roots3n4.c

Graphics Gems 源码 a collection of algorithms, programs, and mathematical techniques for the computer

/*
 *  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 ([e-mail])
 *
 *  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.
 */

#include    <math.h>
#ifndef M_PI
#define M_PI          3.14159265358979323846
#endif
extern double   sqrt(), cbrt(), cos(), acos();

/* epsilon surrounding for near zero values */

#define     EQN_EPS     1e-9
#define	    IsZero(x)	((x) > -EQN_EPS && (x) < EQN_EPS)

#ifdef NOCBRT
#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))
#endif

int SolveQuadric(c, s)
    double c[ 3 ];
    double s[ 2 ];
{
    double p, q, D;

    /* normal form: x^2 + px + q = 0 */

    p = c[ 1 ] / (2 * c[ 2 ]);
    q = c[ 0 ] / c[ 2 ];

    D = p * p - q;

    if (IsZero(D))
    {
	s[ 0 ] = - p;
	return 1;
    }
    else if (D < 0)
    {
	return 0;
    }
    else if (D > 0)
    {
	double sqrt_D = sqrt(D);

	s[ 0 ] =   sqrt_D - p;
	s[ 1 ] = - sqrt_D - p;
	return 2;
    }
}


int SolveCubic(c, s)
    double c[ 4 ];
    double s[ 3 ];
{
    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 */

    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;
	    num = 1;
	}
	else /* one single and one double solution */
	{
	    double u = cbrt(-q);
	    s[ 0 ] = 2 * u;
	    s[ 1 ] = - u;
	    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 SolveQuartic(c, s)
    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 = SolveCubic(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;

	(void) SolveCubic(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 = SolveQuadric(coeffs, s);

	coeffs[ 0 ]= z + u;
	coeffs[ 1 ] = q < 0 ? v : -v;
	coeffs[ 2 ] = 1;

	num += SolveQuadric(coeffs, s + num);
    }

    /* resubstitute */

    sub = 1.0/4 * A;

    for (i = 0; i < num; ++i)
	s[ i ] -= sub;

    return num;
}