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// // SPDX-License-Identifier: BSD-3-Clause // Copyright Contributors to the OpenEXR Project. // // // Functions to solve linear, quadratic or cubic equations // // Note: It is possible that an equation has real solutions, but that // the solutions (or some intermediate result) are not representable. // In this case, either some of the solutions returned are invalid // (nan or infinity), or, if floating-point exceptions have been // enabled, an exception is thrown. // #ifndef INCLUDED_IMATHROOTS_H #define INCLUDED_IMATHROOTS_H #include "ImathMath.h" #include "ImathNamespace.h" #include /// @cond Doxygen_Suppress // If CUDA or HIP (AMD's Heterogeneous-computing Interface for // Portability), use the thrust complex library #if defined(__CUDACC__) || defined(__HIP__) # include # define COMPLEX_NAMESPACE thrust #else # define COMPLEX_NAMESPACE std #endif /// @endcond IMATH_INTERNAL_NAMESPACE_HEADER_ENTER /// /// Solve for x in the linear equation: /// /// a * x + b == 0 /// /// @return 1 if the equation has a solution, 0 if there is no /// solution, and -1 if all real numbers are solutions. template IMATH_HOSTDEVICE IMATH_CONSTEXPR14 int solveLinear (T a, T b, T& x); /// /// Solve for x in the quadratic equation: /// /// a * x*x + b * x + c == 0 /// /// @return 2 if the equation has two solutions, 1 if the equation has /// a single solution, 0 if there is no solution, and -1 if all real /// numbers are solutions. template IMATH_HOSTDEVICE IMATH_CONSTEXPR14 int solveQuadratic (T a, T b, T c, T x[2]); template /// /// Solve for x in the normalized cubic equation: /// /// x*x*x + r * x*x + s * x + t == 0 /// /// The equation is solved using Cardano's Formula; even though only /// real solutions are produced, some intermediate results are complex /// (std::complex). /// /// @return 0 if there is no solution, and -1 if all real /// numbers are solutions, otherwise return the number of solutions. IMATH_HOSTDEVICE IMATH_CONSTEXPR14 int solveNormalizedCubic (T r, T s, T t, T x[3]); /// /// Solve for x in the cubic equation: /// /// a * x*x*x + b * x*x + c * x + d == 0 /// /// The equation is solved using Cardano's Formula; even though only /// real solutions are produced, some intermediate results are complex /// (std::complex). /// /// @return 0 if there is no solution, and -1 if all real /// numbers are solutions, otherwise return the number of solutions. template IMATH_HOSTDEVICE IMATH_CONSTEXPR14 int solveCubic (T a, T b, T c, T d, T x[3]); //--------------- // Implementation //--------------- template IMATH_CONSTEXPR14 int solveLinear (T a, T b, T& x) { if (a != 0) { x = -b / a; return 1; } else if (b != 0) { return 0; } else { return -1; } } template IMATH_CONSTEXPR14 int solveQuadratic (T a, T b, T c, T x[2]) { if (a == 0) { return solveLinear (b, c, x[0]); } else { T D = b * b - 4 * a * c; if (D > 0) { T s = std::sqrt (D); T q = -(b + (b > 0 ? 1 : -1) * s) / T (2); x[0] = q / a; x[1] = c / q; return 2; } if (D == 0) { x[0] = -b / (2 * a); return 1; } else { return 0; } } } template IMATH_CONSTEXPR14 int solveNormalizedCubic (T r, T s, T t, T x[3]) { T p = (3 * s - r * r) / 3; T q = 2 * r * r * r / 27 - r * s / 3 + t; T p3 = p / 3; T q2 = q / 2; T D = p3 * p3 * p3 + q2 * q2; if (D == 0 && p3 == 0) { x[0] = -r / 3; x[1] = -r / 3; x[2] = -r / 3; return 1; } if (D > 0) { auto real_root = [] (T a, T x) -> T { T sign = std::copysign (T (1), a); return sign * std::pow (sign * a, T (1) / x); }; T u = real_root (-q / 2 + std::sqrt (D), 3); T v = -p / (T (3) * u); x[0] = u + v - r / 3; return 1; } namespace CN = COMPLEX_NAMESPACE; CN::complex u = CN::pow (-q / 2 + CN::sqrt (CN::complex (D)), T (1) / T (3)); CN::complex v = -p / (T (3) * u); const T sqrt3 = T (1.73205080756887729352744634150587); // enough digits // for long double CN::complex y0 (u + v); CN::complex y1 ( -(u + v) / T (2) + (u - v) / T (2) * CN::complex (0, sqrt3)); CN::complex y2 ( -(u + v) / T (2) - (u - v) / T (2) * CN::complex (0, sqrt3)); if (D == 0) { x[0] = y0.real () - r / 3; x[1] = y1.real () - r / 3; return 2; } else { x[0] = y0.real () - r / 3; x[1] = y1.real () - r / 3; x[2] = y2.real () - r / 3; return 3; } } template IMATH_CONSTEXPR14 int solveCubic (T a, T b, T c, T d, T x[3]) { if (a == 0) { return solveQuadratic (b, c, d, x); } else { return solveNormalizedCubic (b / a, c / a, d / a, x); } } IMATH_INTERNAL_NAMESPACE_HEADER_EXIT #endif // INCLUDED_IMATHROOTS_H