sigma_long_Delta_recurrent(&target_view[0], &err_view[0] if get_err else NULL, maxn, x, z)
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elif method == 'series':
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ewald3_2_sigma_long_Delta_series(&target_view[0], &err_view[0] if get_err else NULL, maxn, x, z)
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else:
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ewald3_2_sigma_long_Delta(&target_view[0], &err_view[0] if get_err else NULL, maxn, x, z)
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if get_err:
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return target_np, err_np
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else:
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return target_np
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def gamma_inc(double a, cdouble x, int m=0):
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14
qpms/ewald.c
14
qpms/ewald.c
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@ -261,7 +261,8 @@ int ewald3_21_xy_sigma_long (
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complex double x; // κ*γ/(2*η)
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complex 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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complex double Gamma_pq[lMax/2+1];
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double Gamma_pq_err[lMax/2+1];
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// CHOOSE POINT BEGIN
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// TODO mayby PGen_next_sph is not the best coordinate system choice here
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@ -299,10 +300,13 @@ int ewald3_21_xy_sigma_long (
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complex double x2 = x*x;
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if(particle_shift.z == 0) {
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for(qpms_l_t j = 0; j <= lMax/2; ++j) {
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qpms_csf_result Gam;
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int retval = complex_gamma_inc_e(0.5-j, x2,
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// we take the branch which is principal for the Re x < 0, Im x < 0 quadrant, cf. [Linton, p. 642 in the middle]
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QPMS_LIKELY(creal(x2) < 0) && !signbit(cimag(x2)) ? -1 : 0, Gamma_pq+j);
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QPMS_LIKELY(creal(x2) < 0) && !signbit(cimag(x2)) ? -1 : 0, &Gam);
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QPMS_ENSURE_SUCCESS_OR(retval, GSL_EUNDRFLW);
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Gamma_pq[j] = Gam.val;
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if(err) Gamma_pq_err[j] = Gam.err;
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}
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if (latdim & LAT1D)
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factor1d = M_SQRT1_2 * .5 * k * gamma_pq;
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@ -310,7 +314,7 @@ int ewald3_21_xy_sigma_long (
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QPMS_UNTESTED;
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// TODO check the branches/phases!
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complex double z = I * k * gamma_pq * particle_shift.z;
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ewald3_2_sigma_long_Delta(Gamma_pq, lMax/2, x, z);
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ewald3_2_sigma_long_Delta(Gamma_pq, err ? Gamma_pq_err : NULL, lMax/2, x, z);
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} else {
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QPMS_NOT_IMPLEMENTED("1D lattices in 3D space outside of the line not implemented");
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}
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@ -335,9 +339,9 @@ int ewald3_21_xy_sigma_long (
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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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kahanadd(&jsum_err, &jsum_err_c, Gamma_pq_err[j] * cabs(summand));
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}
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summand *= Gamma_pq[j].val; // GGG
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summand *= Gamma_pq[j]; // 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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22
qpms/ewald.h
22
qpms/ewald.h
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@ -213,9 +213,27 @@ int ewald32_sr_integral(double r, double k, double n, double eta, double *result
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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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* NOTE: The error part is not yet implemented, NANs are returned instead!
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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(qpms_csf_result *target, int maxn, complex double x, complex double z);
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void ewald3_2_sigma_long_Delta(complex double *target, double *target_err, int maxn, complex double x, 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, 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 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, 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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108
qpms/ewaldsf.c
108
qpms/ewaldsf.c
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@ -292,26 +292,112 @@ int hyperg_2F2_series(const double a, const double b, const double c, const doub
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// The Delta_n factor from [Kambe II], Appendix 3
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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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void ewald3_2_sigma_long_Delta(qpms_csf_result *target, int maxn, complex double x, complex double z) {
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void ewald3_2_sigma_long_Delta_recurrent(complex double *target, double *err, int maxn, complex double x, complex double z) {
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complex double expfac = cexp(-x + 0.25 * z*z / x);
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complex double sqrtx = csqrt(x); // TODO check carefully, which branch is needed
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// These are used in the first two recurrences
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complex double w_plus = Faddeeva_w(+z/(2*sqrtx) + I*sqrtx, 0);
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complex double w_minus = Faddeeva_w(-z/(2*sqrtx) + I*sqrtx, 0);
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QPMS_ASSERT(maxn >= 0);
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if (maxn >= 0) {
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target[0].val = 0.5 * M_SQRTPI * expfac * (w_minus + w_plus);
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target[0].err = NAN; //TODO
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}
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if (maxn >= 1) {
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target[1].val = I / z * M_SQRTPI * expfac * (w_minus - w_plus);
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target[1].err = NAN; //TODO
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}
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if (maxn >= 0)
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target[0] = 0.5 * M_SQRTPI * expfac * (w_minus + w_plus);
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if (maxn >= 1)
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target[1] = I / z * M_SQRTPI * expfac * (w_minus - w_plus);
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for(int n = 1; n < maxn; ++n) { // The rest via recurrence
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// TODO The cpow(x, 0.5 - n) might perhaps better be replaced with a recurrently computed variant
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target[n+1].val = -(4 / (z*z)) * (-(0.5 - n) * target[n].val + target[n-1].val - cpow(x, 0.5 - n) * expfac);
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target[n+1].err = NAN; //TODO
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target[n+1] = -(4 / (z*z)) * (-(0.5 - n) * target[n] + target[n-1] - cpow(x, 0.5 - n) * expfac);
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}
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if (err) {
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// The error estimates for library math functions are based on
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// https://www.gnu.org/software/libc/manual/html_node/Errors-in-Math-Functions.html
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// and are not guaranteed to be extremely precise.
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// The error estimate for Faddeeva's functions is based on the package's authors at
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// http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package
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// "we find that the accuracy is typically at at least 13 significant digits in both the real and imaginary parts"
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// FIXME the error estimate seems might be off by several orders of magnitude (try parameters x = -3, z = 0.5, maxn=20)
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double w_plus_abs = cabs(w_plus), w_minus_abs = cabs(w_minus), expfac_abs = cabs(expfac);
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double w_plus_err = w_plus_abs * 1e-13, w_minus_err = w_minus_abs * 1e-13; // LPTODO argument error contrib.
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double expfac_err = expfac_abs * (4 * DBL_EPSILON); // LPTODO add argument error contrib.
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double z_abs = cabs(z);
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double z_err = z_abs * DBL_EPSILON;
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double x_abs = cabs(x);
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if (maxn >= 0)
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err[0] = 0.5 * M_SQRTPI * (expfac_abs * (w_minus_err + w_plus_err) + (w_minus_abs + w_plus_abs) * expfac_err);
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if (maxn >= 1)
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err[1] = 2 * err[0] / z_abs + cabs(target[1]) * z_err / (z_abs*z_abs);
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for(int n = 1; n < maxn; ++n) {
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err[n+1] = (2 * cabs(target[n+1]) / z_abs + 4 * ((0.5+n) * err[n] + err[n-1] +
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pow(x_abs, 0.5 - n) * (2*DBL_EPSILON * expfac_abs + expfac_err)) // LPTODO not ideal, pow() call is an overkill
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) * z_err / (z_abs*z_abs);
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}
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}
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}
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void ewald3_2_sigma_long_Delta_series(complex double *target, double *err, int maxn, complex double x, complex double z) {
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complex double w = 0.25*z*z;
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double w_abs = cabs(w);
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int maxk;
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if (w_abs == 0)
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maxk = 0; // Delta is equal to the respective incomplete Gamma functions
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else {
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// Estimate a suitable maximum k, using Stirling's formula, so that w**maxk / maxk! is less than DBL_EPSILON
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// This implementation is quite stupid, but it is still cheap compared to the actual computation, so LPTODO better one
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maxk = 1;
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double log_w_abs = log(w_abs);
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while (maxk * (log_w_abs - log(maxk) + 1) >= -DBL_MANT_DIG)
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++maxk;
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}
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// TODO asserts on maxn, maxk
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complex double *Gammas;
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double *Gammas_err = NULL, *Gammas_abs = NULL;
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QPMS_CRASHING_CALLOC(Gammas, maxk+maxn+1, sizeof(*Gammas));
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if(err) {
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QPMS_CRASHING_CALLOC(Gammas_err, maxk+maxn+1, sizeof(*Gammas_err));
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QPMS_CRASHING_MALLOC(Gammas_abs, (maxk+maxn+1) * sizeof(*Gammas_abs));
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}
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for(int j = 0; j <= maxn+maxk; ++j) {
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qpms_csf_result g;
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QPMS_ENSURE_SUCCESS(complex_gamma_inc_e(0.5-j, x, 0 /* TODO branch choice */, &g));
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Gammas[j] = g.val;
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if(err) {
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Gammas_abs[j] = cabs(g.val);
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Gammas_err[j] = g.err;
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}
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}
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for(int n = 0; n <= maxn; ++n) target[n] = 0;
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if(err) for(int n = 0; n <= maxn; ++n) err[n] = 0;
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complex double wpowk_over_fack = 1.;
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double wpowk_over_fack_abs = 1.;
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for(int k = 0; k <= maxk; ++k, wpowk_over_fack *= w/k) { // TODO? Kahan sum, Horner's method?
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// Also TODO? for small n, continue for higher k if possible/needed
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for(int n = 0; n <= maxn; ++n) {
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target[n] += Gammas[n+k] * wpowk_over_fack;
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if(err) {
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// DBL_EPSILON might not be the best estimate here, but...
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err[n] += wpowk_over_fack_abs * Gammas_err[n+k] + DBL_EPSILON * Gammas_abs[n+k];
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wpowk_over_fack_abs *= w_abs / (k+1);
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}
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}
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}
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// TODO add an error estimate for the k-cutoff!!!
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free(Gammas);
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free(Gammas_err);
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free(Gammas_abs);
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}
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void ewald3_2_sigma_long_Delta(complex double *target, double *err, int maxn, complex double x, complex double z) {
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double absz = cabs(z);
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if (absz < 2.) // TODO take into account also the other parameters
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ewald3_2_sigma_long_Delta_series(target, err, maxn, x, z);
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else
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ewald3_2_sigma_long_Delta_recurrent(target, err, maxn, x, z);
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}
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