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File indexing completed on 2024-04-06 12:28:37

 #ifndef RecoTracker_PixelTrackFitting_interface_alpaka_RiemannFit_h
 #define RecoTracker_PixelTrackFitting_interface_alpaka_RiemannFit_h
 
 #include <alpaka/alpaka.hpp>
 
 #include <Eigen/Core>
 #include <Eigen/Eigenvalues>
 
 #include "HeterogeneousCore/AlpakaInterface/interface/config.h"
 #include "RecoTracker/PixelTrackFitting/interface/alpaka/FitUtils.h"
 
 namespace ALPAKA_ACCELERATOR_NAMESPACE::riemannFit {
   using namespace ::riemannFit;
 
   /*!  Compute the Radiation length in the uniform hypothesis
  *
  * The Pixel detector, barrel and forward, is considered as an homogeneous
  * cylinder of material, whose radiation lengths has been derived from the TDR
  * plot that shows that 16cm correspond to 0.06 radiation lengths. Therefore
  * one radiation length corresponds to 16cm/0.06 =~ 267 cm. All radiation
  * lengths are computed using this unique number, in both regions, barrel and
  * endcap.
  *
  * NB: no angle corrections nor projections are computed inside this routine.
  * It is therefore the responsibility of the caller to supply the proper
  * lengths in input. These lengths are the path traveled by the particle along
  * its trajectory, namely the so called S of the helix in 3D space.
  *
  * \param length_values vector of incremental distances that will be translated
  * into radiation length equivalent. Each radiation length i is computed
  * incrementally with respect to the previous length i-1. The first length has
  * no reference point (i.e. it has the dca).
  *
  * \return incremental radiation lengths that correspond to each segment.
  */
 
   template <typename TAcc, typename VNd1, typename VNd2>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE void computeRadLenUniformMaterial(const TAcc& acc,
                                                                    const VNd1& length_values,
                                                                    VNd2& rad_lengths) {
     // Radiation length of the pixel detector in the uniform assumption, with
     // 0.06 rad_len at 16 cm
     constexpr double xx_0_inv = 0.06 / 16.;
     uint n = length_values.rows();
     rad_lengths(0) = length_values(0) * xx_0_inv;
     for (uint j = 1; j < n; ++j) {
       rad_lengths(j) = alpaka::math::abs(acc, length_values(j) - length_values(j - 1)) * xx_0_inv;
     }
   }
 
   /*!
     \brief Compute the covariance matrix along cartesian S-Z of points due to
     multiple Coulomb scattering to be used in the line_fit, for the barrel
     and forward cases.
     The input covariance matrix is in the variables s-z, original and
     unrotated.
     The multiple scattering component is computed in the usual linear
     approximation, using the 3D path which is computed as the squared root of
     the squared sum of the s and z components passed in.
     Internally a rotation by theta is performed and the covariance matrix
     returned is the one in the direction orthogonal to the rotated S3D axis,
     i.e. along the rotated Z axis.
     The choice of the rotation is not arbitrary, but derived from the fact that
     putting the horizontal axis along the S3D direction allows the usage of the
     ordinary least squared fitting techiques with the trivial parametrization y
     = mx + q, avoiding the patological case with m = +/- inf, that would
     correspond to the case at eta = 0.
  */
 
   template <typename TAcc, typename V4, typename VNd1, typename VNd2, int N>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE auto scatterCovLine(const TAcc& acc,
                                                      Matrix2d const* cov_sz,
                                                      const V4& fast_fit,
                                                      VNd1 const& s_arcs,
                                                      VNd2 const& z_values,
                                                      const double theta,
                                                      const double bField,
                                                      MatrixNd<N>& ret) {
 #ifdef RFIT_DEBUG
     riemannFit::printIt(&s_arcs, "Scatter_cov_line - s_arcs: ");
 #endif
     constexpr uint n = N;
     double p_t = alpaka::math::min(acc, 20., fast_fit(2) * bField);  // limit pt to avoid too small error!!!
     double p_2 = p_t * p_t * (1. + 1. / sqr(fast_fit(3)));
     VectorNd<N> rad_lengths_S;
     // See documentation at http://eigen.tuxfamily.org/dox/group__TutorialArrayClass.html
     // Basically, to perform cwise operations on Matrices and Vectors, you need
     // to transform them into Array-like objects.
     VectorNd<N> s_values = s_arcs.array() * s_arcs.array() + z_values.array() * z_values.array();
     s_values = s_values.array().sqrt();
     computeRadLenUniformMaterial(acc, s_values, rad_lengths_S);
     VectorNd<N> sig2_S;
     sig2_S = .000225 / p_2 * (1. + 0.038 * rad_lengths_S.array().log()).abs2() * rad_lengths_S.array();
 #ifdef RFIT_DEBUG
     riemannFit::printIt(cov_sz, "Scatter_cov_line - cov_sz: ");
 #endif
     Matrix2Nd<N> tmp = Matrix2Nd<N>::Zero();
     for (uint k = 0; k < n; ++k) {
       tmp(k, k) = cov_sz[k](0, 0);
       tmp(k + n, k + n) = cov_sz[k](1, 1);
       tmp(k, k + n) = tmp(k + n, k) = cov_sz[k](0, 1);
     }
     for (uint k = 0; k < n; ++k) {
       for (uint l = k; l < n; ++l) {
         for (uint i = 0; i < uint(alpaka::math::min(acc, k, l)); ++i) {
           tmp(k + n, l + n) += alpaka::math::abs(acc, s_values(k) - s_values(i)) *
                                alpaka::math::abs(acc, s_values(l) - s_values(i)) * sig2_S(i);
         }
         tmp(l + n, k + n) = tmp(k + n, l + n);
       }
     }
     // We are interested only in the errors orthogonal to the rotated s-axis
     // which, in our formalism, are in the lower square matrix.
 #ifdef RFIT_DEBUG
     riemannFit::printIt(&tmp, "Scatter_cov_line - tmp: ");
 #endif
     ret = tmp.block(n, n, n, n);
   }
 
   /*!
     \brief Compute the covariance matrix (in radial coordinates) of points in
     the transverse plane due to multiple Coulomb scattering.
     \param p2D 2D points in the transverse plane.
     \param fast_fit fast_fit Vector4d result of the previous pre-fit
     structured in this form:(X0, Y0, R, Tan(Theta))).
     \param B magnetic field use to compute p
     \return scatter_cov_rad errors due to multiple scattering.
     \warning input points must be ordered radially from the detector center
     (from inner layer to outer ones; points on the same layer must ordered too).
     \details Only the tangential component is computed (the radial one is
     negligible).
  */
   template <typename TAcc, typename M2xN, typename V4, int N>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE MatrixNd<N> scatter_cov_rad(
       const TAcc& acc, const M2xN& p2D, const V4& fast_fit, VectorNd<N> const& rad, double B) {
     constexpr uint n = N;
     double p_t = alpaka::math::min(acc, 20., fast_fit(2) * B);  // limit pt to avoid too small error!!!
     double p_2 = p_t * p_t * (1. + 1. / sqr(fast_fit(3)));
     double theta = atan(fast_fit(3));
     theta = theta < 0. ? theta + M_PI : theta;
     VectorNd<N> s_values;
     VectorNd<N> rad_lengths;
     const Vector2d oVec(fast_fit(0), fast_fit(1));
 
     // associated Jacobian, used in weights and errors computation
     for (uint i = 0; i < n; ++i) {  // x
       Vector2d pVec = p2D.block(0, i, 2, 1) - oVec;
       const double cross = cross2D(acc, -oVec, pVec);
       const double dot = (-oVec).dot(pVec);
       const double tempAtan2 = atan2(cross, dot);
       s_values(i) = alpaka::math::abs(acc, tempAtan2 * fast_fit(2));
     }
     computeRadLenUniformMaterial(acc, s_values * sqrt(1. + 1. / sqr(fast_fit(3))), rad_lengths);
     MatrixNd<N> scatter_cov_rad = MatrixNd<N>::Zero();
     VectorNd<N> sig2 = (1. + 0.038 * rad_lengths.array().log()).abs2() * rad_lengths.array();
     sig2 *= 0.000225 / (p_2 * sqr(sin(theta)));
     for (uint k = 0; k < n; ++k) {
       for (uint l = k; l < n; ++l) {
         for (uint i = 0; i < uint(alpaka::math::min(acc, k, l)); ++i) {
           scatter_cov_rad(k, l) += (rad(k) - rad(i)) * (rad(l) - rad(i)) * sig2(i);
         }
         scatter_cov_rad(l, k) = scatter_cov_rad(k, l);
       }
     }
 #ifdef RFIT_DEBUG
     riemannFit::printIt(&scatter_cov_rad, "Scatter_cov_rad - scatter_cov_rad: ");
 #endif
     return scatter_cov_rad;
   }
 
   /*!
     \brief Transform covariance matrix from radial (only tangential component)
     to Cartesian coordinates (only transverse plane component).
     \param p2D 2D points in the transverse plane.
     \param cov_rad covariance matrix in radial coordinate.
     \return cov_cart covariance matrix in Cartesian coordinates.
 */
 
   template <typename TAcc, typename M2xN, int N>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE Matrix2Nd<N> cov_radtocart(const TAcc& acc,
                                                             const M2xN& p2D,
                                                             const MatrixNd<N>& cov_rad,
                                                             const VectorNd<N>& rad) {
 #ifdef RFIT_DEBUG
     printf("Address of p2D: %p\n", &p2D);
 #endif
     printIt(&p2D, "cov_radtocart - p2D:");
     constexpr uint n = N;
     Matrix2Nd<N> cov_cart = Matrix2Nd<N>::Zero();
     VectorNd<N> rad_inv = rad.cwiseInverse();
     printIt(&rad_inv, "cov_radtocart - rad_inv:");
     for (uint i = 0; i < n; ++i) {
       for (uint j = i; j < n; ++j) {
         cov_cart(i, j) = cov_rad(i, j) * p2D(1, i) * rad_inv(i) * p2D(1, j) * rad_inv(j);
         cov_cart(i + n, j + n) = cov_rad(i, j) * p2D(0, i) * rad_inv(i) * p2D(0, j) * rad_inv(j);
         cov_cart(i, j + n) = -cov_rad(i, j) * p2D(1, i) * rad_inv(i) * p2D(0, j) * rad_inv(j);
         cov_cart(i + n, j) = -cov_rad(i, j) * p2D(0, i) * rad_inv(i) * p2D(1, j) * rad_inv(j);
         cov_cart(j, i) = cov_cart(i, j);
         cov_cart(j + n, i + n) = cov_cart(i + n, j + n);
         cov_cart(j + n, i) = cov_cart(i, j + n);
         cov_cart(j, i + n) = cov_cart(i + n, j);
       }
     }
     return cov_cart;
   }
 
   /*!
     \brief Transform covariance matrix from Cartesian coordinates (only
     transverse plane component) to radial coordinates (both radial and
     tangential component but only diagonal terms, correlation between different
     point are not managed).
     \param p2D 2D points in transverse plane.
     \param cov_cart covariance matrix in Cartesian coordinates.
     \return cov_rad covariance matrix in raidal coordinate.
     \warning correlation between different point are not computed.
 */
   template <typename TAcc, typename M2xN, int N>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE VectorNd<N> cov_carttorad(const TAcc& acc,
                                                            const M2xN& p2D,
                                                            const Matrix2Nd<N>& cov_cart,
                                                            const VectorNd<N>& rad) {
     constexpr uint n = N;
     VectorNd<N> cov_rad;
     const VectorNd<N> rad_inv2 = rad.cwiseInverse().array().square();
     for (uint i = 0; i < n; ++i) {
       //!< in case you have (0,0) to avoid dividing by 0 radius
       if (rad(i) < 1.e-4)
         cov_rad(i) = cov_cart(i, i);
       else {
         cov_rad(i) = rad_inv2(i) * (cov_cart(i, i) * sqr(p2D(1, i)) + cov_cart(i + n, i + n) * sqr(p2D(0, i)) -
                                     2. * cov_cart(i, i + n) * p2D(0, i) * p2D(1, i));
       }
     }
     return cov_rad;
   }
 
   /*!
     \brief Transform covariance matrix from Cartesian coordinates (only
     transverse plane component) to coordinates system orthogonal to the
     pre-fitted circle in each point.
     Further information in attached documentation.
     \param p2D 2D points in transverse plane.
     \param cov_cart covariance matrix in Cartesian coordinates.
     \param fast_fit fast_fit Vector4d result of the previous pre-fit
     structured in this form:(X0, Y0, R, tan(theta))).
     \return cov_rad covariance matrix in the pre-fitted circle's
     orthogonal system.
 */
   template <typename TAcc, typename M2xN, typename V4, int N>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE VectorNd<N> cov_carttorad_prefit(
       const TAcc& acc, const M2xN& p2D, const Matrix2Nd<N>& cov_cart, V4& fast_fit, const VectorNd<N>& rad) {
     constexpr uint n = N;
     VectorNd<N> cov_rad;
     for (uint i = 0; i < n; ++i) {
       //!< in case you have (0,0) to avoid dividing by 0 radius
       if (rad(i) < 1.e-4)
         cov_rad(i) = cov_cart(i, i);  // TO FIX
       else {
         Vector2d a = p2D.col(i);
         Vector2d b = p2D.col(i) - fast_fit.head(2);
         const double x2 = a.dot(b);
         const double y2 = cross2D(acc, a, b);
         const double tan_c = -y2 / x2;
         const double tan_c2 = sqr(tan_c);
         cov_rad(i) =
             1. / (1. + tan_c2) * (cov_cart(i, i) + cov_cart(i + n, i + n) * tan_c2 + 2 * cov_cart(i, i + n) * tan_c);
       }
     }
     return cov_rad;
   }
 
   /*!
     \brief Compute the points' weights' vector for the circle fit when multiple
     scattering is managed.
     Further information in attached documentation.
     \param cov_rad_inv covariance matrix inverse in radial coordinated
     (or, beter, pre-fitted circle's orthogonal system).
     \return weight VectorNd points' weights' vector.
     \bug I'm not sure this is the right way to compute the weights for non
     diagonal cov matrix. Further investigation needed.
 */
 
   template <typename TAcc, int N>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE VectorNd<N> weightCircle(const TAcc& acc, const MatrixNd<N>& cov_rad_inv) {
     return cov_rad_inv.colwise().sum().transpose();
   }
 
   /*!
     \brief Find particle q considering the  sign of cross product between
     particles velocity (estimated by the first 2 hits) and the vector radius
     between the first hit and the center of the fitted circle.
     \param p2D 2D points in transverse plane.
     \param par_uvr result of the circle fit in this form: (X0,Y0,R).
     \return q int 1 or -1.
 */
   template <typename TAcc, typename M2xN>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE int32_t charge(const TAcc& acc, const M2xN& p2D, const Vector3d& par_uvr) {
     return ((p2D(0, 1) - p2D(0, 0)) * (par_uvr.y() - p2D(1, 0)) - (p2D(1, 1) - p2D(1, 0)) * (par_uvr.x() - p2D(0, 0)) >
             0)
                ? -1
                : 1;
   }
 
   /*!
     \brief Compute the eigenvector associated to the minimum eigenvalue.
     \param A the Matrix you want to know eigenvector and eigenvalue.
     \param chi2 the double were the chi2-related quantity will be stored.
     \return the eigenvector associated to the minimum eigenvalue.
     \warning double precision is needed for a correct assessment of chi2.
     \details The minimus eigenvalue is related to chi2.
     We exploit the fact that the matrix is symmetrical and small (2x2 for line
     fit and 3x3 for circle fit), so the SelfAdjointEigenSolver from Eigen
     library is used, with the computedDirect  method (available only for 2x2
     and 3x3 Matrix) wich computes eigendecomposition of given matrix using a
     fast closed-form algorithm.
     For this optimization the matrix type must be known at compiling time.
 */
   template <typename TAcc>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE Vector3d min_eigen3D(const TAcc& acc, const Matrix3d& A, double& chi2) {
 #ifdef RFIT_DEBUG
     printf("min_eigen3D - enter\n");
 #endif
     Eigen::SelfAdjointEigenSolver<Matrix3d> solver(3);
     solver.computeDirect(A);
     int min_index;
     chi2 = solver.eigenvalues().minCoeff(&min_index);
 #ifdef RFIT_DEBUG
     printf("min_eigen3D - exit\n");
 #endif
     return solver.eigenvectors().col(min_index);
   }
 
   /*!
     \brief A faster version of min_eigen3D() where double precision is not
     needed.
     \param A the Matrix you want to know eigenvector and eigenvalue.
     \param chi2 the double were the chi2-related quantity will be stored
     \return the eigenvector associated to the minimum eigenvalue.
     \detail The computedDirect() method of SelfAdjointEigenSolver for 3x3 Matrix
     indeed, use trigonometry function (it solves a third degree equation) which
     speed up in  single precision.
 */
 
   template <typename TAcc>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE Vector3d min_eigen3D_fast(const TAcc& acc, const Matrix3d& A) {
     Eigen::SelfAdjointEigenSolver<Matrix3f> solver(3);
     solver.computeDirect(A.cast<float>());
     int min_index;
     solver.eigenvalues().minCoeff(&min_index);
     return solver.eigenvectors().col(min_index).cast<double>();
   }
 
   /*!
     \brief 2D version of min_eigen3D().
     \param aMat the Matrix you want to know eigenvector and eigenvalue.
     \param chi2 the double were the chi2-related quantity will be stored
     \return the eigenvector associated to the minimum eigenvalue.
     \detail The computedDirect() method of SelfAdjointEigenSolver for 2x2 Matrix
     do not use special math function (just sqrt) therefore it doesn't speed up
     significantly in single precision.
 */
   template <typename TAcc>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE Vector2d min_eigen2D(const TAcc& acc, const Matrix2d& aMat, double& chi2) {
     Eigen::SelfAdjointEigenSolver<Matrix2d> solver(2);
     solver.computeDirect(aMat);
     int min_index;
     chi2 = solver.eigenvalues().minCoeff(&min_index);
     return solver.eigenvectors().col(min_index);
   }
 
   /*!
     \brief A very fast helix fit: it fits a circle by three points (first, middle
     and last point) and a line by two points (first and last).
     \param hits points to be fitted
     \return result in this form: (X0,Y0,R,tan(theta)).
     \warning points must be passed ordered (from internal layer to external) in
     order to maximize accuracy and do not mistake tan(theta) sign.
     \details This fast fit is used as pre-fit which is needed for:
     - weights estimation and chi2 computation in line fit (fundamental);
     - weights estimation and chi2 computation in circle fit (useful);
     - computation of error due to multiple scattering.
 */
 
   template <typename TAcc, typename M3xN, typename V4>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE void fastFit(const TAcc& acc, const M3xN& hits, V4& result) {
     constexpr uint32_t N = M3xN::ColsAtCompileTime;
     constexpr auto n = N;  // get the number of hits
     printIt(&hits, "Fast_fit - hits: ");
 
     // CIRCLE FIT
     // Make segments between middle-to-first(b) and last-to-first(c) hits
     const Vector2d bVec = hits.block(0, n / 2, 2, 1) - hits.block(0, 0, 2, 1);
     const Vector2d cVec = hits.block(0, n - 1, 2, 1) - hits.block(0, 0, 2, 1);
     printIt(&bVec, "Fast_fit - b: ");
     printIt(&cVec, "Fast_fit - c: ");
     // Compute their lengths
     auto b2 = bVec.squaredNorm();
     auto c2 = cVec.squaredNorm();
     // The algebra has been verified (MR). The usual approach has been followed:
     // * use an orthogonal reference frame passing from the first point.
     // * build the segments (chords)
     // * build orthogonal lines through mid points
     // * make a system and solve for X0 and Y0.
     // * add the initial point
     bool flip = abs(bVec.x()) < abs(bVec.y());
     auto bx = flip ? bVec.y() : bVec.x();
     auto by = flip ? bVec.x() : bVec.y();
     auto cx = flip ? cVec.y() : cVec.x();
     auto cy = flip ? cVec.x() : cVec.y();
     //!< in case b.x is 0 (2 hits with same x)
     auto div = 2. * (cx * by - bx * cy);
     // if aligned TO FIX
     auto y0 = (cx * b2 - bx * c2) / div;
     auto x0 = (0.5 * b2 - y0 * by) / bx;
     result(0) = hits(0, 0) + (flip ? y0 : x0);
     result(1) = hits(1, 0) + (flip ? x0 : y0);
     result(2) = sqrt(sqr(x0) + sqr(y0));
     printIt(&result, "Fast_fit - result: ");
 
     // LINE FIT
     const Vector2d dVec = hits.block(0, 0, 2, 1) - result.head(2);
     const Vector2d eVec = hits.block(0, n - 1, 2, 1) - result.head(2);
     printIt(&eVec, "Fast_fit - e: ");
     printIt(&dVec, "Fast_fit - d: ");
     // Compute the arc-length between first and last point: L = R * theta = R * atan (tan (Theta) )
     auto dr = result(2) * atan2(cross2D(acc, dVec, eVec), dVec.dot(eVec));
     // Simple difference in Z between last and first hit
     auto dz = hits(2, n - 1) - hits(2, 0);
 
     result(3) = (dr / dz);
 
 #ifdef RFIT_DEBUG
     printf("Fast_fit: [%f, %f, %f, %f]\n", result(0), result(1), result(2), result(3));
 #endif
   }
 
   /*!
     \brief Fit a generic number of 2D points with a circle using Riemann-Chernov
     algorithm. Covariance matrix of fitted parameter is optionally computed.
     Multiple scattering (currently only in barrel layer) is optionally handled.
     \param hits2D 2D points to be fitted.
     \param hits_cov2D covariance matrix of 2D points.
     \param fast_fit pre-fit result in this form: (X0,Y0,R,tan(theta)).
     (tan(theta) is not used).
     \param bField magnetic field
     \param error flag for error computation.
     \param scattering flag for multiple scattering
     \return circle circle_fit:
     -par parameter of the fitted circle in this form (X0,Y0,R); \n
     -cov covariance matrix of the fitted parameter (not initialized if
     error = false); \n
     -q charge of the particle; \n
     -chi2.
     \warning hits must be passed ordered from inner to outer layer (double hits
     on the same layer must be ordered too) so that multiple scattering is
     treated properly.
     \warning Multiple scattering for barrel is still not tested.
     \warning Multiple scattering for endcap hits is not handled (yet). Do not
     fit endcap hits with scattering = true !
     \bug for small pt (<0.3 Gev/c) chi2 could be slightly underestimated.
     \bug further investigation needed for error propagation with multiple
     scattering.
 */
   template <typename TAcc, typename M2xN, typename V4, int N>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE CircleFit circleFit(const TAcc& acc,
                                                      const M2xN& hits2D,
                                                      const Matrix2Nd<N>& hits_cov2D,
                                                      const V4& fast_fit,
                                                      const VectorNd<N>& rad,
                                                      const double bField,
                                                      const bool error) {
 #ifdef RFIT_DEBUG
     printf("circle_fit - enter\n");
 #endif
     // INITIALIZATION
     Matrix2Nd<N> vMat = hits_cov2D;
     constexpr uint n = N;
     printIt(&hits2D, "circle_fit - hits2D:");
     printIt(&hits_cov2D, "circle_fit - hits_cov2D:");
 
 #ifdef RFIT_DEBUG
     printf("circle_fit - WEIGHT COMPUTATION\n");
 #endif
     // WEIGHT COMPUTATION
     VectorNd<N> weight;
     MatrixNd<N> gMat;
     double renorm;
     {
       MatrixNd<N> cov_rad = cov_carttorad_prefit(acc, hits2D, vMat, fast_fit, rad).asDiagonal();
       MatrixNd<N> scatterCovRadMat = scatter_cov_rad(acc, hits2D, fast_fit, rad, bField);
       printIt(&scatterCovRadMat, "circle_fit - scatter_cov_rad:");
       printIt(&hits2D, "circle_fit - hits2D bis:");
 #ifdef RFIT_DEBUG
       printf("Address of hits2D: a) %p\n", &hits2D);
 #endif
       vMat += cov_radtocart(acc, hits2D, scatterCovRadMat, rad);
       printIt(&vMat, "circle_fit - V:");
       cov_rad += scatterCovRadMat;
       printIt(&cov_rad, "circle_fit - cov_rad:");
       math::cholesky::invert(cov_rad, gMat);
       // gMat = cov_rad.inverse();
       renorm = gMat.sum();
       gMat *= 1. / renorm;
       weight = weightCircle(acc, gMat);
     }
     printIt(&weight, "circle_fit - weight:");
 
     // SPACE TRANSFORMATION
 #ifdef RFIT_DEBUG
     printf("circle_fit - SPACE TRANSFORMATION\n");
 #endif
 
     // center
 #ifdef RFIT_DEBUG
     printf("Address of hits2D: b) %p\n", &hits2D);
 #endif
     const Vector2d hCentroid = hits2D.rowwise().mean();  // centroid
     printIt(&hCentroid, "circle_fit - h_:");
     Matrix3xNd<N> p3D;
     p3D.block(0, 0, 2, n) = hits2D.colwise() - hCentroid;
     printIt(&p3D, "circle_fit - p3D: a)");
     Vector2Nd<N> mc;  // centered hits, used in error computation
     mc << p3D.row(0).transpose(), p3D.row(1).transpose();
     printIt(&mc, "circle_fit - mc(centered hits):");
 
     // scale
     const double tempQ = mc.squaredNorm();
     const double tempS = sqrt(n * 1. / tempQ);  // scaling factor
     p3D.block(0, 0, 2, n) *= tempS;
 
     // project on paraboloid
     p3D.row(2) = p3D.block(0, 0, 2, n).colwise().squaredNorm();
     printIt(&p3D, "circle_fit - p3D: b)");
 
 #ifdef RFIT_DEBUG
     printf("circle_fit - COST FUNCTION\n");
 #endif
     // COST FUNCTION
 
     // compute
     Vector3d r0;
     r0.noalias() = p3D * weight;  // center of gravity
     const Matrix3xNd<N> xMat = p3D.colwise() - r0;
     Matrix3d aMat = xMat * gMat * xMat.transpose();
     printIt(&aMat, "circle_fit - A:");
 
 #ifdef RFIT_DEBUG
     printf("circle_fit - MINIMIZE\n");
 #endif
     // minimize
     double chi2;
     Vector3d vVec = min_eigen3D(acc, aMat, chi2);
 #ifdef RFIT_DEBUG
     printf("circle_fit - AFTER MIN_EIGEN\n");
 #endif
     printIt(&vVec, "v BEFORE INVERSION");
     vVec *= (vVec(2) > 0) ? 1 : -1;  // TO FIX dovrebbe essere N(3)>0
     printIt(&vVec, "v AFTER INVERSION");
     // This hack to be able to run on GPU where the automatic assignment to a
     // double from the vector multiplication is not working.
 #ifdef RFIT_DEBUG
     printf("circle_fit - AFTER MIN_EIGEN 1\n");
 #endif
     Eigen::Matrix<double, 1, 1> cm;
 #ifdef RFIT_DEBUG
     printf("circle_fit - AFTER MIN_EIGEN 2\n");
 #endif
     cm = -vVec.transpose() * r0;
 #ifdef RFIT_DEBUG
     printf("circle_fit - AFTER MIN_EIGEN 3\n");
 #endif
     const double tempC = cm(0, 0);
 
 #ifdef RFIT_DEBUG
     printf("circle_fit - COMPUTE CIRCLE PARAMETER\n");
 #endif
     // COMPUTE CIRCLE PARAMETER
 
     // auxiliary quantities
     const double tempH = sqrt(1. - sqr(vVec(2)) - 4. * tempC * vVec(2));
     const double v2x2_inv = 1. / (2. * vVec(2));
     const double s_inv = 1. / tempS;
     Vector3d par_uvr;  // used in error propagation
     par_uvr << -vVec(0) * v2x2_inv, -vVec(1) * v2x2_inv, tempH * v2x2_inv;
 
     CircleFit circle;
     circle.par << par_uvr(0) * s_inv + hCentroid(0), par_uvr(1) * s_inv + hCentroid(1), par_uvr(2) * s_inv;
     circle.qCharge = charge(acc, hits2D, circle.par);
     circle.chi2 = abs(chi2) * renorm / sqr(2 * vVec(2) * par_uvr(2) * tempS);
     printIt(&circle.par, "circle_fit - CIRCLE PARAMETERS:");
     printIt(&circle.cov, "circle_fit - CIRCLE COVARIANCE:");
 #ifdef RFIT_DEBUG
     printf("circle_fit - CIRCLE CHARGE: %d\n", circle.qCharge);
 #endif
 
 #ifdef RFIT_DEBUG
     printf("circle_fit - ERROR PROPAGATION\n");
 #endif
     // ERROR PROPAGATION
     if (error) {
 #ifdef RFIT_DEBUG
       printf("circle_fit - ERROR PRPAGATION ACTIVATED\n");
 #endif
       ArrayNd<N> vcsMat[2][2];  // cov matrix of center & scaled points
       MatrixNd<N> cMat[3][3];   // cov matrix of 3D transformed points
 #ifdef RFIT_DEBUG
       printf("circle_fit - ERROR PRPAGATION ACTIVATED 2\n");
 #endif
       {
         Eigen::Matrix<double, 1, 1> cm;
         Eigen::Matrix<double, 1, 1> cm2;
         cm = mc.transpose() * vMat * mc;
         const double tempC2 = cm(0, 0);
         Matrix2Nd<N> tempVcsMat;
         tempVcsMat.template triangularView<Eigen::Upper>() =
             (sqr(tempS) * vMat + sqr(sqr(tempS)) * 1. / (4. * tempQ * n) *
                                      (2. * vMat.squaredNorm() + 4. * tempC2) *  // mc.transpose() * V * mc) *
                                      (mc * mc.transpose()));
 
         printIt(&tempVcsMat, "circle_fit - Vcs:");
         cMat[0][0] = tempVcsMat.block(0, 0, n, n).template selfadjointView<Eigen::Upper>();
         vcsMat[0][1] = tempVcsMat.block(0, n, n, n);
         cMat[1][1] = tempVcsMat.block(n, n, n, n).template selfadjointView<Eigen::Upper>();
         vcsMat[1][0] = vcsMat[0][1].transpose();
         printIt(&tempVcsMat, "circle_fit - Vcs:");
       }
 
       {
         const ArrayNd<N> t0 = (VectorXd::Constant(n, 1.) * p3D.row(0));
         const ArrayNd<N> t1 = (VectorXd::Constant(n, 1.) * p3D.row(1));
         const ArrayNd<N> t00 = p3D.row(0).transpose() * p3D.row(0);
         const ArrayNd<N> t01 = p3D.row(0).transpose() * p3D.row(1);
         const ArrayNd<N> t11 = p3D.row(1).transpose() * p3D.row(1);
         const ArrayNd<N> t10 = t01.transpose();
         vcsMat[0][0] = cMat[0][0];
         cMat[0][1] = vcsMat[0][1];
         cMat[0][2] = 2. * (vcsMat[0][0] * t0 + vcsMat[0][1] * t1);
         vcsMat[1][1] = cMat[1][1];
         cMat[1][2] = 2. * (vcsMat[1][0] * t0 + vcsMat[1][1] * t1);
         MatrixNd<N> tmp;
         tmp.template triangularView<Eigen::Upper>() =
             (2. * (vcsMat[0][0] * vcsMat[0][0] + vcsMat[0][0] * vcsMat[0][1] + vcsMat[1][1] * vcsMat[1][0] +
                    vcsMat[1][1] * vcsMat[1][1]) +
              4. * (vcsMat[0][0] * t00 + vcsMat[0][1] * t01 + vcsMat[1][0] * t10 + vcsMat[1][1] * t11))
                 .matrix();
         cMat[2][2] = tmp.template selfadjointView<Eigen::Upper>();
       }
       printIt(&cMat[0][0], "circle_fit - C[0][0]:");
 
       Matrix3d c0Mat;  // cov matrix of center of gravity (r0.x,r0.y,r0.z)
       for (uint i = 0; i < 3; ++i) {
         for (uint j = i; j < 3; ++j) {
           Eigen::Matrix<double, 1, 1> tmp;
           tmp = weight.transpose() * cMat[i][j] * weight;
           // Workaround to get things working in GPU
           const double tempC = tmp(0, 0);
           c0Mat(i, j) = tempC;  //weight.transpose() * C[i][j] * weight;
           c0Mat(j, i) = c0Mat(i, j);
         }
       }
       printIt(&c0Mat, "circle_fit - C0:");
 
       const MatrixNd<N> wMat = weight * weight.transpose();
       const MatrixNd<N> hMat = MatrixNd<N>::Identity().rowwise() - weight.transpose();
       const MatrixNx3d<N> s_v = hMat * p3D.transpose();
       printIt(&wMat, "circle_fit - W:");
       printIt(&hMat, "circle_fit - H:");
       printIt(&s_v, "circle_fit - s_v:");
 
       MatrixNd<N> dMat[3][3];  // cov(s_v)
       dMat[0][0] = (hMat * cMat[0][0] * hMat.transpose()).cwiseProduct(wMat);
       dMat[0][1] = (hMat * cMat[0][1] * hMat.transpose()).cwiseProduct(wMat);
       dMat[0][2] = (hMat * cMat[0][2] * hMat.transpose()).cwiseProduct(wMat);
       dMat[1][1] = (hMat * cMat[1][1] * hMat.transpose()).cwiseProduct(wMat);
       dMat[1][2] = (hMat * cMat[1][2] * hMat.transpose()).cwiseProduct(wMat);
       dMat[2][2] = (hMat * cMat[2][2] * hMat.transpose()).cwiseProduct(wMat);
       dMat[1][0] = dMat[0][1].transpose();
       dMat[2][0] = dMat[0][2].transpose();
       dMat[2][1] = dMat[1][2].transpose();
       printIt(&dMat[0][0], "circle_fit - D_[0][0]:");
 
       constexpr uint nu[6][2] = {{0, 0}, {0, 1}, {0, 2}, {1, 1}, {1, 2}, {2, 2}};
 
       Matrix6d eMat;  // cov matrix of the 6 independent elements of A
       for (uint a = 0; a < 6; ++a) {
         const uint i = nu[a][0], j = nu[a][1];
         for (uint b = a; b < 6; ++b) {
           const uint k = nu[b][0], l = nu[b][1];
           VectorNd<N> t0(n);
           VectorNd<N> t1(n);
           if (l == k) {
             t0 = 2. * dMat[j][l] * s_v.col(l);
             if (i == j)
               t1 = t0;
             else
               t1 = 2. * dMat[i][l] * s_v.col(l);
           } else {
             t0 = dMat[j][l] * s_v.col(k) + dMat[j][k] * s_v.col(l);
             if (i == j)
               t1 = t0;
             else
               t1 = dMat[i][l] * s_v.col(k) + dMat[i][k] * s_v.col(l);
           }
 
           if (i == j) {
             Eigen::Matrix<double, 1, 1> cm;
             cm = s_v.col(i).transpose() * (t0 + t1);
             // Workaround to get things working in GPU
             const double tempC = cm(0, 0);
             eMat(a, b) = 0. + tempC;
           } else {
             Eigen::Matrix<double, 1, 1> cm;
             cm = (s_v.col(i).transpose() * t0) + (s_v.col(j).transpose() * t1);
             // Workaround to get things working in GPU
             const double tempC = cm(0, 0);
             eMat(a, b) = 0. + tempC;  //(s_v.col(i).transpose() * t0) + (s_v.col(j).transpose() * t1);
           }
           if (b != a)
             eMat(b, a) = eMat(a, b);
         }
       }
       printIt(&eMat, "circle_fit - E:");
 
       Eigen::Matrix<double, 3, 6> j2Mat;  // Jacobian of min_eigen() (numerically computed)
       for (uint a = 0; a < 6; ++a) {
         const uint i = nu[a][0], j = nu[a][1];
         Matrix3d delta = Matrix3d::Zero();
         delta(i, j) = delta(j, i) = abs(aMat(i, j) * epsilon);
         j2Mat.col(a) = min_eigen3D_fast(acc, aMat + delta);
         const int sign = (j2Mat.col(a)(2) > 0) ? 1 : -1;
         j2Mat.col(a) = (j2Mat.col(a) * sign - vVec) / delta(i, j);
       }
       printIt(&j2Mat, "circle_fit - J2:");
 
       Matrix4d cvcMat;  // joint cov matrix of (v0,v1,v2,c)
       {
         Matrix3d t0 = j2Mat * eMat * j2Mat.transpose();
         Vector3d t1 = -t0 * r0;
         cvcMat.block(0, 0, 3, 3) = t0;
         cvcMat.block(0, 3, 3, 1) = t1;
         cvcMat.block(3, 0, 1, 3) = t1.transpose();
         Eigen::Matrix<double, 1, 1> cm1;
         Eigen::Matrix<double, 1, 1> cm3;
         cm1 = (vVec.transpose() * c0Mat * vVec);
         //      cm2 = (c0Mat.cwiseProduct(t0)).sum();
         cm3 = (r0.transpose() * t0 * r0);
         // Workaround to get things working in GPU
         const double tempC = cm1(0, 0) + (c0Mat.cwiseProduct(t0)).sum() + cm3(0, 0);
         cvcMat(3, 3) = tempC;
         // (v.transpose() * c0Mat * v) + (c0Mat.cwiseProduct(t0)).sum() + (r0.transpose() * t0 * r0);
       }
       printIt(&cvcMat, "circle_fit - Cvc:");
 
       Eigen::Matrix<double, 3, 4> j3Mat;  // Jacobian (v0,v1,v2,c)->(X0,Y0,R)
       {
         const double t = 1. / tempH;
         j3Mat << -v2x2_inv, 0, vVec(0) * sqr(v2x2_inv) * 2., 0, 0, -v2x2_inv, vVec(1) * sqr(v2x2_inv) * 2., 0,
             vVec(0) * v2x2_inv * t, vVec(1) * v2x2_inv * t,
             -tempH * sqr(v2x2_inv) * 2. - (2. * tempC + vVec(2)) * v2x2_inv * t, -t;
       }
       printIt(&j3Mat, "circle_fit - J3:");
 
       const RowVector2Nd<N> Jq = mc.transpose() * tempS * 1. / n;  // var(q)
       printIt(&Jq, "circle_fit - Jq:");
 
       Matrix3d cov_uvr = j3Mat * cvcMat * j3Mat.transpose() * sqr(s_inv)  // cov(X0,Y0,R)
                          + (par_uvr * par_uvr.transpose()) * (Jq * vMat * Jq.transpose());
 
       circle.cov = cov_uvr;
     }
 
     printIt(&circle.cov, "Circle cov:");
 #ifdef RFIT_DEBUG
     printf("circle_fit - exit\n");
 #endif
     return circle;
   }
 
   /*!  \brief Perform an ordinary least square fit in the s-z plane to compute
  * the parameters cotTheta and Zip.
  *
  * The fit is performed in the rotated S3D-Z' plane, following the formalism of
  * Frodesen, Chapter 10, p. 259.
  *
  * The system has been rotated to both try to use the combined errors in s-z
  * along Z', as errors in the Y direction and to avoid the patological case of
  * degenerate lines with angular coefficient m = +/- inf.
  *
  * The rotation is using the information on the theta angle computed in the
  * fast fit. The rotation is such that the S3D axis will be the X-direction,
  * while the rotated Z-axis will be the Y-direction. This pretty much follows
  * what is done in the same fit in the Broken Line approach.
  */
 
   template <typename TAcc, typename M3xN, typename M6xN, typename V4>
   ALPAKA_FN_ACC ALPAKA_FN_INLINE LineFit lineFit(const TAcc& acc,
                                                  const M3xN& hits,
                                                  const M6xN& hits_ge,
                                                  const CircleFit& circle,
                                                  const V4& fast_fit,
                                                  const double bField,
                                                  const bool error) {
     constexpr uint32_t N = M3xN::ColsAtCompileTime;
     constexpr auto n = N;
     double theta = -circle.qCharge * atan(fast_fit(3));
     theta = theta < 0. ? theta + M_PI : theta;
 
     // Prepare the Rotation Matrix to rotate the points
     Eigen::Matrix<double, 2, 2> rot;
     rot << sin(theta), cos(theta), -cos(theta), sin(theta);
 
     // PROJECTION ON THE CILINDER
     //
     // p2D will be:
     // [s1, s2, s3, ..., sn]
     // [z1, z2, z3, ..., zn]
     // s values will be ordinary x-values
     // z values will be ordinary y-values
 
     Matrix2xNd<N> p2D = Matrix2xNd<N>::Zero();
     Eigen::Matrix<double, 2, 6> jxMat;
 
 #ifdef RFIT_DEBUG
     printf("Line_fit - B: %g\n", bField);
     printIt(&hits, "Line_fit points: ");
     printIt(&hits_ge, "Line_fit covs: ");
     printIt(&rot, "Line_fit rot: ");
 #endif
     // x & associated Jacobian
     // cfr https://indico.cern.ch/event/663159/contributions/2707659/attachments/1517175/2368189/Riemann_fit.pdf
     // Slide 11
     // a ==> -o i.e. the origin of the circle in XY plane, negative
     // b ==> p i.e. distances of the points wrt the origin of the circle.
     const Vector2d oVec(circle.par(0), circle.par(1));
 
     // associated Jacobian, used in weights and errors computation
     Matrix6d covMat = Matrix6d::Zero();
     Matrix2d cov_sz[N];
     for (uint i = 0; i < n; ++i) {
       Vector2d pVec = hits.block(0, i, 2, 1) - oVec;
       const double cross = cross2D(acc, -oVec, pVec);
       const double dot = (-oVec).dot(pVec);
       // atan2(cross, dot) give back the angle in the transverse plane so tha the
       // final equation reads: x_i = -q*R*theta (theta = angle returned by atan2)
       const double tempQAtan2 = -circle.qCharge * atan2(cross, dot);
       //    p2D.coeffRef(1, i) = atan2_ * circle.par(2);
       p2D(0, i) = tempQAtan2 * circle.par(2);
 
       // associated Jacobian, used in weights and errors- computation
       const double temp0 = -circle.qCharge * circle.par(2) * 1. / (sqr(dot) + sqr(cross));
       double d_X0 = 0., d_Y0 = 0., d_R = 0.;  // good approximation for big pt and eta
       if (error) {
         d_X0 = -temp0 * ((pVec(1) + oVec(1)) * dot - (pVec(0) - oVec(0)) * cross);
         d_Y0 = temp0 * ((pVec(0) + oVec(0)) * dot - (oVec(1) - pVec(1)) * cross);
         d_R = tempQAtan2;
       }
       const double d_x = temp0 * (oVec(1) * dot + oVec(0) * cross);
       const double d_y = temp0 * (-oVec(0) * dot + oVec(1) * cross);
       jxMat << d_X0, d_Y0, d_R, d_x, d_y, 0., 0., 0., 0., 0., 0., 1.;
 
       covMat.block(0, 0, 3, 3) = circle.cov;
       covMat(3, 3) = hits_ge.col(i)[0];                 // x errors
       covMat(4, 4) = hits_ge.col(i)[2];                 // y errors
       covMat(5, 5) = hits_ge.col(i)[5];                 // z errors
       covMat(3, 4) = covMat(4, 3) = hits_ge.col(i)[1];  // cov_xy
       covMat(3, 5) = covMat(5, 3) = hits_ge.col(i)[3];  // cov_xz
       covMat(4, 5) = covMat(5, 4) = hits_ge.col(i)[4];  // cov_yz
       Matrix2d tmp = jxMat * covMat * jxMat.transpose();
       cov_sz[i].noalias() = rot * tmp * rot.transpose();
     }
     // Math of d_{X0,Y0,R,x,y} all verified by hand
     p2D.row(1) = hits.row(2);
 
     // The following matrix will contain errors orthogonal to the rotated S
     // component only, with the Multiple Scattering properly treated!!
     MatrixNd<N> cov_with_ms;
     scatterCovLine(acc, cov_sz, fast_fit, p2D.row(0), p2D.row(1), theta, bField, cov_with_ms);
 #ifdef RFIT_DEBUG
     printIt(cov_sz, "line_fit - cov_sz:");
     printIt(&cov_with_ms, "line_fit - cov_with_ms: ");
 #endif
 
     // Rotate Points with the shape [2, n]
     Matrix2xNd<N> p2D_rot = rot * p2D;
 
 #ifdef RFIT_DEBUG
     printf("Fast fit Tan(theta): %g\n", fast_fit(3));
     printf("Rotation angle: %g\n", theta);
     printIt(&rot, "Rotation Matrix:");
     printIt(&p2D, "Original Hits(s,z):");
     printIt(&p2D_rot, "Rotated hits(S3D, Z'):");
     printIt(&rot, "Rotation Matrix:");
 #endif
 
     // Build the A Matrix
     Matrix2xNd<N> aMat;
     aMat << MatrixXd::Ones(1, n), p2D_rot.row(0);  // rotated s values
 
 #ifdef RFIT_DEBUG
     printIt(&aMat, "A Matrix:");
 #endif
 
     // Build A^T V-1 A, where V-1 is the covariance of only the Y components.
     MatrixNd<N> vyInvMat;
     math::cholesky::invert(cov_with_ms, vyInvMat);
     // MatrixNd<N> vyInvMat = cov_with_ms.inverse();
     Eigen::Matrix<double, 2, 2> covParamsMat = aMat * vyInvMat * aMat.transpose();
     // Compute the Covariance Matrix of the fit parameters
     math::cholesky::invert(covParamsMat, covParamsMat);
 
     // Now Compute the Parameters in the form [2,1]
     // The first component is q.
     // The second component is m.
     Eigen::Matrix<double, 2, 1> sol = covParamsMat * aMat * vyInvMat * p2D_rot.row(1).transpose();
 
 #ifdef RFIT_DEBUG
     printIt(&sol, "Rotated solutions:");
 #endif
 
     // We need now to transfer back the results in the original s-z plane
     const auto sinTheta = sin(theta);
     const auto cosTheta = cos(theta);
     auto common_factor = 1. / (sinTheta - sol(1, 0) * cosTheta);
     Eigen::Matrix<double, 2, 2> jMat;
     jMat << 0., common_factor * common_factor, common_factor, sol(0, 0) * cosTheta * common_factor * common_factor;
 
     double tempM = common_factor * (sol(1, 0) * sinTheta + cosTheta);
     double tempQ = common_factor * sol(0, 0);
     auto cov_mq = jMat * covParamsMat * jMat.transpose();
 
     VectorNd<N> res = p2D_rot.row(1).transpose() - aMat.transpose() * sol;
     double chi2 = res.transpose() * vyInvMat * res;
 
     LineFit line;
     line.par << tempM, tempQ;
     line.cov << cov_mq;
     line.chi2 = chi2;
 
 #ifdef RFIT_DEBUG
     printf("Common_factor: %g\n", common_factor);
     printIt(&jMat, "Jacobian:");
     printIt(&sol, "Rotated solutions:");
     printIt(&covParamsMat, "Cov_params:");
     printIt(&cov_mq, "Rotated Covariance Matrix:");
     printIt(&(line.par), "Real Parameters:");
     printIt(&(line.cov), "Real Covariance Matrix:");
     printf("Chi2: %g\n", chi2);
 #endif
 
     return line;
   }
 
 }  // namespace ALPAKA_ACCELERATOR_NAMESPACE::riemannFit
 
 namespace riemannFit {
   /*!
     \brief Helix fit by three step:
     -fast pre-fit (see Fast_fit() for further info); \n
     -circle fit of hits projected in the transverse plane by Riemann-Chernov
         algorithm (see Circle_fit() for further info); \n
     -line fit of hits projected on cylinder surface by orthogonal distance
         regression (see Line_fit for further info). \n
     Points must be passed ordered (from inner to outer layer).
     \param hits Matrix3xNd hits coordinates in this form: \n
         |x0|x1|x2|...|xn| \n
         |y0|y1|y2|...|yn| \n
         |z0|z1|z2|...|zn|
     \param hits_cov Matrix3Nd covariance matrix in this form (()->cov()): \n
    |(x0,x0)|(x1,x0)|(x2,x0)|.|(y0,x0)|(y1,x0)|(y2,x0)|.|(z0,x0)|(z1,x0)|(z2,x0)| \n
    |(x0,x1)|(x1,x1)|(x2,x1)|.|(y0,x1)|(y1,x1)|(y2,x1)|.|(z0,x1)|(z1,x1)|(z2,x1)| \n
    |(x0,x2)|(x1,x2)|(x2,x2)|.|(y0,x2)|(y1,x2)|(y2,x2)|.|(z0,x2)|(z1,x2)|(z2,x2)| \n
        .       .       .    .    .       .       .    .    .       .       .     \n
    |(x0,y0)|(x1,y0)|(x2,y0)|.|(y0,y0)|(y1,y0)|(y2,x0)|.|(z0,y0)|(z1,y0)|(z2,y0)| \n
    |(x0,y1)|(x1,y1)|(x2,y1)|.|(y0,y1)|(y1,y1)|(y2,x1)|.|(z0,y1)|(z1,y1)|(z2,y1)| \n
    |(x0,y2)|(x1,y2)|(x2,y2)|.|(y0,y2)|(y1,y2)|(y2,x2)|.|(z0,y2)|(z1,y2)|(z2,y2)| \n
        .       .       .    .    .       .       .    .    .       .       .     \n
    |(x0,z0)|(x1,z0)|(x2,z0)|.|(y0,z0)|(y1,z0)|(y2,z0)|.|(z0,z0)|(z1,z0)|(z2,z0)| \n
    |(x0,z1)|(x1,z1)|(x2,z1)|.|(y0,z1)|(y1,z1)|(y2,z1)|.|(z0,z1)|(z1,z1)|(z2,z1)| \n
    |(x0,z2)|(x1,z2)|(x2,z2)|.|(y0,z2)|(y1,z2)|(y2,z2)|.|(z0,z2)|(z1,z2)|(z2,z2)|
    \param bField magnetic field in the center of the detector in Gev/cm/c
    unit, in order to perform pt calculation.
    \param error flag for error computation.
    \param scattering flag for multiple scattering treatment.
    (see Circle_fit() documentation for further info).
    \warning see Circle_fit(), Line_fit() and Fast_fit() warnings.
    \bug see Circle_fit(), Line_fit() and Fast_fit() bugs.
 */
 
   template <int N>
   class helixFit {
   public:
     template <typename TAcc>
     ALPAKA_FN_ACC ALPAKA_FN_INLINE void operator()(const TAcc& acc,
                                                    const Matrix3xNd<N>* hits,
                                                    const Eigen::Matrix<float, 6, N>* hits_ge,
                                                    const double bField,
                                                    const bool error,
                                                    HelixFit* helix) const {
       constexpr uint n = N;
       VectorNd<4> rad = (hits->block(0, 0, 2, n).colwise().norm());
 
       // Fast_fit gives back (X0, Y0, R, theta) w/o errors, using only 3 points.
       Vector4d fast_fit;
       ALPAKA_ACCELERATOR_NAMESPACE::riemannFit::fastFit(acc, *hits, fast_fit);
       riemannFit::Matrix2Nd<N> hits_cov = MatrixXd::Zero(2 * n, 2 * n);
       ALPAKA_ACCELERATOR_NAMESPACE::riemannFit::loadCovariance2D(acc, *hits_ge, hits_cov);
       CircleFit circle = ALPAKA_ACCELERATOR_NAMESPACE::riemannFit::circleFit(
           acc, hits->block(0, 0, 2, n), hits_cov, fast_fit, rad, bField, error);
       LineFit line =
           ALPAKA_ACCELERATOR_NAMESPACE::riemannFit::lineFit(acc, *hits, *hits_ge, circle, fast_fit, bField, error);
 
       ALPAKA_ACCELERATOR_NAMESPACE::riemannFit::par_uvrtopak(acc, circle, bField, error);
 
       helix->par << circle.par, line.par;
       if (error) {
         helix->cov = MatrixXd::Zero(5, 5);
         helix->cov.block(0, 0, 3, 3) = circle.cov;
         helix->cov.block(3, 3, 2, 2) = line.cov;
       }
       helix->qCharge = circle.qCharge;
       helix->chi2_circle = circle.chi2;
       helix->chi2_line = line.chi2;
     }
   };
 
 }  // namespace riemannFit
 
 #endif  // RecoTracker_PixelTrackFitting_interface_alpaka_RiemannFit_h

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