--- /dev/null
+// (C) 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)
+#ifndef BOOST_MATH_TOOLS_QUARTIC_ROOTS_HPP
+#define BOOST_MATH_TOOLS_QUARTIC_ROOTS_HPP
+#include <array>
+#include <cmath>
+#include <boost/math/tools/cubic_roots.hpp>
+
+namespace boost::math::tools {
+
+namespace detail {
+
+// Make sure the nans are always at the back of the array:
+template<typename Real>
+bool comparator(Real r1, Real r2) {
+ using std::isnan;
+ if (isnan(r1)) { return false; }
+ if (isnan(r2)) { return true; }
+ return r1 < r2;
+}
+
+template<typename Real>
+std::array<Real, 4> polish_and_sort(Real a, Real b, Real c, Real d, Real e, std::array<Real, 4>& roots) {
+ // Polish the roots with a Halley iterate.
+ using std::fma;
+ using std::abs;
+ for (auto &r : roots) {
+ Real df = fma(4*a, r, 3*b);
+ df = fma(df, r, 2*c);
+ df = fma(df, r, d);
+ Real d2f = fma(12*a, r, 6*b);
+ d2f = fma(d2f, r, 2*c);
+ Real f = fma(a, r, b);
+ f = fma(f,r,c);
+ f = fma(f,r,d);
+ f = fma(f,r,e);
+ Real denom = 2*df*df - f*d2f;
+ if (abs(denom) > (std::numeric_limits<Real>::min)())
+ {
+ r -= 2*f*df/denom;
+ }
+ }
+ std::sort(roots.begin(), roots.end(), detail::comparator<Real>);
+ return roots;
+}
+
+}
+// Solves ax^4 + bx^3 + cx^2 + dx + e = 0.
+// Only returns the real roots, as these are the only roots of interest in ray intersection problems.
+// Follows Graphics Gems V: https://github.com/erich666/GraphicsGems/blob/master/gems/Roots3And4.c
+template<typename Real>
+std::array<Real, 4> quartic_roots(Real a, Real b, Real c, Real d, Real e) {
+ using std::abs;
+ using std::sqrt;
+ auto nan = std::numeric_limits<Real>::quiet_NaN();
+ std::array<Real, 4> roots{nan, nan, nan, nan};
+ if (abs(a) <= (std::numeric_limits<Real>::min)()) {
+ auto cbrts = cubic_roots(b, c, d, e);
+ roots[0] = cbrts[0];
+ roots[1] = cbrts[1];
+ roots[2] = cbrts[2];
+ if (b == 0 && c == 0 && d == 0 && e == 0) {
+ roots[3] = 0;
+ }
+ return detail::polish_and_sort(a, b, c, d, e, roots);
+ }
+ if (abs(e) <= (std::numeric_limits<Real>::min)()) {
+ auto v = cubic_roots(a, b, c, d);
+ roots[0] = v[0];
+ roots[1] = v[1];
+ roots[2] = v[2];
+ roots[3] = 0;
+ return detail::polish_and_sort(a, b, c, d, e, roots);
+ }
+ // Now solve x^4 + Ax^3 + Bx^2 + Cx + D = 0.
+ Real A = b/a;
+ Real B = c/a;
+ Real C = d/a;
+ Real D = e/a;
+ Real Asq = A*A;
+ // Let x = y - A/4:
+ // Mathematica: Expand[(y - A/4)^4 + A*(y - A/4)^3 + B*(y - A/4)^2 + C*(y - A/4) + D]
+ // We now solve the depressed quartic y^4 + py^2 + qy + r = 0.
+ Real p = B - 3*Asq/8;
+ Real q = C - A*B/2 + Asq*A/8;
+ Real r = D - A*C/4 + Asq*B/16 - 3*Asq*Asq/256;
+ if (abs(r) <= (std::numeric_limits<Real>::min)()) {
+ auto [r1, r2, r3] = cubic_roots(Real(1), Real(0), p, q);
+ r1 -= A/4;
+ r2 -= A/4;
+ r3 -= A/4;
+ roots[0] = r1;
+ roots[1] = r2;
+ roots[2] = r3;
+ roots[3] = -A/4;
+ return detail::polish_and_sort(a, b, c, d, e, roots);
+ }
+ // Biquadratic case:
+ if (abs(q) <= (std::numeric_limits<Real>::min)()) {
+ auto [r1, r2] = quadratic_roots(Real(1), p, r);
+ if (r1 >= 0) {
+ Real rtr = sqrt(r1);
+ roots[0] = rtr - A/4;
+ roots[1] = -rtr - A/4;
+ }
+ if (r2 >= 0) {
+ Real rtr = sqrt(r2);
+ roots[2] = rtr - A/4;
+ roots[3] = -rtr - A/4;
+ }
+ return detail::polish_and_sort(a, b, c, d, e, roots);
+ }
+
+ // Now split the depressed quartic into two quadratics:
+ // y^4 + py^2 + qy + r = (y^2 + sy + u)(y^2 - sy + v) = y^4 + (v+u-s^2)y^2 + s(v - u)y + uv
+ // So p = v+u-s^2, q = s(v - u), r = uv.
+ // Then (v+u)^2 - (v-u)^2 = 4uv = 4r = (p+s^2)^2 - q^2/s^2.
+ // Multiply through by s^2 to get s^2(p+s^2)^2 - q^2 - 4rs^2 = 0, which is a cubic in s^2.
+ // Then we let z = s^2, to get
+ // z^3 + 2pz^2 + (p^2 - 4r)z - q^2 = 0.
+ auto z_roots = cubic_roots(Real(1), 2*p, p*p - 4*r, -q*q);
+ // z = s^2, so s = sqrt(z).
+ // No real roots:
+ if (z_roots.back() <= 0) {
+ return roots;
+ }
+ Real s = sqrt(z_roots.back());
+
+ // s is nonzero, because we took care of the biquadratic case.
+ Real v = (p + s*s + q/s)/2;
+ Real u = v - q/s;
+ // Now solve y^2 + sy + u = 0:
+ auto [root0, root1] = quadratic_roots(Real(1), s, u);
+
+ // Now solve y^2 - sy + v = 0:
+ auto [root2, root3] = quadratic_roots(Real(1), -s, v);
+ roots[0] = root0;
+ roots[1] = root1;
+ roots[2] = root2;
+ roots[3] = root3;
+
+ for (auto& r : roots) {
+ r -= A/4;
+ }
+ return detail::polish_and_sort(a, b, c, d, e, roots);
+}
+
+}
+#endif
projects / ceph.git / blobdiff