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 // This file is part of the ACTS project.
 //
 // Copyright (C) 2016 CERN for the benefit of the ACTS project
 //
 // This Source Code Form is subject to the terms of the Mozilla Public
 // License, v. 2.0. If a copy of the MPL was not distributed with this
 // file, You can obtain one at https://mozilla.org/MPL/2.0/.
 
 #include "Acts/Definitions/Direction.hpp"
 #include "Acts/EventData/TransformationHelpers.hpp"
 #include "Acts/Propagator/ConstrainedStep.hpp"
 #include "Acts/Propagator/EigenStepperError.hpp"
 #include "Acts/Propagator/detail/CovarianceEngine.hpp"
 
 #include <limits>
 
 template <typename E>
 Acts::EigenStepper<E>::EigenStepper(
     std::shared_ptr<const MagneticFieldProvider> bField)
     : m_bField(std::move(bField)) {}
 
 template <typename E>
 auto Acts::EigenStepper<E>::makeState(const Options& options) const -> State {
   State state{options, m_bField->makeCache(options.magFieldContext)};
   return state;
 }
 
 template <typename E>
 void Acts::EigenStepper<E>::initialize(State& state,
                                        const BoundTrackParameters& par) const {
   initialize(state, par.parameters(), par.covariance(),
              par.particleHypothesis(), par.referenceSurface());
 }
 
 template <typename E>
 void Acts::EigenStepper<E>::initialize(State& state,
                                        const BoundVector& boundParams,
                                        const std::optional<BoundMatrix>& cov,
                                        ParticleHypothesis particleHypothesis,
                                        const Surface& surface) const {
   FreeVector freeParams = transformBoundToFreeParameters(
       surface, state.options.geoContext, boundParams);
 
   state.particleHypothesis = particleHypothesis;
 
   state.pathAccumulated = 0;
   state.nSteps = 0;
   state.nStepTrials = 0;
   state.stepSize = ConstrainedStep(state.options.maxStepSize);
   state.previousStepSize = 0;
   state.statistics = StepperStatistics();
 
   state.pars = freeParams;
 
   // Init the jacobian matrix if needed
   state.covTransport = cov.has_value();
   if (state.covTransport) {
     state.cov = *cov;
     state.jacToGlobal = surface.boundToFreeJacobian(
         state.options.geoContext, freeParams.segment<3>(eFreePos0),
         freeParams.segment<3>(eFreeDir0));
     state.jacobian = BoundMatrix::Identity();
     state.jacTransport = FreeMatrix::Identity();
     state.derivative = FreeVector::Zero();
   }
 }
 
 template <typename E>
 auto Acts::EigenStepper<E>::boundState(
     State& state, const Surface& surface, bool transportCov,
     const FreeToBoundCorrection& freeToBoundCorrection) const
     -> Result<BoundState> {
   return detail::boundState(
       state.options.geoContext, surface, state.cov, state.jacobian,
       state.jacTransport, state.derivative, state.jacToGlobal, state.pars,
       state.particleHypothesis, state.covTransport && transportCov,
       state.pathAccumulated, freeToBoundCorrection);
 }
 
 template <typename E>
 bool Acts::EigenStepper<E>::prepareCurvilinearState(State& state) const {
   // test whether the accumulated path has still its initial value.
   if (state.pathAccumulated != 0) {
     return true;
   }
 
   // if no step was executed the path length derivates have not been
   // computed but are needed to compute the curvilinear covariance. The
   // derivates are given by k1 for a zero step width.
   // First Runge-Kutta point (at current position)
   auto& sd = state.stepData;
   auto pos = position(state);
   auto fieldRes = getField(state, pos);
   if (!fieldRes.ok()) {
     return false;
   }
 
   sd.B_first = *fieldRes;
   if (!state.extension.template k<0>(state, *this, nullptr, sd.k1, sd.B_first,
                                      sd.kQoP)) {
     return false;
   }
 
   // dr/ds :
   state.derivative.template head<3>() =
       state.pars.template segment<3>(eFreeDir0);
   // d (dr/ds) / ds :
   state.derivative.template segment<3>(4) = sd.k1;
   // to set dt/ds :
   state.extension.finalize(state, *this, nullptr, state.pathAccumulated);
   return true;
 }
 
 template <typename E>
 auto Acts::EigenStepper<E>::curvilinearState(
     State& state, bool transportCov) const -> CurvilinearState {
   return detail::curvilinearState(
       state.cov, state.jacobian, state.jacTransport, state.derivative,
       state.jacToGlobal, state.pars, state.particleHypothesis,
       state.covTransport && transportCov, state.pathAccumulated);
 }
 
 template <typename E>
 void Acts::EigenStepper<E>::update(State& state, const FreeVector& freeParams,
                                    const BoundVector& /*boundParams*/,
                                    const Covariance& covariance,
                                    const Surface& surface) const {
   state.pars = freeParams;
   state.cov = covariance;
   state.jacToGlobal = surface.boundToFreeJacobian(
       state.options.geoContext, freeParams.template segment<3>(eFreePos0),
       freeParams.template segment<3>(eFreeDir0));
 }
 
 template <typename E>
 void Acts::EigenStepper<E>::update(State& state, const Vector3& uposition,
                                    const Vector3& udirection, double qOverP,
                                    double time) const {
   state.pars.template segment<3>(eFreePos0) = uposition;
   state.pars.template segment<3>(eFreeDir0) = udirection;
   state.pars[eFreeTime] = time;
   state.pars[eFreeQOverP] = qOverP;
 }
 
 template <typename E>
 void Acts::EigenStepper<E>::transportCovarianceToCurvilinear(
     State& state) const {
   detail::transportCovarianceToCurvilinear(state.cov, state.jacobian,
                                            state.jacTransport, state.derivative,
                                            state.jacToGlobal, direction(state));
 }
 
 template <typename E>
 void Acts::EigenStepper<E>::transportCovarianceToBound(
     State& state, const Surface& surface,
     const FreeToBoundCorrection& freeToBoundCorrection) const {
   detail::transportCovarianceToBound(
       state.options.geoContext, surface, state.cov, state.jacobian,
       state.jacTransport, state.derivative, state.jacToGlobal, state.pars,
       freeToBoundCorrection);
 }
 
 template <typename E>
 Acts::Result<double> Acts::EigenStepper<E>::step(
     State& state, Direction propDir, const IVolumeMaterial* material) const {
   // Runge-Kutta integrator state
   auto& sd = state.stepData;
 
   double errorEstimate = 0;
   double h2 = 0;
   double half_h = 0;
 
   auto pos = position(state);
   auto dir = direction(state);
 
   // First Runge-Kutta point (at current position)
   auto fieldRes = getField(state, pos);
   if (!fieldRes.ok()) {
     return fieldRes.error();
   }
   sd.B_first = *fieldRes;
   if (!state.extension.template k<0>(state, *this, material, sd.k1, sd.B_first,
                                      sd.kQoP)) {
     return 0.;
   }
 
   const auto calcStepSizeScaling = [&](const double errorEstimate_) -> double {
     // For details about these values see ATL-SOFT-PUB-2009-001
     constexpr double lower = 0.25;
     constexpr double upper = 4.0;
     // This is given by the order of the Runge-Kutta method
     constexpr double exponent = 0.25;
 
     double x = state.options.stepTolerance / errorEstimate_;
 
     if constexpr (exponent == 0.25) {
       // This is 3x faster than std::pow
       x = std::sqrt(std::sqrt(x));
     } else {
       x = std::pow(x, exponent);
     }
 
     return std::clamp(x, lower, upper);
   };
 
   const auto isErrorTolerable = [&](const double errorEstimate_) {
     // For details about these values see ATL-SOFT-PUB-2009-001
     constexpr double marginFactor = 4.0;
 
     return errorEstimate_ <= marginFactor * state.options.stepTolerance;
   };
 
   // The following functor starts to perform a Runge-Kutta step of a certain
   // size, going up to the point where it can return an estimate of the local
   // integration error. The results are stated in the local variables above,
   // allowing integration to continue once the error is deemed satisfactory
   const auto tryRungeKuttaStep = [&](const double h) -> Result<bool> {
     // helpers because bool and std::error_code are ambiguous
     constexpr auto success = &Result<bool>::success;
     constexpr auto failure = &Result<bool>::failure;
 
     // State the square and half of the step size
     h2 = h * h;
     half_h = h * 0.5;
 
     // Second Runge-Kutta point
     const Vector3 pos1 = pos + half_h * dir + h2 * 0.125 * sd.k1;
     auto field = getField(state, pos1);
     if (!field.ok()) {
       return failure(field.error());
     }
     sd.B_middle = *field;
 
     if (!state.extension.template k<1>(state, *this, material, sd.k2,
                                        sd.B_middle, sd.kQoP, half_h, sd.k1)) {
       return success(false);
     }
 
     // Third Runge-Kutta point
     if (!state.extension.template k<2>(state, *this, material, sd.k3,
                                        sd.B_middle, sd.kQoP, half_h, sd.k2)) {
       return success(false);
     }
 
     // Last Runge-Kutta point
     const Vector3 pos2 = pos + h * dir + h2 * 0.5 * sd.k3;
     field = getField(state, pos2);
     if (!field.ok()) {
       return failure(field.error());
     }
     sd.B_last = *field;
     if (!state.extension.template k<3>(state, *this, material, sd.k4, sd.B_last,
                                        sd.kQoP, h, sd.k3)) {
       return success(false);
     }
 
     // Compute and check the local integration error estimate
     errorEstimate =
         h2 * ((sd.k1 - sd.k2 - sd.k3 + sd.k4).template lpNorm<1>() +
               std::abs(sd.kQoP[0] - sd.kQoP[1] - sd.kQoP[2] + sd.kQoP[3]));
     // Protect against division by zero
     errorEstimate = std::max(1e-20, errorEstimate);
 
     return success(isErrorTolerable(errorEstimate));
   };
 
   const double initialH = state.stepSize.value() * propDir;
   double h = initialH;
   std::size_t nStepTrials = 0;
   // Select and adjust the appropriate Runge-Kutta step size as given
   // ATL-SOFT-PUB-2009-001
   while (true) {
     ++nStepTrials;
     ++state.statistics.nAttemptedSteps;
 
     auto res = tryRungeKuttaStep(h);
     if (!res.ok()) {
       return res.error();
     }
     if (!!res.value()) {
       break;
     }
 
     ++state.statistics.nRejectedSteps;
 
     const double stepSizeScaling = calcStepSizeScaling(errorEstimate);
     h *= stepSizeScaling;
 
     // If step size becomes too small the particle remains at the initial
     // place
     if (std::abs(h) < std::abs(state.options.stepSizeCutOff)) {
       // Not moving due to too low momentum needs an aborter
       return EigenStepperError::StepSizeStalled;
     }
 
     // If the parameter is off track too much or given stepSize is not
     // appropriate
     if (nStepTrials > state.options.maxRungeKuttaStepTrials) {
       // Too many trials, have to abort
       return EigenStepperError::StepSizeAdjustmentFailed;
     }
   }
 
   // When doing error propagation, update the associated Jacobian matrix
   if (state.covTransport) {
     // using the direction before updated below
 
     // The step transport matrix in global coordinates
     FreeMatrix D;
     if (!state.extension.finalize(state, *this, material, h, D)) {
       return EigenStepperError::StepInvalid;
     }
 
     // See the documentation of Acts::blockedMult for a description of blocked
     // matrix multiplication. However, we can go one further. Let's assume that
     // some of these sub-matrices are zero matrices 0₈ and identity matrices
     // I₈, namely:
     //
     // D₁₁ = I₈, J₁₁ = I₈, D₂₁ = 0₈, J₂₁ = 0₈
     //
     // Which gives:
     //
     // K₁₁ = I₈  * I₈  + D₁₂ * 0₈  = I₈
     // K₁₂ = I₈  * J₁₂ + D₁₂ * J₂₂ = J₁₂ + D₁₂ * J₂₂
     // K₂₁ = 0₈  * I₈  + D₂₂ * 0₈  = 0₈
     // K₂₂ = 0₈  * J₁₂ + D₂₂ * J₂₂ = D₂₂ * J₂₂
     //
     // Furthermore, we're constructing K in place of J, and since
     // K₁₁ = I₈ = J₁₁ and K₂₁ = 0₈ = D₂₁, we don't actually need to touch those
     // sub-matrices at all!
     assert((D.topLeftCorner<4, 4>().isIdentity()));
     assert((D.bottomLeftCorner<4, 4>().isZero()));
     assert((state.jacTransport.template topLeftCorner<4, 4>().isIdentity()));
     assert((state.jacTransport.template bottomLeftCorner<4, 4>().isZero()));
 
     state.jacTransport.template topRightCorner<4, 4>() +=
         D.topRightCorner<4, 4>() *
         state.jacTransport.template bottomRightCorner<4, 4>();
     state.jacTransport.template bottomRightCorner<4, 4>() =
         (D.bottomRightCorner<4, 4>() *
          state.jacTransport.template bottomRightCorner<4, 4>())
             .eval();
   } else {
     if (!state.extension.finalize(state, *this, material, h)) {
       return EigenStepperError::StepInvalid;
     }
   }
 
   // Update the track parameters according to the equations of motion
   state.pars.template segment<3>(eFreePos0) +=
       h * dir + h2 / 6. * (sd.k1 + sd.k2 + sd.k3);
   state.pars.template segment<3>(eFreeDir0) +=
       h / 6. * (sd.k1 + 2. * (sd.k2 + sd.k3) + sd.k4);
   (state.pars.template segment<3>(eFreeDir0)).normalize();
 
   if (state.covTransport) {
     // using the updated direction
     state.derivative.template head<3>() =
         state.pars.template segment<3>(eFreeDir0);
     state.derivative.template segment<3>(4) = sd.k4;
   }
 
   state.pathAccumulated += h;
   ++state.nSteps;
   state.nStepTrials += nStepTrials;
 
   ++state.statistics.nSuccessfulSteps;
   if (propDir != Direction::fromScalarZeroAsPositive(initialH)) {
     ++state.statistics.nReverseSteps;
   }
   state.statistics.pathLength += h;
   state.statistics.absolutePathLength += std::abs(h);
 
   const double stepSizeScaling = calcStepSizeScaling(errorEstimate);
   const double nextAccuracy = std::abs(h * stepSizeScaling);
   const double previousAccuracy = std::abs(state.stepSize.accuracy());
   const double initialStepLength = std::abs(initialH);
   if (nextAccuracy < initialStepLength || nextAccuracy > previousAccuracy) {
     state.stepSize.setAccuracy(nextAccuracy);
   }
 
   return h;
 }
 
 template <typename E>
 void Acts::EigenStepper<E>::setIdentityJacobian(State& state) const {
   state.jacobian = BoundMatrix::Identity();
 }

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