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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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