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src/BulletSoftBody/poly34.cpp
447 строк
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Xuchen Han
clang format PhysicsServerCommandProcessor and BulletSoftBody
18 июн 2020, 03:20
18 июн 2020, 03:20
f23e745
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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 //-----------------------------------------------------------------------------