/* Copyright 2002-2021 CS GROUP
    * Licensed to CS GROUP (CS) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *   http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
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   */
  package org.orekit.orbits;
  
  
  import java.util.List;
  import java.util.stream.Collectors;
  import java.util.stream.Stream;
  
  import org.hipparchus.CalculusFieldElement;
  import org.hipparchus.analysis.differentiation.FieldUnivariateDerivative1;
  import org.hipparchus.analysis.interpolation.FieldHermiteInterpolator;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.FieldSinCos;
  import org.hipparchus.util.MathArrays;
  import org.hipparchus.util.MathUtils;
  import org.hipparchus.util.Precision;
  import org.orekit.errors.OrekitException;
  import org.orekit.errors.OrekitIllegalArgumentException;
  import org.orekit.errors.OrekitInternalError;
  import org.orekit.errors.OrekitMessages;
  import org.orekit.frames.Frame;
  import org.orekit.time.FieldAbsoluteDate;
  import org.orekit.utils.FieldPVCoordinates;
  import org.orekit.utils.TimeStampedFieldPVCoordinates;
  
  
  /**
   * This class handles traditional Keplerian orbital parameters.
  
   * <p>
   * The parameters used internally are the classical Keplerian elements:
   *   <pre>
   *     a
   *     e
   *     i
   *     ω
   *     Ω
   *     v
   *   </pre>
   * where ω stands for the Perigee Argument, Ω stands for the
   * Right Ascension of the Ascending Node and v stands for the true anomaly.
   * <p>
   * This class supports hyperbolic orbits, using the convention that semi major
   * axis is negative for such orbits (and of course eccentricity is greater than 1).
   * </p>
   * <p>
   * When orbit is either equatorial or circular, some Keplerian elements
   * (more precisely ω and Ω) become ambiguous so this class should not
   * be used for such orbits. For this reason, {@link EquinoctialOrbit equinoctial
   * orbits} is the recommended way to represent orbits.
   * </p>
  
   * <p>
   * The instance <code>KeplerianOrbit</code> is guaranteed to be immutable.
   * </p>
   * @see     Orbit
   * @see    CircularOrbit
   * @see    CartesianOrbit
   * @see    EquinoctialOrbit
   * @author Luc Maisonobe
   * @author Guylaine Prat
   * @author Fabien Maussion
   * @author V&eacute;ronique Pommier-Maurussane
   * @author Andrea Antolino
   * @since 9.0
   */
  public class FieldKeplerianOrbit<T extends CalculusFieldElement<T>> extends FieldOrbit<T> {
  
      /** Name of the eccentricity parameter. */
      private static final String ECCENTRICITY = "eccentricity";
  
      /** First coefficient to compute Kepler equation solver starter. */
      private static final double A;
  
      /** Second coefficient to compute Kepler equation solver starter. */
      private static final double B;
  
      static {
          final double k1 = 3 * FastMath.PI + 2;
          final double k2 = FastMath.PI - 1;
          final double k3 = 6 * FastMath.PI - 1;
          A  = 3 * k2 * k2 / k1;
         B  = k3 * k3 / (6 * k1);
     }
 
     /** Semi-major axis (m). */
     private final T a;
 
     /** Eccentricity. */
     private final T e;
 
     /** Inclination (rad). */
     private final T i;
 
     /** Perigee Argument (rad). */
     private final T pa;
 
     /** Right Ascension of Ascending Node (rad). */
     private final T raan;
 
     /** True anomaly (rad). */
     private final T v;
 
     /** Semi-major axis derivative (m/s). */
     private final T aDot;
 
     /** Eccentricity derivative. */
     private final T eDot;
 
     /** Inclination derivative (rad/s). */
     private final T iDot;
 
     /** Perigee Argument derivative (rad/s). */
     private final T paDot;
 
     /** Right Ascension of Ascending Node derivative (rad/s). */
     private final T raanDot;
 
     /** True anomaly derivative (rad/s). */
     private final T vDot;
 
     /** Partial Cartesian coordinates (position and velocity are valid, acceleration may be missing). */
     private FieldPVCoordinates<T> partialPV;
 
     /** Identity element. */
     private final T one;
 
     /** Zero element. */
     private final T zero;
 
     /** Third Canonical Vector. */
     private final FieldVector3D<T> PLUS_K;
 
     /** Creates a new instance.
      * @param a  semi-major axis (m), negative for hyperbolic orbits
      * @param e eccentricity (positive or equal to 0)
      * @param i inclination (rad)
      * @param pa perigee argument (ω, rad)
      * @param raan right ascension of ascending node (Ω, rad)
      * @param anomaly mean, eccentric or true anomaly (rad)
      * @param type type of anomaly
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame} or a and e don't match for hyperbolic orbits,
      * or v is out of range for hyperbolic orbits
      */
     public FieldKeplerianOrbit(final T a, final T e, final T i,
                                final T pa, final T raan,
                                final T anomaly, final PositionAngle type,
                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         this(a, e, i, pa, raan, anomaly,
              null, null, null, null, null, null,
              type, frame, date, mu);
     }
 
     /** Creates a new instance.
      * @param a  semi-major axis (m), negative for hyperbolic orbits
      * @param e eccentricity (positive or equal to 0)
      * @param i inclination (rad)
      * @param pa perigee argument (ω, rad)
      * @param raan right ascension of ascending node (Ω, rad)
      * @param anomaly mean, eccentric or true anomaly (rad)
      * @param aDot  semi-major axis derivative, null if unknown (m/s)
      * @param eDot eccentricity derivative, null if unknown
      * @param iDot inclination derivative, null if unknown (rad/s)
      * @param paDot perigee argument derivative, null if unknown (rad/s)
      * @param raanDot right ascension of ascending node derivative, null if unknown (rad/s)
      * @param anomalyDot mean, eccentric or true anomaly derivative, null if unknown (rad/s)
      * @param type type of anomaly
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame} or a and e don't match for hyperbolic orbits,
      * or v is out of range for hyperbolic orbits
      */
     public FieldKeplerianOrbit(final T a, final T e, final T i,
                                final T pa, final T raan, final T anomaly,
                                final T aDot, final T eDot, final T iDot,
                                final T paDot, final T raanDot, final T anomalyDot,
                                final PositionAngle type,
                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         super(frame, date, mu);
         if (a.multiply(e.negate().add(1)).getReal() < 0) {
             throw new OrekitIllegalArgumentException(OrekitMessages.ORBIT_A_E_MISMATCH_WITH_CONIC_TYPE, a.getReal(), e.getReal());
         }
 
         // Checking eccentricity range
         checkParameterRangeInclusive(ECCENTRICITY, e.getReal(), 0.0, Double.POSITIVE_INFINITY);
 
         this.a       =    a;
         this.aDot    =    aDot;
         this.e       =    e;
         this.eDot    =    eDot;
         this.i       =    i;
         this.iDot    =    iDot;
         this.pa      =    pa;
         this.paDot   =    paDot;
         this.raan    =    raan;
         this.raanDot =    raanDot;
 
         /** Identity element. */
         this.one = a.getField().getOne();
 
         /** Zero element. */
         this.zero = a.getField().getZero();
 
         /**Third canonical vector. */
         this.PLUS_K = FieldVector3D.getPlusK(a.getField());
 
         if (hasDerivatives()) {
             final FieldUnivariateDerivative1<T> eUD        = new FieldUnivariateDerivative1<>(e, eDot);
             final FieldUnivariateDerivative1<T> anomalyUD  = new FieldUnivariateDerivative1<>(anomaly,  anomalyDot);
             final FieldUnivariateDerivative1<T> vUD;
             switch (type) {
                 case MEAN :
                     vUD = (a.getReal() < 0) ?
                           hyperbolicEccentricToTrue(meanToHyperbolicEccentric(anomalyUD, eUD), eUD) :
                           ellipticEccentricToTrue(meanToEllipticEccentric(anomalyUD, eUD), eUD);
                     break;
                 case ECCENTRIC :
                     vUD = (a.getReal() < 0) ?
                           hyperbolicEccentricToTrue(anomalyUD, eUD) :
                           ellipticEccentricToTrue(anomalyUD, eUD);
                     break;
                 case TRUE :
                     vUD = anomalyUD;
                     break;
                 default : // this should never happen
                     throw new OrekitInternalError(null);
             }
             this.v    = vUD.getValue();
             this.vDot = vUD.getDerivative(1);
         } else {
             switch (type) {
                 case MEAN :
 
                     this.v = (a.getReal() < 0) ?
                              hyperbolicEccentricToTrue(meanToHyperbolicEccentric(anomaly, e), e) :
                              ellipticEccentricToTrue(meanToEllipticEccentric(anomaly, e), e);
 
                     break;
                 case ECCENTRIC :
                     this.v = (a.getReal() < 0) ?
                              hyperbolicEccentricToTrue(anomaly, e) :
                              ellipticEccentricToTrue(anomaly, e);
 
                     break;
                 case TRUE :
                     this.v = anomaly;
                     break;
                 default : // this should never happen
                     throw new OrekitInternalError(null);
             }
             this.vDot = null;
         }
 
         // check true anomaly range
         if (e.multiply(v.cos()).add(1).getReal() <= 0) {
             final double vMax = e.reciprocal().negate().acos().getReal();
             throw new OrekitIllegalArgumentException(OrekitMessages.ORBIT_ANOMALY_OUT_OF_HYPERBOLIC_RANGE,
                                                      v.getReal(), e.getReal(), -vMax, vMax);
         }
 
         this.partialPV = null;
 
     }
 
     /** Constructor from Cartesian parameters.
      *
      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
      * use {@code mu} and the position to compute the acceleration, including
      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
      *
      * @param pvCoordinates the PVCoordinates of the satellite
      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame}
      */
     public FieldKeplerianOrbit(final TimeStampedFieldPVCoordinates<T> pvCoordinates,
                                final Frame frame, final T mu)
         throws IllegalArgumentException {
         this(pvCoordinates, frame, mu, hasNonKeplerianAcceleration(pvCoordinates, mu));
     }
 
     /** Constructor from Cartesian parameters.
      *
      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
      * use {@code mu} and the position to compute the acceleration, including
      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
      *
      * @param pvCoordinates the PVCoordinates of the satellite
      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param mu central attraction coefficient (m³/s²)
      * @param reliableAcceleration if true, the acceleration is considered to be reliable
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame}
      */
     private FieldKeplerianOrbit(final TimeStampedFieldPVCoordinates<T> pvCoordinates,
                                 final Frame frame, final T mu,
                                 final boolean reliableAcceleration)
         throws IllegalArgumentException {
 
         super(pvCoordinates, frame, mu);
 
         // identity element
         this.one = pvCoordinates.getPosition().getX().getField().getOne();
 
         // zero element
         this.zero = one.getField().getZero();
 
         // third canonical vector
         this.PLUS_K = FieldVector3D.getPlusK(one.getField());
 
         // compute inclination
         final FieldVector3D<T> momentum = pvCoordinates.getMomentum();
         final T m2 = momentum.getNormSq();
 
         i = FieldVector3D.angle(momentum, PLUS_K);
         // compute right ascension of ascending node
         raan = FieldVector3D.crossProduct(PLUS_K, momentum).getAlpha();
         // preliminary computations for parameters depending on orbit shape (elliptic or hyperbolic)
         final FieldVector3D<T> pvP     = pvCoordinates.getPosition();
         final FieldVector3D<T> pvV     = pvCoordinates.getVelocity();
         final FieldVector3D<T> pvA     = pvCoordinates.getAcceleration();
 
         final T   r2      = pvP.getNormSq();
         final T   r       = r2.sqrt();
         final T   V2      = pvV.getNormSq();
         final T   rV2OnMu = r.multiply(V2).divide(mu);
 
         // compute semi-major axis (will be negative for hyperbolic orbits)
         a = r.divide(rV2OnMu.negate().add(2.0));
         final T muA = a.multiply(mu);
 
         // compute true anomaly
         if (a.getReal() > 0) {
             // elliptic or circular orbit
             final T eSE = FieldVector3D.dotProduct(pvP, pvV).divide(muA.sqrt());
             final T eCE = rV2OnMu.subtract(1);
             e = (eSE.multiply(eSE).add(eCE.multiply(eCE))).sqrt();
             v = ellipticEccentricToTrue(eSE.atan2(eCE), e);
         } else {
             // hyperbolic orbit
             final T eSH = FieldVector3D.dotProduct(pvP, pvV).divide(muA.negate().sqrt());
             final T eCH = rV2OnMu.subtract(1);
             e = (m2.negate().divide(muA).add(1)).sqrt();
             v = hyperbolicEccentricToTrue((eCH.add(eSH)).divide(eCH.subtract(eSH)).log().divide(2), e);
         }
 
         // Checking eccentricity range
         checkParameterRangeInclusive(ECCENTRICITY, e.getReal(), 0.0, Double.POSITIVE_INFINITY);
 
         // compute perigee argument
         final FieldVector3D<T> node = new FieldVector3D<>(raan, zero);
         final T px = FieldVector3D.dotProduct(pvP, node);
         final T py = FieldVector3D.dotProduct(pvP, FieldVector3D.crossProduct(momentum, node)).divide(m2.sqrt());
         pa = py.atan2(px).subtract(v);
 
         partialPV = pvCoordinates;
 
         if (reliableAcceleration) {
             // we have a relevant acceleration, we can compute derivatives
 
             final T[][] jacobian = MathArrays.buildArray(a.getField(), 6, 6);
             getJacobianWrtCartesian(PositionAngle.MEAN, jacobian);
 
             final FieldVector3D<T> keplerianAcceleration    = new FieldVector3D<>(r.multiply(r2).reciprocal().multiply(mu.negate()), pvP);
             final FieldVector3D<T> nonKeplerianAcceleration = pvA.subtract(keplerianAcceleration);
             final T   aX                       = nonKeplerianAcceleration.getX();
             final T   aY                       = nonKeplerianAcceleration.getY();
             final T   aZ                       = nonKeplerianAcceleration.getZ();
             aDot    = jacobian[0][3].multiply(aX).add(jacobian[0][4].multiply(aY)).add(jacobian[0][5].multiply(aZ));
             eDot    = jacobian[1][3].multiply(aX).add(jacobian[1][4].multiply(aY)).add(jacobian[1][5].multiply(aZ));
             iDot    = jacobian[2][3].multiply(aX).add(jacobian[2][4].multiply(aY)).add(jacobian[2][5].multiply(aZ));
             paDot   = jacobian[3][3].multiply(aX).add(jacobian[3][4].multiply(aY)).add(jacobian[3][5].multiply(aZ));
             raanDot = jacobian[4][3].multiply(aX).add(jacobian[4][4].multiply(aY)).add(jacobian[4][5].multiply(aZ));
 
             // in order to compute true anomaly derivative, we must compute
             // mean anomaly derivative including Keplerian motion and convert to true anomaly
             final T MDot = getKeplerianMeanMotion().
                            add(jacobian[5][3].multiply(aX)).add(jacobian[5][4].multiply(aY)).add(jacobian[5][5].multiply(aZ));
             final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
             final FieldUnivariateDerivative1<T> MUD = new FieldUnivariateDerivative1<>(getMeanAnomaly(), MDot);
             final FieldUnivariateDerivative1<T> vUD = (a.getReal() < 0) ?
                                             FieldKeplerianOrbit.hyperbolicEccentricToTrue(FieldKeplerianOrbit.meanToHyperbolicEccentric(MUD, eUD), eUD) :
                                             FieldKeplerianOrbit.ellipticEccentricToTrue(FieldKeplerianOrbit.meanToEllipticEccentric(MUD, eUD), eUD);
             vDot = vUD.getDerivative(1);
 
         } else {
             // acceleration is either almost zero or NaN,
             // we assume acceleration was not known
             // we don't set up derivatives
             aDot    = null;
             eDot    = null;
             iDot    = null;
             paDot   = null;
             raanDot = null;
             vDot    = null;
         }
 
     }
 
     /** Constructor from Cartesian parameters.
      *
      * <p> The acceleration provided in {@code FieldPVCoordinates} is accessible using
      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
      * use {@code mu} and the position to compute the acceleration, including
      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
      *
      * @param FieldPVCoordinates the PVCoordinates of the satellite
      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if frame is not a {@link
      * Frame#isPseudoInertial pseudo-inertial frame}
      */
     public FieldKeplerianOrbit(final FieldPVCoordinates<T> FieldPVCoordinates,
                                final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         this(new TimeStampedFieldPVCoordinates<>(date, FieldPVCoordinates), frame, mu);
     }
 
     /** Constructor from any kind of orbital parameters.
      * @param op orbital parameters to copy
      */
     public FieldKeplerianOrbit(final FieldOrbit<T> op) {
         this(op.getPVCoordinates(), op.getFrame(), op.getMu(), op.hasDerivatives());
     }
 
     /** {@inheritDoc} */
     public OrbitType getType() {
         return OrbitType.KEPLERIAN;
     }
 
     /** {@inheritDoc} */
     public T getA() {
         return a;
     }
 
     /** {@inheritDoc} */
     public T getADot() {
         return aDot;
     }
 
     /** {@inheritDoc} */
     public T getE() {
         return e;
     }
 
     /** {@inheritDoc} */
     public T getEDot() {
         return eDot;
     }
 
     /** {@inheritDoc} */
     public T getI() {
         return i;
     }
 
     /** {@inheritDoc} */
     public T getIDot() {
         return iDot;
     }
 
     /** Get the perigee argument.
      * @return perigee argument (rad)
      */
     public T getPerigeeArgument() {
         return pa;
     }
 
     /** Get the perigee argument derivative.
      * <p>
      * If the orbit was created without derivatives, the value returned is null.
      * </p>
      * @return perigee argument derivative (rad/s)
      */
     public T getPerigeeArgumentDot() {
         return paDot;
     }
 
     /** Get the right ascension of the ascending node.
      * @return right ascension of the ascending node (rad)
      */
     public T getRightAscensionOfAscendingNode() {
         return raan;
     }
 
     /** Get the right ascension of the ascending node derivative.
      * <p>
      * If the orbit was created without derivatives, the value returned is null.
      * </p>
      * @return right ascension of the ascending node derivative (rad/s)
      */
     public T getRightAscensionOfAscendingNodeDot() {
         return raanDot;
     }
 
     /** Get the true anomaly.
      * @return true anomaly (rad)
      */
     public T getTrueAnomaly() {
         return v;
     }
 
     /** Get the true anomaly derivative.
      * <p>
      * If the orbit was created without derivatives, the value returned is null.
      * </p>
      * @return true anomaly derivative (rad/s)
      */
     public T getTrueAnomalyDot() {
         return vDot;
     }
 
     /** Get the eccentric anomaly.
      * @return eccentric anomaly (rad)
      */
     public T getEccentricAnomaly() {
         return (a.getReal() < 0) ? trueToHyperbolicEccentric(v, e) : trueToEllipticEccentric(v, e);
     }
 
     /** Get the eccentric anomaly derivative.
      * <p>
      * If the orbit was created without derivatives, the value returned is null.
      * </p>
      * @return eccentric anomaly derivative (rad/s)
      */
     public T getEccentricAnomalyDot() {
 
         if (!hasDerivatives()) {
             return null;
         }
 
         final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
         final FieldUnivariateDerivative1<T> vUD = new FieldUnivariateDerivative1<>(v, vDot);
         final FieldUnivariateDerivative1<T> EUD = (a.getReal() < 0) ?
                                                 trueToHyperbolicEccentric(vUD, eUD) :
                                                 trueToEllipticEccentric(vUD, eUD);
         return EUD.getDerivative(1);
 
     }
 
     /** Get the mean anomaly.
      * @return mean anomaly (rad)
      */
     public T getMeanAnomaly() {
         return (a.getReal() < 0) ?
                hyperbolicEccentricToMean(trueToHyperbolicEccentric(v, e), e) :
                ellipticEccentricToMean(trueToEllipticEccentric(v, e), e);
     }
 
     /** Get the mean anomaly derivative.
      * <p>
      * If the orbit was created without derivatives, the value returned is null.
      * </p>
      * @return mean anomaly derivative (rad/s)
      */
     public T getMeanAnomalyDot() {
 
         if (!hasDerivatives()) {
             return null;
         }
 
         final FieldUnivariateDerivative1<T> eUD = new FieldUnivariateDerivative1<>(e, eDot);
         final FieldUnivariateDerivative1<T> vUD = new FieldUnivariateDerivative1<>(v, vDot);
         final FieldUnivariateDerivative1<T> MUD = (a.getReal() < 0) ?
                                                 hyperbolicEccentricToMean(trueToHyperbolicEccentric(vUD, eUD), eUD) :
                                                 ellipticEccentricToMean(trueToEllipticEccentric(vUD, eUD), eUD);
         return MUD.getDerivative(1);
 
     }
 
     /** Get the anomaly.
      * @param type type of the angle
      * @return anomaly (rad)
      */
     public T getAnomaly(final PositionAngle type) {
         return (type == PositionAngle.MEAN) ? getMeanAnomaly() :
                                               ((type == PositionAngle.ECCENTRIC) ? getEccentricAnomaly() :
                                                                                    getTrueAnomaly());
     }
 
     /** Get the anomaly derivative.
      * <p>
      * If the orbit was created without derivatives, the value returned is null.
      * </p>
      * @param type type of the angle
      * @return anomaly derivative (rad/s)
      */
     public T getAnomalyDot(final PositionAngle type) {
         return (type == PositionAngle.MEAN) ? getMeanAnomalyDot() :
                                               ((type == PositionAngle.ECCENTRIC) ? getEccentricAnomalyDot() :
                                                                                    getTrueAnomalyDot());
     }
 
     /** {@inheritDoc} */
     @Override
     public boolean hasDerivatives() {
         return aDot != null;
     }
 
     /** Computes the true anomaly from the elliptic eccentric anomaly.
      * @param E eccentric anomaly (rad)
      * @param e eccentricity
      * @param <T> type of the field elements
      * @return v the true anomaly
      */
     public static <T extends CalculusFieldElement<T>> T ellipticEccentricToTrue(final T E, final T e) {
         final T beta = e.divide(e.multiply(e).negate().add(1).sqrt().add(1));
         final FieldSinCos<T> scE = FastMath.sinCos(E);
         return E.add(beta.multiply(scE.sin()).divide(beta.multiply(scE.cos()).subtract(1).negate()).atan().multiply(2));
     }
 
     /** Computes the elliptic eccentric anomaly from the true anomaly.
      * @param v true anomaly (rad)
      * @param e eccentricity
      * @param <T> type of the field elements
      * @return E the elliptic eccentric anomaly
      */
     public static <T extends CalculusFieldElement<T>> T trueToEllipticEccentric(final T v, final T e) {
         final T beta = e.divide(e.multiply(e).negate().add(1).sqrt().add(1));
         final FieldSinCos<T> scv = FastMath.sinCos(v);
         return v.subtract((beta.multiply(scv.sin()).divide(beta.multiply(scv.cos()).add(1))).atan().multiply(2));
     }
 
     /** Computes the true anomaly from the hyperbolic eccentric anomaly.
      * @param H hyperbolic eccentric anomaly (rad)
      * @param e eccentricity
      * @param <T> type of the field elements
      * @return v the true anomaly
      */
     public static <T extends CalculusFieldElement<T>> T hyperbolicEccentricToTrue(final T H, final T e) {
         final T s    = e.add(1).divide(e.subtract(1)).sqrt();
         final T tanH = H.multiply(0.5).tanh();
         return s.multiply(tanH).atan().multiply(2);
     }
 
     /** Computes the hyperbolic eccentric anomaly from the true anomaly.
      * @param v true anomaly (rad)
      * @param e eccentricity
      * @param <T> type of the field elements
      * @return H the hyperbolic eccentric anomaly
      */
     public static <T extends CalculusFieldElement<T>> T trueToHyperbolicEccentric(final T v, final T e) {
         final FieldSinCos<T> scv = FastMath.sinCos(v);
         final T sinhH = e.multiply(e).subtract(1).sqrt().multiply(scv.sin()).divide(e.multiply(scv.cos()).add(1));
         return sinhH.asinh();
     }
 
     /** Computes the mean anomaly from the hyperbolic eccentric anomaly.
      * @param H hyperbolic eccentric anomaly (rad)
      * @param e eccentricity
      * @param <T> type of the field elements
      * @return M the mean anomaly
      */
     public static <T extends CalculusFieldElement<T>> T hyperbolicEccentricToMean(final T H, final T e) {
         return e.multiply(H.sinh()).subtract(H);
     }
 
     /** Computes the elliptic eccentric anomaly from the mean anomaly.
      * <p>
      * The algorithm used here for solving Kepler equation has been published
      * in: "Procedures for  solving Kepler's Equation", A. W. Odell and
      * R. H. Gooding, Celestial Mechanics 38 (1986) 307-334
      * </p>
      * @param M mean anomaly (rad)
      * @param e eccentricity
      * @param <T> type of the field elements
      * @return E the eccentric anomaly
      */
     public static <T extends CalculusFieldElement<T>> T meanToEllipticEccentric(final T M, final T e) {
         // reduce M to [-PI PI) interval
         final T reducedM = MathUtils.normalizeAngle(M, M.getField().getZero());
 
         // compute start value according to A. W. Odell and R. H. Gooding S12 starter
         T E;
         if (reducedM.abs().getReal() < 1.0 / 6.0) {
             if (FastMath.abs(reducedM.getReal()) < Precision.SAFE_MIN) {
                 // this is an Orekit change to the S12 starter, mainly used when T is some kind of derivative structure.
                 // If reducedM is 0.0, the derivative of cbrt is infinite which induces NaN appearing later in
                 // the computation. As in this case E and M are almost equal, we initialize E with reducedM
                 E = reducedM;
             } else {
                 // this is the standard S12 starter
                 E = reducedM.add(e.multiply( (reducedM.multiply(6).cbrt()).subtract(reducedM)));
             }
         } else {
             final T pi = e.getPi();
             if (reducedM.getReal() < 0) {
                 final T w = reducedM.add(pi);
                 E = reducedM.add(e.multiply(w.multiply(A).divide(w.negate().add(B)).subtract(pi).subtract(reducedM)));
             } else {
                 final T w = reducedM.negate().add(pi);
                 E = reducedM.add(e.multiply(w.multiply(A).divide(w.negate().add(B)).negate().subtract(reducedM).add(pi)));
             }
         }
 
         final T e1 = e.negate().add(1);
         final boolean noCancellationRisk = (e1.getReal() + E.getReal() * E.getReal() / 6) >= 0.1;
 
         // perform two iterations, each consisting of one Halley step and one Newton-Raphson step
         for (int j = 0; j < 2; ++j) {
 
             final T f;
             T fd;
             final FieldSinCos<T> scE = FastMath.sinCos(E);
             final T fdd  = e.multiply(scE.sin());
             final T fddd = e.multiply(scE.cos());
 
             if (noCancellationRisk) {
 
                 f  = (E.subtract(fdd)).subtract(reducedM);
                 fd = fddd.negate().add(1);
             } else {
 
 
                 f  = eMeSinE(E, e).subtract(reducedM);
                 final T s = E.multiply(0.5).sin();
                 fd = e1.add(e.multiply(s).multiply(s).multiply(2));
             }
             final T dee = f.multiply(fd).divide(f.multiply(fdd).multiply(0.5).subtract(fd.multiply(fd)));
 
             // update eccentric anomaly, using expressions that limit underflow problems
             final T w = fd.add(dee.multiply(fdd.add(dee.multiply(fddd.divide(3)))).multiply(0.5));
             fd = fd.add(dee.multiply(fdd.add(dee.multiply(fddd).multiply(0.5))));
             E = E.subtract(f.subtract(dee.multiply(fd.subtract(w))).divide(fd));
 
         }
 
         // expand the result back to original range
         E = E.add(M).subtract(reducedM);
         return E;
     }
 
     /** Accurate computation of E - e sin(E).
      * <p>
      * This method is used when E is close to 0 and e close to 1,
      * i.e. near the perigee of almost parabolic orbits
      * </p>
      * @param E eccentric anomaly
      * @param e eccentricity
      * @param <T> Type of the field elements
      * @return E - e sin(E)
      */
     private static <T extends CalculusFieldElement<T>> T eMeSinE(final T E, final T e) {
 
         T x = (e.negate().add(1)).multiply(E.sin());
         final T mE2 = E.negate().multiply(E);
         T term = E;
         double d    = 0;
         // the inequality test below IS intentional and should NOT be replaced by a check with a small tolerance
         for (T x0 = E.getField().getZero().add(Double.NaN); !Double.valueOf(x.getReal()).equals(Double.valueOf(x0.getReal()));) {
             d += 2;
             term = term.multiply(mE2.divide(d * (d + 1)));
             x0 = x;
             x = x.subtract(term);
         }
         return x;
     }
 
     /** Computes the hyperbolic eccentric anomaly from the mean anomaly.
      * <p>
      * The algorithm used here for solving hyperbolic Kepler equation is
      * Danby's iterative method (3rd order) with Vallado's initial guess.
      * </p>
      * @param M mean anomaly (rad)
      * @param e eccentricity
      * @param <T> Type of the field elements
      * @return H the hyperbolic eccentric anomaly
      */
     public static <T extends CalculusFieldElement<T>> T meanToHyperbolicEccentric(final T M, final T e) {
 
         // Resolution of hyperbolic Kepler equation for Keplerian parameters
 
         // Field value of pi
         final T pi = e.getPi();
 
         // Initial guess
         T H;
         if (e.getReal() < 1.6) {
             if (-pi.getReal() < M.getReal() && M.getReal() < 0. || M.getReal() > pi.getReal()) {
                 H = M.subtract(e);
             } else {
                 H = M.add(e);
             }
         } else {
             if (e.getReal() < 3.6 && M.abs().getReal() > pi.getReal()) {
                 H = M.subtract(e.copySign(M));
             } else {
                 H = M.divide(e.subtract(1));
             }
         }
 
         // Iterative computation
         int iter = 0;
         do {
             final T f3  = e.multiply(H.cosh());
             final T f2  = e.multiply(H.sinh());
             final T f1  = f3.subtract(1);
             final T f0  = f2.subtract(H).subtract(M);
             final T f12 = f1.multiply(2);
             final T d   = f0.divide(f12);
             final T fdf = f1.subtract(d.multiply(f2));
             final T ds  = f0.divide(fdf);
 
             final T shift = f0.divide(fdf.add(ds.multiply(ds.multiply(f3.divide(6)))));
 
             H = H.subtract(shift);
 
             if ((shift.abs().getReal()) <= 1.0e-12) {
                 return H;
             }
 
         } while (++iter < 50);
 
         throw new MathIllegalArgumentException(OrekitMessages.UNABLE_TO_COMPUTE_HYPERBOLIC_ECCENTRIC_ANOMALY,
                                                iter);
     }
 
     /** Computes the mean anomaly from the elliptic eccentric anomaly.
      * @param E eccentric anomaly (rad)
      * @param e eccentricity
      * @param <T> type of the field elements
      * @return M the mean anomaly
      */
     public static <T extends CalculusFieldElement<T>> T ellipticEccentricToMean(final T E, final T e) {
         return E.subtract(e.multiply(E.sin()));
     }
 
     /** {@inheritDoc} */
     public T getEquinoctialEx() {
         return e.multiply(pa.add(raan).cos());
     }
 
     /** {@inheritDoc} */
     public T getEquinoctialExDot() {
 
         if (!hasDerivatives()) {
             return null;
         }
 
         final FieldUnivariateDerivative1<T> eUD    = new FieldUnivariateDerivative1<>(e,    eDot);
         final FieldUnivariateDerivative1<T> paUD   = new FieldUnivariateDerivative1<>(pa,   paDot);
         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
         return eUD.multiply(paUD.add(raanUD).cos()).getDerivative(1);
 
     }
 
     /** {@inheritDoc} */
     public T getEquinoctialEy() {
         return  e.multiply((pa.add(raan)).sin());
     }
 
     /** {@inheritDoc} */
     public T getEquinoctialEyDot() {
 
         if (!hasDerivatives()) {
             return null;
         }
 
         final FieldUnivariateDerivative1<T> eUD    = new FieldUnivariateDerivative1<>(e,    eDot);
         final FieldUnivariateDerivative1<T> paUD   = new FieldUnivariateDerivative1<>(pa,   paDot);
         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
         return eUD.multiply(paUD.add(raanUD).sin()).getDerivative(1);
 
     }
 
     /** {@inheritDoc} */
     public T getHx() {
         // Check for equatorial retrograde orbit
         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
             return this.zero.add(Double.NaN);
         }
         return  raan.cos().multiply(i.divide(2).tan());
     }
 
     /** {@inheritDoc} */
     public T getHxDot() {
 
         if (!hasDerivatives()) {
             return null;
         }
 
         // Check for equatorial retrograde orbit
         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
             return this.zero.add(Double.NaN);
         }
 
         final FieldUnivariateDerivative1<T> iUD    = new FieldUnivariateDerivative1<>(i,    iDot);
         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
         return raanUD.cos().multiply(iUD.multiply(0.5).tan()).getDerivative(1);
 
     }
 
     /** {@inheritDoc} */
     public T getHy() {
         // Check for equatorial retrograde orbit
         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
             return this.zero.add(Double.NaN);
         }
         return  raan.sin().multiply(i.divide(2).tan());
     }
 
     /** {@inheritDoc} */
     public T getHyDot() {
 
         if (!hasDerivatives()) {
             return null;
         }
 
         // Check for equatorial retrograde orbit
         if (FastMath.abs(i.subtract(i.getPi()).getReal()) < 1.0e-10) {
             return this.zero.add(Double.NaN);
         }
 
         final FieldUnivariateDerivative1<T> iUD    = new FieldUnivariateDerivative1<>(i,    iDot);
         final FieldUnivariateDerivative1<T> raanUD = new FieldUnivariateDerivative1<>(raan, raanDot);
         return raanUD.sin().multiply(iUD.multiply(0.5).tan()).getDerivative(1);
 
     }
 
     /** {@inheritDoc} */
     public T getLv() {
         return pa.add(raan).add(v);
     }
 
     /** {@inheritDoc} */
     public T getLvDot() {
         return hasDerivatives() ?
                paDot.add(raanDot).add(vDot) :
                null;
     }
 
     /** {@inheritDoc} */
     public T getLE() {
         return pa.add(raan).add(getEccentricAnomaly());
     }
 
     /** {@inheritDoc} */
     public T getLEDot() {
         return hasDerivatives() ?
                paDot.add(raanDot).add(getEccentricAnomalyDot()) :
                null;
     }
 
     /** {@inheritDoc} */
     public T getLM() {
         return pa.add(raan).add(getMeanAnomaly());
     }
 
     /** {@inheritDoc} */
     public T getLMDot() {
         return hasDerivatives() ?
                paDot.add(raanDot).add(getMeanAnomalyDot()) :
                null;
     }
 
     /** Compute position and velocity but not acceleration.
      */
     private void computePVWithoutA() {
 
         if (partialPV != null) {
             // already computed
             return;
         }
 
         // preliminary variables
         final FieldSinCos<T> scRaan = FastMath.sinCos(raan);
         final FieldSinCos<T> scPa   = FastMath.sinCos(pa);
         final FieldSinCos<T> scI    = FastMath.sinCos(i);
         final T cosRaan = scRaan.cos();
         final T sinRaan = scRaan.sin();
         final T cosPa   = scPa.cos();
         final T sinPa   = scPa.sin();
         final T cosI    = scI.cos();
         final T sinI    = scI.sin();
         final T crcp    = cosRaan.multiply(cosPa);
         final T crsp    = cosRaan.multiply(sinPa);
         final T srcp    = sinRaan.multiply(cosPa);
         final T srsp    = sinRaan.multiply(sinPa);
 
         // reference axes defining the orbital plane
         final FieldVector3D<T> p = new FieldVector3D<>(crcp.subtract(cosI.multiply(srsp)),  srcp.add(cosI.multiply(crsp)), sinI.multiply(sinPa));
         final FieldVector3D<T> q = new FieldVector3D<>(crsp.add(cosI.multiply(srcp)).negate(), cosI.multiply(crcp).subtract(srsp), sinI.multiply(cosPa));
 
         if (a.getReal() > 0) {
 
             // elliptical case
 
             // elliptic eccentric anomaly
             final T uME2             = e.negate().add(1).multiply(e.add(1));
             final T s1Me2            = uME2.sqrt();
             final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
             final T cosE             = scE.cos();
             final T sinE             = scE.sin();
 
             // coordinates of position and velocity in the orbital plane
             final T x      = a.multiply(cosE.subtract(e));
             final T y      = a.multiply(sinE).multiply(s1Me2);
             final T factor = FastMath.sqrt(getMu().divide(a)).divide(e.negate().multiply(cosE).add(1));
             final T xDot   = sinE.negate().multiply(factor);
             final T yDot   = cosE.multiply(s1Me2).multiply(factor);
 
             final FieldVector3D<T> position = new FieldVector3D<>(x, p, y, q);
             final FieldVector3D<T> velocity = new FieldVector3D<>(xDot, p, yDot, q);
             partialPV = new FieldPVCoordinates<>(position, velocity);
 
         } else {
 
             // hyperbolic case
 
             // compute position and velocity factors
             final FieldSinCos<T> scV = FastMath.sinCos(v);
             final T sinV             = scV.sin();
             final T cosV             = scV.cos();
             final T f                = a.multiply(e.multiply(e).negate().add(1));
             final T posFactor        = f.divide(e.multiply(cosV).add(1));
             final T velFactor        = FastMath.sqrt(getMu().divide(f));
 
             final FieldVector3D<T> position     = new FieldVector3D<>(posFactor.multiply(cosV), p, posFactor.multiply(sinV), q);
             final FieldVector3D<T> velocity     = new FieldVector3D<>(velFactor.multiply(sinV).negate(), p, velFactor.multiply(e.add(cosV)), q);
             partialPV = new FieldPVCoordinates<>(position, velocity);
 
         }
 
     }
 
     /** Compute non-Keplerian part of the acceleration from first time derivatives.
      * <p>
      * This method should be called only when {@link #hasDerivatives()} returns true.
      * </p>
      * @return non-Keplerian part of the acceleration
      */
     private FieldVector3D<T> nonKeplerianAcceleration() {
 
         final T[][] dCdP = MathArrays.buildArray(a.getField(), 6, 6);
         getJacobianWrtParameters(PositionAngle.MEAN, dCdP);
 
         final T nonKeplerianMeanMotion = getMeanAnomalyDot().subtract(getKeplerianMeanMotion());
         final T nonKeplerianAx =     dCdP[3][0].multiply(aDot).
                                  add(dCdP[3][1].multiply(eDot)).
                                  add(dCdP[3][2].multiply(iDot)).
                                  add(dCdP[3][3].multiply(paDot)).
                                  add(dCdP[3][4].multiply(raanDot)).
                                  add(dCdP[3][5].multiply(nonKeplerianMeanMotion));
         final T nonKeplerianAy =     dCdP[4][0].multiply(aDot).
                                  add(dCdP[4][1].multiply(eDot)).
                                  add(dCdP[4][2].multiply(iDot)).
                                  add(dCdP[4][3].multiply(paDot)).
                                  add(dCdP[4][4].multiply(raanDot)).
                                  add(dCdP[4][5].multiply(nonKeplerianMeanMotion));
         final T nonKeplerianAz =     dCdP[5][0].multiply(aDot).
                                  add(dCdP[5][1].multiply(eDot)).
                                  add(dCdP[5][2].multiply(iDot)).
                                  add(dCdP[5][3].multiply(paDot)).
                                  add(dCdP[5][4].multiply(raanDot)).
                                  add(dCdP[5][5].multiply(nonKeplerianMeanMotion));
 
         return new FieldVector3D<>(nonKeplerianAx, nonKeplerianAy, nonKeplerianAz);
 
     }
 
     /** {@inheritDoc} */
     protected TimeStampedFieldPVCoordinates<T> initPVCoordinates() {
 
         // position and velocity
         computePVWithoutA();
 
         // acceleration
         final T r2 = partialPV.getPosition().getNormSq();
         final FieldVector3D<T> keplerianAcceleration = new FieldVector3D<>(r2.multiply(FastMath.sqrt(r2)).reciprocal().multiply(getMu().negate()),
                                                                            partialPV.getPosition());
         final FieldVector3D<T> acceleration = hasDerivatives() ?
                                               keplerianAcceleration.add(nonKeplerianAcceleration()) :
                                               keplerianAcceleration;
 
         return new TimeStampedFieldPVCoordinates<>(getDate(), partialPV.getPosition(), partialPV.getVelocity(), acceleration);
 
     }
 
     /** {@inheritDoc} */
     public FieldKeplerianOrbit<T> shiftedBy(final double dt) {
         return shiftedBy(getDate().getField().getZero().add(dt));
     }
 
     /** {@inheritDoc} */
     public FieldKeplerianOrbit<T> shiftedBy(final T dt) {
 
         // use Keplerian-only motion
         final FieldKeplerianOrbit<T> keplerianShifted = new FieldKeplerianOrbit<>(a, e, i, pa, raan,
                                                                                   getKeplerianMeanMotion().multiply(dt).add(getMeanAnomaly()),
                                                                                   PositionAngle.MEAN, getFrame(), getDate().shiftedBy(dt), getMu());
 
         if (hasDerivatives()) {
 
             // extract non-Keplerian acceleration from first time derivatives
             final FieldVector3D<T> nonKeplerianAcceleration = nonKeplerianAcceleration();
 
             // add quadratic effect of non-Keplerian acceleration to Keplerian-only shift
             keplerianShifted.computePVWithoutA();
             final FieldVector3D<T> fixedP   = new FieldVector3D<>(one, keplerianShifted.partialPV.getPosition(),
                                                                   dt.multiply(dt).multiply(0.5), nonKeplerianAcceleration);
             final T   fixedR2 = fixedP.getNormSq();
             final T   fixedR  = fixedR2.sqrt();
             final FieldVector3D<T> fixedV  = new FieldVector3D<>(one, keplerianShifted.partialPV.getVelocity(),
                                                                  dt, nonKeplerianAcceleration);
             final FieldVector3D<T> fixedA  = new FieldVector3D<>(fixedR2.multiply(fixedR).reciprocal().multiply(getMu().negate()),
                                                                  keplerianShifted.partialPV.getPosition(),
                                                                  one, nonKeplerianAcceleration);
 
             // build a new orbit, taking non-Keplerian acceleration into account
             return new FieldKeplerianOrbit<>(new TimeStampedFieldPVCoordinates<>(keplerianShifted.getDate(),
                                                                                  fixedP, fixedV, fixedA),
                                              keplerianShifted.getFrame(), keplerianShifted.getMu());
 
         } else {
             // Keplerian-only motion is all we can do
             return keplerianShifted;
         }
 
     }
 
     /** {@inheritDoc}
      * <p>
      * The interpolated instance is created by polynomial Hermite interpolation
      * on Keplerian elements, without derivatives (which means the interpolation
      * falls back to Lagrange interpolation only).
      * </p>
      * <p>
      * As this implementation of interpolation is polynomial, it should be used only
      * with small samples (about 10-20 points) in order to avoid <a
      * href="http://en.wikipedia.org/wiki/Runge%27s_phenomenon">Runge's phenomenon</a>
      * and numerical problems (including NaN appearing).
      * </p>
      * <p>
      * If orbit interpolation on large samples is needed, using the {@link
      * org.orekit.propagation.analytical.Ephemeris} class is a better way than using this
      * low-level interpolation. The Ephemeris class automatically handles selection of
      * a neighboring sub-sample with a predefined number of point from a large global sample
      * in a thread-safe way.
      * </p>
      */
     public FieldKeplerianOrbit<T> interpolate(final FieldAbsoluteDate<T> date, final Stream<FieldOrbit<T>> sample) {
 
         // first pass to check if derivatives are available throughout the sample
         final List<FieldOrbit<T>> list = sample.collect(Collectors.toList());
         boolean useDerivatives = true;
         for (final FieldOrbit<T> orbit : list) {
             useDerivatives = useDerivatives && orbit.hasDerivatives();
         }
 
         // set up an interpolator
         final FieldHermiteInterpolator<T> interpolator = new FieldHermiteInterpolator<>();
 
         // second pass to feed interpolator
         FieldAbsoluteDate<T> previousDate = null;
         T                    previousPA   = zero.add(Double.NaN);
         T                    previousRAAN = zero.add(Double.NaN);
         T                    previousM    = zero.add(Double.NaN);
         for (final FieldOrbit<T> orbit : list) {
             final FieldKeplerianOrbit<T> kep = (FieldKeplerianOrbit<T>) OrbitType.KEPLERIAN.convertType(orbit);
             final T continuousPA;
             final T continuousRAAN;
             final T continuousM;
             if (previousDate == null) {
                 continuousPA   = kep.getPerigeeArgument();
                 continuousRAAN = kep.getRightAscensionOfAscendingNode();
                 continuousM    = kep.getMeanAnomaly();
             } else {
                 final T dt      = kep.getDate().durationFrom(previousDate);
                 final T keplerM = previousM.add(kep.getKeplerianMeanMotion().multiply(dt));
                 continuousPA   = MathUtils.normalizeAngle(kep.getPerigeeArgument(), previousPA);
                 continuousRAAN = MathUtils.normalizeAngle(kep.getRightAscensionOfAscendingNode(), previousRAAN);
                 continuousM    = MathUtils.normalizeAngle(kep.getMeanAnomaly(), keplerM);
             }
             previousDate = kep.getDate();
             previousPA   = continuousPA;
             previousRAAN = continuousRAAN;
             previousM    = continuousM;
             final T[] toAdd = MathArrays.buildArray(getA().getField(), 6);
             toAdd[0] = kep.getA();
             toAdd[1] = kep.getE();
             toAdd[2] = kep.getI();
             toAdd[3] = continuousPA;
             toAdd[4] = continuousRAAN;
             toAdd[5] = continuousM;
             if (useDerivatives) {
                 final T[] toAddDot = MathArrays.buildArray(one.getField(), 6);
                 toAddDot[0] = kep.getADot();
                 toAddDot[1] = kep.getEDot();
                 toAddDot[2] = kep.getIDot();
                 toAddDot[3] = kep.getPerigeeArgumentDot();
                 toAddDot[4] = kep.getRightAscensionOfAscendingNodeDot();
                 toAddDot[5] = kep.getMeanAnomalyDot();
                 interpolator.addSamplePoint(kep.getDate().durationFrom(date),
                                             toAdd, toAddDot);
             } else {
                 interpolator.addSamplePoint(this.zero.add(kep.getDate().durationFrom(date)),
                                             toAdd);
             }
         }
 
         // interpolate
         final T[][] interpolated = interpolator.derivatives(zero, 1);
 
         // build a new interpolated instance
         return new FieldKeplerianOrbit<>(interpolated[0][0], interpolated[0][1], interpolated[0][2],
                                          interpolated[0][3], interpolated[0][4], interpolated[0][5],
                                          interpolated[1][0], interpolated[1][1], interpolated[1][2],
                                          interpolated[1][3], interpolated[1][4], interpolated[1][5],
                                          PositionAngle.MEAN, getFrame(), date, getMu());
 
     }
 
     /** {@inheritDoc} */
     protected T[][] computeJacobianMeanWrtCartesian() {
         if (a.getReal() > 0) {
             return computeJacobianMeanWrtCartesianElliptical();
         } else {
             return computeJacobianMeanWrtCartesianHyperbolic();
         }
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianMeanWrtCartesianElliptical() {
 
         final T[][] jacobian = MathArrays.buildArray(getA().getField(), 6, 6);
 
         // compute various intermediate parameters
         computePVWithoutA();
         final FieldVector3D<T> position = partialPV.getPosition();
         final FieldVector3D<T> velocity = partialPV.getVelocity();
         final FieldVector3D<T> momentum = partialPV.getMomentum();
         final T v2         = velocity.getNormSq();
         final T r2         = position.getNormSq();
         final T r          = r2.sqrt();
         final T r3         = r.multiply(r2);
 
         final T px         = position.getX();
         final T py         = position.getY();
         final T pz         = position.getZ();
         final T vx         = velocity.getX();
         final T vy         = velocity.getY();
         final T vz         = velocity.getZ();
         final T mx         = momentum.getX();
         final T my         = momentum.getY();
         final T mz         = momentum.getZ();
 
         final T mu         = getMu();
         final T sqrtMuA    = FastMath.sqrt(a.multiply(mu));
         final T sqrtAoMu   = FastMath.sqrt(a.divide(mu));
         final T a2         = a.multiply(a);
         final T twoA       = a.multiply(2);
         final T rOnA       = r.divide(a);
 
         final T oMe2       = e.multiply(e).negate().add(1);
         final T epsilon    = oMe2.sqrt();
         final T sqrtRec    = epsilon.reciprocal();
 
         final FieldSinCos<T> scI  = FastMath.sinCos(i);
         final FieldSinCos<T> scPA = FastMath.sinCos(pa);
         final T cosI       = scI.cos();
         final T sinI       = scI.sin();
         final T cosPA      = scPA.cos();
         final T sinPA      = scPA.sin();
 
         final T pv         = FieldVector3D.dotProduct(position, velocity);
         final T cosE       = a.subtract(r).divide(a.multiply(e));
         final T sinE       = pv.divide(e.multiply(sqrtMuA));
 
         // da
         final FieldVector3D<T> vectorAR = new FieldVector3D<>(a2.multiply(2).divide(r3), position);
         final FieldVector3D<T> vectorARDot = velocity.scalarMultiply(a2.multiply(mu.divide(2.).reciprocal()));
         fillHalfRow(this.one, vectorAR,    jacobian[0], 0);
         fillHalfRow(this.one, vectorARDot, jacobian[0], 3);
 
         // de
         final T factorER3 = pv.divide(twoA);
         final FieldVector3D<T> vectorER   = new FieldVector3D<>(cosE.multiply(v2).divide(r.multiply(mu)), position,
                                                                 sinE.divide(sqrtMuA), velocity,
                                                                 factorER3.negate().multiply(sinE).divide(sqrtMuA), vectorAR);
         final FieldVector3D<T> vectorERDot = new FieldVector3D<>(sinE.divide(sqrtMuA), position,
                                                                  cosE.multiply(mu.divide(2.).reciprocal()).multiply(r), velocity,
                                                                  factorER3.negate().multiply(sinE).divide(sqrtMuA), vectorARDot);
         fillHalfRow(this.one, vectorER,    jacobian[1], 0);
         fillHalfRow(this.one, vectorERDot, jacobian[1], 3);
 
         // dE / dr (Eccentric anomaly)
         final T coefE = cosE.divide(e.multiply(sqrtMuA));
         final FieldVector3D<T>  vectorEAnR =
             new FieldVector3D<>(sinE.negate().multiply(v2).divide(e.multiply(r).multiply(mu)), position, coefE, velocity,
                                 factorER3.negate().multiply(coefE), vectorAR);
 
         // dE / drDot
         final FieldVector3D<T>  vectorEAnRDot =
             new FieldVector3D<>(sinE.multiply(-2).multiply(r).divide(e.multiply(mu)), velocity, coefE, position,
                                 factorER3.negate().multiply(coefE), vectorARDot);
 
         // precomputing some more factors
         final T s1 = sinE.negate().multiply(pz).divide(r).subtract(cosE.multiply(vz).multiply(sqrtAoMu));
         final T s2 = cosE.negate().multiply(pz).divide(r3);
         final T s3 = sinE.multiply(vz).divide(sqrtMuA.multiply(-2));
         final T t1 = sqrtRec.multiply(cosE.multiply(pz).divide(r).subtract(sinE.multiply(vz).multiply(sqrtAoMu)));
         final T t2 = sqrtRec.multiply(sinE.negate().multiply(pz).divide(r3));
         final T t3 = sqrtRec.multiply(cosE.subtract(e)).multiply(vz).divide(sqrtMuA.multiply(2));
         final T t4 = sqrtRec.multiply(e.multiply(sinI).multiply(cosPA).multiply(sqrtRec).subtract(vz.multiply(sqrtAoMu)));
         final FieldVector3D<T> s = new FieldVector3D<>(cosE.divide(r), this.PLUS_K,
                                                        s1,       vectorEAnR,
                                                        s2,       position,
                                                        s3,       vectorAR);
         final FieldVector3D<T> sDot = new FieldVector3D<>(sinE.negate().multiply(sqrtAoMu), this.PLUS_K,
                                                           s1,               vectorEAnRDot,
                                                           s3,               vectorARDot);
         final FieldVector3D<T> t =
             new FieldVector3D<>(sqrtRec.multiply(sinE).divide(r), this.PLUS_K).add(new FieldVector3D<>(t1, vectorEAnR,
                                                                                                        t2, position,
                                                                                                        t3, vectorAR,
                                                                                                        t4, vectorER));
         final FieldVector3D<T> tDot = new FieldVector3D<>(sqrtRec.multiply(cosE.subtract(e)).multiply(sqrtAoMu), this.PLUS_K,
                                                           t1,                                                    vectorEAnRDot,
                                                           t3,                                                    vectorARDot,
                                                           t4,                                                    vectorERDot);
 
         // di
         final T factorI1 = sinI.negate().multiply(sqrtRec).divide(sqrtMuA);
         final T i1 =  factorI1;
         final T i2 =  factorI1.negate().multiply(mz).divide(twoA);
         final T i3 =  factorI1.multiply(mz).multiply(e).divide(oMe2);
         final T i4 = cosI.multiply(sinPA);
         final T i5 = cosI.multiply(cosPA);
         fillHalfRow(i1, new FieldVector3D<>(vy, vx.negate(), this.zero), i2, vectorAR, i3, vectorER, i4, s, i5, t,
                     jacobian[2], 0);
         fillHalfRow(i1, new FieldVector3D<>(py.negate(), px, this.zero), i2, vectorARDot, i3, vectorERDot, i4, sDot, i5, tDot,
                     jacobian[2], 3);
 
         // dpa
         fillHalfRow(cosPA.divide(sinI), s,    sinPA.negate().divide(sinI), t,    jacobian[3], 0);
         fillHalfRow(cosPA.divide(sinI), sDot, sinPA.negate().divide(sinI), tDot, jacobian[3], 3);
 
         // dRaan
         final T factorRaanR = (a.multiply(mu).multiply(oMe2).multiply(sinI).multiply(sinI)).reciprocal();
         fillHalfRow( factorRaanR.negate().multiply(my), new FieldVector3D<>(zero, vz, vy.negate()),
                      factorRaanR.multiply(mx), new FieldVector3D<>(vz.negate(), zero,  vx),
                      jacobian[4], 0);
         fillHalfRow(factorRaanR.negate().multiply(my), new FieldVector3D<>(zero, pz.negate(),  py),
                      factorRaanR.multiply(mx), new FieldVector3D<>(pz, zero, px.negate()),
                      jacobian[4], 3);
 
         // dM
         fillHalfRow(rOnA, vectorEAnR,    sinE.negate(), vectorER,    jacobian[5], 0);
         fillHalfRow(rOnA, vectorEAnRDot, sinE.negate(), vectorERDot, jacobian[5], 3);
 
         return jacobian;
 
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianMeanWrtCartesianHyperbolic() {
 
         final T[][] jacobian = MathArrays.buildArray(getA().getField(), 6, 6);
 
         // compute various intermediate parameters
         computePVWithoutA();
         final FieldVector3D<T> position = partialPV.getPosition();
         final FieldVector3D<T> velocity = partialPV.getVelocity();
         final FieldVector3D<T> momentum = partialPV.getMomentum();
         final T r2         = position.getNormSq();
         final T r          = r2.sqrt();
         final T r3         = r.multiply(r2);
 
         final T x          = position.getX();
         final T y          = position.getY();
         final T z          = position.getZ();
         final T vx         = velocity.getX();
         final T vy         = velocity.getY();
         final T vz         = velocity.getZ();
         final T mx         = momentum.getX();
         final T my         = momentum.getY();
         final T mz         = momentum.getZ();
 
         final T mu         = getMu();
         final T absA       = a.negate();
         final T sqrtMuA    = absA.multiply(mu).sqrt();
         final T a2         = a.multiply(a);
         final T rOa        = r.divide(absA);
 
         final FieldSinCos<T> scI = FastMath.sinCos(i);
         final T cosI       = scI.cos();
         final T sinI       = scI.sin();
 
         final T pv         = FieldVector3D.dotProduct(position, velocity);
 
         // da
         final FieldVector3D<T> vectorAR = new FieldVector3D<>(a2.multiply(-2).divide(r3), position);
         final FieldVector3D<T> vectorARDot = velocity.scalarMultiply(a2.multiply(-2).divide(mu));
         fillHalfRow(this.one.negate(), vectorAR,    jacobian[0], 0);
         fillHalfRow(this.one.negate(), vectorARDot, jacobian[0], 3);
 
         // differentials of the momentum
         final T m      = momentum.getNorm();
         final T oOm    = m.reciprocal();
         final FieldVector3D<T> dcXP = new FieldVector3D<>(this.zero, vz, vy.negate());
         final FieldVector3D<T> dcYP = new FieldVector3D<>(vz.negate(),   this.zero,  vx);
         final FieldVector3D<T> dcZP = new FieldVector3D<>( vy, vx.negate(),   this.zero);
         final FieldVector3D<T> dcXV = new FieldVector3D<>(  this.zero,  z.negate(),   y);
         final FieldVector3D<T> dcYV = new FieldVector3D<>(  z,   this.zero,  x.negate());
         final FieldVector3D<T> dcZV = new FieldVector3D<>( y.negate(),   x,   this.zero);
         final FieldVector3D<T> dCP  = new FieldVector3D<>(mx.multiply(oOm), dcXP, my.multiply(oOm), dcYP, mz.multiply(oOm), dcZP);
         final FieldVector3D<T> dCV  = new FieldVector3D<>(mx.multiply(oOm), dcXV, my.multiply(oOm), dcYV, mz.multiply(oOm), dcZV);
 
         // dp
         final T mOMu   = m.divide(mu);
         final FieldVector3D<T> dpP  = new FieldVector3D<>(mOMu.multiply(2), dCP);
         final FieldVector3D<T> dpV  = new FieldVector3D<>(mOMu.multiply(2), dCV);
 
         // de
         final T p      = m.multiply(mOMu);
         final T moO2ae = absA.multiply(2).multiply(e).reciprocal();
         final T m2OaMu = p.negate().divide(absA);
         fillHalfRow(moO2ae, dpP, m2OaMu.multiply(moO2ae), vectorAR,    jacobian[1], 0);
         fillHalfRow(moO2ae, dpV, m2OaMu.multiply(moO2ae), vectorARDot, jacobian[1], 3);
 
         // di
         final T cI1 = m.multiply(sinI).reciprocal();
         final T cI2 = cosI.multiply(cI1);
         fillHalfRow(cI2, dCP, cI1.negate(), dcZP, jacobian[2], 0);
         fillHalfRow(cI2, dCV, cI1.negate(), dcZV, jacobian[2], 3);
 
 
         // dPA
         final T cP1     =  y.multiply(oOm);
         final T cP2     =  x.negate().multiply(oOm);
         final T cP3     =  mx.multiply(cP1).add(my.multiply(cP2)).negate();
         final T cP4     =  cP3.multiply(oOm);
         final T cP5     =  r2.multiply(sinI).multiply(sinI).negate().reciprocal();
         final T cP6     = z.multiply(cP5);
         final T cP7     = cP3.multiply(cP5);
         final FieldVector3D<T> dacP  = new FieldVector3D<>(cP1, dcXP, cP2, dcYP, cP4, dCP, oOm, new FieldVector3D<>(my.negate(), mx, this.zero));
         final FieldVector3D<T> dacV  = new FieldVector3D<>(cP1, dcXV, cP2, dcYV, cP4, dCV);
         final FieldVector3D<T> dpoP  = new FieldVector3D<>(cP6, dacP, cP7, this.PLUS_K);
         final FieldVector3D<T> dpoV  = new FieldVector3D<>(cP6, dacV);
 
         final T re2     = r2.multiply(e).multiply(e);
         final T recOre2 = p.subtract(r).divide(re2);
         final T resOre2 = pv.multiply(mOMu).divide(re2);
         final FieldVector3D<T> dreP  = new FieldVector3D<>(mOMu, velocity, pv.divide(mu), dCP);
         final FieldVector3D<T> dreV  = new FieldVector3D<>(mOMu, position, pv.divide(mu), dCV);
         final FieldVector3D<T> davP  = new FieldVector3D<>(resOre2.negate(), dpP, recOre2, dreP, resOre2.divide(r), position);
         final FieldVector3D<T> davV  = new FieldVector3D<>(resOre2.negate(), dpV, recOre2, dreV);
         fillHalfRow(this.one, dpoP, this.one.negate(), davP, jacobian[3], 0);
         fillHalfRow(this.one, dpoV, this.one.negate(), davV, jacobian[3], 3);
 
         // dRAAN
         final T cO0 = cI1.multiply(cI1);
         final T cO1 =  mx.multiply(cO0);
         final T cO2 =  my.negate().multiply(cO0);
         fillHalfRow(cO1, dcYP, cO2, dcXP, jacobian[4], 0);
         fillHalfRow(cO1, dcYV, cO2, dcXV, jacobian[4], 3);
 
         // dM
         final T s2a    = pv.divide(absA.multiply(2));
         final T oObux  = m.multiply(m).add(absA.multiply(mu)).sqrt().reciprocal();
         final T scasbu = pv.multiply(oObux);
         final FieldVector3D<T> dauP = new FieldVector3D<>(sqrtMuA.reciprocal(), velocity, s2a.negate().divide(sqrtMuA), vectorAR);
         final FieldVector3D<T> dauV = new FieldVector3D<>(sqrtMuA.reciprocal(), position, s2a.negate().divide(sqrtMuA), vectorARDot);
         final FieldVector3D<T> dbuP = new FieldVector3D<>(oObux.multiply(mu.divide(2.)), vectorAR,    m.multiply(oObux), dCP);
         final FieldVector3D<T> dbuV = new FieldVector3D<>(oObux.multiply(mu.divide(2.)), vectorARDot, m.multiply(oObux), dCV);
         final FieldVector3D<T> dcuP = new FieldVector3D<>(oObux, velocity, scasbu.negate().multiply(oObux), dbuP);
         final FieldVector3D<T> dcuV = new FieldVector3D<>(oObux, position, scasbu.negate().multiply(oObux), dbuV);
         fillHalfRow(this.one, dauP, e.negate().divide(rOa.add(1)), dcuP, jacobian[5], 0);
         fillHalfRow(this.one, dauV, e.negate().divide(rOa.add(1)), dcuV, jacobian[5], 3);
 
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     protected T[][] computeJacobianEccentricWrtCartesian() {
         if (a.getReal() > 0) {
             return computeJacobianEccentricWrtCartesianElliptical();
         } else {
             return computeJacobianEccentricWrtCartesianHyperbolic();
         }
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianEccentricWrtCartesianElliptical() {
 
         // start by computing the Jacobian with mean angle
         final T[][] jacobian = computeJacobianMeanWrtCartesianElliptical();
 
         // Differentiating the Kepler equation M = E - e sin E leads to:
         // dM = (1 - e cos E) dE - sin E de
         // which is inverted and rewritten as:
         // dE = a/r dM + sin E a/r de
         final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
         final T aOr              = e.negate().multiply(scE.cos()).add(1).reciprocal();
 
         // update anomaly row
         final T[] eRow           = jacobian[1];
         final T[] anomalyRow     = jacobian[5];
         for (int j = 0; j < anomalyRow.length; ++j) {
             anomalyRow[j] = aOr.multiply(anomalyRow[j].add(scE.sin().multiply(eRow[j])));
         }
 
         return jacobian;
 
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianEccentricWrtCartesianHyperbolic() {
 
         // start by computing the Jacobian with mean angle
         final T[][] jacobian = computeJacobianMeanWrtCartesianHyperbolic();
 
         // Differentiating the Kepler equation M = e sinh H - H leads to:
         // dM = (e cosh H - 1) dH + sinh H de
         // which is inverted and rewritten as:
         // dH = 1 / (e cosh H - 1) dM - sinh H / (e cosh H - 1) de
         final T H      = getEccentricAnomaly();
         final T coshH  = H.cosh();
         final T sinhH  = H.sinh();
         final T absaOr = e.multiply(coshH).subtract(1).reciprocal();
         // update anomaly row
         final T[] eRow       = jacobian[1];
         final T[] anomalyRow = jacobian[5];
 
         for (int j = 0; j < anomalyRow.length; ++j) {
             anomalyRow[j] = absaOr.multiply(anomalyRow[j].subtract(sinhH.multiply(eRow[j])));
 
         }
 
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     protected T[][] computeJacobianTrueWrtCartesian() {
         if (a.getReal() > 0) {
             return computeJacobianTrueWrtCartesianElliptical();
         } else {
             return computeJacobianTrueWrtCartesianHyperbolic();
         }
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianTrueWrtCartesianElliptical() {
 
         // start by computing the Jacobian with eccentric angle
         final T[][] jacobian = computeJacobianEccentricWrtCartesianElliptical();
         // Differentiating the eccentric anomaly equation sin E = sqrt(1-e^2) sin v / (1 + e cos v)
         // and using cos E = (e + cos v) / (1 + e cos v) to get rid of cos E leads to:
         // dE = [sqrt (1 - e^2) / (1 + e cos v)] dv - [sin E / (1 - e^2)] de
         // which is inverted and rewritten as:
         // dv = sqrt (1 - e^2) a/r dE + [sin E / sqrt (1 - e^2)] a/r de
         final T e2               = e.multiply(e);
         final T oMe2             = e2.negate().add(1);
         final T epsilon          = oMe2.sqrt();
         final FieldSinCos<T> scE = FastMath.sinCos(getEccentricAnomaly());
         final T aOr              = e.multiply(scE.cos()).negate().add(1).reciprocal();
         final T aFactor          = epsilon.multiply(aOr);
         final T eFactor          = scE.sin().multiply(aOr).divide(epsilon);
 
         // update anomaly row
         final T[] eRow           = jacobian[1];
         final T[] anomalyRow     = jacobian[5];
         for (int j = 0; j < anomalyRow.length; ++j) {
             anomalyRow[j] = aFactor.multiply(anomalyRow[j]).add(eFactor.multiply(eRow[j]));
         }
         return jacobian;
 
     }
 
     /** Compute the Jacobian of the orbital parameters with respect to the Cartesian parameters.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of parameter i of the orbit with
      * respect to Cartesian coordinate j (x for j=0, y for j=1, z for j=2, xDot for j=3,
      * yDot for j=4, zDot for j=5).
      * </p>
      * @return 6x6 Jacobian matrix
      */
     private T[][] computeJacobianTrueWrtCartesianHyperbolic() {
 
         // start by computing the Jacobian with eccentric angle
         final T[][] jacobian = computeJacobianEccentricWrtCartesianHyperbolic();
 
         // Differentiating the eccentric anomaly equation sinh H = sqrt(e^2-1) sin v / (1 + e cos v)
         // and using cosh H = (e + cos v) / (1 + e cos v) to get rid of cosh H leads to:
         // dH = [sqrt (e^2 - 1) / (1 + e cos v)] dv + [sinh H / (e^2 - 1)] de
         // which is inverted and rewritten as:
         // dv = sqrt (1 - e^2) a/r dH - [sinh H / sqrt (e^2 - 1)] a/r de
         final T e2       = e.multiply(e);
         final T e2Mo     = e2.subtract(1);
         final T epsilon  = e2Mo.sqrt();
         final T H        = getEccentricAnomaly();
         final T coshH    = H.cosh();
         final T sinhH    = H.sinh();
         final T aOr      = e.multiply(coshH).subtract(1).reciprocal();
         final T aFactor  = epsilon.multiply(aOr);
         final T eFactor  = sinhH .multiply(aOr).divide(epsilon);
 
         // update anomaly row
         final T[] eRow           = jacobian[1];
         final T[] anomalyRow     = jacobian[5];
         for (int j = 0; j < anomalyRow.length; ++j) {
             anomalyRow[j] = aFactor.multiply(anomalyRow[j]).subtract(eFactor.multiply(eRow[j]));
         }
 
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     public void addKeplerContribution(final PositionAngle type, final T gm,
                                       final T[] pDot) {
         final T oMe2;
         final T ksi;
         final T absA = a.abs();
         final T n    = absA.reciprocal().multiply(gm).sqrt().divide(absA);
         switch (type) {
             case MEAN :
                 pDot[5] = pDot[5].add(n);
                 break;
             case ECCENTRIC :
                 oMe2 = e.multiply(e).negate().add(1).abs();
                 ksi  = e.multiply(v.cos()).add(1);
                 pDot[5] = pDot[5].add( n.multiply(ksi).divide(oMe2));
                 break;
             case TRUE :
                 oMe2 = e.multiply(e).negate().add(1).abs();
                 ksi  = e.multiply(v.cos()).add(1);
                 pDot[5] = pDot[5].add(n.multiply(ksi).multiply(ksi).divide(oMe2.multiply(oMe2.sqrt())));
                 break;
             default :
                 throw new OrekitInternalError(null);
         }
     }
 
     /**  Returns a string representation of this Keplerian parameters object.
      * @return a string representation of this object
      */
     public String toString() {
         return new StringBuffer().append("Keplerian parameters: ").append('{').
                                   append("a: ").append(a.getReal()).
                                   append("; e: ").append(e.getReal()).
                                   append("; i: ").append(FastMath.toDegrees(i.getReal())).
                                   append("; pa: ").append(FastMath.toDegrees(pa.getReal())).
                                   append("; raan: ").append(FastMath.toDegrees(raan.getReal())).
                                   append("; v: ").append(FastMath.toDegrees(v.getReal())).
                                   append(";}").toString();
     }
 
     /** Check if the given parameter is within an acceptable range.
      * The bounds are inclusive: an exception is raised when either of those conditions are met:
      * <ul>
      *     <li>The parameter is strictly greater than upperBound</li>
      *     <li>The parameter is strictly lower than lowerBound</li>
      * </ul>
      * <p>
      * In either of these cases, an OrekitException is raised.
      * </p>
      * @param parameterName name of the parameter
      * @param parameter value of the parameter
      * @param lowerBound lower bound of the acceptable range (inclusive)
      * @param upperBound upper bound of the acceptable range (inclusive)
      */
     private void checkParameterRangeInclusive(final String parameterName, final double parameter,
                                               final double lowerBound, final double upperBound) {
         if (parameter < lowerBound || parameter > upperBound) {
             throw new OrekitException(OrekitMessages.INVALID_PARAMETER_RANGE, parameterName,
                                       parameter, lowerBound, upperBound);
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public KeplerianOrbit toOrbit() {
         if (hasDerivatives()) {
             return new KeplerianOrbit(a.getReal(), e.getReal(), i.getReal(),
                                       pa.getReal(), raan.getReal(), v.getReal(),
                                       aDot.getReal(), eDot.getReal(), iDot.getReal(),
                                       paDot.getReal(), raanDot.getReal(), vDot.getReal(),
                                       PositionAngle.TRUE,
                                       getFrame(), getDate().toAbsoluteDate(), getMu().getReal());
         } else {
             return new KeplerianOrbit(a.getReal(), e.getReal(), i.getReal(),
                                       pa.getReal(), raan.getReal(), v.getReal(),
                                       PositionAngle.TRUE,
                                       getFrame(), getDate().toAbsoluteDate(), getMu().getReal());
         }
     }
 
 
 }