TrackFwd.cxx

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// Copyright 2019-2020 CERN and copyright holders of ALICE O2.
// See https://alice-o2.web.cern.ch/copyright for details of the copyright holders.
// All rights not expressly granted are reserved.
//
// This software is distributed under the terms of the GNU General Public
// License v3 (GPL Version 3), copied verbatim in the file "COPYING".
//
// In applying this license CERN does not waive the privileges and immunities
// granted to it by virtue of its status as an Intergovernmental Organization
// or submit itself to any jurisdiction.
 
#include "ReconstructionDataFormats/TrackFwd.h"
#include "ReconstructionDataFormats/TrackParametrization.h"
#include "ReconstructionDataFormats/TrackParametrizationWithError.h"
#include "Math/MatrixFunctions.h"
#include <GPUCommonLogger.h>
 
namespace o2
{
namespace track
{
using namespace std;
 
//_________________________________________________________________________
TrackParCovFwd::TrackParCovFwd(const Double_t z, const SMatrix5& parameters, const SMatrix55Sym& covariances, const Double_t chi2)
{
  setZ(z);
  setParameters(parameters);
  setCovariances(covariances);
  setTrackChi2(chi2);
}
 
//__________________________________________________________________________
void TrackParFwd::propagateParamToZlinear(double zEnd)
{
  // Track parameters linearly extrapolated to the plane at "zEnd".
 
  if (getZ() == zEnd) {
    return; // nothing to be done if same z
  }
 
  // Compute track parameters
  auto dZ = (zEnd - getZ());
  auto phi0 = getPhi();
  auto [sinphi0, cosphi0] = o2::math_utils::sincosd(phi0);
  auto invtanl0 = 1.0 / getTgl();
  auto n = dZ * invtanl0;
  mParameters(0) += n * cosphi0;
  mParameters(1) += n * sinphi0;
  mZ = zEnd;
}
 
//__________________________________________________________________________
void TrackParCovFwd::propagateToZlinear(double zEnd)
{
  // Track parameters and their covariances linearly extrapolated to the plane at "zEnd".
 
  // Calculate the jacobian related to the track parameters extrapolated to "zEnd"
  auto dZ = (zEnd - getZ());
  auto phi0 = getPhi();
  auto tanl0 = getTanl();
  auto invtanl0 = 1.0 / tanl0;
  auto [sinphi0, cosphi0] = o2::math_utils::sincosd(phi0);
  auto n = dZ * invtanl0;
  auto m = n * invtanl0;
 
  // Extrapolate track parameters to "zEnd"
  mParameters(0) += n * cosphi0;
  mParameters(1) += n * sinphi0;
  setZ(zEnd);
 
  // Calculate Jacobian
  SMatrix55Std jacob = ROOT::Math::SMatrixIdentity();
  jacob(0, 2) = -n * sinphi0;
  jacob(0, 3) = -m * cosphi0;
  jacob(1, 2) = n * cosphi0;
  jacob(1, 3) = -m * sinphi0;
 
  // Extrapolate track parameter covariances to "zEnd"
  setCovariances(ROOT::Math::Similarity(jacob, mCovariances));
}
 
//__________________________________________________________________________
void TrackParFwd::propagateParamToZquadratic(double zEnd, double zField)
{
  // Track parameters extrapolated to the plane at "zEnd" considering a helix
 
  if (getZ() == zEnd) {
    return; // nothing to be done if same z
  }
 
  // Compute track parameters
  auto dZ = (zEnd - getZ());
  auto phi0 = getPhi();
  auto [sinphi0, cosphi0] = o2::math_utils::sincosd(phi0);
  auto invtanl0 = 1.0 / getTanl();
  auto invqpt0 = getInvQPt();
  auto Hz = std::copysign(1, zField);
  auto k = TMath::Abs(o2::constants::math::B2C * zField);
  auto n = dZ * invtanl0;
  auto theta = -invqpt0 * dZ * k * invtanl0;
 
  mParameters(0) += n * cosphi0 - 0.5 * n * theta * Hz * sinphi0;
  mParameters(1) += n * sinphi0 + 0.5 * n * theta * Hz * cosphi0;
  mParameters(2) += Hz * theta;
  setZ(zEnd);
}
 
//__________________________________________________________________________
void TrackParCovFwd::propagateToZquadratic(double zEnd, double zField)
{
  // Extrapolate track parameters and covariances matrix to "zEnd"
  // using quadratic track model
 
  if (getZ() == zEnd) {
    return; // nothing to be done if same z
  }
 
  // Compute track parameters
  auto dZ = (zEnd - getZ());
  auto phi0 = getPhi();
  auto [sinphi0, cosphi0] = o2::math_utils::sincosd(phi0);
  auto invtanl0 = 1.0 / getTanl();
  auto invqpt0 = getInvQPt();
  auto Hz = std::copysign(1, zField);
  auto k = TMath::Abs(o2::constants::math::B2C * zField);
  auto n = dZ * invtanl0;
  auto m = n * invtanl0;
  auto theta = -invqpt0 * dZ * k * invtanl0;
 
  // Extrapolate track parameters to "zEnd"
  mParameters(0) += n * cosphi0 - 0.5 * n * theta * Hz * sinphi0;
  mParameters(1) += n * sinphi0 + 0.5 * n * theta * Hz * cosphi0;
  mParameters(2) += Hz * theta;
  mZ = zEnd;
 
  // Calculate Jacobian
  SMatrix55Std jacob = ROOT::Math::SMatrixIdentity();
  jacob(0, 2) = -n * theta * 0.5 * Hz * cosphi0 - n * sinphi0;
  jacob(0, 3) = Hz * m * theta * sinphi0 - m * cosphi0;
  jacob(0, 4) = k * m * 0.5 * Hz * dZ * sinphi0;
  jacob(1, 2) = -n * theta * 0.5 * Hz * sinphi0 + n * cosphi0;
  jacob(1, 3) = -Hz * m * theta * cosphi0 - m * sinphi0;
  jacob(1, 4) = -k * m * 0.5 * Hz * dZ * cosphi0;
  jacob(2, 3) = -Hz * theta * invtanl0;
  jacob(2, 4) = -Hz * k * n;
 
  // Extrapolate track parameter covariances to "zEnd"
  setCovariances(ROOT::Math::Similarity(jacob, mCovariances));
}
 
//__________________________________________________________________________
void TrackParFwd::propagateParamToZhelix(double zEnd, double zField)
{
  // Track parameters extrapolated to the plane at "zEnd"
  // using helix track model
 
  if (getZ() == zEnd) {
    return; // nothing to be done if same z
  }
 
  // Compute track parameters
  auto dZ = (zEnd - getZ());
  auto phi0 = getPhi();
  auto tanl0 = getTanl();
  auto invtanl0 = 1.0 / tanl0;
  auto invqpt0 = getInvQPt();
  auto qpt0 = 1.0 / invqpt0;
  auto [sinphi0, cosphi0] = o2::math_utils::sincosd(phi0);
 
  auto k = TMath::Abs(o2::constants::math::B2C * zField);
  auto invk = 1.0 / k;
  auto theta = -invqpt0 * dZ * k * invtanl0;
  auto [sintheta, costheta] = o2::math_utils::sincosd(theta);
  auto Hz = std::copysign(1, zField);
  auto Y = sinphi0 * qpt0 * invk;
  auto X = cosphi0 * qpt0 * invk;
  auto YC = Y * costheta;
  auto YS = Y * sintheta;
  auto XC = X * costheta;
  auto XS = X * sintheta;
 
  // Extrapolate track parameters to "zEnd"
  mParameters(0) += Hz * (Y - YC) - XS;
  mParameters(1) += Hz * (-X + XC) - YS;
  mParameters(2) += Hz * theta;
  mZ = zEnd;
}
 
//__________________________________________________________________________
void TrackParCovFwd::propagateToZhelix(double zEnd, double zField)
{
  // Extrapolate track parameters and covariances matrix to "zEnd"
  // using helix track model
 
  auto dZ = (zEnd - getZ());
  auto phi0 = getPhi();
  auto tanl0 = getTanl();
  auto invtanl0 = 1.0 / tanl0;
  auto invqpt0 = getInvQPt();
  auto qpt0 = 1.0 / invqpt0;
  auto [sinphi0, cosphi0] = o2::math_utils::sincosd(phi0);
  auto k = TMath::Abs(o2::constants::math::B2C * zField);
  auto invk = 1.0 / k;
  auto theta = -invqpt0 * dZ * k * invtanl0;
  auto [sintheta, costheta] = o2::math_utils::sincosd(theta);
  auto Hz = std::copysign(1, zField);
  auto L = qpt0 * qpt0 * invk;
  auto N = dZ * invtanl0 * qpt0;
  auto O = sintheta * cosphi0;
  auto P = sinphi0 * costheta;
  auto R = sinphi0 * sintheta;
  auto S = cosphi0 * costheta;
  auto Y = sinphi0 * qpt0 * invk;
  auto X = cosphi0 * qpt0 * invk;
  auto YC = Y * costheta;
  auto YS = Y * sintheta;
  auto XC = X * costheta;
  auto XS = X * sintheta;
  auto T = qpt0 * costheta;
  auto U = qpt0 * sintheta;
  auto V = qpt0;
  auto n = dZ * invtanl0;
  auto m = n * invtanl0;
 
  // Extrapolate track parameters to "zEnd"
  mParameters(0) += Hz * (Y - YC) - XS;
  mParameters(1) += Hz * (-X + XC) - YS;
  mParameters(2) += Hz * theta;
  mZ = zEnd;
 
  // Calculate Jacobian
  SMatrix55Std jacob = ROOT::Math::SMatrixIdentity();
  jacob(0, 2) = Hz * X - Hz * XC + YS;
  jacob(0, 3) = Hz * R * m - S * m;
  jacob(0, 4) = -Hz * N * R + Hz * T * Y - Hz * V * Y + N * S + U * X;
  jacob(1, 2) = Hz * Y - Hz * YC - XS;
  jacob(1, 3) = -Hz * O * m - P * m;
  jacob(1, 4) = Hz * N * O - Hz * T * X + Hz * V * X + N * P + U * Y;
  jacob(2, 3) = -Hz * theta * invtanl0;
  jacob(2, 4) = -Hz * k * n;
 
  // Extrapolate track parameter covariances to "zEnd"
  setCovariances(ROOT::Math::Similarity(jacob, mCovariances));
}
 
//__________________________________________________________________________
void TrackParCovFwd::propagateToZ(double zEnd, double zField)
{
  // Security for zero B field
  if (zField == 0.0) {
    propagateToZlinear(zEnd);
    return;
  }
 
  // Extrapolate track parameters and covariances matrix to "zEnd"
  // Parameters: helix track model; Error propagation: Quadratic
 
  auto dZ = (zEnd - getZ());
  auto phi0 = getPhi();
  auto tanl0 = getTanl();
  auto invtanl0 = 1.0 / tanl0;
  auto invqpt0 = getInvQPt();
  auto qpt0 = 1.0 / invqpt0;
  auto [sinphi0, cosphi0] = o2::math_utils::sincosd(phi0);
  auto k = TMath::Abs(o2::constants::math::B2C * zField);
  auto invk = 1.0 / k;
  auto theta = -invqpt0 * dZ * k * invtanl0;
  auto [sintheta, costheta] = o2::math_utils::sincosd(theta);
  auto Hz = std::copysign(1, zField);
  auto Y = sinphi0 * qpt0 * invk;
  auto X = cosphi0 * qpt0 * invk;
  auto YC = Y * costheta;
  auto YS = Y * sintheta;
  auto XC = X * costheta;
  auto XS = X * sintheta;
  auto n = dZ * invtanl0;
  auto m = n * invtanl0;
 
  // Extrapolate track parameters to "zEnd"
  // Helix
  mParameters(0) += Hz * (Y - YC) - XS;
  mParameters(1) += Hz * (-X + XC) - YS;
  mParameters(2) += Hz * theta;
  mZ = zEnd;
 
  // Jacobian (quadratic)
  SMatrix55Std jacob = ROOT::Math::SMatrixIdentity();
  jacob(0, 2) = -n * theta * 0.5 * Hz * cosphi0 - n * sinphi0;
  jacob(0, 3) = Hz * m * theta * sinphi0 - m * cosphi0;
  jacob(0, 4) = k * m * 0.5 * Hz * dZ * sinphi0;
  jacob(1, 2) = -n * theta * 0.5 * Hz * sinphi0 + n * cosphi0;
  jacob(1, 3) = -Hz * m * theta * cosphi0 - m * sinphi0;
  jacob(1, 4) = -k * m * 0.5 * Hz * dZ * cosphi0;
  jacob(2, 3) = -Hz * theta * invtanl0;
  jacob(2, 4) = -Hz * k * n;
 
  // Extrapolate track parameter covariances to "zEnd"
  setCovariances(ROOT::Math::Similarity(jacob, mCovariances));
}
 
//__________________________________________________________________________
bool TrackParCovFwd::update(const std::array<float, 2>& p, const std::array<float, 2>& cov)
{
 
  using SVector2 = ROOT::Math::SVector<double, 2>;
  using SMatrix22 = ROOT::Math::SMatrix<double, 2>;
  using SMatrix25 = ROOT::Math::SMatrix<double, 2, 5>;
  using SMatrix52 = ROOT::Math::SMatrix<double, 5, 2>;
 
  SMatrix55Sym I = ROOT::Math::SMatrixIdentity();
  SMatrix25 H_k;
  SMatrix22 V_k;
  SVector2 m_k(p[0], p[1]), r_k_kminus1;
  V_k(0, 0) = cov[0];
  V_k(1, 1) = cov[1];
  H_k(0, 0) = 1.0;
  H_k(1, 1) = 1.0;
 
  // Covariance of residuals
  SMatrix22 invResCov = (V_k + ROOT::Math::Similarity(H_k, mCovariances));
  invResCov.Invert();
 
  // Kalman Gain Matrix
  SMatrix52 K_k = mCovariances * ROOT::Math::Transpose(H_k) * invResCov;
 
  // Update Parameters
  r_k_kminus1 = m_k - H_k * mParameters; // Residuals of prediction
  mParameters = mParameters + K_k * r_k_kminus1;
 
  // Update covariances Matrix
  SMatrix55Std updatedCov;
  auto& CP = mCovariances;
  auto& sigmax2 = cov[0];
  auto& sigmay2 = cov[1];
  auto A = 1. / (sigmax2 * sigmay2 + sigmax2 * CP(1, 1) + sigmay2 * CP(0, 0) + CP(0, 0) * CP(1, 1) - CP(0, 1) * CP(0, 1));
  auto AX = A * sigmax2;
  auto AY = A * sigmay2;
  auto B = sigmax2 * sigmay2;
  auto C = (sigmax2 + CP(0, 0)) * (sigmay2 + CP(1, 1));
  auto D = 1 / (-C + CP(0, 1) * CP(0, 1));
  auto E = sigmax2 + CP(0, 0);
  auto F = sigmay2 + CP(1, 1);
  auto G = -C + CP(0, 1) * CP(0, 1);
 
  // Explicit evaluation of "updatedCov = (I - K_k * H_k) * mCovariances"
  updatedCov(0, 0) = AX * (sigmay2 * CP(0, 0) + CP(0, 0) * CP(1, 1) - CP(0, 1) * CP(0, 1));
  updatedCov(0, 1) = AX * sigmay2 * CP(0, 1);
  updatedCov(0, 2) = AX * (sigmay2 * CP(0, 2) - CP(0, 1) * CP(1, 2) + CP(0, 2) * CP(1, 1));
  updatedCov(0, 3) = AX * (sigmay2 * CP(0, 3) - CP(0, 1) * CP(1, 3) + CP(0, 3) * CP(1, 1));
  updatedCov(0, 4) = AX * (sigmay2 * CP(0, 4) - CP(0, 1) * CP(1, 4) + CP(0, 4) * CP(1, 1));
  updatedCov(1, 1) = AY * (sigmax2 * CP(1, 1) + CP(0, 0) * CP(1, 1) - CP(0, 1) * CP(0, 1));
  updatedCov(1, 2) = AY * (sigmax2 * CP(1, 2) + CP(0, 0) * CP(1, 2) - CP(0, 1) * CP(0, 2));
  updatedCov(1, 3) = AY * (sigmax2 * CP(1, 3) + CP(0, 0) * CP(1, 3) - CP(0, 1) * CP(0, 3));
  updatedCov(1, 4) = AY * (sigmax2 * CP(1, 4) + CP(0, 0) * CP(1, 4) - CP(0, 1) * CP(0, 4));
  updatedCov(2, 2) = D * (G * CP(2, 2) - CP(0, 2) * (-F * CP(0, 2) + CP(0, 1) * CP(1, 2)) - CP(1, 2) * (-E * CP(1, 2) + CP(0, 1) * CP(0, 2)));
  updatedCov(2, 3) = D * (G * CP(2, 3) - CP(0, 2) * (-F * CP(0, 3) + CP(0, 1) * CP(1, 3)) - CP(1, 2) * (-E * CP(1, 3) + CP(0, 1) * CP(0, 3)));
  updatedCov(2, 4) = D * (G * CP(2, 4) - CP(0, 2) * (-F * CP(0, 4) + CP(0, 1) * CP(1, 4)) - CP(1, 2) * (-E * CP(1, 4) + CP(0, 1) * CP(0, 4)));
  updatedCov(3, 3) = D * (G * CP(3, 3) - CP(0, 3) * (-F * CP(0, 3) + CP(0, 1) * CP(1, 3)) - CP(1, 3) * (-E * CP(1, 3) + CP(0, 1) * CP(0, 3)));
  updatedCov(3, 4) = D * (G * CP(3, 4) - CP(0, 3) * (-F * CP(0, 4) + CP(0, 1) * CP(1, 4)) - CP(1, 3) * (-E * CP(1, 4) + CP(0, 1) * CP(0, 4)));
  updatedCov(4, 4) = D * (G * CP(4, 4) - CP(0, 4) * (-F * CP(0, 4) + CP(0, 1) * CP(1, 4)) - CP(1, 4) * (-E * CP(1, 4) + CP(0, 1) * CP(0, 4)));
 
  mCovariances(0, 0) = updatedCov(0, 0);
  mCovariances(0, 1) = updatedCov(0, 1);
  mCovariances(0, 2) = updatedCov(0, 2);
  mCovariances(0, 3) = updatedCov(0, 3);
  mCovariances(0, 4) = updatedCov(0, 4);
  mCovariances(1, 1) = updatedCov(1, 1);
  mCovariances(1, 2) = updatedCov(1, 2);
  mCovariances(1, 3) = updatedCov(1, 3);
  mCovariances(1, 4) = updatedCov(1, 4);
  mCovariances(2, 2) = updatedCov(2, 2);
  mCovariances(2, 3) = updatedCov(2, 3);
  mCovariances(2, 4) = updatedCov(2, 4);
  mCovariances(3, 3) = updatedCov(3, 3);
  mCovariances(3, 4) = updatedCov(3, 4);
  mCovariances(4, 4) = updatedCov(4, 4);
 
  auto addChi2Track = ROOT::Math::Similarity(r_k_kminus1, invResCov);
  mTrackChi2 += addChi2Track;
 
  return true;
}
 
//__________________________________________________________________________
void TrackParCovFwd::addMCSEffect(double x_over_X0)
{
 
  if (x_over_X0 == 0) { // Nothing to do
    return;
  }
 
  auto phi0 = getPhi();
  auto tanl0 = getTanl();
  auto invtanl0 = 1.0 / tanl0;
  auto invqpt0 = getInvQPt();
 
  auto [sinphi0, cosphi0] = o2::math_utils::sincosd(phi0);
 
  auto csclambda = TMath::Abs(TMath::Sqrt(1 + tanl0 * tanl0) * invtanl0);
  auto pathLengthOverX0 = x_over_X0 * csclambda; //
 
  // Angular dispersion square of the track (variance) in a plane perpendicular to the trajectory
  auto sigmathetasq = 0.0136 * getInverseMomentum();
  sigmathetasq *= sigmathetasq * pathLengthOverX0;
 
  // Get covariance matrix
  SMatrix55Sym newParamCov(getCovariances());
 
  auto A = tanl0 * tanl0 + 1;
 
  newParamCov(2, 2) += sigmathetasq * A;
 
  newParamCov(3, 3) += sigmathetasq * A * A;
 
  newParamCov(4, 4) += sigmathetasq * tanl0 * tanl0 * invqpt0 * invqpt0;
 
  // Set new covariances
  setCovariances(newParamCov);
}
 
//_______________________________________________________
void TrackParFwd::getCircleParams(float bz, o2::math_utils::CircleXY<float>& c, float& sna, float& csa) const
{
  c.rC = getCurvature(bz);
  constexpr double MinCurv = 1e-6;
  if (std::abs(c.rC) > MinCurv) {
    c.rC = 1.f / getCurvature(bz);
    double sn = getSnp(), cs = std::sqrt((1.f - sn) * (1.f + sn));
    c.xC = getX() - sn * c.rC; // center in tracking
    c.yC = getY() + cs * c.rC; // frame. Note: r is signed!!!
    c.rC = std::abs(c.rC);
  } else {
    c.rC = 0.f; // signal straight line
    c.xC = getX();
    c.yC = getY();
  }
}
//________________________________________________________________
bool TrackParCovFwd::propagateToVtxhelixWithMCS(double z, const std::array<float, 2>& p, const std::array<float, 2>& cov, double field, double x_over_X0)
{
  // Propagate fwd track to vertex using helix model, adding MCS effects
  addMCSEffect(x_over_X0);
  propagateToZhelix(z, field);
  return update(p, cov);
}
//________________________________________________________________
bool TrackParCovFwd::propagateToVtxlinearWithMCS(double z, const std::array<float, 2>& p, const std::array<float, 2>& cov, double x_over_X0)
{
  // Propagate fwd track to vertex using linear model, adding MCS effects
  addMCSEffect(x_over_X0);
  propagateToZlinear(z);
  return update(p, cov);
}
 
bool TrackParCovFwd::getCovXYZPxPyPzGlo(std::array<float, 21>& cv) const
{
  //---------------------------------------------------------------------
  // This function returns the global covariance matrix of the fwdtrack params
  //
  // Cov(x,x) ... :   cv[0]
  // Cov(y,x) ... :   cv[1]  cv[2]
  // Cov(z,x) ... :   cv[3]  cv[4]  cv[5]
  // Cov(px,x)... :   cv[6]  cv[7]  cv[8]  cv[9]
  // Cov(py,x)... :   cv[10] cv[11] cv[12] cv[13] cv[14]
  // Cov(pz,x)... :   cv[15] cv[16] cv[17] cv[18] cv[19] cv[20]
  //---------------------------------------------------------------------
  auto pt = getPt();
  auto cp = std::sqrt((1. - getSnp()) * (1. + getSnp()));
  auto sp = getSnp();
  auto tgl = getTgl();
 
  auto px = pt * std::sqrt((1. - getSnp()) * (1. + getSnp()));
  auto py = pt * getSnp();
  auto pz = pt * getTgl();
  auto q = getCharge();
 
  cv[0] = mCovariances(0, 0);
  cv[1] = mCovariances(1, 0);
  cv[2] = mCovariances(1, 1);
  cv[3] = 0;
  cv[4] = 0;
  cv[5] = 0;
  cv[6] = -mCovariances(0, 2) * py - mCovariances(0, 4) * px * pt * q;
  cv[7] = -mCovariances(1, 2) * py - mCovariances(1, 4) * px * pt * q;
  cv[8] = 0;
  cv[9] = 2 * mCovariances(2, 4) * px * py * q * pt + mCovariances(2, 2) * py * py + mCovariances(4, 4) * px * px * pt * pt;
  cv[10] = mCovariances(0, 2) * px - mCovariances(0, 4) * py * pt * q;
  cv[11] = mCovariances(1, 2) * px - mCovariances(1, 4) * py * pt * q;
  cv[12] = 0;
  cv[13] = mCovariances(2, 4) * (py * py - px * px) * q * pt - mCovariances(2, 2) * px * py + mCovariances(4, 4) * px * py * pt * pt;
  cv[14] = -2 * mCovariances(2, 4) * px * py * q * pt + mCovariances(2, 2) * px * px + mCovariances(4, 4) * py * py * pt * pt;
  cv[15] = mCovariances(0, 3) * pt - mCovariances(0, 4) * pt * pz * q;
  cv[16] = mCovariances(1, 3) * pt - mCovariances(1, 4) * pt * pz * q;
  cv[17] = 0;
  cv[18] = -mCovariances(2, 3) * py * pt - mCovariances(3, 4) * px * q * pt * pt + mCovariances(2, 4) * py * pz * q * pt + mCovariances(4, 4) * px * pz * pt * pt;
  cv[19] = mCovariances(2, 3) * px * pt - mCovariances(3, 4) * q * pt * pt * py - mCovariances(2, 4) * px * pz * q * pt + mCovariances(4, 4) * py * pz * pt * pt;
  cv[20] = -2 * mCovariances(3, 4) * pz * q * pt * pt + mCovariances(3, 3) * pt * pt + mCovariances(4, 4) * pz * pz * pt * pt;
 
  return true;
}
 
//________________________________________________________________
 
void TrackParCovFwd::propagateToDCAhelix(double zField, const std::array<double, 3>& p, std::array<double, 3>& dca)
{
  // Computing DCA of fwd track w.r.t vertex in helix track model, using Newton-Raphson minimization
 
  auto x0 = mParameters(0);
  auto y0 = mParameters(1);
  auto z0 = mZ;
  auto phi0 = mParameters(2);
  auto tanl = mParameters(3);
  auto qOverPt = mParameters(4);
  auto k = TMath::Abs(o2::constants::math::B2C * zField);
  auto qpt = 1.0 / qOverPt;
  auto qR = qpt / std::fabs(k);
  auto invtanl = 1.0 / tanl;
  auto Hz = std::copysign(1, zField);
 
  auto xPV = p[0];
  auto yPV = p[1];
  auto zPV = p[2];
 
  auto qRtanl = qR * tanl;
  auto invqRtanl = 1.0 / qRtanl;
  auto [sinp, cosp] = o2::math_utils::sincosd(phi0);
 
  auto z = zPV;
  double tol = 1e-4;
  int max_iter = 10;
  int iter = 0;
 
  while (iter++ < max_iter) {
    double theta = (z0 - z) * invqRtanl;
    double phi_theta = phi0 + Hz * theta;
    double sin_phi_theta = sin(phi_theta);
    double cos_phi_theta = cos(phi_theta);
 
    double DX = x0 - Hz * qR * (sin_phi_theta - sinp) - xPV;
    double DY = y0 + Hz * qR * (cos_phi_theta - cosp) - yPV;
    double DZ = z - zPV;
 
    double dD2_dZ =
      2 * DX * cos_phi_theta * invtanl +
      2 * DY * sin_phi_theta * invtanl +
      2 * DZ;
 
    double d2D2_dZ2 =
      2 * invtanl * invtanl +
      2 * invtanl * (DX * Hz * sin_phi_theta - DY * Hz * cos_phi_theta) * invqRtanl +
      2;
 
    double z_new = z - dD2_dZ / d2D2_dZ2;
 
    if (std::abs(z_new - z) < tol) {
      z = z_new;
      this->propagateToZhelix(z, zField);
      dca[0] = this->getX() - xPV;
      dca[1] = this->getY() - yPV;
      dca[2] = this->getZ() - zPV;
      LOG(debug) << "Converged after " << iter << " iterations for vertex X=" << p[0] << ", Y=" << p[1] << ", Z = " << p[2];
      return;
    }
    z = z_new;
  }
  LOG(debug) << "Failed to converge after " << iter << " iterations for vertex X=" << p[0] << ", Y=" << p[1] << ", Z = " << p[2];
  return;
}
 
template <typename T>
void TrackParFwd::toBarrelTrackPar(TrackParametrization<T>& t) const
{
  // we select the barrel frame with alpha = phi, then by construction the snp is 0
  auto alpha = getPhi();
  auto csa = TMath::Cos(alpha), sna = TMath::Sin(alpha);
  t.setAlpha(alpha);
  t.setX(csa * getX() + sna * getY());
  t.setY(-sna * getX() + csa * getY());
  t.setZ(getZ());
  t.setSnp(0);
  t.setTgl(getTanl());
  t.setQ2Pt(getInvQPt());
}
 
template <typename T>
void TrackParCovFwd::toBarrelTrackParCov(TrackParametrizationWithError<T>& t) const
{
  // We select the barrel frame with alpha = phi, then by construction the snp is 0
  auto alpha = getPhi();
  auto csa = TMath::Cos(alpha), sna = TMath::Sin(alpha);
  t.setAlpha(alpha);
  t.setX(csa * getX() + sna * getY());
  t.setY(-sna * getX() + csa * getY());
  t.setZ(getZ());
  t.setSnp(0);
  t.setTgl(getTgl());
  t.setQ2Pt(getInvQPt());
  /*
    The standard Jacobian d{barrel_param} / d{fwd_param} should be augmented by the "forward" uncertainty
    in X_barrel translated to uncertainty in Z, i.e:
    A fwd param variation delta_x, delta_y leads to barrel frame coordinate variaion
    delta_xb =  csa delta_x + sna delta_y
    delta_yb = -sna delta_x + csa delta_y
    with dx_b/dz = csp/tgL = 1/tgL, dy_b/dz = snp/tgL = 0  (for phi 0 in the tracking frame) the variation of delta_xb would require
    a shift in delta_zb = -tgL delta_xb to stat at the same X.
    So, for alpha=phi (-> snp=0) choice the full Jacobian fwd->barrel is:
    / -sna       csa      0 0 0  \
    | -tgL*csa  -tgL*sna  0 0 0  |
    |  0          0       1 0 0  |
    |  0          0       0 1 0  |
    \  0          0       0 0 1 /
  */
  const T a1 = -sna;
  const T a2 = csa;
  const T b1 = -getTgl() * csa;
  const T b2 = -getTgl() * sna;
  const T cphi = 1;
  const auto& C = getCovariances();
  typename TrackParametrizationWithError<T>::covMat_t covBarrel = {
    T(a1 * a1 * C(0, 0) + 2 * a1 * a2 * C(0, 1) + a2 * a2 * C(1, 1)),         // kSigY2
    T(a1 * b1 * C(0, 0) + (a1 * b2 + a2 * b1) * C(0, 1) + a2 * b2 * C(1, 1)), // kSigZY
    T(b1 * b1 * C(0, 0) + 2 * b1 * b2 * C(0, 1) + b2 * b2 * C(1, 1)),         // kSigZ2
    T(csa * (a1 * C(0, 2) + a2 * C(1, 2))),                                   // kSigSnpY
    T(csa * (b1 * C(0, 2) + b2 * C(1, 2))),                                   // kSigSnpZ
    T(csa * csa * C(2, 2)),                                                   // kSigSnp2
    T(a1 * C(0, 3) + a2 * C(1, 3)),                                           // kSigTglY
    T(b1 * C(0, 3) + b2 * C(1, 3)),                                           // kSigTglZ
    T(csa * C(2, 3)),                                                         // kSigTglSnp
    T(C(3, 3)),                                                               // kSigTgl2
    T(a1 * C(0, 4) + a2 * C(1, 4)),                                           // kSigQ2PtY
    T(b1 * C(0, 4) + b2 * C(1, 4)),                                           // kSigQ2PtZ
    T(csa * C(2, 4)),                                                         // kSigQ2PtSnp
    T(C(3, 4)),                                                               // kSigQ2PtTgl
    T(C(4, 4))                                                                // kSigQ2Pt2
  };
  t.setCov(covBarrel);
}
 
template void TrackParFwd::toBarrelTrackPar<float>(TrackParametrization<float>&) const;
template void TrackParFwd::toBarrelTrackPar<double>(TrackParametrization<double>&) const;
template void TrackParCovFwd::toBarrelTrackParCov<float>(TrackParametrizationWithError<float>&) const;
template void TrackParCovFwd::toBarrelTrackParCov<double>(TrackParametrizationWithError<double>&) const;
 
} // namespace track
} // namespace o2