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qpms/qpms/ewaldsf.c

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#include "ewald.h"
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_expint.h>
#include <gsl/gsl_sf_exp.h>
#include <gsl/gsl_sf_result.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_machine.h> // Maybe I should rather use DBL_EPSILON instead of GSL_DBL_EPSILON.
#include "kahansum.h"
#include <math.h>
#include <complex.h>
//#include <gsl/gsl_integration.h>
#include <gsl/gsl_errno.h>
#include <float.h>
#include <stdbool.h>
#include <Faddeeva.h>
#include "tiny_inlines.h"
// Some magic constants
#ifndef COMPLEXPART_REL_ZERO_LIMIT
#define COMPLEXPART_REL_ZERO_LIMIT 1e-14
#endif
#ifndef DELTA_RECURRENT_EXPOVERFLOW_LIMIT
#define DELTA_RECURRENT_EXPOVERFLOW_LIMIT 10.
#endif
gsl_error_handler_t IgnoreUnderflowsGSLErrorHandler;
void IgnoreUnderflowsGSLErrorHandler (const char * reason,
const char * file,
const int line,
const int gsl_errno) {
if (gsl_errno == GSL_EUNDRFLW)
return;
gsl_stream_printf ("ERROR", file, line, reason);
fflush(stdout);
fprintf (stderr, "Underflow-ignoring error handler invoked.\n");
fflush(stderr);
abort();
}
// DLMF 8.7.3 (latter expression) for complex second argument
// BTW if a is large negative, it might take a while to evaluate.
// This can't be used for non-positive integer a due to
// Г(a) in the formula.
int cx_gamma_inc_series_e(const double a, const complex double z, qpms_csf_result * result) {
if (a <= 0 && a == (int) a) {
result->val = NAN + NAN*I;
result->err = NAN;
GSL_ERROR("Undefined for non-positive integer values", GSL_EDOM);
}
gsl_sf_result fullgamma;
int retval = gsl_sf_gamma_e(a, &fullgamma);
if (GSL_EUNDRFLW == retval)
result->err += DBL_MIN;
else if (GSL_SUCCESS != retval){
result->val = NAN + NAN*I; result->err = NAN;
return retval;
}
complex double sumprefac = cpow(z, a) * cexp(-z);
double sumprefac_abs = cabs(sumprefac);
complex double sum, sumc; ckahaninit(&sum, &sumc);
double err, errc; kahaninit(&err, &errc);
bool breakswitch = false;
for (int k = 0; (!breakswitch) && (a + k + 1 <= GSL_SF_GAMMA_XMAX); ++k) {
gsl_sf_result fullgamma_ak;
if (GSL_SUCCESS != (retval = gsl_sf_gamma_e(a+k+1, &fullgamma_ak))) {
result->val = NAN + NAN*I; result->err = NAN;
return retval;
}
complex double summand = - cpow(z, k) / fullgamma_ak.val; // TODO test branch selection here with cimag(z) = -0.0
ckahanadd(&sum, &sumc, summand);
double summanderr = fabs(fullgamma_ak.err * cabs(summand / fullgamma_ak.val));
// TODO add also the rounding error
kahanadd(&err, &errc, summanderr);
// TODO put some smarter cutoff break here?
if (a + k >= 18 && (cabs(summand) < err || cabs(summand) < DBL_EPSILON))
breakswitch = true;
}
sum *= sumprefac; // Not sure if not breaking the Kahan summation here
sumc *= sumprefac;
err *= sumprefac_abs;
errc *= sumprefac_abs;
ckahanadd(&sum, &sumc, 1.);
kahanadd(&err, &errc, DBL_EPSILON);
result->err = cabs(sum) * fullgamma.err + err * fabs(fullgamma.val);
result->val = sum * fullgamma.val; // + sumc*fullgamma.val???
if (breakswitch)
return GSL_SUCCESS;
else GSL_ERROR("Overflow; the absolute value of the z argument is probably too large.", GSL_ETOL); // maybe different error code...
}
/* Continued fraction which occurs in evaluation
* of Q(a,z) or Gamma(a,z).
* Borrowed from GSL and adapted for complex z.
*
* 1 (1-a)/z 1/z (2-a)/z 2/z (3-a)/z
* F(a,z) = ---- ------- ----- -------- ----- -------- ...
* 1 + 1 + 1 + 1 + 1 + 1 +
*
*/
static int
cx_gamma_inc_F_CF(const double a, const complex double z, qpms_csf_result * result)
{
const int nmax = 5000;
const double small = DBL_EPSILON * DBL_EPSILON * DBL_EPSILON;
complex double hn = 1.0; /* convergent */
complex double Cn = 1.0 / small;
complex double Dn = 1.0;
int n;
/* n == 1 has a_1, b_1, b_0 independent of a,z,
so that has been done by hand */
for ( n = 2 ; n < nmax ; n++ )
{
complex double an;
complex double delta;
if(n % 2)
an = 0.5*(n-1)/z;
else
an = (0.5*n-a)/z;
Dn = 1.0 + an * Dn;
if ( cabs(Dn) < small )
Dn = small;
Cn = 1.0 + an/Cn;
if ( cabs(Cn) < small )
Cn = small;
Dn = 1.0 / Dn;
delta = Cn * Dn;
hn *= delta;
if(cabs(delta-1.0) < DBL_EPSILON) break;
}
result->val = hn;
result->err = 2.0*GSL_DBL_EPSILON * cabs(hn);
result->err += GSL_DBL_EPSILON * (2.0 + 0.5*n) * cabs(result->val);
if(n == nmax)
GSL_ERROR ("error in CF for F(a,x)", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
// Incomplete gamma fuction with complex second argument as continued fraction.
int cx_gamma_inc_CF_e(const double a, const complex double z, qpms_csf_result *result)
{
qpms_csf_result F;
gsl_sf_result pre;
const complex double am1lgz = (a-1.0)*clog(z); // TODO check branches
const int stat_F = cx_gamma_inc_F_CF(a, z, &F);
const int stat_E = gsl_sf_exp_err_e(creal(am1lgz - z), GSL_DBL_EPSILON*cabs(am1lgz), &pre);
complex double cpre = pre.val * cexp(I*cimag(am1lgz - z));// TODO add the error estimate for this
//complex double cpre = cpow(z, a-1) * cexp(-z);
result->val = F.val * cpre;
result->err = fabs(F.err * pre.val) + fabs(F.val * pre.err);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_F, stat_E);
}
// Incomplete gamma function for complex second argument.
int complex_gamma_inc_e(double a, complex double x, int m, qpms_csf_result *result) {
int retval;
if (creal(x) >= 0 &&
(0 == fabs(cimag(x)) || // x is real positive; just use the real fun
fabs(cimag(x)) < fabs(creal(x)) * COMPLEXPART_REL_ZERO_LIMIT)) {
gsl_sf_result real_gamma_inc_result;
retval = gsl_sf_gamma_inc_e(a, creal(x), &real_gamma_inc_result);
result->val = real_gamma_inc_result.val;
result->err = real_gamma_inc_result.err;
} else if (creal(x) >= 0 && cabs(x) > 0.5)
retval = cx_gamma_inc_CF_e(a, x, result);
else if (QPMS_LIKELY(a > 0 || fmod(a, 1.0)))
retval = cx_gamma_inc_series_e(a, x, result);
else
/* FIXME cx_gamma_inc_series_e() probably fails for non-positive integer a.
* This does not matter for 2D lattices in 3D space,
* but it might cause problems in the other cases.
*/
QPMS_NOT_IMPLEMENTED("Incomplete Gamma function with non-positive integer a.");
if (m) { // Non-principal branch.
/* This might be sub-optimal, as Γ(a) has probably been already evaluated
* somewhere in the functions called above. */
gsl_sf_result fullgamma;
int retval_fg = gsl_sf_gamma_e(a, &fullgamma);
if (GSL_EUNDRFLW == retval_fg)
fullgamma.err += DBL_MIN;
else if (GSL_SUCCESS != retval_fg){
result->val = NAN + NAN*I; result->err = NAN;
return GSL_ERROR_SELECT_2(retval_fg, retval);
}
complex double f = cexp(2*m*M_PI*a*I);
result->val *= f;
f = -f + 1;
result->err += cabs(f) * fullgamma.err;
result->val += f * fullgamma.val;
}
return retval;
}
// Exponential integral for complex argument; !UNTESTED! and probably not needed, as I expressed everything in terms of inc. gammas anyways.
int complex_expint_n_e(int n, complex double x, qpms_csf_result *result) {
if (creal(x) >= 0 &&
(0 == fabs(cimag(x)) || // x is real positive; just use the real fun
fabs(cimag(x)) < fabs(creal(x)) * COMPLEXPART_REL_ZERO_LIMIT)) {
gsl_sf_result real_expint_result;
int retval = gsl_sf_expint_En_e(n, creal(x), &real_expint_result);
result->val = real_expint_result.val;
result->err = real_expint_result.err;
return retval;
} else {
int retval = complex_gamma_inc_e(-n+1, x, 0, result);
complex double f = cpow(x, 2*n-2);
result->val *= f;
result->err *= cabs(f);
return retval;
}
}
// inspired by GSL's hyperg_2F1_series
int hyperg_2F2_series(const double a, const double b, const double c, const double d,
const double x, gsl_sf_result *result
)
{
double sum_pos = 1.0;
double sum_neg = 0.0;
double del_pos = 1.0;
double del_neg = 0.0;
double del = 1.0;
double del_prev;
double k = 0.0;
int i = 0;
if(fabs(c) < GSL_DBL_EPSILON || fabs(d) < GSL_DBL_EPSILON) {
result->val = NAN;
result->err = INFINITY;
GSL_ERROR ("error", GSL_EDOM);
}
do {
if(++i > 30000) {
result->val = sum_pos - sum_neg;
result->err = del_pos + del_neg;
result->err += 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += 2.0 * GSL_DBL_EPSILON * (2.0*sqrt(k)+1.0) * fabs(result->val);
GSL_ERROR ("error", GSL_EMAXITER);
}
del_prev = del;
del *= (a+k)*(b+k) * x / ((c+k) * (d+k) * (k+1.0)); /* Gauss series */
if(del > 0.0) {
del_pos = del;
sum_pos += del;
}
else if(del == 0.0) {
/* Exact termination (a or b was a negative integer).
*/
del_pos = 0.0;
del_neg = 0.0;
break;
}
else {
del_neg = -del;
sum_neg -= del;
}
/*
* This stopping criteria is taken from the thesis
* "Computation of Hypergeometic Functions" by J. Pearson, pg. 31
* (http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf)
* and fixes bug #45926
*/
if (fabs(del_prev / (sum_pos - sum_neg)) < GSL_DBL_EPSILON &&
fabs(del / (sum_pos - sum_neg)) < GSL_DBL_EPSILON)
break;
k += 1.0;
} while(fabs((del_pos + del_neg)/(sum_pos-sum_neg)) > GSL_DBL_EPSILON);
result->val = sum_pos - sum_neg;
result->err = del_pos + del_neg;
result->err += 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += 2.0 * GSL_DBL_EPSILON * (2.0*sqrt(k) + 1.0) * fabs(result->val);
return GSL_SUCCESS;
}
// Complex square root with branch selection
static inline complex double csqrt_branch(complex double x, int xbranch) {
return csqrt(x) * min1pow(xbranch);
}
// The Delta_n factor from [Kambe II], Appendix 3
// \f[ \Delta_n = \int_n^\infty t^{-1/2 - n} \exp(-t + z^2/(4t))\ud t \f]
/* If |Im z| is big, Faddeeva_z might cause double overflow. In such case,
* use bigimz = true to use a slightly different formula to initialise
* the first two elements.
*
* The actual choice is done outside this function in order to enable
* testing/comparison of the results.
*/
void ewald3_2_sigma_long_Delta_recurrent(complex double *target, double *err,
int maxn, complex double x, int xbranch, complex double z, bool bigimz) {
complex double expfac = cexp(-x + 0.25 * z*z / x);
complex double sqrtx = csqrt_branch(x, xbranch); // TODO check carefully, which branch is needed
double w_plus_abs = NAN, w_minus_abs = NAN; // Used only if err != NULL
QPMS_ASSERT(maxn >= 0);
// These are used to fill the first two elements for recurrence
if(!bigimz) {
complex double w_plus = Faddeeva_w(+z/(2*sqrtx) + I*sqrtx, 0);
complex double w_minus = Faddeeva_w(-z/(2*sqrtx) + I*sqrtx, 0);
if (maxn >= 0)
target[0] = 0.5 * M_SQRTPI * expfac * (w_minus + w_plus);
if (maxn >= 1)
target[1] = I / z * M_SQRTPI * expfac * (w_minus - w_plus);
if(err) { w_plus_abs = cabs(w_plus); w_minus_abs = cabs(w_minus); }
} else {
/* A different strategy to avoid double overflow using the formula
* w(y) = exp(-y*y) * erfc(-I*y):
* Labeling
* ž_± = ±z/(2*sqrtx) + I * sqrtx
* and expfac = exp(ž**2), where ž**2 = -x + (z*z/4/x),
* we have
* expfac * w(ž_±) = exp(ž**2 - ž_±**2) erfc(-I * ž_±)
* = exp(∓ I * z) erfc(-I * ž_±)
*/
complex double w_plus_n = cexp(- I * z) * Faddeeva_erfc(-I * (+z/(2*sqrtx) + I*sqrtx), 0);
complex double w_minus_n = cexp(+ I * z) * Faddeeva_erfc(-I * (-z/(2*sqrtx) + I*sqrtx), 0);
if (maxn >= 0)
target[0] = 0.5 * M_SQRTPI * (w_minus_n + w_plus_n);
if (maxn >= 1)
target[1] = I / z * M_SQRTPI * (w_minus_n - w_plus_n);
}
for(int n = 1; n < maxn; ++n) { // The rest via recurrence
// TODO The cpow(x, 0.5 - n) might perhaps better be replaced with a recurrently computed variant
target[n+1] = -(4 / (z*z)) * (-(0.5 - n) * target[n] + target[n-1] - sqrtx * cpow(x, -n) * expfac);
if(isnan(creal(target[n+1])) || isnan(cimag(target[n+1]))) {
QPMS_WARN("Encountered NaN.");
}
}
if (err) {
// The error estimates for library math functions are based on
// https://www.gnu.org/software/libc/manual/html_node/Errors-in-Math-Functions.html
// and are not guaranteed to be extremely precise.
// The error estimate for Faddeeva's functions is based on the package's authors at
// http://ab-initio.mit.edu/wiki/index.php/Faddeeva_Package
// "we find that the accuracy is typically at at least 13 significant digits in both the real and imaginary parts"
// FIXME the error estimate seems might be off by several orders of magnitude (try parameters x = -3, z = 0.5, maxn=20)
// FIXME the error estimate does not take into account the alternative recurrence init. formula (bigimz)
double expfac_abs = cabs(expfac);
double w_plus_err = w_plus_abs * 1e-13, w_minus_err = w_minus_abs * 1e-13; // LPTODO argument error contrib.
double expfac_err = expfac_abs * (4 * DBL_EPSILON); // LPTODO add argument error contrib.
double z_abs = cabs(z);
double z_err = z_abs * DBL_EPSILON;
double x_abs = cabs(x);
if (maxn >= 0)
err[0] = 0.5 * M_SQRTPI * (expfac_abs * (w_minus_err + w_plus_err) + (w_minus_abs + w_plus_abs) * expfac_err);
if (maxn >= 1)
err[1] = 2 * err[0] / z_abs + cabs(target[1]) * z_err / (z_abs*z_abs);
for(int n = 1; n < maxn; ++n) {
err[n+1] = (2 * cabs(target[n+1]) / z_abs + 4 * ((0.5+n) * err[n] + err[n-1] +
pow(x_abs, 0.5 - n) * (2*DBL_EPSILON * expfac_abs + expfac_err)) // LPTODO not ideal, pow() call is an overkill
) * z_err / (z_abs*z_abs);
}
}
}
void ewald3_2_sigma_long_Delta_series(complex double *target, double *err,
int maxn, complex double x, int xbranch, complex double z) {
complex double w = 0.25*z*z;
double w_abs = cabs(w);
int maxk;
if (w_abs == 0)
maxk = 0; // Delta is equal to the respective incomplete Gamma functions
else {
// Estimate a suitable maximum k, using Stirling's formula, so that w**maxk / maxk! is less than DBL_EPSILON
// This implementation is quite stupid, but it is still cheap compared to the actual computation, so LPTODO better one
maxk = 1;
double log_w_abs = log(w_abs);
while (maxk * (log_w_abs - log(maxk) + 1) >= -DBL_MANT_DIG)
++maxk;
}
// TODO asserts on maxn, maxk
complex double *Gammas;
double *Gammas_err = NULL, *Gammas_abs = NULL;
QPMS_CRASHING_CALLOC(Gammas, maxk+maxn+1, sizeof(*Gammas));
if(err) {
QPMS_CRASHING_CALLOC(Gammas_err, maxk+maxn+1, sizeof(*Gammas_err));
QPMS_CRASHING_MALLOC(Gammas_abs, (maxk+maxn+1) * sizeof(*Gammas_abs));
}
for(int j = 0; j <= maxn+maxk; ++j) {
qpms_csf_result g;
QPMS_ENSURE_SUCCESS(complex_gamma_inc_e(0.5-j, x, xbranch, &g));
Gammas[j] = g.val;
if(err) {
Gammas_abs[j] = cabs(g.val);
Gammas_err[j] = g.err;
}
}
for(int n = 0; n <= maxn; ++n) target[n] = 0;
if(err) for(int n = 0; n <= maxn; ++n) err[n] = 0;
complex double wpowk_over_fack = 1.;
double wpowk_over_fack_abs = 1.;
for(int k = 0; k <= maxk; ++k, wpowk_over_fack *= w/k) { // TODO? Kahan sum, Horner's method?
// Also TODO? for small n, continue for higher k if possible/needed
for(int n = 0; n <= maxn; ++n) {
target[n] += Gammas[n+k] * wpowk_over_fack;
if(err) {
// DBL_EPSILON might not be the best estimate here, but...
err[n] += wpowk_over_fack_abs * Gammas_err[n+k] + DBL_EPSILON * Gammas_abs[n+k];
wpowk_over_fack_abs *= w_abs / (k+1);
}
}
}
// TODO add an error estimate for the k-cutoff!!!
free(Gammas);
free(Gammas_err);
free(Gammas_abs);
}
void ewald3_2_sigma_long_Delta(complex double *target, double *err,
int maxn, complex double x, int xbranch, complex double z) {
double absz = cabs(z);
if (absz < 2.) // TODO take into account also the other parameters
ewald3_2_sigma_long_Delta_series(target, err, maxn, x, xbranch, z);
else
ewald3_2_sigma_long_Delta_recurrent(target, err, maxn, x, xbranch, z,
fabs(cimag(z)) > DELTA_RECURRENT_EXPOVERFLOW_LIMIT);
}
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