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