ReferenceTrajectory.cc

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//  Author     : Gero Flucke (based on code by Edmund Widl replacing ORCA's TkReferenceTrack)
//  date       : 2006/09/17
//  last update: $Date: 2012/12/25 16:42:04 $
//  by         : $Author: innocent $

#include <memory>
#include <limits>
#include <cmath>
#include <cstdlib>

#include "Alignment/ReferenceTrajectories/interface/ReferenceTrajectory.h"

#include "DataFormats/GeometrySurface/interface/Surface.h"
#include "DataFormats/GeometrySurface/interface/Plane.h"

#include "FWCore/MessageLogger/interface/MessageLogger.h"

#include "DataFormats/CLHEP/interface/AlgebraicObjects.h"
#include "DataFormats/GeometrySurface/interface/LocalError.h"
#include "DataFormats/GeometryVector/interface/LocalPoint.h"
#include "Geometry/CommonDetUnit/interface/TrackerGeomDet.h"

#include "DataFormats/TrajectoryState/interface/LocalTrajectoryParameters.h"
#include "DataFormats/TrackingRecHit/interface/KfComponentsHolder.h"

#include "TrackingTools/AnalyticalJacobians/interface/AnalyticalCurvilinearJacobian.h"
#include "TrackingTools/AnalyticalJacobians/interface/JacobianLocalToCurvilinear.h"
#include "TrackingTools/AnalyticalJacobians/interface/JacobianCurvilinearToLocal.h"

#include "TrackingTools/GeomPropagators/interface/AnalyticalPropagator.h"
#include "TrackPropagation/RungeKutta/interface/defaultRKPropagator.h"

#include "TrackingTools/TrajectoryState/interface/TrajectoryStateOnSurface.h"
#include "TrackingTools/TrajectoryParametrization/interface/GlobalTrajectoryParameters.h"

#include "TrackingTools/MaterialEffects/interface/MultipleScatteringUpdator.h"
#include "TrackingTools/MaterialEffects/interface/EnergyLossUpdator.h"
#include "TrackingTools/MaterialEffects/interface/CombinedMaterialEffectsUpdator.h"
#include <TrackingTools/PatternTools/interface/TSCPBuilderNoMaterial.h>
#include <TrackingTools/PatternTools/interface/TSCBLBuilderNoMaterial.h>
#include "TrackingTools/TrajectoryState/interface/TrajectoryStateClosestToPoint.h"
#include "TrackingTools/TrajectoryState/interface/TrajectoryStateClosestToBeamLine.h"

#include "MagneticField/Engine/interface/MagneticField.h"

#include "Alignment/ReferenceTrajectories/interface/BeamSpotTransientTrackingRecHit.h"
#include "Alignment/ReferenceTrajectories/interface/BeamSpotGeomDet.h"

//__________________________________________________________________________________
using namespace gbl;

ReferenceTrajectory::ReferenceTrajectory(const TrajectoryStateOnSurface &refTsos,
                                         const TransientTrackingRecHit::ConstRecHitContainer &recHits,
                                         const MagneticField *magField,
                                         const reco::BeamSpot &beamSpot,
                                         const ReferenceTrajectoryBase::Config &config)
    : ReferenceTrajectoryBase(
          (config.materialEffects >= brokenLinesCoarse) ? 1 : refTsos.localParameters().mixedFormatVector().kSize,
          (config.useBeamSpot) ? recHits.size() + 1 : recHits.size(),
          (config.materialEffects >= brokenLinesCoarse)
              ? 2 * ((config.useBeamSpot) ? recHits.size() + 1 : recHits.size())
              : ((config.materialEffects == breakPoints)
                     ? 2 * ((config.useBeamSpot) ? recHits.size() + 1 : recHits.size()) - 2
                     : 0),
          (config.materialEffects >= brokenLinesCoarse)
              ? 2 * ((config.useBeamSpot) ? recHits.size() + 1 : recHits.size()) - 4
              : ((config.materialEffects == breakPoints)
                     ? 2 * ((config.useBeamSpot) ? recHits.size() + 1 : recHits.size()) - 2
                     : 0)),
      mass_(config.mass),
      materialEffects_(config.materialEffects),
      propDir_(config.propDir),
      useBeamSpot_(config.useBeamSpot),
      includeAPEs_(config.includeAPEs),
      allowZeroMaterial_(config.allowZeroMaterial) {
  // no check against magField == 0
  theParameters = asHepVector<5>(refTsos.localParameters().mixedFormatVector());

  if (config.hitsAreReverse) {
    TransientTrackingRecHit::ConstRecHitContainer fwdRecHits;
    fwdRecHits.reserve(recHits.size());
    for (TransientTrackingRecHit::ConstRecHitContainer::const_reverse_iterator it = recHits.rbegin();
         it != recHits.rend();
         ++it) {
      fwdRecHits.push_back(*it);
    }
    theValidityFlag = this->construct(refTsos, fwdRecHits, magField, beamSpot);
  } else {
    theValidityFlag = this->construct(refTsos, recHits, magField, beamSpot);
  }
}

//__________________________________________________________________________________

ReferenceTrajectory::ReferenceTrajectory(unsigned int nPar,
                                         unsigned int nHits,
                                         const ReferenceTrajectoryBase::Config &config)
    : ReferenceTrajectoryBase((config.materialEffects >= brokenLinesCoarse) ? 1 : nPar,
                              nHits,
                              (config.materialEffects >= brokenLinesCoarse)
                                  ? 2 * nHits
                                  : ((config.materialEffects == breakPoints) ? 2 * nHits - 2 : 0),
                              (config.materialEffects >= brokenLinesCoarse)
                                  ? 2 * nHits - 4
                                  : ((config.materialEffects == breakPoints) ? 2 * nHits - 2 : 0)),
      mass_(config.mass),
      materialEffects_(config.materialEffects),
      propDir_(config.propDir),
      useBeamSpot_(config.useBeamSpot),
      includeAPEs_(config.includeAPEs),
      allowZeroMaterial_(config.allowZeroMaterial) {}

//__________________________________________________________________________________

bool ReferenceTrajectory::construct(const TrajectoryStateOnSurface &refTsos,
                                    const TransientTrackingRecHit::ConstRecHitContainer &recHits,
                                    const MagneticField *magField,
                                    const reco::BeamSpot &beamSpot) {
  TrajectoryStateOnSurface theRefTsos = refTsos;

  const SurfaceSide surfaceSide = this->surfaceSide(propDir_);
  // auto_ptr to avoid memory leaks in case of not reaching delete at end of method:
  std::unique_ptr<MaterialEffectsUpdator> aMaterialEffectsUpdator(this->createUpdator(materialEffects_, mass_));
  if (!aMaterialEffectsUpdator.get())
    return false;  // empty auto_ptr

  AlgebraicMatrix fullJacobian(theParameters.num_row(), theParameters.num_row());
  std::vector<AlgebraicMatrix> allJacobians;
  allJacobians.reserve(theNumberOfHits);

  TransientTrackingRecHit::ConstRecHitPointer previousHitPtr;
  TrajectoryStateOnSurface previousTsos;
  AlgebraicSymMatrix previousChangeInCurvature(theParameters.num_row(), 1);
  std::vector<AlgebraicSymMatrix> allCurvatureChanges;
  allCurvatureChanges.reserve(theNumberOfHits);

  const LocalTrajectoryError zeroErrors(0., 0., 0., 0., 0.);

  std::vector<AlgebraicMatrix> allProjections;
  allProjections.reserve(theNumberOfHits);
  std::vector<AlgebraicSymMatrix> allDeltaParameterCovs;
  allDeltaParameterCovs.reserve(theNumberOfHits);

  // CHK
  std::vector<AlgebraicMatrix> allLocalToCurv;
  allLocalToCurv.reserve(theNumberOfHits);
  std::vector<double> allSteps;
  allSteps.reserve(theNumberOfHits);
  std::vector<AlgebraicMatrix> allCurvlinJacobians;
  allCurvlinJacobians.reserve(theNumberOfHits);

  AlgebraicMatrix firstCurvlinJacobian(5, 5, 1);

  unsigned int iRow = 0;

  theNomField = magField->nominalValue();  // nominal magnetic field in kGauss
  // local storage vector of all rechits (including rechit for beam spot in case it is used)
  TransientTrackingRecHit::ConstRecHitContainer allRecHits;

  if (useBeamSpot_ && propDir_ == alongMomentum) {
    GlobalPoint bs(beamSpot.x0(), beamSpot.y0(), beamSpot.z0());

    TrajectoryStateClosestToBeamLine tsctbl(TSCBLBuilderNoMaterial()(*(refTsos.freeState()), beamSpot));
    if (!tsctbl.isValid()) {
      edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::construct"
                                 << "TrajectoryStateClostestToBeamLine invalid. Skip track.";
      return false;
    }

    FreeTrajectoryState pcaFts = tsctbl.trackStateAtPCA();
    GlobalVector bd(beamSpot.dxdz(), beamSpot.dydz(), 1.0);

    //propagation FIXME: Should use same propagator everywhere...
    AnalyticalPropagator propagator(magField);
    std::pair<TrajectoryStateOnSurface, double> tsosWithPath = propagator.propagateWithPath(pcaFts, refTsos.surface());

    if (!tsosWithPath.first.isValid())
      return false;

    GlobalVector momDir(pcaFts.momentum());
    GlobalVector perpDir(bd.cross(momDir));
    Plane::RotationType rotation(perpDir, bd);

    BeamSpotGeomDet *bsGeom = new BeamSpotGeomDet(Plane::build(bs, rotation));

    // There is also a constructor taking the magentic field. Use this one instead?
    theRefTsos = TrajectoryStateOnSurface(pcaFts, bsGeom->surface());

    TransientTrackingRecHit::ConstRecHitPointer bsRecHit(
        new BeamSpotTransientTrackingRecHit(beamSpot, bsGeom, theRefTsos.freeState()->momentum().phi()));
    allRecHits.push_back(bsRecHit);
  }

  // copy all rechits to the local storage vector
  TransientTrackingRecHit::ConstRecHitContainer::const_iterator itRecHit;
  for (itRecHit = recHits.begin(); itRecHit != recHits.end(); ++itRecHit) {
    const TransientTrackingRecHit::ConstRecHitPointer &hitPtr = *itRecHit;
    allRecHits.push_back(hitPtr);
  }

  for (itRecHit = allRecHits.begin(); itRecHit != allRecHits.end(); ++itRecHit) {
    const TransientTrackingRecHit::ConstRecHitPointer &hitPtr = *itRecHit;
    theRecHits.push_back(hitPtr);

    if (0 == iRow) {
      // compute the derivatives of the reference-track's parameters w.r.t. the initial ones
      // derivative of the initial reference-track parameters w.r.t. themselves is of course the identity
      fullJacobian = AlgebraicMatrix(theParameters.num_row(), theParameters.num_row(), 1);
      allJacobians.push_back(fullJacobian);
      theTsosVec.push_back(theRefTsos);
      const JacobianLocalToCurvilinear startTrafo(hitPtr->det()->surface(), theRefTsos.localParameters(), *magField);
      const AlgebraicMatrix localToCurvilinear = asHepMatrix<5>(startTrafo.jacobian());
      if (materialEffects_ <= breakPoints) {
        theInnerTrajectoryToCurvilinear = asHepMatrix<5>(startTrafo.jacobian());
        theInnerLocalToTrajectory = AlgebraicMatrix(5, 5, 1);
      }
      allLocalToCurv.push_back(localToCurvilinear);
      allSteps.push_back(0.);
      allCurvlinJacobians.push_back(firstCurvlinJacobian);

    } else {
      AlgebraicMatrix nextJacobian;
      AlgebraicMatrix nextCurvlinJacobian;
      double nextStep = 0.;
      TrajectoryStateOnSurface nextTsos;

      if (!this->propagate(previousHitPtr->det()->surface(),
                           previousTsos,
                           hitPtr->det()->surface(),
                           nextTsos,
                           nextJacobian,
                           nextCurvlinJacobian,
                           nextStep,
                           magField)) {
        return false;  // stop if problem...// no delete aMaterialEffectsUpdator needed
      }

      allJacobians.push_back(nextJacobian);
      fullJacobian = nextJacobian * previousChangeInCurvature * fullJacobian;
      theTsosVec.push_back(nextTsos);

      const JacobianLocalToCurvilinear startTrafo(hitPtr->det()->surface(), nextTsos.localParameters(), *magField);
      const AlgebraicMatrix localToCurvilinear = asHepMatrix<5>(startTrafo.jacobian());
      allLocalToCurv.push_back(localToCurvilinear);
      if (nextStep == 0.) {
        edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::construct"
                                   << "step 0. from id " << previousHitPtr->geographicalId() << " to "
                                   << hitPtr->det()->geographicalId() << ".";
        // brokenLinesFine will not work, brokenLinesCoarse combines close by layers
        if (materialEffects_ == brokenLinesFine) {
          edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::construct"
                                     << "Skip track.";
          return false;
        }
      }
      allSteps.push_back(nextStep);
      allCurvlinJacobians.push_back(nextCurvlinJacobian);
    }

    // take material effects into account. since trajectory-state is constructed with errors equal zero,
    // the updated state contains only the uncertainties due to interactions in the current layer.
    const TrajectoryStateOnSurface tmpTsos(
        theTsosVec.back().localParameters(), zeroErrors, theTsosVec.back().surface(), magField, surfaceSide);
    const TrajectoryStateOnSurface updatedTsos = aMaterialEffectsUpdator->updateState(tmpTsos, propDir_);

    if (!updatedTsos.isValid())
      return false;  // no delete aMaterialEffectsUpdator needed

    if (theTsosVec.back().localParameters().charge()) {
      previousChangeInCurvature[0][0] = updatedTsos.localParameters().signedInverseMomentum() /
                                        theTsosVec.back().localParameters().signedInverseMomentum();
    }

    // get multiple-scattering covariance-matrix
    allDeltaParameterCovs.push_back(asHepMatrix<5>(updatedTsos.localError().matrix()));
    allCurvatureChanges.push_back(previousChangeInCurvature);

    // projection-matrix tsos-parameters -> measurement-coordinates
    allProjections.push_back(this->getHitProjectionMatrix(hitPtr));
    // set start-parameters for next propagation. trajectory-state without error
    //  - no error propagation needed here.
    previousHitPtr = hitPtr;
    previousTsos = TrajectoryStateOnSurface(updatedTsos.globalParameters(), updatedTsos.surface(), surfaceSide);

    if (materialEffects_ < brokenLinesCoarse) {
      this->fillDerivatives(allProjections.back(), fullJacobian, iRow);
    }

    AlgebraicVector mixedLocalParams = asHepVector<5>(theTsosVec.back().localParameters().mixedFormatVector());
    this->fillTrajectoryPositions(allProjections.back(), mixedLocalParams, iRow);
    if (useRecHit(hitPtr))
      this->fillMeasurementAndError(hitPtr, iRow, updatedTsos);

    iRow += nMeasPerHit;
  }  // end of loop on hits

  bool msOK = true;
  switch (materialEffects_) {
    case none:
      break;
    case multipleScattering:
    case energyLoss:
    case combined:
      msOK = this->addMaterialEffectsCov(allJacobians, allProjections, allCurvatureChanges, allDeltaParameterCovs);
      break;
    case breakPoints:
      msOK = this->addMaterialEffectsBp(
          allJacobians, allProjections, allCurvatureChanges, allDeltaParameterCovs, allLocalToCurv);
      break;
    case brokenLinesCoarse:
      msOK = this->addMaterialEffectsBrl(
          allProjections, allDeltaParameterCovs, allLocalToCurv, allSteps, refTsos.globalParameters());
      break;
    case brokenLinesFine:
      msOK = this->addMaterialEffectsBrl(allCurvlinJacobians,
                                         allProjections,
                                         allCurvatureChanges,
                                         allDeltaParameterCovs,
                                         allLocalToCurv,
                                         refTsos.globalParameters());
      break;
    case localGBL:
      msOK = this->addMaterialEffectsLocalGbl(allJacobians, allProjections, allCurvatureChanges, allDeltaParameterCovs);
      break;
    case curvlinGBL:
      msOK = this->addMaterialEffectsCurvlinGbl(
          allCurvlinJacobians, allProjections, allCurvatureChanges, allDeltaParameterCovs, allLocalToCurv);
  }
  if (!msOK)
    return false;

  if (refTsos.hasError()) {
    AlgebraicSymMatrix parameterCov = asHepMatrix<5>(refTsos.localError().matrix());
    AlgebraicMatrix parDeriv;
    if (theNumberOfVirtualPars > 0) {
      parDeriv = theDerivatives.sub(1, nMeasPerHit * allJacobians.size(), 1, theParameters.num_row());
    } else {
      parDeriv = theDerivatives;
    }
    theTrajectoryPositionCov = parameterCov.similarity(parDeriv);
  } else {
    theTrajectoryPositionCov = AlgebraicSymMatrix(theDerivatives.num_row(), 1);
  }

  return true;
}

//__________________________________________________________________________________

MaterialEffectsUpdator *ReferenceTrajectory::createUpdator(MaterialEffects materialEffects, double mass) const {
  switch (materialEffects) {
      // MultipleScatteringUpdator doesn't change the trajectory-state
      // during update and can therefore be used if material effects should be ignored:
    case none:
    case multipleScattering:
      return new MultipleScatteringUpdator(mass);
    case energyLoss:
      return new EnergyLossUpdator(mass);
    case combined:
      return new CombinedMaterialEffectsUpdator(mass);
    case breakPoints:
      return new CombinedMaterialEffectsUpdator(mass);
    case brokenLinesCoarse:
    case brokenLinesFine:
    case localGBL:
    case curvlinGBL:
      return new CombinedMaterialEffectsUpdator(mass);
  }

  return nullptr;
}

//__________________________________________________________________________________

bool ReferenceTrajectory::propagate(const Plane &previousSurface,
                                    const TrajectoryStateOnSurface &previousTsos,
                                    const Plane &newSurface,
                                    TrajectoryStateOnSurface &newTsos,
                                    AlgebraicMatrix &newJacobian,
                                    AlgebraicMatrix &newCurvlinJacobian,
                                    double &nextStep,
                                    const MagneticField *magField) const {
  // propagate to next layer
  //AnalyticalPropagator aPropagator(magField, propDir_);
  // Hard coded RungeKutta instead Analytical (avoid bias in TEC), but
  // work around TrackPropagation/RungeKutta/interface/RKTestPropagator.h and
  // http://www.parashift.com/c++-faq-lite/strange-inheritance.html#faq-23.9
  defaultRKPropagator::Product rkprod(magField, propDir_);  //double tolerance = 5.e-5)
  Propagator &aPropagator = rkprod.propagator;
  const std::pair<TrajectoryStateOnSurface, double> tsosWithPath =
      aPropagator.propagateWithPath(previousTsos, newSurface);

  // stop if propagation wasn't successful
  if (!tsosWithPath.first.isValid())
    return false;

  nextStep = tsosWithPath.second;
  // calculate derivative of reference-track parameters on the actual layer w.r.t. the ones
  // on the previous layer (both in global coordinates)
  const AnalyticalCurvilinearJacobian aJacobian(previousTsos.globalParameters(),
                                                tsosWithPath.first.globalPosition(),
                                                tsosWithPath.first.globalMomentum(),
                                                tsosWithPath.second);
  const AlgebraicMatrix curvilinearJacobian = asHepMatrix<5, 5>(aJacobian.jacobian());

  // jacobian of the track parameters on the previous layer for local->global transformation
  const JacobianLocalToCurvilinear startTrafo(previousSurface, previousTsos.localParameters(), *magField);
  const AlgebraicMatrix localToCurvilinear = asHepMatrix<5>(startTrafo.jacobian());

  // jacobian of the track parameters on the actual layer for global->local transformation
  const JacobianCurvilinearToLocal endTrafo(newSurface, tsosWithPath.first.localParameters(), *magField);
  const AlgebraicMatrix curvilinearToLocal = asHepMatrix<5>(endTrafo.jacobian());

  // compute derivative of reference-track parameters on the actual layer w.r.t. the ones on
  // the previous layer (both in their local representation)
  newCurvlinJacobian = curvilinearJacobian;
  newJacobian = curvilinearToLocal * curvilinearJacobian * localToCurvilinear;
  newTsos = tsosWithPath.first;

  return true;
}

//__________________________________________________________________________________

void ReferenceTrajectory::fillMeasurementAndError(const TransientTrackingRecHit::ConstRecHitPointer &hitPtr,
                                                  unsigned int iRow,
                                                  const TrajectoryStateOnSurface &updatedTsos) {
  // get the measurements and their errors, use information updated with tsos if improving
  // (GF: Also for measurements or only for errors or do the former not change?)
  // GF 10/2008: I doubt that it makes sense to update the hit with the tsos here:
  //             That is an analytical extrapolation and not the best guess of the real
  //             track state on the module, but the latter should be better to get the best
  //             hit uncertainty estimate!

  // FIXME FIXME  CLONE
  const auto &newHitPtr = hitPtr;
  //  TransientTrackingRecHit::ConstRecHitPointer newHitPtr(hitPtr->canImproveWithTrack() ?
  //                            hitPtr->clone(updatedTsos) : hitPtr);

  const LocalPoint localMeasurement = newHitPtr->localPosition();
  const LocalError localMeasurementCov = newHitPtr->localPositionError();  // CPE+APE

  theMeasurements[iRow] = localMeasurement.x();
  theMeasurements[iRow + 1] = localMeasurement.y();
  theMeasurementsCov[iRow][iRow] = localMeasurementCov.xx();
  theMeasurementsCov[iRow][iRow + 1] = localMeasurementCov.xy();
  theMeasurementsCov[iRow + 1][iRow + 1] = localMeasurementCov.yy();

  if (!includeAPEs_) {
    // subtract APEs (if existing) from covariance matrix
    auto det = static_cast<const TrackerGeomDet *>(newHitPtr->det());
    const auto localAPE = det->localAlignmentError();
    if (localAPE.valid()) {
      theMeasurementsCov[iRow][iRow] -= localAPE.xx();
      theMeasurementsCov[iRow][iRow + 1] -= localAPE.xy();
      theMeasurementsCov[iRow + 1][iRow + 1] -= localAPE.yy();
    }
  }
}

//__________________________________________________________________________________

void ReferenceTrajectory::fillDerivatives(const AlgebraicMatrix &projection,
                                          const AlgebraicMatrix &fullJacobian,
                                          unsigned int iRow) {
  // derivatives of the local coordinates of the reference track w.r.t. to the inital track-parameters
  const AlgebraicMatrix projectedJacobian(projection * fullJacobian);
  for (int i = 0; i < parameters().num_row(); ++i) {
    for (int j = 0; j < projectedJacobian.num_row(); ++j) {
      theDerivatives[iRow + j][i] = projectedJacobian[j][i];
    }
  }
}

//__________________________________________________________________________________

void ReferenceTrajectory::fillTrajectoryPositions(const AlgebraicMatrix &projection,
                                                  const AlgebraicVector &mixedLocalParams,
                                                  unsigned int iRow) {
  // get the local coordinates of the reference trajectory
  const AlgebraicVector localPosition(projection * mixedLocalParams);
  for (int i = 0; i < localPosition.num_row(); ++i) {
    theTrajectoryPositions[iRow + i] = localPosition[i];
  }
}

//__________________________________________________________________________________

bool ReferenceTrajectory::addMaterialEffectsCov(const std::vector<AlgebraicMatrix> &allJacobians,
                                                const std::vector<AlgebraicMatrix> &allProjections,
                                                const std::vector<AlgebraicSymMatrix> &allCurvatureChanges,
                                                const std::vector<AlgebraicSymMatrix> &allDeltaParameterCovs) {
  // the uncertainty due to multiple scattering is 'transferred' to the error matrix of the hits

  // GF: Needs update once hit dimension is not hardcoded as nMeasPerHit!

  AlgebraicSymMatrix materialEffectsCov(nMeasPerHit * allJacobians.size(), 0);

  // additional covariance-matrix of the measurements due to material-effects (single measurement)
  AlgebraicSymMatrix deltaMaterialEffectsCov;

  // additional covariance-matrix of the parameters due to material-effects
  AlgebraicSymMatrix paramMaterialEffectsCov(allDeltaParameterCovs[0]);  //initialization
  // error-propagation to state after energy loss
  //GFback  paramMaterialEffectsCov = paramMaterialEffectsCov.similarity(allCurvatureChanges[0]);

  AlgebraicMatrix tempParameterCov;
  AlgebraicMatrix tempMeasurementCov;

  for (unsigned int k = 1; k < allJacobians.size(); ++k) {
    // error-propagation to next layer
    paramMaterialEffectsCov = paramMaterialEffectsCov.similarity(allJacobians[k]);

    // get dependences for the measurements
    deltaMaterialEffectsCov = paramMaterialEffectsCov.similarity(allProjections[k]);
    materialEffectsCov[nMeasPerHit * k][nMeasPerHit * k] = deltaMaterialEffectsCov[0][0];
    materialEffectsCov[nMeasPerHit * k][nMeasPerHit * k + 1] = deltaMaterialEffectsCov[0][1];
    materialEffectsCov[nMeasPerHit * k + 1][nMeasPerHit * k] = deltaMaterialEffectsCov[1][0];
    materialEffectsCov[nMeasPerHit * k + 1][nMeasPerHit * k + 1] = deltaMaterialEffectsCov[1][1];

    // GFback add uncertainties for the following layers due to scattering at this layer
    paramMaterialEffectsCov += allDeltaParameterCovs[k];
    // end GFback
    tempParameterCov = paramMaterialEffectsCov;

    // compute "inter-layer-dependencies"
    for (unsigned int l = k + 1; l < allJacobians.size(); ++l) {
      tempParameterCov = allJacobians[l] * allCurvatureChanges[l] * tempParameterCov;
      tempMeasurementCov = allProjections[l] * tempParameterCov * allProjections[k].T();

      materialEffectsCov[nMeasPerHit * l][nMeasPerHit * k] = tempMeasurementCov[0][0];
      materialEffectsCov[nMeasPerHit * k][nMeasPerHit * l] = tempMeasurementCov[0][0];

      materialEffectsCov[nMeasPerHit * l][nMeasPerHit * k + 1] = tempMeasurementCov[0][1];
      materialEffectsCov[nMeasPerHit * k + 1][nMeasPerHit * l] = tempMeasurementCov[0][1];

      materialEffectsCov[nMeasPerHit * l + 1][nMeasPerHit * k] = tempMeasurementCov[1][0];
      materialEffectsCov[nMeasPerHit * k][nMeasPerHit * l + 1] = tempMeasurementCov[1][0];

      materialEffectsCov[nMeasPerHit * l + 1][nMeasPerHit * k + 1] = tempMeasurementCov[1][1];
      materialEffectsCov[nMeasPerHit * k + 1][nMeasPerHit * l + 1] = tempMeasurementCov[1][1];
    }
    // add uncertainties for the following layers due to scattering at this layer
    // GFback paramMaterialEffectsCov += allDeltaParameterCovs[k];
    // error-propagation to state after energy loss
    paramMaterialEffectsCov = paramMaterialEffectsCov.similarity(allCurvatureChanges[k]);
  }
  theMeasurementsCov += materialEffectsCov;

  return true;  // cannot fail
}

//__________________________________________________________________________________

bool ReferenceTrajectory::addMaterialEffectsBp(const std::vector<AlgebraicMatrix> &allJacobians,
                                               const std::vector<AlgebraicMatrix> &allProjections,
                                               const std::vector<AlgebraicSymMatrix> &allCurvatureChanges,
                                               const std::vector<AlgebraicSymMatrix> &allDeltaParameterCovs,
                                               const std::vector<AlgebraicMatrix> &allLocalToCurv) {
  //CHK: add material effects using break points
  int offsetPar = theNumberOfPars;
  int offsetMeas = nMeasPerHit * allJacobians.size();
  int ierr = 0;

  AlgebraicMatrix tempJacobian;
  AlgebraicMatrix MSprojection(2, 5);
  MSprojection[0][1] = 1;
  MSprojection[1][2] = 1;
  AlgebraicSymMatrix tempMSCov;
  AlgebraicSymMatrix tempMSCovProj;
  AlgebraicMatrix tempMSJacProj;

  for (unsigned int k = 1; k < allJacobians.size(); ++k) {
    // CHK
    int kbp = k - 1;
    tempJacobian = allJacobians[k] * allCurvatureChanges[k];
    tempMSCov = allDeltaParameterCovs[k - 1].similarity(allLocalToCurv[k - 1]);
    tempMSCovProj = tempMSCov.similarity(MSprojection);
    theMeasurementsCov[offsetMeas + nMeasPerHit * kbp][offsetMeas + nMeasPerHit * kbp] = tempMSCovProj[0][0];
    theMeasurementsCov[offsetMeas + nMeasPerHit * kbp + 1][offsetMeas + nMeasPerHit * kbp + 1] = tempMSCovProj[1][1];
    theDerivatives[offsetMeas + nMeasPerHit * kbp][offsetPar + 2 * kbp] = 1.0;
    theDerivatives[offsetMeas + nMeasPerHit * kbp + 1][offsetPar + 2 * kbp + 1] = 1.0;

    tempMSJacProj = (allProjections[k] * (tempJacobian * allLocalToCurv[k - 1].inverse(ierr))) * MSprojection.T();
    if (ierr) {
      edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBp"
                                 << "Inversion 1 for break points failed: " << ierr;
      return false;
    }
    theDerivatives[nMeasPerHit * k][offsetPar + 2 * kbp] = tempMSJacProj[0][0];
    theDerivatives[nMeasPerHit * k][offsetPar + 2 * kbp + 1] = tempMSJacProj[0][1];
    theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * kbp] = tempMSJacProj[1][0];
    theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * kbp + 1] = tempMSJacProj[1][1];

    for (unsigned int l = k + 1; l < allJacobians.size(); ++l) {
      // CHK
      int kbp = k - 1;
      tempJacobian = allJacobians[l] * allCurvatureChanges[l] * tempJacobian;
      tempMSJacProj = (allProjections[l] * (tempJacobian * allLocalToCurv[k - 1].inverse(ierr))) * MSprojection.T();
      if (ierr) {
        edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBp"
                                   << "Inversion 2 for break points failed: " << ierr;
        return false;
      }
      theDerivatives[nMeasPerHit * l][offsetPar + 2 * kbp] = tempMSJacProj[0][0];
      theDerivatives[nMeasPerHit * l][offsetPar + 2 * kbp + 1] = tempMSJacProj[0][1];
      theDerivatives[nMeasPerHit * l + 1][offsetPar + 2 * kbp] = tempMSJacProj[1][0];
      theDerivatives[nMeasPerHit * l + 1][offsetPar + 2 * kbp + 1] = tempMSJacProj[1][1];
    }
  }

  return true;
}

//__________________________________________________________________________________

bool ReferenceTrajectory::addMaterialEffectsBrl(const std::vector<AlgebraicMatrix> &allCurvlinJacobians,
                                                const std::vector<AlgebraicMatrix> &allProjections,
                                                const std::vector<AlgebraicSymMatrix> &allCurvatureChanges,
                                                const std::vector<AlgebraicSymMatrix> &allDeltaParameterCovs,
                                                const std::vector<AlgebraicMatrix> &allLocalToCurv,
                                                const GlobalTrajectoryParameters &gtp) {
  //CHK: add material effects using broken lines
  //fine: use exact Jacobians, all detectors
  //broken lines: pair of offsets (u1,u2) = (xt,yt) (in curvilinear frame (q/p,lambda,phi,xt,yt)) at each layer
  //              scattering angles (alpha1,alpha2) = (cosLambda*dPhi, dLambda) (cosLambda cancels in Chi2)
  //              DU' = (dU'/dU)*DU + (dU'/dAlpha)*DAlpha + (dU'/dQbyp)*DQbyp (propagation of U)
  //                  = J*DU + S*DAlpha + d*DQbyp
  //           => DAlpha = S^-1 (DU' - J*DU - d*DQbyp)

  int offsetPar = theNumberOfPars;
  int offsetMeas = nMeasPerHit * allCurvlinJacobians.size();
  int ierr = 0;

  GlobalVector p = gtp.momentum();
  double cosLambda = sqrt((p.x() * p.x() + p.y() * p.y()) / (p.x() * p.x() + p.y() * p.y() + p.z() * p.z()));

  // transformations Curvilinear <-> BrokenLines
  AlgebraicMatrix QbypToCurv(5, 1);    // dCurv/dQbyp
  QbypToCurv[0][0] = 1.;               // dQbyp/dQbyp
  AlgebraicMatrix AngleToCurv(5, 2);   // dCurv/dAlpha
  AngleToCurv[1][1] = 1.;              // dlambda/dalpha2
  AngleToCurv[2][0] = 1. / cosLambda;  // dphi/dalpha1
  AlgebraicMatrix CurvToAngle(2, 5);   // dAlpha/dCurv
  CurvToAngle[1][1] = 1.;              // dalpha2/dlambda
  CurvToAngle[0][2] = cosLambda;       // dalpha1/dphi
  AlgebraicMatrix OffsetToCurv(5, 2);  // dCurv/dU
  OffsetToCurv[3][0] = 1.;             // dxt/du1
  OffsetToCurv[4][1] = 1.;             // dyt/du2
  AlgebraicMatrix CurvToOffset(2, 5);  // dU/dCurv
  CurvToOffset[0][3] = 1.;             // du1/dxt
  CurvToOffset[1][4] = 1.;             // du2/dyt

  // transformations  trajectory to components (Qbyp, U1, U2)
  AlgebraicMatrix TrajToQbyp(1, 5);
  TrajToQbyp[0][0] = 1.;
  AlgebraicMatrix TrajToOff1(2, 5);
  TrajToOff1[0][1] = 1.;
  TrajToOff1[1][2] = 1.;
  AlgebraicMatrix TrajToOff2(2, 5);
  TrajToOff2[0][3] = 1.;
  TrajToOff2[1][4] = 1.;

  AlgebraicMatrix JacOffsetToAngleC, JacQbypToAngleC;
  AlgebraicMatrix JacCurvToOffsetL, JacOffsetToOffsetL, JacAngleToOffsetL, JacQbypToOffsetL, JacOffsetToAngleL;
  AlgebraicMatrix JacCurvToOffsetN, JacOffsetToOffsetN, JacAngleToOffsetN, JacQbypToOffsetN, JacOffsetToAngleN;

  // transformation from trajectory to curvilinear parameters

  JacCurvToOffsetN = CurvToOffset * allCurvlinJacobians[1];  // (dU'/dCurv') * (dCurv'/dCurv) @ 2nd point
  JacOffsetToOffsetN = JacCurvToOffsetN * OffsetToCurv;  // J: (dU'/dU)     = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dU)
  JacAngleToOffsetN =
      JacCurvToOffsetN * AngleToCurv;                // S: (dU'/dAlpha) = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dAlpha)
  JacQbypToOffsetN = JacCurvToOffsetN * QbypToCurv;  // d: (dU'/dQbyp)  = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dQbyp)
  JacOffsetToAngleN = JacAngleToOffsetN.inverse(ierr);  // W
  if (ierr) {
    edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBrl"
                               << "Inversion 1 for fine broken lines failed: " << ierr;
    return false;
  }
  JacOffsetToAngleC = -(JacOffsetToAngleN * JacOffsetToOffsetN);  // (dAlpha/dU)
  JacQbypToAngleC = -(JacOffsetToAngleN * JacQbypToOffsetN);      // (dAlpha/dQbyp)
  // (dAlpha/dTraj) = (dAlpha/dQbyp) * (dQbyp/dTraj) + (dAlpha/dU1) * (dU1/dTraj) + (dAlpha/dU2) * (dU2/dTraj)
  AlgebraicMatrix JacTrajToAngle =
      JacQbypToAngleC * TrajToQbyp + JacOffsetToAngleC * TrajToOff1 + JacOffsetToAngleN * TrajToOff2;
  // (dCurv/dTraj) = (dCurv/dQbyp) * (dQbyp/dTraj) + (dCurv/dAlpha) * (dAlpha/dTraj) + (dCurv/dU) * (dU/dTraj)
  theInnerTrajectoryToCurvilinear = QbypToCurv * TrajToQbyp + AngleToCurv * JacTrajToAngle + OffsetToCurv * TrajToOff1;
  theInnerLocalToTrajectory = theInnerTrajectoryToCurvilinear.inverse(ierr) * allLocalToCurv[0];
  if (ierr) {
    edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBrl"
                               << "Inversion 2 for fine broken lines failed: " << ierr;
    return false;
  }

  AlgebraicMatrix tempJacobian(allCurvatureChanges[0]);
  AlgebraicSymMatrix tempMSCov;
  AlgebraicSymMatrix tempMSCovProj;
  AlgebraicMatrix tempJacL, tempJacN;
  AlgebraicMatrix JacOffsetToMeas;

  // measurements from hits
  for (unsigned int k = 0; k < allCurvlinJacobians.size(); ++k) {
    //  (dMeas/dU) = (dMeas/dLoc) * (dLoc/dCurv) * (dCurv/dU)
    JacOffsetToMeas = (allProjections[k] * allLocalToCurv[k].inverse(ierr)) * OffsetToCurv;
    if (ierr) {
      edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBrl"
                                 << "Inversion 3 for fine broken lines failed: " << ierr;
      return false;
    }
    theDerivatives[nMeasPerHit * k][offsetPar + 2 * k] = JacOffsetToMeas[0][0];
    theDerivatives[nMeasPerHit * k][offsetPar + 2 * k + 1] = JacOffsetToMeas[0][1];
    theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * k] = JacOffsetToMeas[1][0];
    theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * k + 1] = JacOffsetToMeas[1][1];
  }

  // measurement of MS kink
  for (unsigned int k = 1; k < allCurvlinJacobians.size() - 1; ++k) {
    // CHK
    int iMsMeas = k - 1;
    int l = k - 1;  // last hit
    int n = k + 1;  // next hit

    // amount of multiple scattering in layer k  (angular error perp to direction)
    tempMSCov = allDeltaParameterCovs[k].similarity(allLocalToCurv[k]);
    tempMSCovProj = tempMSCov.similarity(CurvToAngle);
    theMeasurementsCov[offsetMeas + nMeasPerHit * iMsMeas][offsetMeas + nMeasPerHit * iMsMeas] = tempMSCovProj[1][1];
    theMeasurementsCov[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetMeas + nMeasPerHit * iMsMeas + 1] =
        tempMSCovProj[0][0];

    // transformation matices for offsets ( l <- k -> n )
    tempJacL = allCurvlinJacobians[k] * tempJacobian;
    JacCurvToOffsetL = CurvToOffset * tempJacL.inverse(ierr);  // (dU'/dCurv') * (dCurv'/dCurv) @ last point

    if (ierr) {
      edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBrl"
                                 << "Inversion 4 for fine broken lines failed: " << ierr;
      return false;
    }
    JacOffsetToOffsetL =
        JacCurvToOffsetL * OffsetToCurv;  // J-: (dU'/dU)     = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dU)
    JacAngleToOffsetL =
        JacCurvToOffsetL * AngleToCurv;  // S-: (dU'/dAlpha) = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dAlpha)
    JacQbypToOffsetL =
        JacCurvToOffsetL * QbypToCurv;  // d-: (dU'/dQbyp)  = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dQbyp)
    JacOffsetToAngleL = -JacAngleToOffsetL.inverse(ierr);  // W-
    if (ierr) {
      edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBrl"
                                 << "Inversion 5 for fine broken lines failed: " << ierr;
      return false;
    }
    tempJacobian = tempJacobian * allCurvatureChanges[k];
    tempJacN = allCurvlinJacobians[n] * tempJacobian;
    JacCurvToOffsetN = CurvToOffset * tempJacN;  // (dU'/dCurv') * (dCurv'/dCurv) @ next point
    JacOffsetToOffsetN =
        JacCurvToOffsetN * OffsetToCurv;  // J+: (dU'/dU)     = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dU)
    JacAngleToOffsetN =
        JacCurvToOffsetN * AngleToCurv;  // S+: (dU'/dAlpha) = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dAlpha)
    JacQbypToOffsetN =
        JacCurvToOffsetN * QbypToCurv;  // d+: (dU'/dQbyp)  = (dU'/dCurv') * (dCurv'/dCurv) * (dCurv/dQbyp)
    JacOffsetToAngleN = JacAngleToOffsetN.inverse(ierr);  // W+
    if (ierr) {
      edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBrl"
                                 << "Inversion 6 for fine broken lines failed: " << ierr;
      return false;
    }
    JacOffsetToAngleC = -(JacOffsetToAngleL * JacOffsetToOffsetL + JacOffsetToAngleN * JacOffsetToOffsetN);
    JacQbypToAngleC = -(JacOffsetToAngleL * JacQbypToOffsetL + JacOffsetToAngleN * JacQbypToOffsetN);

    // bending
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][0] = JacQbypToAngleC[0][0];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][0] = JacQbypToAngleC[1][0];
    // last layer
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * l] = JacOffsetToAngleL[0][0];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * l + 1] = JacOffsetToAngleL[0][1];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * l] = JacOffsetToAngleL[1][0];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * l + 1] = JacOffsetToAngleL[1][1];
    // current layer
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * k] = JacOffsetToAngleC[0][0];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * k + 1] = JacOffsetToAngleC[0][1];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * k] = JacOffsetToAngleC[1][0];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * k + 1] = JacOffsetToAngleC[1][1];

    // next layer
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * n] = JacOffsetToAngleN[0][0];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * n + 1] = JacOffsetToAngleN[0][1];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * n] = JacOffsetToAngleN[1][0];
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * n + 1] = JacOffsetToAngleN[1][1];
  }

  return true;
}

//__________________________________________________________________________________

bool ReferenceTrajectory::addMaterialEffectsBrl(const std::vector<AlgebraicMatrix> &allProjections,
                                                const std::vector<AlgebraicSymMatrix> &allDeltaParameterCovs,
                                                const std::vector<AlgebraicMatrix> &allLocalToCurv,
                                                const std::vector<double> &allSteps,
                                                const GlobalTrajectoryParameters &gtp,
                                                const double minStep) {
  //CHK: add material effects using broken lines
  //BrokenLinesCoarse: combine close by detectors,
  //                   use approximate Jacobians from Steps (limit Qbyp -> 0),
  //                   bending only in RPhi (B=(0,0,Bz)), no energy loss correction

  int offsetPar = theNumberOfPars;
  int offsetMeas = nMeasPerHit * allSteps.size();
  int ierr = 0;

  GlobalVector p = gtp.momentum();
  double cosLambda = sqrt((p.x() * p.x() + p.y() * p.y()) / (p.x() * p.x() + p.y() * p.y() + p.z() * p.z()));
  double bFac = -gtp.magneticFieldInInverseGeV(gtp.position()).mag();

  // transformation from trajectory to curvilinear parameters at refTsos
  double delta(1.0 / allSteps[1]);
  theInnerTrajectoryToCurvilinear[0][0] = 1;
  theInnerTrajectoryToCurvilinear[1][2] = -delta;
  theInnerTrajectoryToCurvilinear[1][4] = delta;
  theInnerTrajectoryToCurvilinear[2][0] = -0.5 * bFac / delta;
  theInnerTrajectoryToCurvilinear[2][1] = -delta / cosLambda;
  theInnerTrajectoryToCurvilinear[2][3] = delta / cosLambda;
  theInnerTrajectoryToCurvilinear[3][1] = 1;
  theInnerTrajectoryToCurvilinear[4][2] = 1;
  theInnerLocalToTrajectory = theInnerTrajectoryToCurvilinear.inverse(ierr) * allLocalToCurv[0];
  if (ierr) {
    edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBrl"
                               << "Inversion 1 for coarse broken lines failed: " << ierr;
    return false;
  }

  AlgebraicMatrix CurvToAngle(2, 5);   // dAlpha/dCurv
  CurvToAngle[1][1] = 1.;              // dalpha2/dlambda
  CurvToAngle[0][2] = cosLambda;       // dalpha1/dphi
  AlgebraicMatrix OffsetToCurv(5, 2);  // dCurv/dU
  OffsetToCurv[3][0] = 1.;             // dxt/du1
  OffsetToCurv[4][1] = 1.;             // dyt/du2

  AlgebraicSymMatrix tempMSCov;
  AlgebraicSymMatrix tempMSCovProj;
  AlgebraicMatrix JacOffsetToMeas;

  // combine closeby detectors into single plane
  std::vector<unsigned int> first(allSteps.size());
  std::vector<unsigned int> last(allSteps.size());
  std::vector<unsigned int> plane(allSteps.size());
  std::vector<double> sPlane(allSteps.size());
  unsigned int nPlane = 0;
  double sTot = 0;

  for (unsigned int k = 1; k < allSteps.size(); ++k) {
    sTot += allSteps[k];
    if (fabs(allSteps[k]) > minStep) {
      nPlane += 1;
      first[nPlane] = k;
    }
    last[nPlane] = k;
    plane[k] = nPlane;
    sPlane[nPlane] += sTot;
  }
  if (nPlane < 2)
    return false;  // pathological cases: need at least 2 planes

  theNumberOfVirtualPars = 2 * (nPlane + 1);
  theNumberOfVirtualMeas = 2 * (nPlane - 1);  // unsigned underflow for nPlane == 0...
  for (unsigned int k = 0; k <= nPlane; ++k) {
    sPlane[k] /= (double)(last[k] - first[k] + 1);
  }

  // measurements from hits
  sTot = 0;
  for (unsigned int k = 0; k < allSteps.size(); ++k) {
    sTot += allSteps[k];
    //  (dMeas/dU) = (dMeas/dLoc) * (dLoc/dCurv) * (dCurv/dU)
    JacOffsetToMeas = (allProjections[k] * allLocalToCurv[k].inverse(ierr)) * OffsetToCurv;
    if (ierr) {
      edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsBrl"
                                 << "Inversion 2 for coarse broken lines failed: " << ierr;
      return false;
    }

    unsigned int iPlane = plane[k];
    if (last[iPlane] == first[iPlane]) {  // single plane
      theDerivatives[nMeasPerHit * k][offsetPar + 2 * iPlane] = JacOffsetToMeas[0][0];
      theDerivatives[nMeasPerHit * k][offsetPar + 2 * iPlane + 1] = JacOffsetToMeas[0][1];
      theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * iPlane] = JacOffsetToMeas[1][0];
      theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * iPlane + 1] = JacOffsetToMeas[1][1];
    } else {                // combined plane: (linear) interpolation
      unsigned int jPlane;  // neighbor plane for interpolation
      if (fabs(sTot) < fabs(sPlane[iPlane])) {
        jPlane = (iPlane > 0) ? iPlane - 1 : 1;
      } else {
        jPlane = (iPlane < nPlane) ? iPlane + 1 : nPlane - 1;
      }
      // interpolation weights
      double sDiff = sPlane[iPlane] - sPlane[jPlane];
      double iFrac = (sTot - sPlane[jPlane]) / sDiff;
      double jFrac = 1.0 - iFrac;
      theDerivatives[nMeasPerHit * k][offsetPar + 2 * iPlane] = JacOffsetToMeas[0][0] * iFrac;
      theDerivatives[nMeasPerHit * k][offsetPar + 2 * iPlane + 1] = JacOffsetToMeas[0][1] * iFrac;
      theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * iPlane] = JacOffsetToMeas[1][0] * iFrac;
      theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * iPlane + 1] = JacOffsetToMeas[1][1] * iFrac;
      theDerivatives[nMeasPerHit * k][offsetPar + 2 * jPlane] = JacOffsetToMeas[0][0] * jFrac;
      theDerivatives[nMeasPerHit * k][offsetPar + 2 * jPlane + 1] = JacOffsetToMeas[0][1] * jFrac;
      theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * jPlane] = JacOffsetToMeas[1][0] * jFrac;
      theDerivatives[nMeasPerHit * k + 1][offsetPar + 2 * jPlane + 1] = JacOffsetToMeas[1][1] * jFrac;
      // 2nd order neglected
      // theDerivatives[nMeasPerHit*k  ][                   0] = -0.5*bFac*sDiff*iFrac*sDiff*jFrac*cosLambda;
    }
  }

  // measurement of MS kink
  for (unsigned int i = 1; i < nPlane; ++i) {
    // CHK
    int iMsMeas = i - 1;
    int l = i - 1;  // last hit
    int n = i + 1;  // next hit

    // amount of multiple scattering in plane k
    for (unsigned int k = first[i]; k <= last[i]; ++k) {
      tempMSCov = allDeltaParameterCovs[k].similarity(allLocalToCurv[k]);
      tempMSCovProj = tempMSCov.similarity(CurvToAngle);
      theMeasurementsCov[offsetMeas + nMeasPerHit * iMsMeas][offsetMeas + nMeasPerHit * iMsMeas] += tempMSCovProj[0][0];
      theMeasurementsCov[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetMeas + nMeasPerHit * iMsMeas + 1] +=
          tempMSCovProj[1][1];
    }
    // broken line measurements for layer k, correlations between both planes neglected
    double stepK = sPlane[i] - sPlane[l];
    double stepN = sPlane[n] - sPlane[i];
    double deltaK(1.0 / stepK);
    double deltaN(1.0 / stepN);
    // bending (only in RPhi)
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][0] = -0.5 * bFac * (stepK + stepN) * cosLambda;
    // last layer
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * l] = deltaK;
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * l + 1] = deltaK;
    // current layer
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * i] = -(deltaK + deltaN);
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * i + 1] = -(deltaK + deltaN);
    // next layer
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas][offsetPar + 2 * n] = deltaN;
    theDerivatives[offsetMeas + nMeasPerHit * iMsMeas + 1][offsetPar + 2 * n + 1] = deltaN;
  }

  return true;
}

//__________________________________________________________________________________

bool ReferenceTrajectory::addMaterialEffectsLocalGbl(const std::vector<AlgebraicMatrix> &allJacobians,
                                                     const std::vector<AlgebraicMatrix> &allProjections,
                                                     const std::vector<AlgebraicSymMatrix> &allCurvatureChanges,
                                                     const std::vector<AlgebraicSymMatrix> &allDeltaParameterCovs) {
  //CHK: add material effects using general broken lines, no initial kinks
  // local track parameters are defined in the TSO system

  // Minimum precision to use a measurement.
  // Measurements with smaller values are rejected and with larger values are accepted
  // (ending up in the MP2 binary files and used for alignment).
  // The precision for the measurement along the strips is 12./Length^2. Thus:
  // - for the Phase 0 Strips modules (Length ~ 10 cm) is 0.12 => rejected
  // - for the Phase 2 Strips in PS modules (Length ~ 2.4 cm) is 2.08 => accepted
  // - for the Phase 2 Strips in 2S modules (Length ~ 5 cm) is 0.48 => accepted
  const double minPrec = 0.3;

  AlgebraicMatrix OffsetToLocal(5, 2);  // dLocal/dU
  OffsetToLocal[3][0] = 1.;
  OffsetToLocal[4][1] = 1.;
  AlgebraicMatrix SlopeToLocal(5, 2);  // dLocal/dU'
  SlopeToLocal[1][0] = 1.;
  SlopeToLocal[2][1] = 1.;

  // GBL uses Eigen matrices as interface
  Eigen::Matrix2d covariance, scatPrecision, proLocalToMeas;
  Matrix5d jacPointToPoint;
  auto identity = Matrix5d::Identity();
  Eigen::Vector2d measurement, measPrecDiag;
  auto scatterer = Eigen::Vector2d::Zero();

  //bool initialKinks = (allCurvlinKinks.size()>0);

  // measurements and scatterers from hits
  unsigned int numHits = allJacobians.size();
  std::vector<GblPoint> GblPointList;
  GblPointList.reserve(numHits);
  for (unsigned int k = 0; k < numHits; ++k) {
    // GBL point to point jacobian
    clhep2eigen(allJacobians[k] * allCurvatureChanges[k], jacPointToPoint);

    // GBL point
    GblPoint aGblPoint(jacPointToPoint);

    // GBL projection from local to measurement system
    clhep2eigen(allProjections[k] * OffsetToLocal, proLocalToMeas);

    // GBL measurement (residuum to initial trajectory)
    clhep2eigen(theMeasurements.sub(2 * k + 1, 2 * k + 2) - theTrajectoryPositions.sub(2 * k + 1, 2 * k + 2),
                measurement);

    // GBL measurement covariance matrix
    clhep2eigen(theMeasurementsCov.sub(2 * k + 1, 2 * k + 2), covariance);

    // GBL add measurement to point
    if (std::abs(covariance(0, 1)) < std::numeric_limits<double>::epsilon()) {
      // covariance matrix is diagonal, independent measurements
      for (unsigned int row = 0; row < 2; ++row) {
        measPrecDiag(row) = (0. < covariance(row, row) ? 1.0 / covariance(row, row) : 0.);
      }
      aGblPoint.addMeasurement(proLocalToMeas, measurement, measPrecDiag, minPrec);
    } else {
      // covariance matrix needs diagonalization
      aGblPoint.addMeasurement(proLocalToMeas, measurement, covariance.inverse(), minPrec);
    }

    // GBL multiple scattering (full matrix in local system)
    clhep2eigen(allDeltaParameterCovs[k].similarityT(SlopeToLocal), scatPrecision);
    if (!(scatPrecision.colPivHouseholderQr().isInvertible())) {
      if (!allowZeroMaterial_) {
        throw cms::Exception("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsLocalGbl"
                                          << "\nEncountered singular scatter covariance-matrix without allowing "
                                          << "for zero material.";
      }
    } else {
      // GBL add scatterer to point
      aGblPoint.addScatterer(scatterer, Eigen::Matrix2d(scatPrecision.inverse()));
    }
    // add point to list
    GblPointList.push_back(aGblPoint);
  }
  // add list of points and transformation local to fit (=local) system at first hit
  theGblInput.push_back(std::make_pair(GblPointList, identity));

  return true;
}

//__________________________________________________________________________________

bool ReferenceTrajectory::addMaterialEffectsCurvlinGbl(const std::vector<AlgebraicMatrix> &allCurvlinJacobians,
                                                       const std::vector<AlgebraicMatrix> &allProjections,
                                                       const std::vector<AlgebraicSymMatrix> &allCurvatureChanges,
                                                       const std::vector<AlgebraicSymMatrix> &allDeltaParameterCovs,
                                                       const std::vector<AlgebraicMatrix> &allLocalToCurv) {
  //CHK: add material effects using general broken lines
  // local track parameters are defined in the curvilinear system

  // Minimum precision to use a measurement.
  // Measurements with smaller values are rejected and with larger values are accepted
  // (ending up in the MP2 binary files and used for alignment).
  // The precision for the measurement along the strips is 12./Length^2. Thus:
  // - for the Phase 0 Strips modules (Length ~ 10 cm) is 0.12 => rejected
  // - for the Phase 2 Strips in PS modules (Length ~ 2.4 cm) is 2.08 => accepted
  // - for the Phase 2 Strips in 2S modules (Length ~ 5 cm) is 0.48 => accepted
  const double minPrec = 0.3;
  int ierr = 0;

  AlgebraicMatrix OffsetToCurv(5, 2);  // dCurv/dU
  OffsetToCurv[3][0] = 1.;             // dxt/du1
  OffsetToCurv[4][1] = 1.;             // dyt/du2

  AlgebraicMatrix JacOffsetToMeas, tempMSCov;

  // GBL uses Eigen matrices as interface
  Eigen::Matrix2d covariance, proLocalToMeas;
  Matrix5d jacPointToPoint, firstLocalToCurv;
  Eigen::Vector2d measurement, measPrecDiag, scatPrecDiag;
  auto scatterer = Eigen::Vector2d::Zero();

  // measurements and scatterers from hits
  unsigned int numHits = allCurvlinJacobians.size();
  std::vector<GblPoint> GblPointList;
  GblPointList.reserve(numHits);
  for (unsigned int k = 0; k < numHits; ++k) {
    //  (dMeas/dU) = (dMeas/dLoc) * (dLoc/dCurv) * (dCurv/dU)
    JacOffsetToMeas = (allProjections[k] * allLocalToCurv[k].inverse(ierr)) * OffsetToCurv;
    if (ierr) {
      edm::LogError("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsGbl"
                                 << "Inversion 1 for general broken lines failed: " << ierr;
      return false;
    }

    // GBL point to point jacobian
    clhep2eigen(allCurvlinJacobians[k] * allCurvatureChanges[k], jacPointToPoint);

    // GBL point
    GblPoint aGblPoint(jacPointToPoint);

    // GBL projection from local to measurement system
    clhep2eigen(JacOffsetToMeas, proLocalToMeas);

    // GBL measurement (residuum to initial trajectory)
    clhep2eigen(theMeasurements.sub(2 * k + 1, 2 * k + 2) - theTrajectoryPositions.sub(2 * k + 1, 2 * k + 2),
                measurement);

    // GBL measurement covariance matrix
    clhep2eigen(theMeasurementsCov.sub(2 * k + 1, 2 * k + 2), covariance);

    // GBL add measurement to point
    if (std::abs(covariance(0, 1)) < std::numeric_limits<double>::epsilon()) {
      // covariance matrix is diagonal, independent measurements
      for (unsigned int row = 0; row < 2; ++row) {
        measPrecDiag(row) = (0. < covariance(row, row) ? 1.0 / covariance(row, row) : 0.);
      }
      aGblPoint.addMeasurement(proLocalToMeas, measurement, measPrecDiag, minPrec);
    } else {
      // covariance matrix needs diagonalization
      aGblPoint.addMeasurement(proLocalToMeas, measurement, covariance.inverse(), minPrec);
    }

    // GBL multiple scattering (diagonal matrix in curvilinear system)
    tempMSCov = allDeltaParameterCovs[k].similarity(allLocalToCurv[k]);
    for (unsigned int row = 0; row < 2; ++row) {
      scatPrecDiag(row) = 1.0 / tempMSCov[row + 1][row + 1];
    }

    // check for singularity
    bool singularCovariance{false};
    for (int row = 0; row < scatPrecDiag.rows(); ++row) {
      if (!(scatPrecDiag[row] < std::numeric_limits<double>::infinity())) {
        singularCovariance = true;
        break;
      }
    }
    if (singularCovariance && !allowZeroMaterial_) {
      throw cms::Exception("Alignment") << "@SUB=ReferenceTrajectory::addMaterialEffectsCurvlinGbl"
                                        << "\nEncountered singular scatter covariance-matrix without allowing "
                                        << "for zero material.";
    }

    // GBL add scatterer to point
    aGblPoint.addScatterer(scatterer, Eigen::Vector2d(scatPrecDiag));

    // add point to list
    GblPointList.push_back(aGblPoint);
  }
  // add list of points and transformation local to fit (=curvilinear) system at first hit
  clhep2eigen(allLocalToCurv[0], firstLocalToCurv);
  theGblInput.push_back(std::make_pair(GblPointList, firstLocalToCurv));

  return true;
}

//__________________________________________________________________________________
template <typename Derived>
void ReferenceTrajectory::clhep2eigen(const AlgebraicVector &in, Eigen::MatrixBase<Derived> &out) {
  static_assert(Derived::ColsAtCompileTime == 1, "clhep2eigen: 'out' must be of vector type");
  for (int row = 0; row < in.num_row(); ++row) {
    out(row) = in[row];
  }
}

template <typename Derived>
void ReferenceTrajectory::clhep2eigen(const AlgebraicMatrix &in, Eigen::MatrixBase<Derived> &out) {
  for (int row = 0; row < in.num_row(); ++row) {
    for (int col = 0; col < in.num_col(); ++col) {
      out(row, col) = in[row][col];
    }
  }
}

template <typename Derived>
void ReferenceTrajectory::clhep2eigen(const AlgebraicSymMatrix &in, Eigen::MatrixBase<Derived> &out) {
  for (int row = 0; row < in.num_row(); ++row) {
    for (int col = 0; col < in.num_col(); ++col) {
      out(row, col) = in[row][col];
    }
  }
}

//__________________________________________________________________________________

AlgebraicMatrix ReferenceTrajectory::getHitProjectionMatrix(
    const TransientTrackingRecHit::ConstRecHitPointer &hitPtr) const {
  if (this->useRecHit(hitPtr)) {
    // check which templated non-member function to call:
    switch (hitPtr->dimension()) {
      case 1:
        return getHitProjectionMatrixT<1>(hitPtr);
      case 2:
        return getHitProjectionMatrixT<2>(hitPtr);
      case 3:
        return getHitProjectionMatrixT<3>(hitPtr);
      case 4:
        return getHitProjectionMatrixT<4>(hitPtr);
      case 5:
        return getHitProjectionMatrixT<5>(hitPtr);
      default:
        throw cms::Exception("ReferenceTrajectory::getHitProjectionMatrix")
            << "Unexpected hit dimension: " << hitPtr->dimension() << "\n";
    }
  }
  // invalid or (to please compiler) unknown dimension
  return AlgebraicMatrix(2, 5, 0);  // get size from ???
}

//__________________________________________________________________________________

template <unsigned int N>
AlgebraicMatrix ReferenceTrajectory::getHitProjectionMatrixT(
    const TransientTrackingRecHit::ConstRecHitPointer &hitPtr) const {
  // define variables that will be used to setup the KfComponentsHolder
  // (their allocated memory is needed by 'hitPtr->getKfComponents(..)'

  ProjectMatrix<double, 5, N> pf;
  typename AlgebraicROOTObject<N>::Vector r, rMeas;
  typename AlgebraicROOTObject<N, N>::SymMatrix V, VMeas;
  // input for the holder - but dummy is OK here to just get the projection matrix:
  const AlgebraicVector5 dummyPars;
  const AlgebraicSymMatrix55 dummyErr;

  // setup the holder with the correct dimensions and get the values
  KfComponentsHolder holder;
  holder.setup<N>(&r, &V, &pf, &rMeas, &VMeas, dummyPars, dummyErr);
  hitPtr->getKfComponents(holder);

  return asHepMatrix<N, 5>(holder.projection<N>());
}