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-rw-r--r--thirdparty/bullet/BulletSoftBody/poly34.cpp447
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diff --git a/thirdparty/bullet/BulletSoftBody/poly34.cpp b/thirdparty/bullet/BulletSoftBody/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
-//-----------------------------------------------------------------------------
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