448 lines
11 KiB
C++
448 lines
11 KiB
C++
|
// 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
|
||
|
|
//-----------------------------------------------------------------------------
|