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gcikpi.hpp
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gcikpi.hpp
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/* -------------------------------------------------------------------------
* Copyright (c) 2024 Fabio Souto de Azevedo <fabio.azevedo@ufrgs.br>
* Guilherme Fiorin Fornel <gffrnl@gmail.com>
* Matheus Correa dos Santos <matheus.santos@ufrgs.br>
* Esequia Sauter <esequia.sauter@ufrgs.br>
*
* All rights reserved.
*
* Licensed under the MIT License. See LICENSE in the project root for
* license information.
* ------------------------------------------------------------------------- */
#pragma once
#include <cstddef>
#include <cmath>
double Y(std::size_t k, double beta) {
double const f = 1/((k * M_PI) * (k * M_PI));
std::size_t const kpi2 = std::ceil(1/f);
double prod = 1.0;
double sum = 0.0;
std::size_t j;
for (j = 0; (j+2)*(j+1) <= kpi2; j += 2) {
prod *= -(beta - j) * (beta - j - 1) * f;
double const old_sum = sum;
sum += prod;
if (sum == old_sum) {
return sum;
}
}
prod *= -(beta - j) * (beta - j - 1) * f;
return sum + prod;
return sum;
}
class GzedCosineIntegralskPi {
// Constant object members
double alpha;
double deltax;
double sin_v; // std::sin(M_PI * alpha / 2)
double cos_v; // std::cos(M_PI * alpha / 2)
double sinc_v; // std::sinc(M_PI * alpha / 2)/alpha
double sinm1_v; // -sq(std::cos(M_PI * alpha / 2.0))
// / (1.0 + std::sin(M_PI * alpha / 2))
double tgamma_v; // std::tgamma(alpha + 1)
double tgammam1_v; // std::lgamma(alpha) - 1
double pi_alpha; // std::pow(M_PI, alpha)
double const_mult; // std::pow(deltax, -alpha) * alpha / M_PI;
// Constant class members
static constexpr size_t k_small = 10;
// Mathemathical constants
static constexpr double gamma_v = 0.577215664901532861;
static constexpr double pi_v = M_PI;
double mu_from_fit(size_t k);
double mu_from_assymptotics(size_t k);
public:
GzedCosineIntegralskPi(double alpha, double deltax);
double operator()(size_t k) {
if (k == 0) {
return std::pow(pi_v/deltax, alpha) / (alpha + 1.0);
} else if (k <= k_small) {
return mu_from_fit(k);
} else {
return mu_from_assymptotics(k) * const_mult;
}
}
double operator()(size_t k_1, size_t k_2);
private:
//
// Coefficients 1 to 10
//
static double coef_1(double const & ealpha) {
static double const P[] = {
-6.366197723675813440769e-01,
-2.059297839733384303484e-01,
-2.292961545633005969750e-02,
-8.681444663645901367220e-04,
-1.582316408178944320388e-09,
+3.918977599710780955764e-11};
static double const Q[] = {
1.000000000000000000000e+00,
1.147612567595240351553e+00,
4.749649121779834721790e-01,
9.072899514665789767032e-02,
8.127542932033228903687e-03,
2.763512269646208214064e-04};
double const x = ealpha - 1;
double numer{P[5]};
double denom{Q[5]};
for (int i = 4; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_2(double const & ealpha) {
static double const P[] = {
-7.759291740995763658089e-01,
-2.829217703086196658121e-01,
-5.637457780907778450443e-02,
-7.007412223701844205798e-03,
-6.193816376638685183221e-04,
-3.553519937251785659484e-05,
-1.367979103048156252422e-06};
static double const Q[] = {
+1.000000000000000000000e+00,
+7.075928393432178267543e-01,
+7.411465822590414278618e-02,
-1.733490210731306556679e-02,
-7.574067579592451475499e-04,
+2.249821654221352245262e-04,
-9.909862606418638557347e-06};
double const x = ealpha - 1;
double numer{P[6]};
double denom{Q[6]};
for (int i = 5; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_3(double const & ealpha) {
static double const P[] = {
-6.366197723675813431208e-01,
-1.181856265161497932998e-01,
-3.983716909337714229956e-01,
-1.563482404044122387383e-01,
-3.419095330183582834564e-02,
-4.674327221765267860909e-03,
-4.426622442597902427640e-04,
-2.677498711812969262154e-05,
-9.775269296293205906909e-07};
static double const Q[] = {
+1.000000000000000000000e+00,
+4.920020775703081555415e-01,
-6.167273791302243451617e-02,
-2.177844016897405044047e-02,
+3.972871056744789555343e-03,
+6.067168191454400087860e-05,
-6.673454339958298168532e-05,
+6.476978402921136039895e-06,
-2.218498388536791430567e-07};
double const x = ealpha - 1;
double numer{P[8]};
double denom{Q[8]};
for (int i = 7; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_4(double const & ealpha) {
static double const P[] = {
-9.913304792854260644113e-01L,
-3.604465765936957834959e-01,
-1.926408341999085247102e-01,
-6.676082389958020871953e-02,
-1.544232597594299740411e-02,
-2.380213741653500629811e-03,
-2.565051478564905242352e-04,
-1.792922349756083858332e-05,
-7.290004803716587430137e-07};
static double const Q[] = {
+1.000000000000000000000e+00L,
+3.982369752877753372544e-01,
-1.081539473708352530033e-01,
-1.619145220444328044554e-02,
+6.086186982755490323047e-03,
-2.790053161910219408667e-04,
-7.944363051877226811732e-05,
+1.192960335632360789574e-05,
-5.377198840244050502600e-07};
double const x = ealpha - 1;
double numer{P[8]};
double denom{Q[8]};
for (int i = 7; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_5(double const & ealpha) {
static double const P[] = {
-6.366197723675813431116e-01,
-1.132929113788398219819e-01,
-6.386856002702607426713e-01,
-2.319230172068818600686e-01,
-7.107160881892270445913e-02,
-1.683779180893674303425e-02,
-2.999191267001301728827e-03,
-3.661212171405373457705e-04,
-2.891219623822112043671e-05,
-1.196294595409810400987e-06};
static double const Q[] = {
+1.000000000000000000000e+00,
+2.322333051171824434270e-01,
-1.686247406724822322708e-01,
+5.034737292322581875708e-03,
+8.433450741235804247939e-03,
-1.533264413828341522702e-03,
+2.413414174372103081538e-06,
+2.988167850255876128913e-05,
-3.886356922263679390340e-06,
+1.775387177125740664526e-07};
double const x = ealpha - 1;
double numer{P[9]};
double denom{Q[9]};
for (int i = 8; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_6(double const & ealpha) {
static double const P[] = {
-1.119328559217149816456e+00,
-3.774027075730150242996e-01,
-3.036889963674641126427e-01,
-1.128760354600728122843e-01,
-2.851553552537801321030e-02,
-5.334526442559530124315e-03,
-7.815085890729471000657e-04,
-8.652175625909772587552e-05,
-6.617987967027415450828e-06,
-2.872208941784029013527e-07};
static double const Q[] = {
+1.000000000000000000000e+00,
+1.847105866560383586572e-01,
-1.839046179052694885812e-01,
+1.210987308706122274892e-02,
+8.950647649050410760246e-03,
-1.951263799975775258156e-03,
+3.535091652686901164834e-05,
+3.579811918968853779267e-05,
-5.045309440502321649547e-06,
+2.360850467196300692369e-07};
double const x = ealpha - 1;
double numer{P[9]};
double denom{Q[9]};
for (int i = 8; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_7(double const & ealpha) {
static double const P[] = {
-6.366197723675813433915e-01,
-1.313982442799666563455e-01,
-8.192210713297660515892e-01,
-3.093250531212002864565e-01,
-1.161930635137453542677e-01,
-3.267468318586470084177e-02,
-6.417603679860046273259e-03,
-8.619419248285984557777e-04,
-7.474804962830469146716e-05,
-3.718248126938508891003e-06};
static double const Q[] = {
+1.000000000000000000000e+00,
+9.339602775251610107010e-02,
-2.069999684787325933570e-01,
+2.852588834259701921674e-02,
+9.135575279360643552115e-03,
-2.974015018870659194391e-03,
+1.660624005887600592604e-04,
+4.920957378633904081284e-05,
-9.358564500451040259930e-06,
+5.313222025292598314613e-07};
double const x = ealpha - 1;
double numer{P[9]};
double denom{Q[9]};
for (int i = 8; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_8(double const & ealpha) {
static double const P[] = {
-1.210518378665849886399e+00,
-4.705979536760370495798e-01,
-4.318731991300256296135e-01,
-1.795611320887010742946e-01,
-5.165456879556205502871e-02,
-1.159728053603521662630e-02,
-2.002804431097856561209e-03,
-2.602712585795104360641e-04,
-2.262287494211915644076e-05,
-1.165203046489235364453e-06};
static double const Q[] = {
+1.000000000000000000000e+00,
+1.014995029616300718625e-01,
-2.094113759867870555031e-01,
+2.681273948915745606830e-02,
+9.972781039797130578565e-03,
-3.036908541967370299212e-03,
+1.348674477897301571139e-04,
+5.749085315677258350305e-05,
-1.011569660649543222883e-05,
+5.512116802699264269481e-07,
};
double const x = ealpha - 1;
double numer{P[9]};
double denom{Q[9]};
for (int i = 8; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_9(double const & ealpha) {
static double const P[] = {
-6.366197723675813442203e-01,
-1.570064393190070275588e-01,
-9.657821889034147624603e-01,
-3.943420774033056905967e-01,
-1.656581796412698690422e-01,
-5.251279224546032568510e-02,
-1.098591490399947662539e-02,
-1.640883122409694358546e-03,
-1.584710416747311185655e-04,
-9.808245277913587130547e-06};
static double const Q[] = {
+1.000000000000000000000e+00,
+8.364767804663631321282e-03,
-2.282992755732345746752e-01,
+4.688523114789032207492e-02,
+9.550033789972392915305e-03,
-4.511312915271598335823e-03,
+3.713301944162669106982e-04,
+8.044591510960301968304e-05,
-1.939912103338426163548e-05,
+1.281282482064112927509e-06
};
double const x = ealpha - 1;
double numer{P[9]};
double denom{Q[9]};
for (int i = 8; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
static double coef_10(double const & ealpha) {
static double const P[] = {
-1.281368483991673320070e+00,
-5.806619792384043326929e-01,
-5.545544036957355009114e-01,
-2.581853378548760646070e-01,
-8.045577459119633669278e-02,
-2.044372857644134313770e-02,
-3.882942557215586938672e-03,
-5.598000213437109457659e-04,
-5.290328388126457794829e-05,
-3.101987499652022207225e-06};
static double const Q[] = {
+1.000000000000000000000e+00,
+6.042724705199052308765e-02,
-2.314978046932805685846e-01,
+3.603456480649801602474e-02,
+1.254058490890153216257e-02,
-4.381848103520755694525e-03,
+2.075652415696184525883e-04,
+1.041056590982561133958e-04,
-1.991846677612194718337e-05,
+1.183243311078138411900e-06};
double const x = ealpha - 1;
double numer{P[9]};
double denom{Q[9]};
for (int i = 8; i >= 0; --i) {
numer = (numer * x) + P[i];
denom = (denom * x) + Q[i];
}
return numer/denom;
}
};
GzedCosineIntegralskPi::GzedCosineIntegralskPi(double alpha, double deltax)
: alpha(alpha),
deltax(deltax)
{ // NOLINT
if (alpha == 0.0) {
sin_v = 0.0;
cos_v = 1.0;
sinc_v = pi_v / 2.0;
sinm1_v = -1.0;
tgamma_v = 1.0; // gamma(alpha + 1) = 0!
// gamma(0) is not defined
pi_alpha = 1.0; // pi^0 = 1
} else if (alpha == 1.0) {
sin_v = 1.0;
cos_v = 0.0;
sinc_v = 1.0;
sinm1_v = 0.0;
tgamma_v = 1.0; // gamma(alpha + 1) = 1!
tgammam1_v = 0.0; // gamma(alpha) - 1 = 0! - 1
pi_alpha = pi_v;
} else if (alpha == 2.0) {
sin_v = 0.0;
cos_v = -1.0;
sinc_v = 0.0;
sinm1_v = -1.0;
tgamma_v = 2.0; // gamma(alpha + 1) = 2!
tgammam1_v = 0.0; // gamma(alpha) - 1 = 0! - 1
pi_alpha = pi_v * pi_v;
} else {
sin_v = std::sin(pi_v * alpha / 2);
cos_v = std::cos(pi_v * alpha / 2);
sinc_v = sin_v / alpha;
sinm1_v = -cos_v*cos_v/(1.0 + sin_v);
tgamma_v = std::tgamma(alpha + 1);
tgammam1_v = std::expm1(std::lgamma(alpha)); // gamma(alpha) - 1
pi_alpha = std::pow(pi_v, alpha);
}
const_mult = std::pow(deltax, -alpha) * alpha / M_PI;
}
double GzedCosineIntegralskPi::mu_from_assymptotics(std::size_t k) {
double sgn = (k % 2 == 0) ? -1. : 1.;
if (alpha < 1e-18) {
return -(pi_v/2
+ sgn
* (1 + Y(k, -1))/(pi_v*k)
)
* (k % 2 == 0 ? (1.0 - alpha) : 1.0)
/ k;
} else if (alpha == 1 && k % 2 == 0) {
return 0.0;
} else if (alpha == 1) {
return -2.0/(k*k);
} else if (k % 2 != 0 || alpha < 0.5) {
double E_1 = std::pow(k, 1.0 - alpha)
* tgamma_v
* sinc_v;
double E_2 = (1.0 + Y(k, alpha - 1.0))
* (pi_alpha/pi_v);
return -(E_1 + sgn * E_2)/(k*k);
} else {
double E_1 = tgammam1_v * sin_v;
double E_2 = sinm1_v;
double E_3 = -std::expm1((alpha - 1.0) * std::log(k * M_PI));
double E_4 = -Y(k, alpha - 1.0) * pi_alpha /(pi_v*k);
double E_5 = (E_1 + E_2 + E_3) * std::pow(k, -alpha);
return -(E_4 + E_5) / k;
}
}
double GzedCosineIntegralskPi::mu_from_fit(std::size_t k) {
static double (*coef[])(double const &) = {
coef_1, coef_2, coef_3, coef_4, coef_5,
coef_6, coef_7, coef_8, coef_9, coef_10};
double const I_c = coef[k-1](alpha)
* pi_alpha
* std::pow(k*deltax, -alpha)
/ (pi_v*k);
if (k % 2 == 0) {
return alpha * (1.0 - alpha) * I_c;
} else {
return alpha * I_c;
}
}