poly_roots.cpp

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/* +------------------------------------------------------------------------+
   |                     Mobile Robot Programming Toolkit (MRPT)            |
   |                          https://www.mrpt.org/                         |
   |                                                                        |
   | Copyright (c) 2005-2020, Individual contributors, see AUTHORS file     |
   | See: https://www.mrpt.org/Authors - All rights reserved.               |
   | Released under BSD License. See: https://www.mrpt.org/License          |
   +------------------------------------------------------------------------+ */

#include "math-precomp.h"  // Precompiled headers

#include <mrpt/math/poly_roots.h>
#include <cmath>

// Based on:
// poly.cpp : solution of cubic and quartic equation
// (c) Khashin S.I. http://math.ivanovo.ac.ru/dalgebra/Khashin/index.html
// khash2 (at) gmail.com
//

#define TwoPi 6.28318530717958648
const double eps = 1e-14;

//---------------------------------------------------------------------------
// 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 mrpt::math::solve_poly3(double* x, double a, double b, double c) noexcept
{  // solve cubic equation x^3 + a*x^2 + b*x + c
    double a2 = a * a;
    double q = (a2 - 3 * b) / 9;
    double r = (a * (2 * a2 - 9 * b) + 27 * c) / 54;
    double r2 = r * r;
    double q3 = q * q * q;
    double A, B;
    if (r2 < q3)
    {
        double 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(std::abs(r) + sqrt(r2 - q3), 1. / 3);
        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 (std::abs(x[2]) < eps)
        {
            x[2] = x[1];
            return (2);
        }
        return (1);
    }
}  // SolveP3(double *x,double a,double b,double c) {
//---------------------------------------------------------------------------
// a>=0!
void CSqrt(
    double x, double y, double& a, double& b)  // returns:  a+i*s = sqrt(x+i*y)
{
    double 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(
    double* x, double b, double d)  // solve equation x^4 + b*x^2 + d = 0
{
    double D = b * b - 4 * d;
    if (D >= 0)
    {
        double sD = sqrt(D);
        double x1 = (-b + sD) / 2;
        double x2 = (-b - sD) / 2;  // x2 <= x1
        if (x2 >= 0)  // 0 <= x2 <= x1, 4 real roots
        {
            double sx1 = sqrt(x1);
            double 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
        {
            double sx1 = sqrt(-x1);
            double 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
        double sx1 = sqrt(x1);
        double 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
        double 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(double *x, double b, double d)  // solve equation x^4 + b*x^2 d
//---------------------------------------------------------------------------
static void dblSort3(double& a, double& b, double& c)  // make: a <= b <= c
{
    if (a > b) std::swap(a, b);  // now a<=b
    if (c < b)
    {
        std::swap(b, c);  // now a<=b, b<=c
        if (a > b) std::swap(a, b);  // now a<=b
    }
}
//---------------------------------------------------------------------------
int SolveP4De(
    double* x, double b, double c,
    double d)  // solve equation x^4 + b*x^2 + c*x + d
{
    // if( c==0 ) return SolveP4Bi(x,b,d); // After that, c!=0
    if (std::abs(c) < 1e-14 * (std::abs(b) + std::abs(d)))
        return SolveP4Bi(x, b, d);  // After that, c!=0

    int res3 = mrpt::math::solve_poly3(
        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
        {
            double sz1 = sqrt(x[0]);
            double sz2 = sqrt(x[1]);
            double 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
        double sz1 = sqrt(-x[0]);
        double sz2 = sqrt(-x[1]);
        double 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,
    double sz1 = sqrt(x[0]);
    double 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(double *x, double b, double c, double d)    // solve equation
// x^4
// + b*x^2 + c*x + d
//-----------------------------------------------------------------------------
double N4Step(
    double x, double a, double b, double c,
    double d)  // one Newton step for x^4 + a*x^3 + b*x^2 + c*x + d
{
    double fxs = ((4 * x + 3 * a) * x + 2 * b) * x + c;  // f'(x)
    if (fxs == 0) return 1e99;
    double 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 mrpt::math::solve_poly4(
    double* x, double a, double b, double c, double d) noexcept
{  // solve equation x^4 + a*x^3 + b*x^2 + c*x + d by Dekart-Euler method
    // move to a=0:
    double d1 = d + 0.25 * a * (0.25 * b * a - 3. / 64 * a * a * a - c);
    double c1 = c + 0.5 * a * (0.25 * a * a - b);
    double 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)
//-----------------------------------------------------------------------------
static double SolveP5_1(
    double a, double b, double c, double d,
    double e)  // return real root of x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
{
    int cnt;
    if (std::abs(e) < eps) return 0;

    double brd = std::abs(a);  // brd - border of real roots
    if (std::abs(b) > brd) brd = std::abs(b);
    if (std::abs(c) > brd) brd = std::abs(c);
    if (std::abs(d) > brd) brd = std::abs(d);
    if (std::abs(e) > brd) brd = std::abs(e);
    brd++;  // brd - border of real roots

    double x0, f0;  // less, than root
    double x1, f1;  // greater, than root
    double x2, f2, f2s;  // next values, f(x2), f'(x2)
    double dx = 1e8;

    if (e < 0)
    {
        x0 = 0;
        x1 = brd;
        f0 = e;
        f1 = F5(x1);
        x2 = 0.01 * brd;
    }
    else
    {
        x0 = -brd;
        x1 = 0;
        f0 = F5(x0);
        f1 = e;
        x2 = -0.01 * brd;
    }

    if (std::abs(f0) < eps) return x0;
    if (std::abs(f1) < eps) return x1;

    // now x0<x1, f(x0)<0, f(x1)>0
    // Firstly 5 bisections
    for (cnt = 0; cnt < 5; cnt++)
    {
        x2 = (x0 + x1) / 2;  // next point
        f2 = F5(x2);  // f(x2)
        if (std::abs(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
    {
        cnt++;
        if (x2 <= x0 || x2 >= x1) x2 = (x0 + x1) / 2;  // now  x0 < x2 < x1
        f2 = F5(x2);  // f(x2)
        if (std::abs(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 (std::abs(f2s) < eps)
        {
            x2 = 1e99;
            continue;
        }
        dx = f2 / f2s;
        x2 -= dx;
    } while (std::abs(dx) > eps);
    return x2;
}  // SolveP5_1(double a,double b,double c,double d,double e)   // return real
// root of x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
//-----------------------------------------------------------------------------
int mrpt::math::solve_poly5(
    double* x, double a, double b, double c, double d,
    double
        e) noexcept  // solve equation x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
{
    double r = x[0] = SolveP5_1(a, b, c, d, e);
    double a1 = a + r, b1 = b + r * a1, c1 = c + r * b1, d1 = d + r * c1;
    return 1 + solve_poly4(x + 1, a1, b1, c1, d1);
}  // SolveP5(double *x,double a,double b,double c,double d,double e)   // solve
// equation x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
//-----------------------------------------------------------------------------

// a*x^2 + b*x + c = 0
int mrpt::math::solve_poly2(
    double a, double b, double c, double& r1, double& r2) noexcept
{
    if (std::abs(a) < eps)
    {
        // b*x+c=0
        if (std::abs(b) < eps) return 0;
        r1 = -c / b;
        r2 = 1e99;
        return 1;
    }
    else
    {
        double Di = b * b - 4 * a * c;
        if (Di < 0)
        {
            r1 = r2 = 1e99;
            return 0;
        }
        Di = sqrt(Di);
        r1 = (-b + Di) / (2 * a);
        r2 = (-b - Di) / (2 * a);
        // We ensure at output that r1 <= r2
        if (r2 < r1) std::swap(r1, r2);
        return 2;
    }
}