Davismt2.cc

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// #ifndef MT2_BISECT_C
// #define MT2_BISECT_C
/***********************************************************************/
/*                                                                     */
/*              Finding mt2 by Bisection                               */
/*                                                                     */
/*              Authors: Hsin-Chia Cheng, Zhenyu Han                   */
/*              Dec 11, 2008, v1.01a                                   */
/*                                                                     */
/***********************************************************************/
 
/*******************************************************************************
Usage: 
 
1. Define an object of type "mt2":
 
mt2_bisect::mt2 mt2_event;
 
2. Set momenta and the mass of the invisible particle, mn:
 
mt2_event.set_momenta( pa, pb, pmiss );
mt2_event.set_mass( mn );
 
where array pa[0..2], pb[0..2], pmiss[0..2] contains (mass,px,py) 
    for the visible particles and the missing momentum. pmiss[0] is not used. 
    All quantities are given in double.    
 
    3. Use Davismt2::get_mt2() to obtain the value of mt2:
 
double mt2_value = mt2_event.get_mt2();       
 
*******************************************************************************/
 
#include "PhysicsTools/Heppy/interface/Davismt2.h"
// ClassImp(Davismt2);
 
using namespace std;
 
namespace heppy {
 
  /*The user can change the desired precision below, the larger one of the following two definitions is used. Relative precision less than 0.00001 is not guaranteed to be 
  achievable--use with caution*/
 
  const float Davismt2::RELATIVE_PRECISION = 0.00001;
  const float Davismt2::ABSOLUTE_PRECISION = 0.0;
  const float Davismt2::ZERO_MASS = 0.0;
  const float Davismt2::MIN_MASS = 0.1;
  const float Davismt2::SCANSTEP = 0.1;
 
  Davismt2::Davismt2() {
    solved = false;
    momenta_set = false;
    mt2_b = 0.;
    scale = 1.;
    verbose = 1;
  }
 
  Davismt2::~Davismt2() {}
 
  double Davismt2::get_mt2() {
    if (!momenta_set) {
      cout << "Davismt2::get_mt2() ==> Please set momenta first!" << endl;
      return 0;
    }
 
    if (!solved)
      mt2_bisect();
    return mt2_b * scale;
  }
 
  void Davismt2::set_momenta(double* pa0, double* pb0, double* pmiss0) {
    solved = false;  //reset solved tag when momenta are changed.
    momenta_set = true;
 
    ma = fabs(pa0[0]);  // mass cannot be negative
 
    if (ma < ZERO_MASS)
      ma = ZERO_MASS;
 
    pax = pa0[1];
    pay = pa0[2];
    masq = ma * ma;
    Easq = masq + pax * pax + pay * pay;
    Ea = sqrt(Easq);
 
    mb = fabs(pb0[0]);
 
    if (mb < ZERO_MASS)
      mb = ZERO_MASS;
 
    pbx = pb0[1];
    pby = pb0[2];
    mbsq = mb * mb;
    Ebsq = mbsq + pbx * pbx + pby * pby;
    Eb = sqrt(Ebsq);
 
    pmissx = pmiss0[1];
    pmissy = pmiss0[2];
    pmissxsq = pmissx * pmissx;
    pmissysq = pmissy * pmissy;
 
    // set ma>= mb
    if (masq < mbsq) {
      double temp;
      temp = pax;
      pax = pbx;
      pbx = temp;
      temp = pay;
      pay = pby;
      pby = temp;
      temp = Ea;
      Ea = Eb;
      Eb = temp;
      temp = Easq;
      Easq = Ebsq;
      Ebsq = temp;
      temp = masq;
      masq = mbsq;
      mbsq = temp;
      temp = ma;
      ma = mb;
      mb = temp;
    }
    //normalize max{Ea, Eb} to 100
    if (Ea > Eb)
      scale = Ea / 100.;
    else
      scale = Eb / 100.;
 
    if (sqrt(pmissxsq + pmissysq) / 100 > scale)
      scale = sqrt(pmissxsq + pmissysq) / 100;
    //scale = 1;
    double scalesq = scale * scale;
    ma = ma / scale;
    mb = mb / scale;
    masq = masq / scalesq;
    mbsq = mbsq / scalesq;
    pax = pax / scale;
    pay = pay / scale;
    pbx = pbx / scale;
    pby = pby / scale;
    Ea = Ea / scale;
    Eb = Eb / scale;
 
    Easq = Easq / scalesq;
    Ebsq = Ebsq / scalesq;
    pmissx = pmissx / scale;
    pmissy = pmissy / scale;
    pmissxsq = pmissxsq / scalesq;
    pmissysq = pmissysq / scalesq;
    mn = mn_unscale / scale;
    mnsq = mn * mn;
 
    if (ABSOLUTE_PRECISION > 100. * RELATIVE_PRECISION)
      precision = ABSOLUTE_PRECISION;
    else
      precision = 100. * RELATIVE_PRECISION;
  }
 
  void Davismt2::set_mn(double mn0) {
    solved = false;          //reset solved tag when mn is changed.
    mn_unscale = fabs(mn0);  //mass cannot be negative
    mn = mn_unscale / scale;
    mnsq = mn * mn;
  }
 
  void Davismt2::print() {
    cout << " Davismt2::print() ==> pax = " << pax * scale << ";   pay = " << pay * scale << ";   ma = " << ma * scale
         << ";" << endl;
    cout << " Davismt2::print() ==> pbx = " << pbx * scale << ";   pby = " << pby * scale << ";   mb = " << mb * scale
         << ";" << endl;
    cout << " Davismt2::print() ==> pmissx = " << pmissx * scale << ";   pmissy = " << pmissy * scale << ";" << endl;
    cout << " Davismt2::print() ==> mn = " << mn_unscale << ";" << endl;
  }
 
  //special case, the visible particle is massless
  void Davismt2::mt2_massless() {
    //rotate so that pay = 0
    double theta, s, c;
    theta = atan(pay / pax);
    s = sin(theta);
    c = cos(theta);
 
    double pxtemp, pytemp;
    Easq = pax * pax + pay * pay;
    Ebsq = pbx * pbx + pby * pby;
    Ea = sqrt(Easq);
    Eb = sqrt(Ebsq);
 
    pxtemp = pax * c + pay * s;
    pax = pxtemp;
    pay = 0;
    pxtemp = pbx * c + pby * s;
    pytemp = -s * pbx + c * pby;
    pbx = pxtemp;
    pby = pytemp;
    pxtemp = pmissx * c + pmissy * s;
    pytemp = -s * pmissx + c * pmissy;
    pmissx = pxtemp;
    pmissy = pytemp;
 
    a2 = 1 - pbx * pbx / (Ebsq);
    b2 = -pbx * pby / (Ebsq);
    c2 = 1 - pby * pby / (Ebsq);
 
    d21 = (Easq * pbx) / Ebsq;
    d20 = -pmissx + (pbx * (pbx * pmissx + pby * pmissy)) / Ebsq;
    e21 = (Easq * pby) / Ebsq;
    e20 = -pmissy + (pby * (pbx * pmissx + pby * pmissy)) / Ebsq;
    f22 = -(Easq * Easq / Ebsq);
    f21 = -2 * Easq * (pbx * pmissx + pby * pmissy) / Ebsq;
    f20 = mnsq + pmissxsq + pmissysq - (pbx * pmissx + pby * pmissy) * (pbx * pmissx + pby * pmissy) / Ebsq;
 
    double Deltasq0 = 0;
    double Deltasq_low, Deltasq_high;
    int nsols_high, nsols_low;
 
    Deltasq_low = Deltasq0 + precision;
    nsols_low = nsols_massless(Deltasq_low);
 
    if (nsols_low > 1) {
      mt2_b = (double)sqrt(Deltasq0 + mnsq);
      return;
    }
 
    /*   
    if( nsols_massless(Deltasq_high) > 0 )
    {
        mt2_b = (double) sqrt(mnsq+Deltasq0);
        return;
        }*/
 
    //look for when both parablos contain origin
    double Deltasq_high1, Deltasq_high2;
    Deltasq_high1 = 2 * Eb * sqrt(pmissx * pmissx + pmissy * pmissy + mnsq) - 2 * pbx * pmissx - 2 * pby * pmissy;
    Deltasq_high2 = 2 * Ea * mn;
 
    if (Deltasq_high1 < Deltasq_high2)
      Deltasq_high = Deltasq_high2;
    else
      Deltasq_high = Deltasq_high1;
 
    nsols_high = nsols_massless(Deltasq_high);
 
    int foundhigh;
    if (nsols_high == nsols_low) {
      foundhigh = 0;
 
      double minmass, maxmass;
      minmass = mn;
      maxmass = sqrt(mnsq + Deltasq_high);
      for (double mass = minmass + SCANSTEP; mass < maxmass; mass += SCANSTEP) {
        Deltasq_high = mass * mass - mnsq;
 
        nsols_high = nsols_massless(Deltasq_high);
        if (nsols_high > 0) {
          foundhigh = 1;
          Deltasq_low = (mass - SCANSTEP) * (mass - SCANSTEP) - mnsq;
          break;
        }
      }
      if (foundhigh == 0) {
        if (verbose > 0)
          cout << "Davismt2::mt2_massless() ==> Deltasq_high not found at event " << nevt << endl;
 
        mt2_b = (double)sqrt(Deltasq_low + mnsq);
        return;
      }
    }
 
    if (nsols_high == nsols_low) {
      if (verbose > 0)
        cout << "Davismt2::mt2_massless() ==> error: nsols_low=nsols_high=" << nsols_high << endl;
      if (verbose > 0)
        cout << "Davismt2::mt2_massless() ==> Deltasq_high=" << Deltasq_high << endl;
      if (verbose > 0)
        cout << "Davismt2::mt2_massless() ==> Deltasq_low= " << Deltasq_low << endl;
 
      mt2_b = sqrt(mnsq + Deltasq_low);
      return;
    }
    double minmass, maxmass;
    minmass = sqrt(Deltasq_low + mnsq);
    maxmass = sqrt(Deltasq_high + mnsq);
    while (maxmass - minmass > precision) {
      double Delta_mid, midmass, nsols_mid;
      midmass = (minmass + maxmass) / 2.;
      Delta_mid = midmass * midmass - mnsq;
      nsols_mid = nsols_massless(Delta_mid);
      if (nsols_mid != nsols_low)
        maxmass = midmass;
      if (nsols_mid == nsols_low)
        minmass = midmass;
    }
    mt2_b = minmass;
    return;
  }
 
  int Davismt2::nsols_massless(double Dsq) {
    double delta;
    delta = Dsq / (2 * Easq);
    d1 = d11 * delta;
    e1 = e11 * delta;
    f1 = f12 * delta * delta + f10;
    d2 = d21 * delta + d20;
    e2 = e21 * delta + e20;
    f2 = f22 * delta * delta + f21 * delta + f20;
 
    double a, b;
    if (pax > 0)
      a = Ea / Dsq;
    else
      a = -Ea / Dsq;
    if (pax > 0)
      b = -Dsq / (4 * Ea) + mnsq * Ea / Dsq;
    else
      b = Dsq / (4 * Ea) - mnsq * Ea / Dsq;
 
    double A4, A3, A2, A1, A0;
 
    A4 = a * a * a2;
    A3 = 2 * a * b2 / Ea;
    A2 = (2 * a * a2 * b + c2 + 2 * a * d2) / (Easq);
    A1 = (2 * b * b2 + 2 * e2) / (Easq * Ea);
    A0 = (a2 * b * b + 2 * b * d2 + f2) / (Easq * Easq);
 
    long double A3sq;
    A3sq = A3 * A3;
    /*  
        long  double A0sq, A1sq, A2sq, A3sq, A4sq;
    A0sq = A0*A0;
    A1sq = A1*A1;
    A2sq = A2*A2;
    A3sq = A3*A3;
    A4sq = A4*A4;
        */
 
    long double B3, B2, B1, B0;
    B3 = 4 * A4;
    B2 = 3 * A3;
    B1 = 2 * A2;
    B0 = A1;
    long double C2, C1, C0;
    C2 = -(A2 / 2 - 3 * A3sq / (16 * A4));
    C1 = -(3 * A1 / 4. - A2 * A3 / (8 * A4));
    C0 = -A0 + A1 * A3 / (16 * A4);
    long double D1, D0;
    D1 = -B1 - (B3 * C1 * C1 / C2 - B3 * C0 - B2 * C1) / C2;
    D0 = -B0 - B3 * C0 * C1 / (C2 * C2) + B2 * C0 / C2;
 
    long double E0;
    E0 = -C0 - C2 * D0 * D0 / (D1 * D1) + C1 * D0 / D1;
 
    long double t1, t2, t3, t4, t5;
 
    //find the coefficients for the leading term in the Sturm sequence
    t1 = A4;
    t2 = A4;
    t3 = C2;
    t4 = D1;
    t5 = E0;
 
    int nsol;
    nsol = signchange_n(t1, t2, t3, t4, t5) - signchange_p(t1, t2, t3, t4, t5);
    if (nsol < 0)
      nsol = 0;
 
    return nsol;
  }
 
  void Davismt2::mt2_bisect() {
    solved = true;
    cout.precision(11);
 
    //if masses are very small, use code for massless case.
    if (masq < MIN_MASS && mbsq < MIN_MASS) {
      mt2_massless();
      return;
    }
 
    double Deltasq0;
    Deltasq0 = ma * (ma + 2 * mn);  //The minimum mass square to have two ellipses
 
    // find the coefficients for the two quadratic equations when Deltasq=Deltasq0.
 
    a1 = 1 - pax * pax / (Easq);
    b1 = -pax * pay / (Easq);
    c1 = 1 - pay * pay / (Easq);
    d1 = -pax * (Deltasq0 - masq) / (2 * Easq);
    e1 = -pay * (Deltasq0 - masq) / (2 * Easq);
    a2 = 1 - pbx * pbx / (Ebsq);
    b2 = -pbx * pby / (Ebsq);
    c2 = 1 - pby * pby / (Ebsq);
    d2 = -pmissx + pbx * (Deltasq0 - mbsq) / (2 * Ebsq) + pbx * (pbx * pmissx + pby * pmissy) / (Ebsq);
    e2 = -pmissy + pby * (Deltasq0 - mbsq) / (2 * Ebsq) + pby * (pbx * pmissx + pby * pmissy) / (Ebsq);
    f2 = pmissx * pmissx + pmissy * pmissy -
         ((Deltasq0 - mbsq) / (2 * Eb) + (pbx * pmissx + pby * pmissy) / Eb) *
             ((Deltasq0 - mbsq) / (2 * Eb) + (pbx * pmissx + pby * pmissy) / Eb) +
         mnsq;
 
    // find the center of the smaller ellipse
    double x0, y0;
    x0 = (c1 * d1 - b1 * e1) / (b1 * b1 - a1 * c1);
    y0 = (a1 * e1 - b1 * d1) / (b1 * b1 - a1 * c1);
 
    // Does the larger ellipse contain the smaller one?
    double dis = a2 * x0 * x0 + 2 * b2 * x0 * y0 + c2 * y0 * y0 + 2 * d2 * x0 + 2 * e2 * y0 + f2;
 
    if (dis <= 0.01) {
      mt2_b = (double)sqrt(mnsq + Deltasq0);
      return;
    }
 
    /* find the coefficients for the two quadratic equations           */
    /* coefficients for quadratic terms do not change                  */
    /* coefficients for linear and constant terms are polynomials of   */
    /*       delta=(Deltasq-m7sq)/(2 E7sq)                             */
    d11 = -pax;
    e11 = -pay;
    f10 = mnsq;
    f12 = -Easq;
    d21 = (Easq * pbx) / Ebsq;
    d20 = ((masq - mbsq) * pbx) / (2. * Ebsq) - pmissx + (pbx * (pbx * pmissx + pby * pmissy)) / Ebsq;
    e21 = (Easq * pby) / Ebsq;
    e20 = ((masq - mbsq) * pby) / (2. * Ebsq) - pmissy + (pby * (pbx * pmissx + pby * pmissy)) / Ebsq;
    f22 = -Easq * Easq / Ebsq;
    f21 = (-2 * Easq * ((masq - mbsq) / (2. * Eb) + (pbx * pmissx + pby * pmissy) / Eb)) / Eb;
    f20 = mnsq + pmissx * pmissx + pmissy * pmissy -
          ((masq - mbsq) / (2. * Eb) + (pbx * pmissx + pby * pmissy) / Eb) *
              ((masq - mbsq) / (2. * Eb) + (pbx * pmissx + pby * pmissy) / Eb);
 
    //Estimate upper bound of mT2
    //when Deltasq > Deltasq_high1, the larger encloses the center of the smaller
    double p2x0, p2y0;
    double Deltasq_high1;
    p2x0 = pmissx - x0;
    p2y0 = pmissy - y0;
    Deltasq_high1 = 2 * Eb * sqrt(p2x0 * p2x0 + p2y0 * p2y0 + mnsq) - 2 * pbx * p2x0 - 2 * pby * p2y0 + mbsq;
 
    //Another estimate, if both ellipses enclose the origin, Deltasq > mT2
 
    double Deltasq_high2, Deltasq_high21, Deltasq_high22;
    Deltasq_high21 =
        2 * Eb * sqrt(pmissx * pmissx + pmissy * pmissy + mnsq) - 2 * pbx * pmissx - 2 * pby * pmissy + mbsq;
    Deltasq_high22 = 2 * Ea * mn + masq;
 
    if (Deltasq_high21 < Deltasq_high22)
      Deltasq_high2 = Deltasq_high22;
    else
      Deltasq_high2 = Deltasq_high21;
 
    //pick the smaller upper bound
    double Deltasq_high;
    if (Deltasq_high1 < Deltasq_high2)
      Deltasq_high = Deltasq_high1;
    else
      Deltasq_high = Deltasq_high2;
 
    double Deltasq_low;  //lower bound
    Deltasq_low = Deltasq0;
 
    //number of solutions at Deltasq_low should not be larger than zero
    if (nsols(Deltasq_low) > 0) {
      //cout << "Davismt2::mt2_bisect() ==> nsolutions(Deltasq_low) > 0"<<endl;
      mt2_b = (double)sqrt(mnsq + Deltasq0);
      return;
    }
 
    int nsols_high, nsols_low;
 
    nsols_low = nsols(Deltasq_low);
    int foundhigh;
 
    //if nsols_high=nsols_low, we missed the region where the two ellipse overlap
    //if nsols_high=4, also need a scan because we may find the wrong tangent point.
 
    nsols_high = nsols(Deltasq_high);
 
    if (nsols_high == nsols_low || nsols_high == 4) {
      //foundhigh = scan_high(Deltasq_high);
      foundhigh = find_high(Deltasq_high);
      if (foundhigh == 0) {
        if (verbose > 0)
          cout << "Davismt2::mt2_bisect() ==> Deltasq_high not found at event " << nevt << endl;
        mt2_b = sqrt(Deltasq_low + mnsq);
        return;
      }
    }
 
    while (sqrt(Deltasq_high + mnsq) - sqrt(Deltasq_low + mnsq) > precision) {
      double Deltasq_mid, nsols_mid;
      //bisect
      Deltasq_mid = (Deltasq_high + Deltasq_low) / 2.;
      nsols_mid = nsols(Deltasq_mid);
      // if nsols_mid = 4, rescan for Deltasq_high
      if (nsols_mid == 4) {
        Deltasq_high = Deltasq_mid;
        //scan_high(Deltasq_high);
        find_high(Deltasq_high);
        continue;
      }
 
      if (nsols_mid != nsols_low)
        Deltasq_high = Deltasq_mid;
      if (nsols_mid == nsols_low)
        Deltasq_low = Deltasq_mid;
    }
    mt2_b = (double)sqrt(mnsq + Deltasq_high);
    return;
  }
 
  int Davismt2::find_high(double& Deltasq_high) {
    double x0, y0;
    x0 = (c1 * d1 - b1 * e1) / (b1 * b1 - a1 * c1);
    y0 = (a1 * e1 - b1 * d1) / (b1 * b1 - a1 * c1);
    double Deltasq_low = (mn + ma) * (mn + ma) - mnsq;
    do {
      double Deltasq_mid = (Deltasq_high + Deltasq_low) / 2.;
      int nsols_mid = nsols(Deltasq_mid);
      if (nsols_mid == 2) {
        Deltasq_high = Deltasq_mid;
        return 1;
      } else if (nsols_mid == 4) {
        Deltasq_high = Deltasq_mid;
        continue;
      } else if (nsols_mid == 0) {
        d1 = -pax * (Deltasq_mid - masq) / (2 * Easq);
        e1 = -pay * (Deltasq_mid - masq) / (2 * Easq);
        d2 = -pmissx + pbx * (Deltasq_mid - mbsq) / (2 * Ebsq) + pbx * (pbx * pmissx + pby * pmissy) / (Ebsq);
        e2 = -pmissy + pby * (Deltasq_mid - mbsq) / (2 * Ebsq) + pby * (pbx * pmissx + pby * pmissy) / (Ebsq);
        f2 = pmissx * pmissx + pmissy * pmissy -
             ((Deltasq_mid - mbsq) / (2 * Eb) + (pbx * pmissx + pby * pmissy) / Eb) *
                 ((Deltasq_mid - mbsq) / (2 * Eb) + (pbx * pmissx + pby * pmissy) / Eb) +
             mnsq;
        // Does the larger ellipse contain the smaller one?
        double dis = a2 * x0 * x0 + 2 * b2 * x0 * y0 + c2 * y0 * y0 + 2 * d2 * x0 + 2 * e2 * y0 + f2;
        if (dis < 0)
          Deltasq_high = Deltasq_mid;
        else
          Deltasq_low = Deltasq_mid;
      }
 
    } while (Deltasq_high - Deltasq_low > 0.001);
    return 0;
  }
 
  int Davismt2::scan_high(double& Deltasq_high) {
    int foundhigh = 0;
    int nsols_high;
 
    //        double Deltasq_low;
    double tempmass, maxmass;
    tempmass = mn + ma;
    maxmass = sqrt(mnsq + Deltasq_high);
    if (nevt == 32334)
      cout << "Davismt2::scan_high() ==> Deltasq_high = " << Deltasq_high << endl;  // ???
    for (double mass = tempmass + SCANSTEP; mass < maxmass; mass += SCANSTEP) {
      Deltasq_high = mass * mass - mnsq;
      nsols_high = nsols(Deltasq_high);
 
      if (nsols_high > 0) {
        // Deltasq_low = (mass-SCANSTEP)*(mass-SCANSTEP) - mnsq;
        foundhigh = 1;
        break;
      }
    }
    return foundhigh;
  }
 
  int Davismt2::nsols(double Dsq) {
    double delta = (Dsq - masq) / (2 * Easq);
 
    //calculate coefficients for the two quadratic equations
    d1 = d11 * delta;
    e1 = e11 * delta;
    f1 = f12 * delta * delta + f10;
    d2 = d21 * delta + d20;
    e2 = e21 * delta + e20;
    f2 = f22 * delta * delta + f21 * delta + f20;
 
    //obtain the coefficients for the 4th order equation
    //devided by Ea^n to make the variable dimensionless
    long double A4, A3, A2, A1, A0;
 
    A4 = -4 * a2 * b1 * b2 * c1 + 4 * a1 * b2 * b2 * c1 + a2 * a2 * c1 * c1 + 4 * a2 * b1 * b1 * c2 -
         4 * a1 * b1 * b2 * c2 - 2 * a1 * a2 * c1 * c2 + a1 * a1 * c2 * c2;
 
    A3 = (-4 * a2 * b2 * c1 * d1 + 8 * a2 * b1 * c2 * d1 - 4 * a1 * b2 * c2 * d1 - 4 * a2 * b1 * c1 * d2 +
          8 * a1 * b2 * c1 * d2 - 4 * a1 * b1 * c2 * d2 - 8 * a2 * b1 * b2 * e1 + 8 * a1 * b2 * b2 * e1 +
          4 * a2 * a2 * c1 * e1 - 4 * a1 * a2 * c2 * e1 + 8 * a2 * b1 * b1 * e2 - 8 * a1 * b1 * b2 * e2 -
          4 * a1 * a2 * c1 * e2 + 4 * a1 * a1 * c2 * e2) /
         Ea;
 
    A2 = (4 * a2 * c2 * d1 * d1 - 4 * a2 * c1 * d1 * d2 - 4 * a1 * c2 * d1 * d2 + 4 * a1 * c1 * d2 * d2 -
          8 * a2 * b2 * d1 * e1 - 8 * a2 * b1 * d2 * e1 + 16 * a1 * b2 * d2 * e1 + 4 * a2 * a2 * e1 * e1 +
          16 * a2 * b1 * d1 * e2 - 8 * a1 * b2 * d1 * e2 - 8 * a1 * b1 * d2 * e2 - 8 * a1 * a2 * e1 * e2 +
          4 * a1 * a1 * e2 * e2 - 4 * a2 * b1 * b2 * f1 + 4 * a1 * b2 * b2 * f1 + 2 * a2 * a2 * c1 * f1 -
          2 * a1 * a2 * c2 * f1 + 4 * a2 * b1 * b1 * f2 - 4 * a1 * b1 * b2 * f2 - 2 * a1 * a2 * c1 * f2 +
          2 * a1 * a1 * c2 * f2) /
         Easq;
 
    A1 = (-8 * a2 * d1 * d2 * e1 + 8 * a1 * d2 * d2 * e1 + 8 * a2 * d1 * d1 * e2 - 8 * a1 * d1 * d2 * e2 -
          4 * a2 * b2 * d1 * f1 - 4 * a2 * b1 * d2 * f1 + 8 * a1 * b2 * d2 * f1 + 4 * a2 * a2 * e1 * f1 -
          4 * a1 * a2 * e2 * f1 + 8 * a2 * b1 * d1 * f2 - 4 * a1 * b2 * d1 * f2 - 4 * a1 * b1 * d2 * f2 -
          4 * a1 * a2 * e1 * f2 + 4 * a1 * a1 * e2 * f2) /
         (Easq * Ea);
 
    A0 = (-4 * a2 * d1 * d2 * f1 + 4 * a1 * d2 * d2 * f1 + a2 * a2 * f1 * f1 + 4 * a2 * d1 * d1 * f2 -
          4 * a1 * d1 * d2 * f2 - 2 * a1 * a2 * f1 * f2 + a1 * a1 * f2 * f2) /
         (Easq * Easq);
 
    long double A3sq;
    /*
    long  double A0sq, A1sq, A2sq, A3sq, A4sq;
    A0sq = A0*A0;
    A1sq = A1*A1;
    A2sq = A2*A2;
    A4sq = A4*A4;
        */
    A3sq = A3 * A3;
 
    long double B3, B2, B1, B0;
    B3 = 4 * A4;
    B2 = 3 * A3;
    B1 = 2 * A2;
    B0 = A1;
 
    long double C2, C1, C0;
    C2 = -(A2 / 2 - 3 * A3sq / (16 * A4));
    C1 = -(3 * A1 / 4. - A2 * A3 / (8 * A4));
    C0 = -A0 + A1 * A3 / (16 * A4);
 
    long double D1, D0;
    D1 = -B1 - (B3 * C1 * C1 / C2 - B3 * C0 - B2 * C1) / C2;
    D0 = -B0 - B3 * C0 * C1 / (C2 * C2) + B2 * C0 / C2;
 
    long double E0;
    E0 = -C0 - C2 * D0 * D0 / (D1 * D1) + C1 * D0 / D1;
 
    long double t1, t2, t3, t4, t5;
    //find the coefficients for the leading term in the Sturm sequence
    t1 = A4;
    t2 = A4;
    t3 = C2;
    t4 = D1;
    t5 = E0;
 
    //The number of solutions depends on diffence of number of sign changes for x->Inf and x->-Inf
    int nsol;
    nsol = signchange_n(t1, t2, t3, t4, t5) - signchange_p(t1, t2, t3, t4, t5);
 
    //Cannot have negative number of solutions, must be roundoff effect
    if (nsol < 0)
      nsol = 0;
 
    return nsol;
  }
 
  //inline
  int Davismt2::signchange_n(long double t1, long double t2, long double t3, long double t4, long double t5) {
    int nsc;
    nsc = 0;
    if (t1 * t2 > 0)
      nsc++;
    if (t2 * t3 > 0)
      nsc++;
    if (t3 * t4 > 0)
      nsc++;
    if (t4 * t5 > 0)
      nsc++;
    return nsc;
  }
 
  //inline
  int Davismt2::signchange_p(long double t1, long double t2, long double t3, long double t4, long double t5) {
    int nsc;
    nsc = 0;
    if (t1 * t2 < 0)
      nsc++;
    if (t2 * t3 < 0)
      nsc++;
    if (t3 * t4 < 0)
      nsc++;
    if (t4 * t5 < 0)
      nsc++;
    return nsc;
  }
 
}  // namespace heppy