solve_polynom.cxx

#include "solve_polynom.h"
#include <math.h>
/*
 *  Roots3And4.c
 *
 *  Utility functions to find cubic and quartic roots,
 *  coefficients are passed like this:
 *
 *      c[0] + c[1]*x + c[2]*x^2 + c[3]*x^3 + c[4]*x^4 = 0
 *
 *  The functions return the number of non-complex roots and
 *  put the values into the s array.
 *
 *  Author:         Jochen Schwarze (schwarze@isa.de)
 *
 *  Jan 26, 1990    Version for Graphics Gems
 *  Oct 11, 1990    Fixed sign problem for negative q's in SolveQuartic
 *                  (reported by Mark Podlipec),
 *                  Old-style function definitions,
 *                  IsZero() as a macro
 *  Nov 23, 1990    Some systems do not declare acos() and cbrt() in
 *                  <math.h>, though the functions exist in the library.
 *                  If large coefficients are used, EQN_EPS should be
 *                  reduced considerably (e.g. to 1E-30), results will be
 *                  correct but multiple roots might be reported more
 *                  than once.
 */
 
#ifndef M_PI
#define M_PI          3.14159265358979323846
#endif
 
namespace cgv{
    namespace math{
 
 
 
/* epsilon surrounding for near zero values */
 
#define     EQN_EPS     1e-9
#define     IsZero(x)   ((x) > -EQN_EPS && (x) < EQN_EPS)
#define     cbrt(x)     ((x) > 0.0 ? pow((double)(x), 1.0/3.0) : \
                        ((x) < 0.0 ? -pow((double)-(x), 1.0/3.0) : 0.0))
 
int solve_linear(double c[2],double s[1])
{
    if(IsZero(c[1]))
        return 0;
    s[0]=-c[0]/c[1];
    return 1;
}
 
int solve_quadric(double c[3], double s[2], bool replicate_multiple_solutions)
{
    double p, q, D;
 
    /* normal form: x^2 + px + q = 0 */
 
    if (IsZero(c[2]))
        return solve_linear(c, s);
 
    p = c[ 1 ] / (2 * c[ 2 ]);
    q = c[ 0 ] / c[ 2 ];
 
    D = p * p - q;
 
    if (IsZero(D)) {
        s[ 0 ] = - p;
        if (replicate_multiple_solutions) {
            s[1] = s[0];
            return 2;
        }
        return 1;
    }
    else if (D < 0) {
        return 0;
    }
    else {
        double sqrt_D = sqrt(D);
        s[ 0 ] =   sqrt_D - p;
        s[ 1 ] = - sqrt_D - p;
        return 2;
    }
}
 
 
int solve_cubic(double c[4], double s[3], bool replicate_multiple_solutions)
{
    int     i, num;
    double  sub;
    double  A, B, C;
    double  sq_A, p, q;
    double  cb_p, D;
 
    /* normal form: x^3 + Ax^2 + Bx + C = 0 */
    if(c[3] == 0)
        return solve_quadric(c,s,replicate_multiple_solutions);
 
    A = c[ 2 ] / c[ 3 ];
    B = c[ 1 ] / c[ 3 ];
    C = c[ 0 ] / c[ 3 ];
 
    /*  substitute x = y - A/3 to eliminate quadric term:
    x^3 +px + q = 0 */
 
    sq_A = A * A;
    p = 1.0/3 * (- 1.0/3 * sq_A + B);
    q = 1.0/2 * (2.0/27 * A * sq_A - 1.0/3 * A * B + C);
 
    /* use Cardano's formula */
 
    cb_p = p * p * p;
    D = q * q + cb_p;
 
    if (IsZero(D)) {
        if (IsZero(q)) /* one triple solution */ {
            s[ 0 ] = 0;
            if (replicate_multiple_solutions) {
                s[1] = s[2] = s[0];
                num = 3;
            }
            else
                num = 1;
        }
        else /* one single and one double solution */ {
            double u = cbrt(-q);
            s[ 0 ] = 2 * u;
            s[ 1 ] = - u;
            if (replicate_multiple_solutions) {
                s[2] = s[1];
                num = 3;
            }
            else
                num = 2;
        }
    }
    else if (D < 0) /* Casus irreducibilis: three real solutions */ {
        double phi = 1.0/3 * acos(-q / sqrt(-cb_p));
        double t = 2 * sqrt(-p);
 
        s[ 0 ] =   t * cos(phi);
        s[ 1 ] = - t * cos(phi + M_PI / 3);
        s[ 2 ] = - t * cos(phi - M_PI / 3);
        num = 3;
    }
    else /* one real solution */ {
        double sqrt_D = sqrt(D);
        double u = cbrt(sqrt_D - q);
        double v = - cbrt(sqrt_D + q);
 
        s[ 0 ] = u + v;
        num = 1;
    }
 
    /* resubstitute */
    sub = 1.0/3 * A;
    for (i = 0; i < num; ++i) s[ i ] -= sub;
    return num;
}
 
 
int solve_quartic(double c[5], double s[4])
{
    double  coeffs[ 4 ];
    double  z, u, v, sub;
    double  A, B, C, D;
    double  sq_A, p, q, r;
    int     i, num;
 
    /* normal form: x^4 + Ax^3 + Bx^2 + Cx + D = 0 */
 
    A = c[ 3 ] / c[ 4 ];
    B = c[ 2 ] / c[ 4 ];
    C = c[ 1 ] / c[ 4 ];
    D = c[ 0 ] / c[ 4 ];
 
    /*  substitute x = y - A/4 to eliminate cubic term:
    x^4 + px^2 + qx + r = 0 */
 
    sq_A = A * A;
    p = - 3.0/8 * sq_A + B;
    q = 1.0/8 * sq_A * A - 1.0/2 * A * B + C;
    r = - 3.0/256*sq_A*sq_A + 1.0/16*sq_A*B - 1.0/4*A*C + D;
 
    if (IsZero(r))
    {
        /* no absolute term: y(y^3 + py + q) = 0 */
 
        coeffs[ 0 ] = q;
        coeffs[ 1 ] = p;
        coeffs[ 2 ] = 0;
        coeffs[ 3 ] = 1;
 
        num = solve_cubic(coeffs, s);
 
        s[ num++ ] = 0;
    }
    else
    {
        /* solve the resolvent cubic ... */
 
        coeffs[ 0 ] = 1.0/2 * r * p - 1.0/8 * q * q;
        coeffs[ 1 ] = - r;
        coeffs[ 2 ] = - 1.0/2 * p;
        coeffs[ 3 ] = 1;
 
        solve_cubic(coeffs, s);
 
        /* ... and take the one real solution ... */
 
        z = s[ 0 ];
 
        /* ... to build two quadric equations */
 
        u = z * z - r;
        v = 2 * z - p;
 
        if (IsZero(u))
            u = 0;
        else if (u > 0)
            u = sqrt(u);
        else
            return 0;
 
        if (IsZero(v))
            v = 0;
        else if (v > 0)
            v = sqrt(v);
        else
            return 0;
 
        coeffs[ 0 ] = z - u;
        coeffs[ 1 ] = q < 0 ? -v : v;
        coeffs[ 2 ] = 1;
 
        num = solve_quadric(coeffs, s);
 
        coeffs[ 0 ]= z + u;
        coeffs[ 1 ] = q < 0 ? v : -v;
        coeffs[ 2 ] = 1;
 
        num += solve_quadric(coeffs, s + num);
    }
 
    /* resubstitute */
 
    sub = 1.0/4 * A;
    for (i = 0; i < num; ++i) s[ i ] -= sub;
    return num;
}
 
}
}