opencv/modules/calib3d/src/polynom_solver.cpp

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#include "precomp.hpp"
#include "polynom_solver.h"
#include <math.h>
#include <iostream>
int solve_deg2(double a, double b, double c, double & x1, double & x2)
{
double delta = b * b - 4 * a * c;
if (delta < 0) return 0;
double inv_2a = 0.5 / a;
if (delta == 0) {
x1 = -b * inv_2a;
x2 = x1;
return 1;
}
double sqrt_delta = sqrt(delta);
x1 = (-b + sqrt_delta) * inv_2a;
x2 = (-b - sqrt_delta) * inv_2a;
return 2;
}
/// Reference : Eric W. Weisstein. "Cubic Equation." From MathWorld--A Wolfram Web Resource.
/// http://mathworld.wolfram.com/CubicEquation.html
/// \return Number of real roots found.
int solve_deg3(double a, double b, double c, double d,
double & x0, double & x1, double & x2)
{
if (a == 0) {
// Solve second order sytem
if (b == 0) {
// Solve first order system
if (c == 0)
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return 0;
x0 = -d / c;
return 1;
}
x2 = 0;
return solve_deg2(b, c, d, x0, x1);
}
// Calculate the normalized form x^3 + a2 * x^2 + a1 * x + a0 = 0
double inv_a = 1. / a;
double b_a = inv_a * b, b_a2 = b_a * b_a;
double c_a = inv_a * c;
double d_a = inv_a * d;
// Solve the cubic equation
double Q = (3 * c_a - b_a2) / 9;
double R = (9 * b_a * c_a - 27 * d_a - 2 * b_a * b_a2) / 54;
double Q3 = Q * Q * Q;
double D = Q3 + R * R;
double b_a_3 = (1. / 3.) * b_a;
if (Q == 0) {
if(R == 0) {
x0 = x1 = x2 = - b_a_3;
return 3;
}
else {
x0 = pow(2 * R, 1 / 3.0) - b_a_3;
return 1;
}
}
if (D <= 0) {
// Three real roots
double theta = acos(R / sqrt(-Q3));
double sqrt_Q = sqrt(-Q);
x0 = 2 * sqrt_Q * cos(theta / 3.0) - b_a_3;
x1 = 2 * sqrt_Q * cos((theta + 2 * CV_PI)/ 3.0) - b_a_3;
x2 = 2 * sqrt_Q * cos((theta + 4 * CV_PI)/ 3.0) - b_a_3;
return 3;
}
// D > 0, only one real root
double AD = pow(fabs(R) + sqrt(D), 1.0 / 3.0) * (R > 0 ? 1 : (R < 0 ? -1 : 0));
double BD = (AD == 0) ? 0 : -Q / AD;
// Calculate the only real root
x0 = AD + BD - b_a_3;
return 1;
}
/// Reference : Eric W. Weisstein. "Quartic Equation." From MathWorld--A Wolfram Web Resource.
/// http://mathworld.wolfram.com/QuarticEquation.html
/// \return Number of real roots found.
int solve_deg4(double a, double b, double c, double d, double e,
double & x0, double & x1, double & x2, double & x3)
{
if (a == 0) {
x3 = 0;
return solve_deg3(b, c, d, e, x0, x1, x2);
}
// Normalize coefficients
double inv_a = 1. / a;
b *= inv_a; c *= inv_a; d *= inv_a; e *= inv_a;
double b2 = b * b, bc = b * c, b3 = b2 * b;
// Solve resultant cubic
double r0, r1, r2;
int n = solve_deg3(1, -c, d * b - 4 * e, 4 * c * e - d * d - b2 * e, r0, r1, r2);
if (n == 0) return 0;
// Calculate R^2
double R2 = 0.25 * b2 - c + r0, R;
if (R2 < 0)
return 0;
R = sqrt(R2);
double inv_R = 1. / R;
int nb_real_roots = 0;
// Calculate D^2 and E^2
double D2, E2;
if (R < 10E-12) {
double temp = r0 * r0 - 4 * e;
if (temp < 0)
D2 = E2 = -1;
else {
double sqrt_temp = sqrt(temp);
D2 = 0.75 * b2 - 2 * c + 2 * sqrt_temp;
E2 = D2 - 4 * sqrt_temp;
}
}
else {
double u = 0.75 * b2 - 2 * c - R2,
v = 0.25 * inv_R * (4 * bc - 8 * d - b3);
D2 = u + v;
E2 = u - v;
}
double b_4 = 0.25 * b, R_2 = 0.5 * R;
if (D2 >= 0) {
double D = sqrt(D2);
nb_real_roots = 2;
double D_2 = 0.5 * D;
x0 = R_2 + D_2 - b_4;
x1 = x0 - D;
}
// Calculate E^2
if (E2 >= 0) {
double E = sqrt(E2);
double E_2 = 0.5 * E;
if (nb_real_roots == 0) {
x0 = - R_2 + E_2 - b_4;
x1 = x0 - E;
nb_real_roots = 2;
}
else {
x2 = - R_2 + E_2 - b_4;
x3 = x2 - E;
nb_real_roots = 4;
}
}
return nb_real_roots;
}