poly34.cpp

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// poly34.cpp : solution of cubic and quartic equation
// (c) Khashin S.I. http://math.ivanovo.ac.ru/dalgebra/Khashin/index.html
// khash2 (at) gmail.com
// Thanks to Alexandr Rakhmanin <rakhmanin (at) gmail.com>
// public domain
//
#include <math.h>
 
#include "poly34.h"  // solution of cubic and quartic equation
#define TwoPi 6.28318530717958648
const btScalar eps = SIMD_EPSILON;
 
//=============================================================================
// _root3, root3 from http://prografix.narod.ru
//=============================================================================
static SIMD_FORCE_INLINE btScalar _root3(btScalar x)
{
    btScalar s = 1.;
    while (x < 1.)
    {
        x *= 8.;
        s *= 0.5;
    }
    while (x > 8.)
    {
        x *= 0.125;
        s *= 2.;
    }
    btScalar r = 1.5;
    r -= 1. / 3. * (r - x / (r * r));
    r -= 1. / 3. * (r - x / (r * r));
    r -= 1. / 3. * (r - x / (r * r));
    r -= 1. / 3. * (r - x / (r * r));
    r -= 1. / 3. * (r - x / (r * r));
    r -= 1. / 3. * (r - x / (r * r));
    return r * s;
}
 
btScalar SIMD_FORCE_INLINE root3(btScalar x)
{
    if (x > 0)
        return _root3(x);
    else if (x < 0)
        return -_root3(-x);
    else
        return 0.;
}
 
// x - array of size 2
// return 2: 2 real roots x[0], x[1]
// return 0: pair of complex roots: x[0]i*x[1]
int SolveP2(btScalar* x, btScalar a, btScalar b)
{  // solve equation x^2 + a*x + b = 0
    btScalar D = 0.25 * a * a - b;
    if (D >= 0)
    {
        D = sqrt(D);
        x[0] = -0.5 * a + D;
        x[1] = -0.5 * a - D;
        return 2;
    }
    x[0] = -0.5 * a;
    x[1] = sqrt(-D);
    return 0;
}
//---------------------------------------------------------------------------
// x - array of size 3
// In case 3 real roots: => x[0], x[1], x[2], return 3
//         2 real roots: x[0], x[1],          return 2
//         1 real root : x[0], x[1]  i*x[2], return 1
int SolveP3(btScalar* x, btScalar a, btScalar b, btScalar c)
{  // solve cubic equation x^3 + a*x^2 + b*x + c = 0
    btScalar a2 = a * a;
    btScalar q = (a2 - 3 * b) / 9;
    if (q < 0)
        q = eps;
    btScalar r = (a * (2 * a2 - 9 * b) + 27 * c) / 54;
    // equation x^3 + q*x + r = 0
    btScalar r2 = r * r;
    btScalar q3 = q * q * q;
    btScalar A, B;
    if (r2 <= (q3 + eps))
    {  //<<-- FIXED!
        btScalar t = r / sqrt(q3);
        if (t < -1)
            t = -1;
        if (t > 1)
            t = 1;
        t = acos(t);
        a /= 3;
        q = -2 * sqrt(q);
        x[0] = q * cos(t / 3) - a;
        x[1] = q * cos((t + TwoPi) / 3) - a;
        x[2] = q * cos((t - TwoPi) / 3) - a;
        return (3);
    }
    else
    {
        //A =-pow(fabs(r)+sqrt(r2-q3),1./3);
        A = -root3(fabs(r) + sqrt(r2 - q3));
        if (r < 0)
            A = -A;
        B = (A == 0 ? 0 : q / A);
 
        a /= 3;
        x[0] = (A + B) - a;
        x[1] = -0.5 * (A + B) - a;
        x[2] = 0.5 * sqrt(3.) * (A - B);
        if (fabs(x[2]) < eps)
        {
            x[2] = x[1];
            return (2);
        }
        return (1);
    }
}  // SolveP3(btScalar *x,btScalar a,btScalar b,btScalar c) {
//---------------------------------------------------------------------------
// a>=0!
void CSqrt(btScalar x, btScalar y, btScalar& a, btScalar& b)  // returns:  a+i*s = sqrt(x+i*y)
{
    btScalar r = sqrt(x * x + y * y);
    if (y == 0)
    {
        r = sqrt(r);
        if (x >= 0)
        {
            a = r;
            b = 0;
        }
        else
        {
            a = 0;
            b = r;
        }
    }
    else
    {  // y != 0
        a = sqrt(0.5 * (x + r));
        b = 0.5 * y / a;
    }
}
//---------------------------------------------------------------------------
int SolveP4Bi(btScalar* x, btScalar b, btScalar d)  // solve equation x^4 + b*x^2 + d = 0
{
    btScalar D = b * b - 4 * d;
    if (D >= 0)
    {
        btScalar sD = sqrt(D);
        btScalar x1 = (-b + sD) / 2;
        btScalar x2 = (-b - sD) / 2;  // x2 <= x1
        if (x2 >= 0)                  // 0 <= x2 <= x1, 4 real roots
        {
            btScalar sx1 = sqrt(x1);
            btScalar sx2 = sqrt(x2);
            x[0] = -sx1;
            x[1] = sx1;
            x[2] = -sx2;
            x[3] = sx2;
            return 4;
        }
        if (x1 < 0)  // x2 <= x1 < 0, two pair of imaginary roots
        {
            btScalar sx1 = sqrt(-x1);
            btScalar sx2 = sqrt(-x2);
            x[0] = 0;
            x[1] = sx1;
            x[2] = 0;
            x[3] = sx2;
            return 0;
        }
        // now x2 < 0 <= x1 , two real roots and one pair of imginary root
        btScalar sx1 = sqrt(x1);
        btScalar sx2 = sqrt(-x2);
        x[0] = -sx1;
        x[1] = sx1;
        x[2] = 0;
        x[3] = sx2;
        return 2;
    }
    else
    {  // if( D < 0 ), two pair of compex roots
        btScalar sD2 = 0.5 * sqrt(-D);
        CSqrt(-0.5 * b, sD2, x[0], x[1]);
        CSqrt(-0.5 * b, -sD2, x[2], x[3]);
        return 0;
    }  // if( D>=0 )
}  // SolveP4Bi(btScalar *x, btScalar b, btScalar d)    // solve equation x^4 + b*x^2 d
//---------------------------------------------------------------------------
#define SWAP(a, b) \
    {              \
        t = b;     \
        b = a;     \
        a = t;     \
    }
static void dblSort3(btScalar& a, btScalar& b, btScalar& c)  // make: a <= b <= c
{
    btScalar t;
    if (a > b)
        SWAP(a, b);  // now a<=b
    if (c < b)
    {
        SWAP(b, c);  // now a<=b, b<=c
        if (a > b)
            SWAP(a, b);  // now a<=b
    }
}
//---------------------------------------------------------------------------
int SolveP4De(btScalar* x, btScalar b, btScalar c, btScalar d)  // solve equation x^4 + b*x^2 + c*x + d
{
    //if( c==0 ) return SolveP4Bi(x,b,d); // After that, c!=0
    if (fabs(c) < 1e-14 * (fabs(b) + fabs(d)))
        return SolveP4Bi(x, b, d);  // After that, c!=0
 
    int res3 = SolveP3(x, 2 * b, b * b - 4 * d, -c * c);  // solve resolvent
    // by Viet theorem:  x1*x2*x3=-c*c not equals to 0, so x1!=0, x2!=0, x3!=0
    if (res3 > 1)  // 3 real roots,
    {
        dblSort3(x[0], x[1], x[2]);  // sort roots to x[0] <= x[1] <= x[2]
        // Note: x[0]*x[1]*x[2]= c*c > 0
        if (x[0] > 0)  // all roots are positive
        {
            btScalar sz1 = sqrt(x[0]);
            btScalar sz2 = sqrt(x[1]);
            btScalar sz3 = sqrt(x[2]);
            // Note: sz1*sz2*sz3= -c (and not equal to 0)
            if (c > 0)
            {
                x[0] = (-sz1 - sz2 - sz3) / 2;
                x[1] = (-sz1 + sz2 + sz3) / 2;
                x[2] = (+sz1 - sz2 + sz3) / 2;
                x[3] = (+sz1 + sz2 - sz3) / 2;
                return 4;
            }
            // now: c<0
            x[0] = (-sz1 - sz2 + sz3) / 2;
            x[1] = (-sz1 + sz2 - sz3) / 2;
            x[2] = (+sz1 - sz2 - sz3) / 2;
            x[3] = (+sz1 + sz2 + sz3) / 2;
            return 4;
        }  // if( x[0] > 0) // all roots are positive
        // now x[0] <= x[1] < 0, x[2] > 0
        // two pair of comlex roots
        btScalar sz1 = sqrt(-x[0]);
        btScalar sz2 = sqrt(-x[1]);
        btScalar sz3 = sqrt(x[2]);
 
        if (c > 0)  // sign = -1
        {
            x[0] = -sz3 / 2;
            x[1] = (sz1 - sz2) / 2;  // x[0]i*x[1]
            x[2] = sz3 / 2;
            x[3] = (-sz1 - sz2) / 2;  // x[2]i*x[3]
            return 0;
        }
        // now: c<0 , sign = +1
        x[0] = sz3 / 2;
        x[1] = (-sz1 + sz2) / 2;
        x[2] = -sz3 / 2;
        x[3] = (sz1 + sz2) / 2;
        return 0;
    }  // if( res3>1 )    // 3 real roots,
    // now resoventa have 1 real and pair of compex roots
    // x[0] - real root, and x[0]>0,
    // x[1]i*x[2] - complex roots,
    // x[0] must be >=0. But one times x[0]=~ 1e-17, so:
    if (x[0] < 0)
        x[0] = 0;
    btScalar sz1 = sqrt(x[0]);
    btScalar szr, szi;
    CSqrt(x[1], x[2], szr, szi);  // (szr+i*szi)^2 = x[1]+i*x[2]
    if (c > 0)                    // sign = -1
    {
        x[0] = -sz1 / 2 - szr;  // 1st real root
        x[1] = -sz1 / 2 + szr;  // 2nd real root
        x[2] = sz1 / 2;
        x[3] = szi;
        return 2;
    }
    // now: c<0 , sign = +1
    x[0] = sz1 / 2 - szr;  // 1st real root
    x[1] = sz1 / 2 + szr;  // 2nd real root
    x[2] = -sz1 / 2;
    x[3] = szi;
    return 2;
}  // SolveP4De(btScalar *x, btScalar b, btScalar c, btScalar d)    // solve equation x^4 + b*x^2 + c*x + d
//-----------------------------------------------------------------------------
btScalar N4Step(btScalar x, btScalar a, btScalar b, btScalar c, btScalar d)  // one Newton step for x^4 + a*x^3 + b*x^2 + c*x + d
{
    btScalar fxs = ((4 * x + 3 * a) * x + 2 * b) * x + c;  // f'(x)
    if (fxs == 0)
        return x;                                       //return 1e99; <<-- FIXED!
    btScalar fx = (((x + a) * x + b) * x + c) * x + d;  // f(x)
    return x - fx / fxs;
}
//-----------------------------------------------------------------------------
// x - array of size 4
// return 4: 4 real roots x[0], x[1], x[2], x[3], possible multiple roots
// return 2: 2 real roots x[0], x[1] and complex x[2]i*x[3],
// return 0: two pair of complex roots: x[0]i*x[1],  x[2]i*x[3],
int SolveP4(btScalar* x, btScalar a, btScalar b, btScalar c, btScalar d)
{  // solve equation x^4 + a*x^3 + b*x^2 + c*x + d by Dekart-Euler method
    // move to a=0:
    btScalar d1 = d + 0.25 * a * (0.25 * b * a - 3. / 64 * a * a * a - c);
    btScalar c1 = c + 0.5 * a * (0.25 * a * a - b);
    btScalar b1 = b - 0.375 * a * a;
    int res = SolveP4De(x, b1, c1, d1);
    if (res == 4)
    {
        x[0] -= a / 4;
        x[1] -= a / 4;
        x[2] -= a / 4;
        x[3] -= a / 4;
    }
    else if (res == 2)
    {
        x[0] -= a / 4;
        x[1] -= a / 4;
        x[2] -= a / 4;
    }
    else
    {
        x[0] -= a / 4;
        x[2] -= a / 4;
    }
    // one Newton step for each real root:
    if (res > 0)
    {
        x[0] = N4Step(x[0], a, b, c, d);
        x[1] = N4Step(x[1], a, b, c, d);
    }
    if (res > 2)
    {
        x[2] = N4Step(x[2], a, b, c, d);
        x[3] = N4Step(x[3], a, b, c, d);
    }
    return res;
}
//-----------------------------------------------------------------------------
#define F5(t) (((((t + a) * t + b) * t + c) * t + d) * t + e)
//-----------------------------------------------------------------------------
btScalar SolveP5_1(btScalar a, btScalar b, btScalar c, btScalar d, btScalar e)  // return real root of x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
{
    int cnt;
    if (fabs(e) < eps)
        return 0;
 
    btScalar brd = fabs(a);  // brd - border of real roots
    if (fabs(b) > brd)
        brd = fabs(b);
    if (fabs(c) > brd)
        brd = fabs(c);
    if (fabs(d) > brd)
        brd = fabs(d);
    if (fabs(e) > brd)
        brd = fabs(e);
    brd++;  // brd - border of real roots
 
    btScalar x0, f0;       // less than root
    btScalar x1, f1;       // greater than root
    btScalar x2, f2, f2s;  // next values, f(x2), f'(x2)
    btScalar dx = 0;
 
    if (e < 0)
    {
        x0 = 0;
        x1 = brd;
        f0 = e;
        f1 = F5(x1);
        x2 = 0.01 * brd;
    }  // positive root
    else
    {
        x0 = -brd;
        x1 = 0;
        f0 = F5(x0);
        f1 = e;
        x2 = -0.01 * brd;
    }  // negative root
 
    if (fabs(f0) < eps)
        return x0;
    if (fabs(f1) < eps)
        return x1;
 
    // now x0<x1, f(x0)<0, f(x1)>0
    // Firstly 10 bisections
    for (cnt = 0; cnt < 10; cnt++)
    {
        x2 = (x0 + x1) / 2;  // next point
        //x2 = x0 - f0*(x1 - x0) / (f1 - f0);        // next point
        f2 = F5(x2);  // f(x2)
        if (fabs(f2) < eps)
            return x2;
        if (f2 > 0)
        {
            x1 = x2;
            f1 = f2;
        }
        else
        {
            x0 = x2;
            f0 = f2;
        }
    }
 
    // At each step:
    // x0<x1, f(x0)<0, f(x1)>0.
    // x2 - next value
    // we hope that x0 < x2 < x1, but not necessarily
    do
    {
        if (cnt++ > 50)
            break;
        if (x2 <= x0 || x2 >= x1)
            x2 = (x0 + x1) / 2;  // now  x0 < x2 < x1
        f2 = F5(x2);             // f(x2)
        if (fabs(f2) < eps)
            return x2;
        if (f2 > 0)
        {
            x1 = x2;
            f1 = f2;
        }
        else
        {
            x0 = x2;
            f0 = f2;
        }
        f2s = (((5 * x2 + 4 * a) * x2 + 3 * b) * x2 + 2 * c) * x2 + d;  // f'(x2)
        if (fabs(f2s) < eps)
        {
            x2 = 1e99;
            continue;
        }
        dx = f2 / f2s;
        x2 -= dx;
    } while (fabs(dx) > eps);
    return x2;
}  // SolveP5_1(btScalar a,btScalar b,btScalar c,btScalar d,btScalar e)    // return real root of x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
//-----------------------------------------------------------------------------
int SolveP5(btScalar* x, btScalar a, btScalar b, btScalar c, btScalar d, btScalar e)  // solve equation x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
{
    btScalar r = x[0] = SolveP5_1(a, b, c, d, e);
    btScalar a1 = a + r, b1 = b + r * a1, c1 = c + r * b1, d1 = d + r * c1;
    return 1 + SolveP4(x + 1, a1, b1, c1, d1);
}  // SolveP5(btScalar *x,btScalar a,btScalar b,btScalar c,btScalar d,btScalar e)    // solve equation x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
//-----------------------------------------------------------------------------