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// (C) Copyright John Maddock 2015.
// Use, modification and distribution are subject to 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 "pch.hpp"
#define BOOST_TEST_MAIN
#include <boost/test/unit_test.hpp>
#include <boost/test/tools/floating_point_comparison.hpp>
#include <boost/math/tools/roots.hpp>
#include <boost/test/results_collector.hpp>
#include <boost/test/unit_test.hpp>
#include <boost/math/special_functions/cbrt.hpp>
#include <boost/math/special_functions/beta.hpp>
#include <iostream>
#include <iomanip>
#include <tuple>
#include "table_type.hpp"
// No derivatives - using TOMS748 internally.
struct cbrt_functor_noderiv
{ // cube root of x using only function - no derivatives.
cbrt_functor_noderiv(double to_find_root_of) : a(to_find_root_of)
{ // Constructor just stores value a to find root of.
}
double operator()(double x)
{
double fx = x*x*x - a; // Difference (estimate x^3 - a).
return fx;
}
private:
double a; // to be 'cube_rooted'.
}; // template <class T> struct cbrt_functor_noderiv
// Using 1st derivative only Newton-Raphson
struct cbrt_functor_deriv
{ // Functor also returning 1st derivative.
cbrt_functor_deriv(double const& to_find_root_of) : a(to_find_root_of)
{ // Constructor stores value a to find root of,
// for example: calling cbrt_functor_deriv<double>(x) to use to get cube root of x.
}
std::pair<double, double> operator()(double const& x)
{ // Return both f(x) and f'(x).
double fx = x*x*x - a; // Difference (estimate x^3 - value).
double dx = 3 * x*x; // 1st derivative = 3x^2.
return std::make_pair(fx, dx); // 'return' both fx and dx.
}
private:
double a; // to be 'cube_rooted'.
};
// Using 1st and 2nd derivatives with Halley algorithm.
struct cbrt_functor_2deriv
{ // Functor returning both 1st and 2nd derivatives.
cbrt_functor_2deriv(double const& to_find_root_of) : a(to_find_root_of)
{ // Constructor stores value a to find root of, for example:
// calling cbrt_functor_2deriv<double>(x) to get cube root of x,
}
std::tuple<double, double, double> operator()(double const& x)
{ // Return both f(x) and f'(x) and f''(x).
double fx = x*x*x - a; // Difference (estimate x^3 - value).
double dx = 3 * x*x; // 1st derivative = 3x^2.
double d2x = 6 * x; // 2nd derivative = 6x.
return std::make_tuple(fx, dx, d2x); // 'return' fx, dx and d2x.
}
private:
double a; // to be 'cube_rooted'.
};
template <class T, class Policy>
struct ibeta_roots_1 // for first order algorithms
{
ibeta_roots_1(T _a, T _b, T t, bool inv = false)
: a(_a), b(_b), target(t), invert(inv) {}
T operator()(const T& x)
{
return boost::math::detail::ibeta_imp(a, b, x, Policy(), invert, true) - target;
}
private:
T a, b, target;
bool invert;
};
template <class T, class Policy>
struct ibeta_roots_2 // for second order algorithms
{
ibeta_roots_2(T _a, T _b, T t, bool inv = false)
: a(_a), b(_b), target(t), invert(inv) {}
boost::math::tuple<T, T> operator()(const T& x)
{
typedef boost::math::lanczos::lanczos<T, Policy> S;
typedef typename S::type L;
T f = boost::math::detail::ibeta_imp(a, b, x, Policy(), invert, true) - target;
T f1 = invert ?
-boost::math::detail::ibeta_power_terms(b, a, 1 - x, x, L(), true, Policy())
: boost::math::detail::ibeta_power_terms(a, b, x, 1 - x, L(), true, Policy());
T y = 1 - x;
if (y == 0)
y = boost::math::tools::min_value<T>() * 8;
f1 /= y * x;
// make sure we don't have a zero derivative:
if (f1 == 0)
f1 = (invert ? -1 : 1) * boost::math::tools::min_value<T>() * 64;
return boost::math::make_tuple(f, f1);
}
private:
T a, b, target;
bool invert;
};
template <class T, class Policy>
struct ibeta_roots_3 // for third order algorithms
{
ibeta_roots_3(T _a, T _b, T t, bool inv = false)
: a(_a), b(_b), target(t), invert(inv) {}
boost::math::tuple<T, T, T> operator()(const T& x)
{
typedef typename boost::math::lanczos::lanczos<T, Policy>::type L;
T f = boost::math::detail::ibeta_imp(a, b, x, Policy(), invert, true) - target;
T f1 = invert ?
-boost::math::detail::ibeta_power_terms(b, a, 1 - x, x, L(), true, Policy())
: boost::math::detail::ibeta_power_terms(a, b, x, 1 - x, L(), true, Policy());
T y = 1 - x;
if (y == 0)
y = boost::math::tools::min_value<T>() * 8;
f1 /= y * x;
T f2 = f1 * (-y * a + (b - 2) * x + 1) / (y * x);
if (invert)
f2 = -f2;
// make sure we don't have a zero derivative:
if (f1 == 0)
f1 = (invert ? -1 : 1) * boost::math::tools::min_value<T>() * 64;
return boost::math::make_tuple(f, f1, f2);
}
private:
T a, b, target;
bool invert;
};
BOOST_AUTO_TEST_CASE( test_main )
{
int newton_limits = static_cast<int>(std::numeric_limits<double>::digits * 0.6);
double arg = 1e-50;
std::uintmax_t iters;
double guess;
double dr;
while(arg < 1e50)
{
double result = boost::math::cbrt(arg);
//
// Start with a really bad guess 5 times below the result:
//
guess = result / 5;
iters = 1000;
// TOMS algo first:
std::pair<double, double> r = boost::math::tools::bracket_and_solve_root(cbrt_functor_noderiv(arg), guess, 2.0, true, boost::math::tools::eps_tolerance<double>(), iters);
BOOST_CHECK_CLOSE_FRACTION((r.first + r.second) / 2, result, std::numeric_limits<double>::epsilon() * 4);
BOOST_CHECK_LE(iters, 14);
// Newton next:
iters = 1000;
dr = boost::math::tools::newton_raphson_iterate(cbrt_functor_deriv(arg), guess, guess / 2, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 12);
// Halley next:
iters = 1000;
dr = boost::math::tools::halley_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 7);
// Schroder next:
iters = 1000;
dr = boost::math::tools::schroder_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 11);
//
// Over again with a bad guess 5 times larger than the result:
//
iters = 1000;
guess = result * 5;
r = boost::math::tools::bracket_and_solve_root(cbrt_functor_noderiv(arg), guess, 2.0, true, boost::math::tools::eps_tolerance<double>(), iters);
BOOST_CHECK_CLOSE_FRACTION((r.first + r.second) / 2, result, std::numeric_limits<double>::epsilon() * 4);
BOOST_CHECK_LE(iters, 14);
// Newton next:
iters = 1000;
dr = boost::math::tools::newton_raphson_iterate(cbrt_functor_deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 12);
// Halley next:
iters = 1000;
dr = boost::math::tools::halley_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 7);
// Schroder next:
iters = 1000;
dr = boost::math::tools::schroder_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 11);
//
// A much better guess, 1% below result:
//
iters = 1000;
guess = result * 0.9;
r = boost::math::tools::bracket_and_solve_root(cbrt_functor_noderiv(arg), guess, 2.0, true, boost::math::tools::eps_tolerance<double>(), iters);
BOOST_CHECK_CLOSE_FRACTION((r.first + r.second) / 2, result, std::numeric_limits<double>::epsilon() * 4);
BOOST_CHECK_LE(iters, 12);
// Newton next:
iters = 1000;
dr = boost::math::tools::newton_raphson_iterate(cbrt_functor_deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 5);
// Halley next:
iters = 1000;
dr = boost::math::tools::halley_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 3);
// Schroder next:
iters = 1000;
dr = boost::math::tools::schroder_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 4);
//
// A much better guess, 1% above result:
//
iters = 1000;
guess = result * 1.1;
r = boost::math::tools::bracket_and_solve_root(cbrt_functor_noderiv(arg), guess, 2.0, true, boost::math::tools::eps_tolerance<double>(), iters);
BOOST_CHECK_CLOSE_FRACTION((r.first + r.second) / 2, result, std::numeric_limits<double>::epsilon() * 4);
BOOST_CHECK_LE(iters, 12);
// Newton next:
iters = 1000;
dr = boost::math::tools::newton_raphson_iterate(cbrt_functor_deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 5);
// Halley next:
iters = 1000;
dr = boost::math::tools::halley_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 3);
// Schroder next:
iters = 1000;
dr = boost::math::tools::schroder_iterate(cbrt_functor_2deriv(arg), guess, result / 10, result * 10, newton_limits, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 2);
BOOST_CHECK_LE(iters, 4);
arg *= 3.5;
}
//
// Test ibeta as this triggers all the pathological cases!
//
#ifndef SC_
#define SC_(x) x
#endif
#define T double
# include "ibeta_small_data.ipp"
for (unsigned i = 0; i < ibeta_small_data.size(); ++i)
{
//
// These inverse tests are thrown off if the output of the
// incomplete beta is too close to 1: basically there is insuffient
// information left in the value we're using as input to the inverse
// to be able to get back to the original value.
//
if (ibeta_small_data[i][5] == 0)
{
iters = 1000;
dr = boost::math::tools::newton_raphson_iterate(ibeta_roots_2<double, boost::math::policies::policy<> >(ibeta_small_data[i][0], ibeta_small_data[i][1], ibeta_small_data[i][5]), 0.5, 0.0, 1.0, 53, iters);
BOOST_CHECK_EQUAL(dr, 0.0);
BOOST_CHECK_LE(iters, 27);
iters = 1000;
dr = boost::math::tools::halley_iterate(ibeta_roots_3<double, boost::math::policies::policy<> >(ibeta_small_data[i][0], ibeta_small_data[i][1], ibeta_small_data[i][5]), 0.5, 0.0, 1.0, 53, iters);
BOOST_CHECK_EQUAL(dr, 0.0);
BOOST_CHECK_LE(iters, 10);
}
else if ((1 - ibeta_small_data[i][5] > 0.001)
&& (fabs(ibeta_small_data[i][5]) > 2 * boost::math::tools::min_value<double>()))
{
iters = 1000;
double result = ibeta_small_data[i][2];
dr = boost::math::tools::newton_raphson_iterate(ibeta_roots_2<double, boost::math::policies::policy<> >(ibeta_small_data[i][0], ibeta_small_data[i][1], ibeta_small_data[i][5]), 0.5, 0.0, 1.0, 53, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 200);
#if defined(BOOST_MSVC) && (BOOST_MSVC == 1600)
BOOST_CHECK_LE(iters, 40);
#else
BOOST_CHECK_LE(iters, 27);
#endif
iters = 1000;
result = ibeta_small_data[i][2];
dr = boost::math::tools::halley_iterate(ibeta_roots_3<double, boost::math::policies::policy<> >(ibeta_small_data[i][0], ibeta_small_data[i][1], ibeta_small_data[i][5]), 0.5, 0.0, 1.0, 53, iters);
BOOST_CHECK_CLOSE_FRACTION(dr, result, std::numeric_limits<double>::epsilon() * 200);
#if defined(__PPC__) || defined(__aarch64__) || (LDBL_MANT_DIG > 100)
BOOST_CHECK_LE(iters, 55);
#else
BOOST_CHECK_LE(iters, 40);
#endif
}
else if (1 == ibeta_small_data[i][5])
{
iters = 1000;
dr = boost::math::tools::newton_raphson_iterate(ibeta_roots_2<double, boost::math::policies::policy<> >(ibeta_small_data[i][0], ibeta_small_data[i][1], ibeta_small_data[i][5]), 0.5, 0.0, 1.0, 53, iters);
BOOST_CHECK_EQUAL(dr, 1.0);
BOOST_CHECK_LE(iters, 27);
iters = 1000;
dr = boost::math::tools::halley_iterate(ibeta_roots_3<double, boost::math::policies::policy<> >(ibeta_small_data[i][0], ibeta_small_data[i][1], ibeta_small_data[i][5]), 0.5, 0.0, 1.0, 53, iters);
BOOST_CHECK_EQUAL(dr, 1.0);
BOOST_CHECK_LE(iters, 10);
}
}
}
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