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/*
* Copyright Nick Thompson, 2021
* 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 "math_unit_test.hpp"
#include <boost/math/tools/cubic_roots.hpp>
#include <random>
#include <cmath>
#include <cfloat>
#ifdef BOOST_HAS_FLOAT128
#include <boost/multiprecision/float128.hpp>
using boost::multiprecision::float128;
#endif
using boost::math::tools::cubic_root_condition_number;
using boost::math::tools::cubic_root_residual;
using boost::math::tools::cubic_roots;
using std::cbrt;
using std::abs;
template <class Real> void test_zero_coefficients() {
Real a = 0;
Real b = 0;
Real c = 0;
Real d = 0;
auto roots = cubic_roots(a, b, c, d);
CHECK_EQUAL(roots[0], Real(0));
CHECK_EQUAL(roots[1], Real(0));
CHECK_EQUAL(roots[2], Real(0));
a = 1;
roots = cubic_roots(a, b, c, d);
CHECK_EQUAL(roots[0], Real(0));
CHECK_EQUAL(roots[1], Real(0));
CHECK_EQUAL(roots[2], Real(0));
a = 1;
d = 1;
// x^3 + 1 = 0:
roots = cubic_roots(a, b, c, d);
CHECK_EQUAL(roots[0], Real(-1));
CHECK_NAN(roots[1]);
CHECK_NAN(roots[2]);
d = -1;
// x^3 - 1 = 0:
roots = cubic_roots(a, b, c, d);
CHECK_EQUAL(roots[0], Real(1));
CHECK_NAN(roots[1]);
CHECK_NAN(roots[2]);
d = -2;
// x^3 - 2 = 0
roots = cubic_roots(a, b, c, d);
CHECK_ULP_CLOSE(roots[0], cbrt(Real(2)), 2);
CHECK_NAN(roots[1]);
CHECK_NAN(roots[2]);
d = -8;
roots = cubic_roots(a, b, c, d);
CHECK_EQUAL(roots[0], Real(2));
CHECK_NAN(roots[1]);
CHECK_NAN(roots[2]);
// (x-1)(x-2)(x-3) = x^3 - 6x^2 + 11x - 6
roots = cubic_roots(Real(1), Real(-6), Real(11), Real(-6));
CHECK_ULP_CLOSE(roots[0], Real(1), 2);
CHECK_ULP_CLOSE(roots[1], Real(2), 2);
CHECK_ULP_CLOSE(roots[2], Real(3), 2);
// Double root:
// (x+1)^2(x-2) = x^3 - 3x - 2:
// Note: This test is unstable wrt to perturbations!
roots = cubic_roots(Real(1), Real(0), Real(-3), Real(-2));
CHECK_ULP_CLOSE(Real(-1), roots[0], 2);
CHECK_ULP_CLOSE(Real(-1), roots[1], 2);
CHECK_ULP_CLOSE(Real(2), roots[2], 2);
std::uniform_real_distribution<Real> dis(-2, 2);
std::mt19937 gen(12345);
// Expected roots
std::array<Real, 3> r;
int trials = 10;
for (int i = 0; i < trials; ++i) {
// Mathematica:
// Expand[(x - r0)*(x - r1)*(x - r2)]
// - r0 r1 r2 + (r0 r1 + r0 r2 + r1 r2) x
// - (r0 + r1 + r2) x^2 + x^3
for (auto &root : r) {
root = static_cast<Real>(dis(gen));
}
std::sort(r.begin(), r.end());
Real a = 1;
Real b = -(r[0] + r[1] + r[2]);
Real c = r[0] * r[1] + r[0] * r[2] + r[1] * r[2];
Real d = -r[0] * r[1] * r[2];
auto roots = cubic_roots(a, b, c, d);
// I could check the condition number here, but this is fine right?
if (!CHECK_ULP_CLOSE(r[0], roots[0], (std::numeric_limits<Real>::digits > 100 ? 120 : 60))) {
std::cerr << " Polynomial x^3 + " << b << "x^2 + " << c << "x + "
<< d << " has roots {";
std::cerr << r[0] << ", " << r[1] << ", " << r[2]
<< "}, but the computed roots are {";
std::cerr << roots[0] << ", " << roots[1] << ", " << roots[2]
<< "}\n";
}
CHECK_ULP_CLOSE(r[1], roots[1], 80);
CHECK_ULP_CLOSE(r[2], roots[2], (std::numeric_limits<Real>::digits > 100 ? 120 : 80));
for (auto root : roots) {
auto res = cubic_root_residual(a, b, c, d, root);
CHECK_LE(abs(res[0]), res[1]);
}
}
}
void test_ill_conditioned() {
// An ill-conditioned root reported by SATovstun:
// "Exact" roots produced with a high-precision calculation on Wolfram Alpha:
// NSolve[x^3 + 10000*x^2 + 200*x +1==0,x]
std::array<double, 3> expected_roots{-9999.97999997,
-0.010010015026300100757327057,
-0.009990014973799899662674923};
auto roots = cubic_roots<double>(1, 10000, 200, 1);
CHECK_ABSOLUTE_ERROR(expected_roots[0], roots[0],
std::numeric_limits<double>::epsilon());
if (!(boost::math::isnan)(roots[1]))
{
// This test is so ill-conditioned, that we can't always get here.
// Test case is Clang C++20 mode on MacOS Arm. Best guess is that
// fma is behaving differently there...
CHECK_ABSOLUTE_ERROR(expected_roots[1], roots[1], 1.01e-5);
CHECK_ABSOLUTE_ERROR(expected_roots[2], roots[2], 1.01e-5);
double cond =
cubic_root_condition_number<double>(1, 10000, 200, 1, roots[1]);
double r1 = expected_roots[1];
// The factor of 10 is a fudge factor to make the test pass.
// Nonetheless, it does show this is basically correct:
CHECK_LE(abs(r1 - roots[1]) / abs(r1),
10 * std::numeric_limits<double>::epsilon() * cond);
cond = cubic_root_condition_number<double>(1, 10000, 200, 1, roots[2]);
double r2 = expected_roots[2];
// The factor of 10 is a fudge factor to make the test pass.
// Nonetheless, it does show this is basically correct:
CHECK_LE(abs(r2 - roots[2]) / abs(r2),
10 * std::numeric_limits<double>::epsilon() * cond);
}
else
{
CHECK_NAN(roots[2]);
}
// See https://github.com/boostorg/math/issues/757:
// The polynomial is ((x+1)^2+1)*(x+1) which has roots -1, and two complex
// roots:
roots = cubic_roots<double>(1, 3, 4, 2);
CHECK_ULP_CLOSE(roots[0], -1.0, 3);
CHECK_NAN(roots[1]);
CHECK_NAN(roots[2]);
return;
}
int main() {
test_zero_coefficients<float>();
test_zero_coefficients<double>();
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
test_zero_coefficients<long double>();
#endif
test_ill_conditioned();
#ifdef BOOST_HAS_FLOAT128
// For some reason, the quadmath is way less accurate than the
// float/double/long double:
// test_zero_coefficients<float128>();
#endif
return boost::math::test::report_errors();
}
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