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example010_hypergeometric_2f1.cpp
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example010_hypergeometric_2f1.cpp
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///////////////////////////////////////////////////////////////////
// Copyright Christopher Kormanyos 2018 - 2022. //
// Distributed under the Boost Software License, //
// Version 1.0. (See accompanying file LICENSE_1_0.txt //
// or copy at http://www.boost.org/LICENSE_1_0.txt) //
///////////////////////////////////////////////////////////////////
#include <algorithm>
#include <array>
#include <cstdint>
#include <numeric>
#include <examples/example_decwide_t.h>
#include <math/wide_decimal/decwide_t.h>
namespace example010_hypergeometric
{
template<typename T>
auto hypergeometric_2f1(const T& AP, const T& BP, const T& CP, const T& ZM) -> T // NOLINT(readability-identifier-naming,bugprone-easily-swappable-parameters)
{
// Implement a rational approximation of hypergeometric_2f1.
// This C++11 code uses a computational scheme similar to
// one shown in Y.L. Luke, "Algorithms for the Comuptation
// of Mathematical Functions", Academic Press, New York, 1977.
// The original Fortran77 source can be found in the subroutine
// R2F1() on pages 167-168. Slight corrections are made here
// in order to repair one or two type-setting errors in the
// original publication.
// Luke's original Fortran77 programs and algorithmic work are fascinating
// since they are generic and scalable --- decades ahead of their time.
// Here, in fact, the calculation is scaled to 1,001 decimal digits
// of precision.
// Retain some stylistic elements and comments from Luke's
// original Fortran77 coding style.
const T Z(-ZM); // NOLINT(readability-identifier-naming)
const T my_zero(0U);
const T my_one (1U);
// C
// C INITIALIZATION :
// C
T SABZ((AP + BP) * Z); // NOLINT(readability-identifier-naming)
const T AB (AP * BP); // NOLINT(readability-identifier-naming)
const T ABZ (AB * Z); // NOLINT(readability-identifier-naming)
const T ABZ1((Z + ABZ) + SABZ); // NOLINT(readability-identifier-naming)
const T ABZ2((ABZ1 + SABZ) + (3U * Z)); // NOLINT(readability-identifier-naming)
std::array<T, 4U> A; // NOLINT(readability-identifier-naming)
std::array<T, 4U> B; // NOLINT(readability-identifier-naming)
B[0U] = my_one;
A[0U] = my_one;
const T CP1(CP + my_one); // NOLINT(readability-identifier-naming)
B[1U] = my_one + (ABZ1 / (CP * 2U));
A[1U] = B[1U] - (ABZ / CP);
const T CT1(2U * CP1); // NOLINT(readability-identifier-naming)
B[2U] = my_one + ((ABZ2 / CT1) * (my_one + ABZ1 / ((-T(6U)) + (CT1 * 3U)))); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
A[2U] = B[2U] - ((ABZ / CP) * (my_one + ((ABZ2 - ABZ1) / CT1)));
SABZ /= 4U;
const T Z2(Z / 2U); // NOLINT(readability-identifier-naming)
std::array<T, 9U> D; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers,readability-identifier-naming)
D[0U] = (((T(7U) / 2U) - AB) * Z2) - SABZ; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[1U] = ABZ1 / 4U; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[2U] = D[1U] - (SABZ * 2U); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[3U] = CP1 + my_one; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[4U] = CP1 * D[3U]; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[5U] = CP * D[4U]; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[6U] = T(3U) / 2U; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[7U] = T(3U) / 4U; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[8U] = D[7U] * Z; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
std::array<T, 3U> G; // NOLINT(readability-identifier-naming)
// C
// C FOR I=3,...,N , THE VALUES A(I) AND B(I) ARE CALCULATED
// C USING THE RECURRENCE RELATIONS BELOW.
// C
for(std::uint_fast16_t XI = UINT16_C(3); XI < UINT16_C(10000); ++XI) // NOLINT(readability-identifier-naming)
{
G[2U] = (D[2U] * D[1U]) / (D[7U] * D[5U]); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[1U] += (D[8U] + SABZ); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[2U] += (D[8U] - SABZ); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
G[2U] *= (D[1U] / D[6U]); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
G[0U] = my_one + ((D[1U] + D[0U]) / (D[6U] * D[3U])); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
G[1U] = D[1U] / (D[4U] * D[6U]); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[7U] += (D[6U] * 2U); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
++D[6U]; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
G[1U] *= ((CP1 - XI) - ((D[2U] + D[0U]) / D[6U])); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
// C -----------------------------------------------------------------
// C THE RECURRENCE RELATIONS FOR A(I) and B(I) ARE AS FOLLOWS
// C -----------------------------------------------------------------
A[3U] = std::inner_product(G.crbegin(), G.crend(), A.cbegin(), my_zero);
B[3U] = std::inner_product(G.crbegin(), G.crend(), B.cbegin(), my_zero);
// Check if the iteration difference (delta) is within
// tolerance and break from the recursion if it is.
// Here we analyze the difference between this iteration
// result and the previous iteration result using:
//
// | |(A2/B2)]| |
// delta = | 1 - |--------| |
// | |(A3/B3) | |
//
// | |(A2*B3)| |
// = | 1 - |-------| |,
// | |(A3*B2)| |
//
// where the absolute value of the ratio in the second term
// is used and the absolute value of delta is used.
using std::fabs;
const T ratio = fabs((A[2U] * B[3U]) / (A[3U] * B[2U]));
const T delta = fabs(my_one - ratio);
if((XI > UINT16_C(7)) && (delta < std::numeric_limits<T>::epsilon()))
{
break;
}
// Shift the arrays for the next recursion.
std::copy(A.cbegin() + 1U, A.cend(), A.begin());
std::copy(B.cbegin() + 1U, B.cend(), B.begin());
D[8U] += Z2; // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[0U] += (D[8U] * 2U); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[5U] += (D[4U] * 3U); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
D[4U] += (D[3U] * 2U); // NOLINT(cppcoreguidelines-avoid-magic-numbers,readability-magic-numbers)
++D[3U];
}
// Return the rational approximation. This is given
// by the ratio of the final recursions of A and B.
return A.back() / B.back();
}
} // namespace example010_hypergeometric
#if defined(WIDE_DECIMAL_NAMESPACE)
auto WIDE_DECIMAL_NAMESPACE::math::wide_decimal::example010_hypergeometric_2f1() -> bool
#else
auto ::math::wide_decimal::example010_hypergeometric_2f1() -> bool
#endif
{
#if defined(WIDE_DECIMAL_NAMESPACE)
using dec1001_t = WIDE_DECIMAL_NAMESPACE::math::wide_decimal::decwide_t<static_cast<std::int32_t>(INT32_C(1001))>;
#else
using dec1001_t = ::math::wide_decimal::decwide_t<static_cast<std::int32_t>(INT32_C(1001))>;
#endif
const dec1001_t a( dec1001_t(2U) / 3U);
const dec1001_t b( dec1001_t(4U) / 3U);
const dec1001_t c( dec1001_t(5U) / 7U);
const dec1001_t z(-dec1001_t(3U) / 4U);
const dec1001_t h2f1 = example010_hypergeometric::hypergeometric_2f1(a, b, c, z);
// N[Hypergeometric2F1[2/3, 4/3, 5/7, -3/4], 1003]
const dec1001_t control
{
"0."
"5010047360876106403820298707781130663700964458315280453481414661084883532272790762746959490255023246"
"0312226542605539888036608718209715889951504568390329053796255363287603231434476268734389439044422167"
"5284089490545688340381318820728928965614986405528607006166515996158859902644742923187058504872539864"
"5526393970602257621956446639141677360052376900435972099485022668952455616438857995487419477325979436"
"0624373454823168201440969762998535751649093599185576558711187071798391688855988414416119107109432755"
"7650459245296569373944014705113399144656127270454166592212540118941954333081250095816519510598056279"
"9701330083670272177694982980992777101459836814865935861581023713422564394525179768543448930391133399"
"2365965485619664248076371405092957089253100001866455395914231855187336610490908971056693326326421884"
"7958097660733256147292859976308158801627280967329287950929961899242779950084734217898887067896870114"
"6800382971696052755503110505432490729940028678881070548295642016866992748303933000057149684544171490"
"241"
};
using std::fabs;
const dec1001_t closeness = fabs(1 - (h2f1 / control));
const auto result_is_ok = (closeness < (std::numeric_limits<dec1001_t>::epsilon() * static_cast<std::uint32_t>(UINT8_C(10))));
return result_is_ok;
}
// Enable this if you would like to activate this main() as a standalone example.
#if defined(WIDE_DECIMAL_STANDALONE_EXAMPLE010_HYPERGEOMETRIC_2F1)
#include <iomanip>
#include <iostream>
auto main() -> int
{
const auto result_is_ok = ::math::wide_decimal::example010_hypergeometric_2f1();
std::cout << "result_is_ok: " << std::boolalpha << result_is_ok << std::endl;
}
#endif