AbstractGaussianContribution.java

  1. /* Copyright 2002-2021 CS GROUP
  2.  * Licensed to CS GROUP (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.propagation.semianalytical.dsst.forces;

  19. import java.lang.reflect.Array;
  20. import java.util.ArrayList;
  21. import java.util.Collections;
  22. import java.util.HashMap;
  23. import java.util.List;
  24. import java.util.Map;
  25. import java.util.Set;

  27. import org.hipparchus.Field;
  28. import org.hipparchus.CalculusFieldElement;
  29. import org.hipparchus.analysis.CalculusFieldUnivariateVectorFunction;
  30. import org.hipparchus.analysis.UnivariateVectorFunction;
  31. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  32. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  33. import org.hipparchus.util.FastMath;
  34. import org.hipparchus.util.FieldSinCos;
  35. import org.hipparchus.util.MathArrays;
  36. import org.hipparchus.util.SinCos;
  37. import org.orekit.attitudes.Attitude;
  38. import org.orekit.attitudes.AttitudeProvider;
  39. import org.orekit.attitudes.FieldAttitude;
  40. import org.orekit.forces.ForceModel;
  41. import org.orekit.orbits.EquinoctialOrbit;
  42. import org.orekit.orbits.FieldEquinoctialOrbit;
  43. import org.orekit.orbits.FieldOrbit;
  44. import org.orekit.orbits.Orbit;
  45. import org.orekit.orbits.OrbitType;
  46. import org.orekit.orbits.PositionAngle;
  47. import org.orekit.propagation.FieldSpacecraftState;
  48. import org.orekit.propagation.PropagationType;
  49. import org.orekit.propagation.SpacecraftState;
  50. import org.orekit.propagation.semianalytical.dsst.utilities.AuxiliaryElements;
  51. import org.orekit.propagation.semianalytical.dsst.utilities.CjSjCoefficient;
  52. import org.orekit.propagation.semianalytical.dsst.utilities.FieldAuxiliaryElements;
  53. import org.orekit.propagation.semianalytical.dsst.utilities.FieldCjSjCoefficient;
  54. import org.orekit.propagation.semianalytical.dsst.utilities.FieldShortPeriodicsInterpolatedCoefficient;
  55. import org.orekit.propagation.semianalytical.dsst.utilities.ShortPeriodicsInterpolatedCoefficient;
  56. import org.orekit.time.AbsoluteDate;
  57. import org.orekit.time.FieldAbsoluteDate;
  58. import org.orekit.utils.FieldTimeSpanMap;
  59. import org.orekit.utils.ParameterDriver;
  60. import org.orekit.utils.TimeSpanMap;

  62. /**
  63.  * Common handling of {@link DSSTForceModel} methods for Gaussian contributions
  64.  * to DSST propagation.
  65.  * <p>
  66.  * This abstract class allows to provide easily a subset of
  67.  * {@link DSSTForceModel} methods for specific Gaussian contributions.
  68.  * </p>
  69.  * <p>
  70.  * This class implements the notion of numerical averaging of the DSST theory.
  71.  * Numerical averaging is mainly used for non-conservative disturbing forces
  72.  * such as atmospheric drag and solar radiation pressure.
  73.  * </p>
  74.  * <p>
  75.  * Gaussian contributions can be expressed as: da<sub>i</sub>/dt =
  76.  * δa<sub>i</sub>/δv . q<br>
  77.  * where:
  78.  * <ul>
  79.  * <li>a<sub>i</sub> are the six equinoctial elements</li>
  80.  * <li>v is the velocity vector</li>
  81.  * <li>q is the perturbing acceleration due to the considered force</li>
  82.  * </ul>
  83.  *
  84.  * <p>
  85.  * The averaging process and other considerations lead to integrate this
  86.  * contribution over the true longitude L possibly taking into account some
  87.  * limits.
  88.  *
  89.  * <p>
  90.  * To create a numerically averaged contribution, one needs only to provide a
  91.  * {@link ForceModel} and to implement in the derived class the methods:
  92.  * {@link #getLLimits(SpacecraftState, AuxiliaryElements)} and
  93.  * {@link #getParametersDriversWithoutMu()}.
  94.  * </p>
  95.  * @author Pascal Parraud
  96.  * @author Bryan Cazabonne (field translation)
  97.  */
  98. public abstract class AbstractGaussianContribution implements DSSTForceModel {

  100.     /**
  101.      * Retrograde factor I.
  102.      * <p>
  103.      * DSST model needs equinoctial orbit as internal representation. Classical
  104.      * equinoctial elements have discontinuities when inclination is close to zero.
  105.      * In this representation, I = +1. <br>
  106.      * To avoid this discontinuity, another representation exists and equinoctial
  107.      * elements can be expressed in a different way, called "retrograde" orbit. This
  108.      * implies I = -1. <br>
  109.      * As Orekit doesn't implement the retrograde orbit, I is always set to +1. But
  110.      * for the sake of consistency with the theory, the retrograde factor has been
  111.      * kept in the formulas.
  112.      * </p>
  113.      */
  114.     private static final int I = 1;

  116.     /**
  117.      * Central attraction scaling factor.
  118.      * <p>
  119.      * We use a power of 2 to avoid numeric noise introduction in the
  120.      * multiplications/divisions sequences.
  121.      * </p>
  122.      */
  123.     private static final double MU_SCALE = FastMath.scalb(1.0, 32);

  125.     /** Available orders for Gauss quadrature. */
  126.     private static final int[] GAUSS_ORDER = { 12, 16, 20, 24, 32, 40, 48 };

  128.     /** Max rank in Gauss quadrature orders array. */
  129.     private static final int MAX_ORDER_RANK = GAUSS_ORDER.length - 1;

  131.     /** Number of points for interpolation. */
  132.     private static final int INTERPOLATION_POINTS = 3;

  134.     /** Maximum value for j index. */
  135.     private static final int JMAX = 12;

  137.     /** Contribution to be numerically averaged. */
  138.     private final ForceModel contribution;

  140.     /** Gauss integrator. */
  141.     private final double threshold;

  143.     /** Gauss integrator. */
  144.     private GaussQuadrature integrator;

  146.     /** Flag for Gauss order computation. */
  147.     private boolean isDirty;

  149.     /** Attitude provider. */
  150.     private AttitudeProvider attitudeProvider;

  152.     /** Prefix for coefficients keys. */
  153.     private final String coefficientsKeyPrefix;

  155.     /** Short period terms. */
  156.     private GaussianShortPeriodicCoefficients gaussianSPCoefs;

  158.     /** Short period terms. */
  159.     private Map<Field<?>, FieldGaussianShortPeriodicCoefficients<?>> gaussianFieldSPCoefs;

  161.     /** Driver for gravitational parameter. */
  162.     private final ParameterDriver gmParameterDriver;

  164.     /**
  165.      * Build a new instance.
  166.      * @param coefficientsKeyPrefix prefix for coefficients keys
  167.      * @param threshold             tolerance for the choice of the Gauss quadrature
  168.      *                              order
  169.      * @param contribution          the {@link ForceModel} to be numerically
  170.      *                              averaged
  171.      * @param mu                    central attraction coefficient
  172.      */
  173.     protected AbstractGaussianContribution(final String coefficientsKeyPrefix, final double threshold,
  174.             final ForceModel contribution, final double mu) {

  176.         gmParameterDriver = new ParameterDriver(DSSTNewtonianAttraction.CENTRAL_ATTRACTION_COEFFICIENT, mu, MU_SCALE,
  177.                 0.0, Double.POSITIVE_INFINITY);

  179.         this.coefficientsKeyPrefix = coefficientsKeyPrefix;
  180.         this.contribution = contribution;
  181.         this.threshold = threshold;
  182.         this.integrator = new GaussQuadrature(GAUSS_ORDER[MAX_ORDER_RANK]);
  183.         this.isDirty = true;

  185.         gaussianFieldSPCoefs = new HashMap<>();
  186.     }

  188.     /** {@inheritDoc} */
  189.     @Override
  190.     public void init(final SpacecraftState initialState, final AbsoluteDate target) {
  191.         // Initialize the numerical force model
  192.         contribution.init(initialState, target);
  193.     }

  195.     /** {@inheritDoc} */
  196.     @Override
  197.     public List<ParameterDriver> getParametersDrivers() {

  199.         // Parameter drivers
  200.         final List<ParameterDriver> drivers = new ArrayList<>();

  202.         // Loop on drivers (without central attraction coefficient driver)
  203.         for (final ParameterDriver driver : getParametersDriversWithoutMu()) {
  204.             drivers.add(driver);
  205.         }

  207.         // We put central attraction coefficient driver at the end of the array
  208.         drivers.add(gmParameterDriver);
  209.         return drivers;

  211.     }

  213.     /**
  214.      * Get the drivers for force model parameters except the one for the central
  215.      * attraction coefficient.
  216.      * <p>
  217.      * The driver for central attraction coefficient is automatically added at the
  218.      * last element of the {@link ParameterDriver} array into
  219.      * {@link #getParametersDrivers()} method.
  220.      * </p>
  221.      * @return drivers for force model parameters
  222.      */
  223.     protected abstract List<ParameterDriver> getParametersDriversWithoutMu();

  225.     /** {@inheritDoc} */
  226.     @Override
  227.     public List<ShortPeriodTerms> initializeShortPeriodTerms(final AuxiliaryElements auxiliaryElements, final PropagationType type,
  228.             final double[] parameters) {

  230.         final List<ShortPeriodTerms> list = new ArrayList<ShortPeriodTerms>();
  231.         gaussianSPCoefs = new GaussianShortPeriodicCoefficients(coefficientsKeyPrefix, JMAX, INTERPOLATION_POINTS,
  232.                 new TimeSpanMap<Slot>(new Slot(JMAX, INTERPOLATION_POINTS)));
  233.         list.add(gaussianSPCoefs);
  234.         return list;

  236.     }

  238.     /** {@inheritDoc} */
  239.     @Override
  240.     public <T extends CalculusFieldElement<T>> List<FieldShortPeriodTerms<T>> initializeShortPeriodTerms(
  241.             final FieldAuxiliaryElements<T> auxiliaryElements, final PropagationType type, final T[] parameters) {

  243.         final Field<T> field = auxiliaryElements.getDate().getField();

  245.         final FieldGaussianShortPeriodicCoefficients<T> fgspc = new FieldGaussianShortPeriodicCoefficients<>(
  246.                 coefficientsKeyPrefix, JMAX, INTERPOLATION_POINTS,
  247.                 new FieldTimeSpanMap<>(new FieldSlot<>(JMAX, INTERPOLATION_POINTS), field));
  248.         gaussianFieldSPCoefs.put(field, fgspc);
  249.         return Collections.singletonList(fgspc);
  250.     }

  252.     /**
  253.      * Performs initialization at each integration step for the current force model.
  254.      * <p>
  255.      * This method aims at being called before mean elements rates computation.
  256.      * </p>
  257.      * @param auxiliaryElements auxiliary elements related to the current orbit
  258.      * @param parameters        parameters values of the force model parameters
  259.      * @return new force model context
  260.      */
  261.     private AbstractGaussianContributionContext initializeStep(final AuxiliaryElements auxiliaryElements,
  262.             final double[] parameters) {
  263.         return new AbstractGaussianContributionContext(auxiliaryElements, parameters);
  264.     }

  266.     /**
  267.      * Performs initialization at each integration step for the current force model.
  268.      * <p>
  269.      * This method aims at being called before mean elements rates computation.
  270.      * </p>
  271.      * @param <T>               type of the elements
  272.      * @param auxiliaryElements auxiliary elements related to the current orbit
  273.      * @param parameters        parameters values of the force model parameters
  274.      * @return new force model context
  275.      */
  276.     private <T extends CalculusFieldElement<T>> FieldAbstractGaussianContributionContext<T> initializeStep(
  277.             final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters) {
  278.         return new FieldAbstractGaussianContributionContext<>(auxiliaryElements, parameters);
  279.     }

  281.     /** {@inheritDoc} */
  282.     @Override
  283.     public double[] getMeanElementRate(final SpacecraftState state, final AuxiliaryElements auxiliaryElements,
  284.             final double[] parameters) {

  286.         // Container for attributes
  287.         final AbstractGaussianContributionContext context = initializeStep(auxiliaryElements, parameters);
  288.         double[] meanElementRate = new double[6];
  289.         // Computes the limits for the integral
  290.         final double[] ll = getLLimits(state, auxiliaryElements);
  291.         // Computes integrated mean element rates if Llow < Lhigh
  292.         if (ll[0] < ll[1]) {
  293.             meanElementRate = getMeanElementRate(state, integrator, ll[0], ll[1], context, parameters);
  294.             if (isDirty) {
  295.                 boolean next = true;
  296.                 for (int i = 0; i < MAX_ORDER_RANK && next; i++) {
  297.                     final double[] meanRates = getMeanElementRate(state, new GaussQuadrature(GAUSS_ORDER[i]), ll[0],
  298.                             ll[1], context, parameters);
  299.                     if (getRatesDiff(meanElementRate, meanRates, context) < threshold) {
  300.                         integrator = new GaussQuadrature(GAUSS_ORDER[i]);
  301.                         next = false;
  302.                     }
  303.                 }
  304.                 isDirty = false;
  305.             }
  306.         }
  307.         return meanElementRate;
  308.     }

  310.     /** {@inheritDoc} */
  311.     @Override
  312.     public <T extends CalculusFieldElement<T>> T[] getMeanElementRate(final FieldSpacecraftState<T> state,
  313.             final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters) {

  315.         // Container for attributes
  316.         final FieldAbstractGaussianContributionContext<T> context = initializeStep(auxiliaryElements, parameters);
  317.         final Field<T> field = state.getDate().getField();

  319.         T[] meanElementRate = MathArrays.buildArray(field, 6);
  320.         // Computes the limits for the integral
  321.         final T[] ll = getLLimits(state, auxiliaryElements);
  322.         // Computes integrated mean element rates if Llow < Lhigh
  323.         if (ll[0].getReal() < ll[1].getReal()) {
  324.             meanElementRate = getMeanElementRate(state, integrator, ll[0], ll[1], context, parameters);
  325.             if (isDirty) {
  326.                 boolean next = true;
  327.                 for (int i = 0; i < MAX_ORDER_RANK && next; i++) {
  328.                     final T[] meanRates = getMeanElementRate(state, new GaussQuadrature(GAUSS_ORDER[i]), ll[0], ll[1],
  329.                             context, parameters);
  330.                     if (getRatesDiff(meanElementRate, meanRates, context).getReal() < threshold) {
  331.                         integrator = new GaussQuadrature(GAUSS_ORDER[i]);
  332.                         next = false;
  333.                     }
  334.                 }
  335.                 isDirty = false;
  336.             }
  337.         }

  339.         return meanElementRate;
  340.     }

  342.     /**
  343.      * Compute the limits in L, the true longitude, for integration.
  344.      *
  345.      * @param state             current state information: date, kinematics,
  346.      *                          attitude
  347.      * @param auxiliaryElements auxiliary elements related to the current orbit
  348.      * @return the integration limits in L
  349.      */
  350.     protected abstract double[] getLLimits(SpacecraftState state, AuxiliaryElements auxiliaryElements);

  352.     /**
  353.      * Compute the limits in L, the true longitude, for integration.
  354.      *
  355.      * @param <T>               type of the elements
  356.      * @param state             current state information: date, kinematics,
  357.      *                          attitude
  358.      * @param auxiliaryElements auxiliary elements related to the current orbit
  359.      * @return the integration limits in L
  360.      */
  361.     protected abstract <T extends CalculusFieldElement<T>> T[] getLLimits(FieldSpacecraftState<T> state,
  362.             FieldAuxiliaryElements<T> auxiliaryElements);

  364.     /**
  365.      * Computes the mean equinoctial elements rates da<sub>i</sub> / dt.
  366.      *
  367.      * @param state      current state
  368.      * @param gauss      Gauss quadrature
  369.      * @param low        lower bound of the integral interval
  370.      * @param high       upper bound of the integral interval
  371.      * @param context    container for attributes
  372.      * @param parameters values of the force model parameters
  373.      * @return the mean element rates
  374.      */
  375.     protected double[] getMeanElementRate(final SpacecraftState state, final GaussQuadrature gauss, final double low,
  376.             final double high, final AbstractGaussianContributionContext context, final double[] parameters) {

  378.         // Auxiliary elements related to the current orbit
  379.         final AuxiliaryElements auxiliaryElements = context.getAuxiliaryElements();

  381.         final double[] meanElementRate = gauss.integrate(new IntegrableFunction(state, true, 0, parameters), low, high);

  383.         // Constant multiplier for integral
  384.         final double coef = 1. / (2. * FastMath.PI * auxiliaryElements.getB());
  385.         // Corrects mean element rates
  386.         for (int i = 0; i < 6; i++) {
  387.             meanElementRate[i] *= coef;
  388.         }
  389.         return meanElementRate;
  390.     }

  392.     /**
  393.      * Computes the mean equinoctial elements rates da<sub>i</sub> / dt.
  394.      *
  395.      * @param <T>        type of the elements
  396.      * @param state      current state
  397.      * @param gauss      Gauss quadrature
  398.      * @param low        lower bound of the integral interval
  399.      * @param high       upper bound of the integral interval
  400.      * @param context    container for attributes
  401.      * @param parameters values of the force model parameters
  402.      * @return the mean element rates
  403.      */
  404.     protected <T extends CalculusFieldElement<T>> T[] getMeanElementRate(final FieldSpacecraftState<T> state,
  405.             final GaussQuadrature gauss, final T low, final T high,
  406.             final FieldAbstractGaussianContributionContext<T> context, final T[] parameters) {

  408.         // Field
  409.         final Field<T> field = context.getA().getField();

  411.         // Auxiliary elements related to the current orbit
  412.         final FieldAuxiliaryElements<T> auxiliaryElements = context.getFieldAuxiliaryElements();

  414.         final T[] meanElementRate = gauss.integrate(new FieldIntegrableFunction<>(state, true, 0, parameters, field),
  415.                 low, high, field);
  416.         // Constant multiplier for integral
  417.         final T coef = auxiliaryElements.getB().multiply(low.getPi()).multiply(2.).reciprocal();
  418.         // Corrects mean element rates
  419.         for (int i = 0; i < 6; i++) {
  420.             meanElementRate[i] = meanElementRate[i].multiply(coef);
  421.         }
  422.         return meanElementRate;
  423.     }

  425.     /**
  426.      * Estimates the weighted magnitude of the difference between 2 sets of
  427.      * equinoctial elements rates.
  428.      *
  429.      * @param meanRef reference rates
  430.      * @param meanCur current rates
  431.      * @param context container for attributes
  432.      * @return estimated magnitude of weighted differences
  433.      */
  434.     private double getRatesDiff(final double[] meanRef, final double[] meanCur,
  435.             final AbstractGaussianContributionContext context) {

  437.         // Auxiliary elements related to the current orbit
  438.         final AuxiliaryElements auxiliaryElements = context.getAuxiliaryElements();

  440.         double maxDiff = FastMath.abs(meanRef[0] - meanCur[0]) / auxiliaryElements.getSma();
  441.         // Corrects mean element rates
  442.         for (int i = 1; i < meanRef.length; i++) {
  443.             maxDiff = FastMath.max(maxDiff, FastMath.abs(meanRef[i] - meanCur[i]));
  444.         }
  445.         return maxDiff;
  446.     }

  448.     /**
  449.      * Estimates the weighted magnitude of the difference between 2 sets of
  450.      * equinoctial elements rates.
  451.      *
  452.      * @param <T>     type of the elements
  453.      * @param meanRef reference rates
  454.      * @param meanCur current rates
  455.      * @param context container for attributes
  456.      * @return estimated magnitude of weighted differences
  457.      */
  458.     private <T extends CalculusFieldElement<T>> T getRatesDiff(final T[] meanRef, final T[] meanCur,
  459.             final FieldAbstractGaussianContributionContext<T> context) {

  461.         // Auxiliary elements related to the current orbit
  462.         final FieldAuxiliaryElements<T> auxiliaryElements = context.getFieldAuxiliaryElements();

  464.         T maxDiff = FastMath.abs(meanRef[0].subtract(meanCur[0])).divide(auxiliaryElements.getSma());
  465.         ;
  466.         // Corrects mean element rates
  467.         for (int i = 1; i < meanRef.length; i++) {
  468.             maxDiff = FastMath.max(maxDiff, FastMath.abs(meanRef[i].subtract(meanCur[i])));
  469.         }
  470.         return maxDiff;
  471.     }

  473.     /** {@inheritDoc} */
  474.     @Override
  475.     public void registerAttitudeProvider(final AttitudeProvider provider) {
  476.         this.attitudeProvider = provider;
  477.     }

  479.     /** {@inheritDoc} */
  480.     @Override
  481.     public void updateShortPeriodTerms(final double[] parameters, final SpacecraftState... meanStates) {

  483.         final Slot slot = gaussianSPCoefs.createSlot(meanStates);
  484.         for (final SpacecraftState meanState : meanStates) {

  486.             // Auxiliary elements related to the current orbit
  487.             final AuxiliaryElements auxiliaryElements = new AuxiliaryElements(meanState.getOrbit(), I);

  489.             // Container of attributes
  490.             final AbstractGaussianContributionContext context = initializeStep(auxiliaryElements, parameters);

  492.             // Compute rhoj and sigmaj
  493.             final double[][] currentRhoSigmaj = computeRhoSigmaCoefficients(meanState.getDate(), auxiliaryElements);

  495.             // Generate the Cij and Sij coefficients
  496.             final FourierCjSjCoefficients fourierCjSj = new FourierCjSjCoefficients(meanState, JMAX, auxiliaryElements,
  497.                     parameters);

  499.             // Generate the Uij and Vij coefficients
  500.             final UijVijCoefficients uijvij = new UijVijCoefficients(currentRhoSigmaj, fourierCjSj, JMAX);

  502.             gaussianSPCoefs.computeCoefficients(meanState, slot, fourierCjSj, uijvij, context.getMeanMotion(),
  503.                     auxiliaryElements.getSma());

  505.         }

  507.     }

  509.     /** {@inheritDoc} */
  510.     @Override
  511.     @SuppressWarnings("unchecked")
  512.     public <T extends CalculusFieldElement<T>> void updateShortPeriodTerms(final T[] parameters,
  513.             final FieldSpacecraftState<T>... meanStates) {

  515.         // Field used by default
  516.         final Field<T> field = meanStates[0].getDate().getField();

  518.         final FieldGaussianShortPeriodicCoefficients<T> fgspc = (FieldGaussianShortPeriodicCoefficients<T>) gaussianFieldSPCoefs
  519.                 .get(field);
  520.         final FieldSlot<T> slot = fgspc.createSlot(meanStates);
  521.         for (final FieldSpacecraftState<T> meanState : meanStates) {

  523.             // Auxiliary elements related to the current orbit
  524.             final FieldAuxiliaryElements<T> auxiliaryElements = new FieldAuxiliaryElements<>(meanState.getOrbit(), I);

  526.             // Container of attributes
  527.             final FieldAbstractGaussianContributionContext<T> context = initializeStep(auxiliaryElements, parameters);

  529.             // Compute rhoj and sigmaj
  530.             final T[][] currentRhoSigmaj = computeRhoSigmaCoefficients(meanState.getDate(), context, field);

  532.             // Generate the Cij and Sij coefficients
  533.             final FieldFourierCjSjCoefficients<T> fourierCjSj = new FieldFourierCjSjCoefficients<>(meanState, JMAX,
  534.                     auxiliaryElements, parameters, field);

  536.             // Generate the Uij and Vij coefficients
  537.             final FieldUijVijCoefficients<T> uijvij = new FieldUijVijCoefficients<>(currentRhoSigmaj, fourierCjSj, JMAX,
  538.                     field);

  540.             fgspc.computeCoefficients(meanState, slot, fourierCjSj, uijvij, context.getMeanMotion(),
  541.                     auxiliaryElements.getSma(), field);

  543.         }

  545.     }

  547.     /**
  548.      * Compute the auxiliary quantities ρ<sub>j</sub> and σ<sub>j</sub>.
  549.      * <p>
  550.      * The expressions used are equations 2.5.3-(4) from the Danielson paper. <br/>
  551.      * ρ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>C<sub>j</sub>(k, h) <br/>
  552.      * σ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>S<sub>j</sub>(k, h) <br/>
  553.      * </p>
  554.      * @param date              current date
  555.      * @param auxiliaryElements auxiliary elements related to the current orbit
  556.      * @return computed coefficients
  557.      */
  558.     private double[][] computeRhoSigmaCoefficients(final AbsoluteDate date, final AuxiliaryElements auxiliaryElements) {
  559.         final double[][] currentRhoSigmaj = new double[2][3 * JMAX + 1];
  560.         final CjSjCoefficient cjsjKH = new CjSjCoefficient(auxiliaryElements.getK(), auxiliaryElements.getH());
  561.         final double b = 1. / (1 + auxiliaryElements.getB());

  563.         // (-b)<sup>j</sup>
  564.         double mbtj = 1;

  566.         for (int j = 1; j <= 3 * JMAX; j++) {

  568.             // Compute current rho and sigma;
  569.             mbtj *= -b;
  570.             final double coef = (1 + j * auxiliaryElements.getB()) * mbtj;
  571.             currentRhoSigmaj[0][j] = coef * cjsjKH.getCj(j);
  572.             currentRhoSigmaj[1][j] = coef * cjsjKH.getSj(j);
  573.         }
  574.         return currentRhoSigmaj;
  575.     }

  577.     /**
  578.      * Compute the auxiliary quantities ρ<sub>j</sub> and σ<sub>j</sub>.
  579.      * <p>
  580.      * The expressions used are equations 2.5.3-(4) from the Danielson paper. <br/>
  581.      * ρ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>C<sub>j</sub>(k, h) <br/>
  582.      * σ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>S<sub>j</sub>(k, h) <br/>
  583.      * </p>
  584.      * @param <T>     type of the elements
  585.      * @param date    current date
  586.      * @param context container for attributes
  587.      * @param field   field used by default
  588.      * @return computed coefficients
  589.      */
  590.     private <T extends CalculusFieldElement<T>> T[][] computeRhoSigmaCoefficients(final FieldAbsoluteDate<T> date,
  591.             final FieldAbstractGaussianContributionContext<T> context, final Field<T> field) {
  592.         // zero
  593.         final T zero = field.getZero();

  595.         final FieldAuxiliaryElements<T> auxiliaryElements = context.getFieldAuxiliaryElements();
  596.         final T[][] currentRhoSigmaj = MathArrays.buildArray(field, 2, 3 * JMAX + 1);
  597.         final FieldCjSjCoefficient<T> cjsjKH = new FieldCjSjCoefficient<>(auxiliaryElements.getK(),
  598.                 auxiliaryElements.getH(), field);
  599.         final T b = auxiliaryElements.getB().add(1.).reciprocal();

  601.         // (-b)<sup>j</sup>
  602.         T mbtj = zero.add(1.);

  604.         for (int j = 1; j <= 3 * JMAX; j++) {

  606.             // Compute current rho and sigma;
  607.             mbtj = mbtj.multiply(b.negate());
  608.             final T coef = mbtj.multiply(auxiliaryElements.getB().multiply(j).add(1.));
  609.             currentRhoSigmaj[0][j] = coef.multiply(cjsjKH.getCj(j));
  610.             currentRhoSigmaj[1][j] = coef.multiply(cjsjKH.getSj(j));
  611.         }
  612.         return currentRhoSigmaj;
  613.     }

  615.     /**
  616.      * Internal class for numerical quadrature.
  617.      * <p>
  618.      * This class is a rewrite of {@link IntegrableFunction} for field elements
  619.      * </p>
  620.      */
  621.     protected class FieldIntegrableFunction<T extends CalculusFieldElement<T>>
  622.             implements CalculusFieldUnivariateVectorFunction<T> {

  624.         /** Current state. */
  625.         private final FieldSpacecraftState<T> state;

  627.         /**
  628.          * Signal that this class is used to compute the values required by the mean
  629.          * element variations or by the short periodic element variations.
  630.          */
  631.         private final boolean meanMode;

  633.         /**
  634.          * The j index.
  635.          * <p>
  636.          * Used only for short periodic variation. Ignored for mean elements variation.
  637.          * </p>
  638.          */
  639.         private final int j;

  641.         /** Container for attributes. */
  642.         private final FieldAbstractGaussianContributionContext<T> context;

  644.         /** Auxiliary Elements. */
  645.         private final FieldAuxiliaryElements<T> auxiliaryElements;

  647.         /** Drivers for solar radiation and atmospheric drag forces. */
  648.         private final T[] parameters;

  650.         /**
  651.          * Build a new instance with a new field.
  652.          * @param state      current state information: date, kinematics, attitude
  653.          * @param meanMode   if true return the value associated to the mean elements
  654.          *                   variation, if false return the values associated to the
  655.          *                   short periodic elements variation
  656.          * @param j          the j index. used only for short periodic variation.
  657.          *                   Ignored for mean elements variation.
  658.          * @param parameters values of the force model parameters
  659.          * @param field      field utilized by default
  660.          */
  661.         public FieldIntegrableFunction(final FieldSpacecraftState<T> state, final boolean meanMode, final int j,
  662.                 final T[] parameters, final Field<T> field) {

  664.             this.meanMode = meanMode;
  665.             this.j = j;
  666.             this.parameters = parameters.clone();
  667.             this.auxiliaryElements = new FieldAuxiliaryElements<>(state.getOrbit(), I);
  668.             this.context = new FieldAbstractGaussianContributionContext<>(auxiliaryElements, this.parameters);
  669.             // remove derivatives from state
  670.             final T[] stateVector = MathArrays.buildArray(field, 6);
  671.             OrbitType.EQUINOCTIAL.mapOrbitToArray(state.getOrbit(), PositionAngle.TRUE, stateVector, null);
  672.             final FieldOrbit<T> fixedOrbit = OrbitType.EQUINOCTIAL.mapArrayToOrbit(stateVector, null,
  673.                     PositionAngle.TRUE, state.getDate(), context.getMu(), state.getFrame());
  674.             this.state = new FieldSpacecraftState<>(fixedOrbit, state.getAttitude(), state.getMass());
  675.         }

  677.         /** {@inheritDoc} */
  678.         @Override
  679.         public T[] value(final T x) {

  681.             // Parameters for array building
  682.             final Field<T> field = auxiliaryElements.getDate().getField();
  683.             final int dimension = 6;

  685.             // Compute the time difference from the true longitude difference
  686.             final T shiftedLm = trueToMean(x);
  687.             final T dLm = shiftedLm.subtract(auxiliaryElements.getLM());
  688.             final T dt = dLm.divide(context.getMeanMotion());

  690.             final FieldSinCos<T> scL = FastMath.sinCos(x);
  691.             final T cosL = scL.cos();
  692.             final T sinL = scL.sin();
  693.             final T roa  = auxiliaryElements.getB().multiply(auxiliaryElements.getB()).divide(auxiliaryElements.getH().multiply(sinL).add(auxiliaryElements.getK().multiply(cosL)).add(1.));
  694.             final T roa2 = roa.multiply(roa);
  695.             final T r = auxiliaryElements.getSma().multiply(roa);
  696.             final T X = r.multiply(cosL);
  697.             final T Y = r.multiply(sinL);
  698.             final T naob = context.getMeanMotion().multiply(auxiliaryElements.getSma())
  699.                     .divide(auxiliaryElements.getB());
  700.             final T Xdot = naob.multiply(auxiliaryElements.getH().add(sinL)).negate();
  701.             final T Ydot = naob.multiply(auxiliaryElements.getK().add(cosL));
  702.             final FieldVector3D<T> vel = new FieldVector3D<>(Xdot, auxiliaryElements.getVectorF(), Ydot,
  703.                     auxiliaryElements.getVectorG());

  705.             // Compute acceleration
  706.             FieldVector3D<T> acc = FieldVector3D.getZero(field);

  708.             // shift the orbit to dt
  709.             final FieldOrbit<T> shiftedOrbit = state.getOrbit().shiftedBy(dt);

  711.             // Recompose an orbit with time held fixed to be compliant with DSST theory
  712.             final FieldOrbit<T> recomposedOrbit = new FieldEquinoctialOrbit<>(shiftedOrbit.getA(),
  713.                     shiftedOrbit.getEquinoctialEx(), shiftedOrbit.getEquinoctialEy(), shiftedOrbit.getHx(),
  714.                     shiftedOrbit.getHy(), shiftedOrbit.getLv(), PositionAngle.TRUE, shiftedOrbit.getFrame(),
  715.                     state.getDate(), context.getMu());

  717.             // Get the corresponding attitude
  718.             final FieldAttitude<T> recomposedAttitude = attitudeProvider.getAttitude(recomposedOrbit,
  719.                     recomposedOrbit.getDate(), recomposedOrbit.getFrame());

  721.             // create shifted SpacecraftState with attitude at specified time
  722.             final FieldSpacecraftState<T> shiftedState = new FieldSpacecraftState<>(recomposedOrbit, recomposedAttitude,
  723.                     state.getMass());

  725.             acc = contribution.acceleration(shiftedState, parameters);

  727.             // Compute the derivatives of the elements by the speed
  728.             final T[] deriv = MathArrays.buildArray(field, dimension);
  729.             // da/dv
  730.             deriv[0] = getAoV(vel).dotProduct(acc);
  731.             // dex/dv
  732.             deriv[1] = getKoV(X, Y, Xdot, Ydot).dotProduct(acc);
  733.             // dey/dv
  734.             deriv[2] = getHoV(X, Y, Xdot, Ydot).dotProduct(acc);
  735.             // dhx/dv
  736.             deriv[3] = getQoV(X).dotProduct(acc);
  737.             // dhy/dv
  738.             deriv[4] = getPoV(Y).dotProduct(acc);
  739.             // dλ/dv
  740.             deriv[5] = getLoV(X, Y, Xdot, Ydot).dotProduct(acc);

  742.             // Compute mean elements rates
  743.             T[] val = null;
  744.             if (meanMode) {
  745.                 val = MathArrays.buildArray(field, dimension);
  746.                 for (int i = 0; i < 6; i++) {
  747.                     // da<sub>i</sub>/dt
  748.                     val[i] = deriv[i].multiply(roa2);
  749.                 }
  750.             } else {
  751.                 val = MathArrays.buildArray(field, dimension * 2);
  752.                 //Compute cos(j*L) and sin(j*L);
  753.                 final FieldSinCos<T> scjL = FastMath.sinCos(x.multiply(j));
  754.                 final T cosjL = j == 1 ? cosL : scjL.cos();
  755.                 final T sinjL = j == 1 ? sinL : scjL.sin();

  757.                 for (int i = 0; i < 6; i++) {
  758.                     // da<sub>i</sub>/dv * cos(jL)
  759.                     val[i] = deriv[i].multiply(cosjL);
  760.                     // da<sub>i</sub>/dv * sin(jL)
  761.                     val[i + 6] = deriv[i].multiply(sinjL);
  762.                 }
  763.             }

  765.             return val;
  766.         }

  768.         /**
  769.          * Converts true longitude to mean longitude.
  770.          * @param x True longitude
  771.          * @return Eccentric longitude
  772.          */
  773.         private T trueToMean(final T x) {
  774.             return eccentricToMean(trueToEccentric(x));
  775.         }

  777.         /**
  778.          * Converts true longitude to eccentric longitude.
  779.          * @param lv True longitude
  780.          * @return Eccentric longitude
  781.          */
  782.         private T trueToEccentric (final T lv) {
  783.             final FieldSinCos<T> sclV = FastMath.sinCos(lv);
  784.             final T cosLv   = sclV.cos();
  785.             final T sinLv   = sclV.sin();
  786.             final T num     = auxiliaryElements.getH().multiply(cosLv).subtract(auxiliaryElements.getK().multiply(sinLv));
  787.             final T den     = auxiliaryElements.getB().add(auxiliaryElements.getK().multiply(cosLv)).add(auxiliaryElements.getH().multiply(sinLv)).add(1.);
  788.             return FastMath.atan(num.divide(den)).multiply(2.).add(lv);
  789.         }

  791.         /**
  792.          * Converts eccentric longitude to mean longitude.
  793.          * @param le Eccentric longitude
  794.          * @return Mean longitude
  795.          */
  796.         private T eccentricToMean (final T le) {
  797.             final FieldSinCos<T> scle = FastMath.sinCos(le);
  798.             return le.subtract(auxiliaryElements.getK().multiply(scle.sin())).add(auxiliaryElements.getH().multiply(scle.cos()));
  799.         }

  801.         /**
  802.          * Compute δa/δv.
  803.          * @param vel satellite velocity
  804.          * @return δa/δv
  805.          */
  806.         private FieldVector3D<T> getAoV(final FieldVector3D<T> vel) {
  807.             return new FieldVector3D<>(context.getTon2a(), vel);
  808.         }

  810.         /**
  811.          * Compute δh/δv.
  812.          * @param X    satellite position component along f, equinoctial reference frame
  813.          *             1st vector
  814.          * @param Y    satellite position component along g, equinoctial reference frame
  815.          *             2nd vector
  816.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  817.          *             1st vector
  818.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  819.          *             2nd vector
  820.          * @return δh/δv
  821.          */
  822.         private FieldVector3D<T> getHoV(final T X, final T Y, final T Xdot, final T Ydot) {
  823.             final T kf = (Xdot.multiply(Y).multiply(2.).subtract(X.multiply(Ydot))).multiply(context.getOoMU());
  824.             final T kg = X.multiply(Xdot).multiply(context.getOoMU());
  825.             final T kw = auxiliaryElements.getK().multiply(
  826.                     auxiliaryElements.getQ().multiply(Y).multiply(I).subtract(auxiliaryElements.getP().multiply(X)))
  827.                     .multiply(context.getOOAB());
  828.             return new FieldVector3D<>(kf, auxiliaryElements.getVectorF(), kg.negate(), auxiliaryElements.getVectorG(),
  829.                     kw, auxiliaryElements.getVectorW());
  830.         }

  832.         /**
  833.          * Compute δk/δv.
  834.          * @param X    satellite position component along f, equinoctial reference frame
  835.          *             1st vector
  836.          * @param Y    satellite position component along g, equinoctial reference frame
  837.          *             2nd vector
  838.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  839.          *             1st vector
  840.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  841.          *             2nd vector
  842.          * @return δk/δv
  843.          */
  844.         private FieldVector3D<T> getKoV(final T X, final T Y, final T Xdot, final T Ydot) {
  845.             final T kf = Y.multiply(Ydot).multiply(context.getOoMU());
  846.             final T kg = (X.multiply(Ydot).multiply(2.).subtract(Xdot.multiply(Y))).multiply(context.getOoMU());
  847.             final T kw = auxiliaryElements.getH().multiply(
  848.                     auxiliaryElements.getQ().multiply(Y).multiply(I).subtract(auxiliaryElements.getP().multiply(X)))
  849.                     .multiply(context.getOOAB());
  850.             return new FieldVector3D<>(kf.negate(), auxiliaryElements.getVectorF(), kg, auxiliaryElements.getVectorG(),
  851.                     kw.negate(), auxiliaryElements.getVectorW());
  852.         }

  854.         /**
  855.          * Compute δp/δv.
  856.          * @param Y satellite position component along g, equinoctial reference frame
  857.          *          2nd vector
  858.          * @return δp/δv
  859.          */
  860.         private FieldVector3D<T> getPoV(final T Y) {
  861.             return new FieldVector3D<>(context.getCo2AB().multiply(Y), auxiliaryElements.getVectorW());
  862.         }

  864.         /**
  865.          * Compute δq/δv.
  866.          * @param X satellite position component along f, equinoctial reference frame
  867.          *          1st vector
  868.          * @return δq/δv
  869.          */
  870.         private FieldVector3D<T> getQoV(final T X) {
  871.             return new FieldVector3D<>(context.getCo2AB().multiply(X).multiply(I), auxiliaryElements.getVectorW());
  872.         }

  874.         /**
  875.          * Compute δλ/δv.
  876.          * @param X    satellite position component along f, equinoctial reference frame
  877.          *             1st vector
  878.          * @param Y    satellite position component along g, equinoctial reference frame
  879.          *             2nd vector
  880.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  881.          *             1st vector
  882.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  883.          *             2nd vector
  884.          * @return δλ/δv
  885.          */
  886.         private FieldVector3D<T> getLoV(final T X, final T Y, final T Xdot, final T Ydot) {
  887.             final FieldVector3D<T> pos = new FieldVector3D<>(X, auxiliaryElements.getVectorF(), Y,
  888.                     auxiliaryElements.getVectorG());
  889.             final FieldVector3D<T> v2 = new FieldVector3D<>(auxiliaryElements.getK(), getHoV(X, Y, Xdot, Ydot),
  890.                     auxiliaryElements.getH().negate(), getKoV(X, Y, Xdot, Ydot));
  891.             return new FieldVector3D<>(context.getOOA().multiply(-2.), pos, context.getOoBpo(), v2,
  892.                     context.getOOA().multiply(auxiliaryElements.getQ().multiply(Y).multiply(I)
  893.                             .subtract(auxiliaryElements.getP().multiply(X))),
  894.                     auxiliaryElements.getVectorW());
  895.         }

  897.     }

  899.     /** Internal class for numerical quadrature. */
  900.     protected class IntegrableFunction implements UnivariateVectorFunction {

  902.         /** Current state. */
  903.         private final SpacecraftState state;

  905.         /**
  906.          * Signal that this class is used to compute the values required by the mean
  907.          * element variations or by the short periodic element variations.
  908.          */
  909.         private final boolean meanMode;

  911.         /**
  912.          * The j index.
  913.          * <p>
  914.          * Used only for short periodic variation. Ignored for mean elements variation.
  915.          * </p>
  916.          */
  917.         private final int j;

  919.         /** Container for attributes. */
  920.         private final AbstractGaussianContributionContext context;

  922.         /** Auxiliary Elements. */
  923.         private final AuxiliaryElements auxiliaryElements;

  925.         /** Drivers for solar radiation and atmospheric drag forces. */
  926.         private final double[] parameters;

  928.         /**
  929.          * Build a new instance.
  930.          * @param state      current state information: date, kinematics, attitude
  931.          * @param meanMode   if true return the value associated to the mean elements
  932.          *                   variation, if false return the values associated to the
  933.          *                   short periodic elements variation
  934.          * @param j          the j index. used only for short periodic variation.
  935.          *                   Ignored for mean elements variation.
  936.          * @param parameters values of the force model parameters
  937.          */
  938.         IntegrableFunction(final SpacecraftState state, final boolean meanMode, final int j,
  939.                 final double[] parameters) {

  941.             this.meanMode = meanMode;
  942.             this.j = j;
  943.             this.parameters = parameters.clone();
  944.             this.auxiliaryElements = new AuxiliaryElements(state.getOrbit(), I);
  945.             this.context = new AbstractGaussianContributionContext(auxiliaryElements, this.parameters);
  946.             // remove derivatives from state
  947.             final double[] stateVector = new double[6];
  948.             OrbitType.EQUINOCTIAL.mapOrbitToArray(state.getOrbit(), PositionAngle.TRUE, stateVector, null);
  949.             final Orbit fixedOrbit = OrbitType.EQUINOCTIAL.mapArrayToOrbit(stateVector, null, PositionAngle.TRUE,
  950.                     state.getDate(), context.getMu(), state.getFrame());
  951.             this.state = new SpacecraftState(fixedOrbit, state.getAttitude(), state.getMass());
  952.         }

  954.         /** {@inheritDoc} */
  955.         @Override
  956.         public double[] value(final double x) {

  958.             // Compute the time difference from the true longitude difference
  959.             final double shiftedLm = trueToMean(x);
  960.             final double dLm = shiftedLm - auxiliaryElements.getLM();
  961.             final double dt = dLm / context.getMeanMotion();

  963.             final SinCos scL  = FastMath.sinCos(x);
  964.             final double cosL = scL.cos();
  965.             final double sinL = scL.sin();
  966.             final double roa  = auxiliaryElements.getB() * auxiliaryElements.getB() / (1. + auxiliaryElements.getH() * sinL + auxiliaryElements.getK() * cosL);
  967.             final double roa2 = roa * roa;
  968.             final double r = auxiliaryElements.getSma() * roa;
  969.             final double X = r * cosL;
  970.             final double Y = r * sinL;
  971.             final double naob = context.getMeanMotion() * auxiliaryElements.getSma() / auxiliaryElements.getB();
  972.             final double Xdot = -naob * (auxiliaryElements.getH() + sinL);
  973.             final double Ydot = naob * (auxiliaryElements.getK() + cosL);
  974.             final Vector3D vel = new Vector3D(Xdot, auxiliaryElements.getVectorF(), Ydot,
  975.                     auxiliaryElements.getVectorG());

  977.             // Compute acceleration
  978.             Vector3D acc = Vector3D.ZERO;

  980.             // shift the orbit to dt
  981.             final Orbit shiftedOrbit = state.getOrbit().shiftedBy(dt);

  983.             // Recompose an orbit with time held fixed to be compliant with DSST theory
  984.             final Orbit recomposedOrbit = new EquinoctialOrbit(shiftedOrbit.getA(), shiftedOrbit.getEquinoctialEx(),
  985.                     shiftedOrbit.getEquinoctialEy(), shiftedOrbit.getHx(), shiftedOrbit.getHy(), shiftedOrbit.getLv(),
  986.                     PositionAngle.TRUE, shiftedOrbit.getFrame(), state.getDate(), context.getMu());

  988.             // Get the corresponding attitude
  989.             final Attitude recomposedAttitude = attitudeProvider.getAttitude(recomposedOrbit, recomposedOrbit.getDate(),
  990.                     recomposedOrbit.getFrame());

  992.             // create shifted SpacecraftState with attitude at specified time
  993.             final SpacecraftState shiftedState = new SpacecraftState(recomposedOrbit, recomposedAttitude,
  994.                     state.getMass());

  996.             acc = contribution.acceleration(shiftedState, parameters);

  998.             // Compute the derivatives of the elements by the speed
  999.             final double[] deriv = new double[6];
  1000.             // da/dv
  1001.             deriv[0] = getAoV(vel).dotProduct(acc);
  1002.             // dex/dv
  1003.             deriv[1] = getKoV(X, Y, Xdot, Ydot).dotProduct(acc);
  1004.             // dey/dv
  1005.             deriv[2] = getHoV(X, Y, Xdot, Ydot).dotProduct(acc);
  1006.             // dhx/dv
  1007.             deriv[3] = getQoV(X).dotProduct(acc);
  1008.             // dhy/dv
  1009.             deriv[4] = getPoV(Y).dotProduct(acc);
  1010.             // dλ/dv
  1011.             deriv[5] = getLoV(X, Y, Xdot, Ydot).dotProduct(acc);

  1013.             // Compute mean elements rates
  1014.             double[] val = null;
  1015.             if (meanMode) {
  1016.                 val = new double[6];
  1017.                 for (int i = 0; i < 6; i++) {
  1018.                     // da<sub>i</sub>/dt
  1019.                     val[i] = roa2 * deriv[i];
  1020.                 }
  1021.             } else {
  1022.                 val = new double[12];
  1023.                 //Compute cos(j*L) and sin(j*L);
  1024.                 final SinCos scjL  = FastMath.sinCos(j * x);
  1025.                 final double cosjL = j == 1 ? cosL : scjL.cos();
  1026.                 final double sinjL = j == 1 ? sinL : scjL.sin();

  1028.                 for (int i = 0; i < 6; i++) {
  1029.                     // da<sub>i</sub>/dv * cos(jL)
  1030.                     val[i] = cosjL * deriv[i];
  1031.                     // da<sub>i</sub>/dv * sin(jL)
  1032.                     val[i + 6] = sinjL * deriv[i];
  1033.                 }
  1034.             }
  1035.             return val;
  1036.         }

  1038.         /**
  1039.          * Converts true longitude to eccentric longitude.
  1040.          * @param lv True longitude
  1041.          * @return Eccentric longitude
  1042.          */
  1043.         private double trueToEccentric (final double lv) {
  1044.             final SinCos scLv    = FastMath.sinCos(lv);
  1045.             final double num     = auxiliaryElements.getH() * scLv.cos() - auxiliaryElements.getK() * scLv.sin();
  1046.             final double den     = auxiliaryElements.getB() + 1. + auxiliaryElements.getK() * scLv.cos() + auxiliaryElements.getH() * scLv.sin();
  1047.             return lv + 2. * FastMath.atan(num / den);
  1048.         }

  1050.         /**
  1051.          * Converts eccentric longitude to mean longitude.
  1052.          * @param le Eccentric longitude
  1053.          * @return Mean longitude
  1054.          */
  1055.         private double eccentricToMean (final double le) {
  1056.             final SinCos scLe = FastMath.sinCos(le);
  1057.             return le - auxiliaryElements.getK() * scLe.sin() + auxiliaryElements.getH() * scLe.cos();
  1058.         }

  1060.         /**
  1061.          * Converts true longitude to mean longitude.
  1062.          * @param lv True longitude
  1063.          * @return Eccentric longitude
  1064.          */
  1065.         private double trueToMean(final double lv) {
  1066.             return eccentricToMean(trueToEccentric(lv));
  1067.         }

  1069.         /**
  1070.          * Compute δa/δv.
  1071.          * @param vel satellite velocity
  1072.          * @return δa/δv
  1073.          */
  1074.         private Vector3D getAoV(final Vector3D vel) {
  1075.             return new Vector3D(context.getTon2a(), vel);
  1076.         }

  1078.         /**
  1079.          * Compute δh/δv.
  1080.          * @param X    satellite position component along f, equinoctial reference frame
  1081.          *             1st vector
  1082.          * @param Y    satellite position component along g, equinoctial reference frame
  1083.          *             2nd vector
  1084.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1085.          *             1st vector
  1086.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1087.          *             2nd vector
  1088.          * @return δh/δv
  1089.          */
  1090.         private Vector3D getHoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1091.             final double kf = (2. * Xdot * Y - X * Ydot) * context.getOoMU();
  1092.             final double kg = X * Xdot * context.getOoMU();
  1093.             final double kw = auxiliaryElements.getK() *
  1094.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOAB();
  1095.             return new Vector3D(kf, auxiliaryElements.getVectorF(), -kg, auxiliaryElements.getVectorG(), kw,
  1096.                     auxiliaryElements.getVectorW());
  1097.         }

  1099.         /**
  1100.          * Compute δk/δv.
  1101.          * @param X    satellite position component along f, equinoctial reference frame
  1102.          *             1st vector
  1103.          * @param Y    satellite position component along g, equinoctial reference frame
  1104.          *             2nd vector
  1105.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1106.          *             1st vector
  1107.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1108.          *             2nd vector
  1109.          * @return δk/δv
  1110.          */
  1111.         private Vector3D getKoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1112.             final double kf = Y * Ydot * context.getOoMU();
  1113.             final double kg = (2. * X * Ydot - Xdot * Y) * context.getOoMU();
  1114.             final double kw = auxiliaryElements.getH() *
  1115.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOAB();
  1116.             return new Vector3D(-kf, auxiliaryElements.getVectorF(), kg, auxiliaryElements.getVectorG(), -kw,
  1117.                     auxiliaryElements.getVectorW());
  1118.         }

  1120.         /**
  1121.          * Compute δp/δv.
  1122.          * @param Y satellite position component along g, equinoctial reference frame
  1123.          *          2nd vector
  1124.          * @return δp/δv
  1125.          */
  1126.         private Vector3D getPoV(final double Y) {
  1127.             return new Vector3D(context.getCo2AB() * Y, auxiliaryElements.getVectorW());
  1128.         }

  1130.         /**
  1131.          * Compute δq/δv.
  1132.          * @param X satellite position component along f, equinoctial reference frame
  1133.          *          1st vector
  1134.          * @return δq/δv
  1135.          */
  1136.         private Vector3D getQoV(final double X) {
  1137.             return new Vector3D(I * context.getCo2AB() * X, auxiliaryElements.getVectorW());
  1138.         }

  1140.         /**
  1141.          * Compute δλ/δv.
  1142.          * @param X    satellite position component along f, equinoctial reference frame
  1143.          *             1st vector
  1144.          * @param Y    satellite position component along g, equinoctial reference frame
  1145.          *             2nd vector
  1146.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1147.          *             1st vector
  1148.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1149.          *             2nd vector
  1150.          * @return δλ/δv
  1151.          */
  1152.         private Vector3D getLoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1153.             final Vector3D pos = new Vector3D(X, auxiliaryElements.getVectorF(), Y, auxiliaryElements.getVectorG());
  1154.             final Vector3D v2 = new Vector3D(auxiliaryElements.getK(), getHoV(X, Y, Xdot, Ydot),
  1155.                     -auxiliaryElements.getH(), getKoV(X, Y, Xdot, Ydot));
  1156.             return new Vector3D(-2. * context.getOOA(), pos, context.getOoBpo(), v2,
  1157.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOA(),
  1158.                     auxiliaryElements.getVectorW());
  1159.         }

  1161.     }

  1163.     /**
  1164.      * Class used to {@link #integrate(UnivariateVectorFunction, double, double)
  1165.      * integrate} a {@link org.hipparchus.analysis.UnivariateVectorFunction
  1166.      * function} of the orbital elements using the Gaussian quadrature rule to get
  1167.      * the acceleration.
  1168.      */
  1169.     protected static class GaussQuadrature {

  1171.         // Points and weights for the available quadrature orders

  1173.         /** Points for quadrature of order 12. */
  1174.         private static final double[] P_12 = { -0.98156063424671910000, -0.90411725637047490000,
  1175.             -0.76990267419430470000, -0.58731795428661740000, -0.36783149899818024000, -0.12523340851146890000,
  1176.             0.12523340851146890000, 0.36783149899818024000, 0.58731795428661740000, 0.76990267419430470000,
  1177.             0.90411725637047490000, 0.98156063424671910000 };

  1179.         /** Weights for quadrature of order 12. */
  1180.         private static final double[] W_12 = { 0.04717533638651220000, 0.10693932599531830000, 0.16007832854334633000,
  1181.             0.20316742672306584000, 0.23349253653835478000, 0.24914704581340286000, 0.24914704581340286000,
  1182.             0.23349253653835478000, 0.20316742672306584000, 0.16007832854334633000, 0.10693932599531830000,
  1183.             0.04717533638651220000 };

  1185.         /** Points for quadrature of order 16. */
  1186.         private static final double[] P_16 = { -0.98940093499164990000, -0.94457502307323260000,
  1187.             -0.86563120238783160000, -0.75540440835500310000, -0.61787624440264380000, -0.45801677765722737000,
  1188.             -0.28160355077925890000, -0.09501250983763745000, 0.09501250983763745000, 0.28160355077925890000,
  1189.             0.45801677765722737000, 0.61787624440264380000, 0.75540440835500310000, 0.86563120238783160000,
  1190.             0.94457502307323260000, 0.98940093499164990000 };

  1192.         /** Weights for quadrature of order 16. */
  1193.         private static final double[] W_16 = { 0.02715245941175405800, 0.06225352393864777000, 0.09515851168249283000,
  1194.             0.12462897125553388000, 0.14959598881657685000, 0.16915651939500256000, 0.18260341504492360000,
  1195.             0.18945061045506847000, 0.18945061045506847000, 0.18260341504492360000, 0.16915651939500256000,
  1196.             0.14959598881657685000, 0.12462897125553388000, 0.09515851168249283000, 0.06225352393864777000,
  1197.             0.02715245941175405800 };

  1199.         /** Points for quadrature of order 20. */
  1200.         private static final double[] P_20 = { -0.99312859918509490000, -0.96397192727791390000,
  1201.             -0.91223442825132600000, -0.83911697182221890000, -0.74633190646015080000, -0.63605368072651510000,
  1202.             -0.51086700195082700000, -0.37370608871541955000, -0.22778585114164507000, -0.07652652113349734000,
  1203.             0.07652652113349734000, 0.22778585114164507000, 0.37370608871541955000, 0.51086700195082700000,
  1204.             0.63605368072651510000, 0.74633190646015080000, 0.83911697182221890000, 0.91223442825132600000,
  1205.             0.96397192727791390000, 0.99312859918509490000 };

  1207.         /** Weights for quadrature of order 20. */
  1208.         private static final double[] W_20 = { 0.01761400713915226400, 0.04060142980038684000, 0.06267204833410904000,
  1209.             0.08327674157670477000, 0.10193011981724048000, 0.11819453196151844000, 0.13168863844917678000,
  1210.             0.14209610931838212000, 0.14917298647260380000, 0.15275338713072600000, 0.15275338713072600000,
  1211.             0.14917298647260380000, 0.14209610931838212000, 0.13168863844917678000, 0.11819453196151844000,
  1212.             0.10193011981724048000, 0.08327674157670477000, 0.06267204833410904000, 0.04060142980038684000,
  1213.             0.01761400713915226400 };

  1215.         /** Points for quadrature of order 24. */
  1216.         private static final double[] P_24 = { -0.99518721999702130000, -0.97472855597130950000,
  1217.             -0.93827455200273270000, -0.88641552700440100000, -0.82000198597390300000, -0.74012419157855440000,
  1218.             -0.64809365193697550000, -0.54542147138883950000, -0.43379350762604520000, -0.31504267969616340000,
  1219.             -0.19111886747361634000, -0.06405689286260563000, 0.06405689286260563000, 0.19111886747361634000,
  1220.             0.31504267969616340000, 0.43379350762604520000, 0.54542147138883950000, 0.64809365193697550000,
  1221.             0.74012419157855440000, 0.82000198597390300000, 0.88641552700440100000, 0.93827455200273270000,
  1222.             0.97472855597130950000, 0.99518721999702130000 };

  1224.         /** Weights for quadrature of order 24. */
  1225.         private static final double[] W_24 = { 0.01234122979998733500, 0.02853138862893380600, 0.04427743881741981000,
  1226.             0.05929858491543691500, 0.07334648141108027000, 0.08619016153195320000, 0.09761865210411391000,
  1227.             0.10744427011596558000, 0.11550566805372553000, 0.12167047292780335000, 0.12583745634682825000,
  1228.             0.12793819534675221000, 0.12793819534675221000, 0.12583745634682825000, 0.12167047292780335000,
  1229.             0.11550566805372553000, 0.10744427011596558000, 0.09761865210411391000, 0.08619016153195320000,
  1230.             0.07334648141108027000, 0.05929858491543691500, 0.04427743881741981000, 0.02853138862893380600,
  1231.             0.01234122979998733500 };

  1233.         /** Points for quadrature of order 32. */
  1234.         private static final double[] P_32 = { -0.99726386184948160000, -0.98561151154526840000,
  1235.             -0.96476225558750640000, -0.93490607593773970000, -0.89632115576605220000, -0.84936761373256990000,
  1236.             -0.79448379596794250000, -0.73218211874028970000, -0.66304426693021520000, -0.58771575724076230000,
  1237.             -0.50689990893222950000, -0.42135127613063540000, -0.33186860228212767000, -0.23928736225213710000,
  1238.             -0.14447196158279646000, -0.04830766568773831000, 0.04830766568773831000, 0.14447196158279646000,
  1239.             0.23928736225213710000, 0.33186860228212767000, 0.42135127613063540000, 0.50689990893222950000,
  1240.             0.58771575724076230000, 0.66304426693021520000, 0.73218211874028970000, 0.79448379596794250000,
  1241.             0.84936761373256990000, 0.89632115576605220000, 0.93490607593773970000, 0.96476225558750640000,
  1242.             0.98561151154526840000, 0.99726386184948160000 };

  1244.         /** Weights for quadrature of order 32. */
  1245.         private static final double[] W_32 = { 0.00701861000947013600, 0.01627439473090571200, 0.02539206530926214200,
  1246.             0.03427386291302141000, 0.04283589802222658600, 0.05099805926237621600, 0.05868409347853559000,
  1247.             0.06582222277636193000, 0.07234579410884862000, 0.07819389578707042000, 0.08331192422694673000,
  1248.             0.08765209300440380000, 0.09117387869576390000, 0.09384439908080441000, 0.09563872007927487000,
  1249.             0.09654008851472784000, 0.09654008851472784000, 0.09563872007927487000, 0.09384439908080441000,
  1250.             0.09117387869576390000, 0.08765209300440380000, 0.08331192422694673000, 0.07819389578707042000,
  1251.             0.07234579410884862000, 0.06582222277636193000, 0.05868409347853559000, 0.05099805926237621600,
  1252.             0.04283589802222658600, 0.03427386291302141000, 0.02539206530926214200, 0.01627439473090571200,
  1253.             0.00701861000947013600 };

  1255.         /** Points for quadrature of order 40. */
  1256.         private static final double[] P_40 = { -0.99823770971055930000, -0.99072623869945710000,
  1257.             -0.97725994998377420000, -0.95791681921379170000, -0.93281280827867660000, -0.90209880696887420000,
  1258.             -0.86595950321225960000, -0.82461223083331170000, -0.77830565142651940000, -0.72731825518992710000,
  1259.             -0.67195668461417960000, -0.61255388966798030000, -0.54946712509512820000, -0.48307580168617870000,
  1260.             -0.41377920437160500000, -0.34199409082575850000, -0.26815218500725370000, -0.19269758070137110000,
  1261.             -0.11608407067525522000, -0.03877241750605081600, 0.03877241750605081600, 0.11608407067525522000,
  1262.             0.19269758070137110000, 0.26815218500725370000, 0.34199409082575850000, 0.41377920437160500000,
  1263.             0.48307580168617870000, 0.54946712509512820000, 0.61255388966798030000, 0.67195668461417960000,
  1264.             0.72731825518992710000, 0.77830565142651940000, 0.82461223083331170000, 0.86595950321225960000,
  1265.             0.90209880696887420000, 0.93281280827867660000, 0.95791681921379170000, 0.97725994998377420000,
  1266.             0.99072623869945710000, 0.99823770971055930000 };

  1268.         /** Weights for quadrature of order 40. */
  1269.         private static final double[] W_40 = { 0.00452127709853309800, 0.01049828453115270400, 0.01642105838190797300,
  1270.             0.02224584919416689000, 0.02793700698002338000, 0.03346019528254786500, 0.03878216797447199000,
  1271.             0.04387090818567333000, 0.04869580763507221000, 0.05322784698393679000, 0.05743976909939157000,
  1272.             0.06130624249292891000, 0.06480401345660108000, 0.06791204581523394000, 0.07061164739128681000,
  1273.             0.07288658239580408000, 0.07472316905796833000, 0.07611036190062619000, 0.07703981816424793000,
  1274.             0.07750594797842482000, 0.07750594797842482000, 0.07703981816424793000, 0.07611036190062619000,
  1275.             0.07472316905796833000, 0.07288658239580408000, 0.07061164739128681000, 0.06791204581523394000,
  1276.             0.06480401345660108000, 0.06130624249292891000, 0.05743976909939157000, 0.05322784698393679000,
  1277.             0.04869580763507221000, 0.04387090818567333000, 0.03878216797447199000, 0.03346019528254786500,
  1278.             0.02793700698002338000, 0.02224584919416689000, 0.01642105838190797300, 0.01049828453115270400,
  1279.             0.00452127709853309800 };

  1281.         /** Points for quadrature of order 48. */
  1282.         private static final double[] P_48 = { -0.99877100725242610000, -0.99353017226635080000,
  1283.             -0.98412458372282700000, -0.97059159254624720000, -0.95298770316043080000, -0.93138669070655440000,
  1284.             -0.90587913671556960000, -0.87657202027424800000, -0.84358826162439350000, -0.80706620402944250000,
  1285.             -0.76715903251574020000, -0.72403413092381470000, -0.67787237963266400000, -0.62886739677651370000,
  1286.             -0.57722472608397270000, -0.52316097472223300000, -0.46690290475095840000, -0.40868648199071680000,
  1287.             -0.34875588629216070000, -0.28736248735545555000, -0.22476379039468908000, -0.16122235606889174000,
  1288.             -0.09700469920946270000, -0.03238017096286937000, 0.03238017096286937000, 0.09700469920946270000,
  1289.             0.16122235606889174000, 0.22476379039468908000, 0.28736248735545555000, 0.34875588629216070000,
  1290.             0.40868648199071680000, 0.46690290475095840000, 0.52316097472223300000, 0.57722472608397270000,
  1291.             0.62886739677651370000, 0.67787237963266400000, 0.72403413092381470000, 0.76715903251574020000,
  1292.             0.80706620402944250000, 0.84358826162439350000, 0.87657202027424800000, 0.90587913671556960000,
  1293.             0.93138669070655440000, 0.95298770316043080000, 0.97059159254624720000, 0.98412458372282700000,
  1294.             0.99353017226635080000, 0.99877100725242610000 };

  1296.         /** Weights for quadrature of order 48. */
  1297.         private static final double[] W_48 = { 0.00315334605230596250, 0.00732755390127620800, 0.01147723457923446900,
  1298.             0.01557931572294386600, 0.01961616045735556700, 0.02357076083932435600, 0.02742650970835688000,
  1299.             0.03116722783279807000, 0.03477722256477045000, 0.03824135106583080600, 0.04154508294346483000,
  1300.             0.04467456085669424000, 0.04761665849249054000, 0.05035903555385448000, 0.05289018948519365000,
  1301.             0.05519950369998416500, 0.05727729210040315000, 0.05911483969839566000, 0.06070443916589384000,
  1302.             0.06203942315989268000, 0.06311419228625403000, 0.06392423858464817000, 0.06446616443595010000,
  1303.             0.06473769681268386000, 0.06473769681268386000, 0.06446616443595010000, 0.06392423858464817000,
  1304.             0.06311419228625403000, 0.06203942315989268000, 0.06070443916589384000, 0.05911483969839566000,
  1305.             0.05727729210040315000, 0.05519950369998416500, 0.05289018948519365000, 0.05035903555385448000,
  1306.             0.04761665849249054000, 0.04467456085669424000, 0.04154508294346483000, 0.03824135106583080600,
  1307.             0.03477722256477045000, 0.03116722783279807000, 0.02742650970835688000, 0.02357076083932435600,
  1308.             0.01961616045735556700, 0.01557931572294386600, 0.01147723457923446900, 0.00732755390127620800,
  1309.             0.00315334605230596250 };

  1311.         /** Node points. */
  1312.         private final double[] nodePoints;

  1314.         /** Node weights. */
  1315.         private final double[] nodeWeights;

  1317.         /** Number of points. */
  1318.         private final int numberOfPoints;

  1320.         /**
  1321.          * Creates a Gauss integrator of the given order.
  1322.          *
  1323.          * @param numberOfPoints Order of the integration rule.
  1324.          */
  1325.         GaussQuadrature(final int numberOfPoints) {

  1327.             this.numberOfPoints = numberOfPoints;

  1329.             switch (numberOfPoints) {
  1330.                 case 12:
  1331.                     this.nodePoints = P_12.clone();
  1332.                     this.nodeWeights = W_12.clone();
  1333.                     break;
  1334.                 case 16:
  1335.                     this.nodePoints = P_16.clone();
  1336.                     this.nodeWeights = W_16.clone();
  1337.                     break;
  1338.                 case 20:
  1339.                     this.nodePoints = P_20.clone();
  1340.                     this.nodeWeights = W_20.clone();
  1341.                     break;
  1342.                 case 24:
  1343.                     this.nodePoints = P_24.clone();
  1344.                     this.nodeWeights = W_24.clone();
  1345.                     break;
  1346.                 case 32:
  1347.                     this.nodePoints = P_32.clone();
  1348.                     this.nodeWeights = W_32.clone();
  1349.                     break;
  1350.                 case 40:
  1351.                     this.nodePoints = P_40.clone();
  1352.                     this.nodeWeights = W_40.clone();
  1353.                     break;
  1354.                 case 48:
  1355.                 default:
  1356.                     this.nodePoints = P_48.clone();
  1357.                     this.nodeWeights = W_48.clone();
  1358.                     break;
  1359.             }

  1361.         }

  1363.         /**
  1364.          * Integrates a given function on the given interval.
  1365.          *
  1366.          * @param f          Function to integrate.
  1367.          * @param lowerBound Lower bound of the integration interval.
  1368.          * @param upperBound Upper bound of the integration interval.
  1369.          * @return the integral of the weighted function.
  1370.          */
  1371.         public double[] integrate(final UnivariateVectorFunction f, final double lowerBound, final double upperBound) {

  1373.             final double[] adaptedPoints = nodePoints.clone();
  1374.             final double[] adaptedWeights = nodeWeights.clone();
  1375.             transform(adaptedPoints, adaptedWeights, lowerBound, upperBound);
  1376.             return basicIntegrate(f, adaptedPoints, adaptedWeights);
  1377.         }

  1379.         /**
  1380.          * Integrates a given function on the given interval.
  1381.          *
  1382.          * @param <T>        the type of the field elements
  1383.          * @param f          Function to integrate.
  1384.          * @param lowerBound Lower bound of the integration interval.
  1385.          * @param upperBound Upper bound of the integration interval.
  1386.          * @param field      field utilized by default
  1387.          * @return the integral of the weighted function.
  1388.          */
  1389.         public <T extends CalculusFieldElement<T>> T[] integrate(final CalculusFieldUnivariateVectorFunction<T> f,
  1390.                 final T lowerBound, final T upperBound, final Field<T> field) {

  1392.             final T zero = field.getZero();

  1394.             final T[] adaptedPoints = MathArrays.buildArray(field, numberOfPoints);
  1395.             final T[] adaptedWeights = MathArrays.buildArray(field, numberOfPoints);

  1397.             for (int i = 0; i < numberOfPoints; i++) {
  1398.                 adaptedPoints[i] = zero.add(nodePoints[i]);
  1399.                 adaptedWeights[i] = zero.add(nodeWeights[i]);
  1400.             }

  1402.             transform(adaptedPoints, adaptedWeights, lowerBound, upperBound);
  1403.             return basicIntegrate(f, adaptedPoints, adaptedWeights, field);
  1404.         }

  1406.         /**
  1407.          * Performs a change of variable so that the integration can be performed on an
  1408.          * arbitrary interval {@code [a, b]}.
  1409.          * <p>
  1410.          * It is assumed that the natural interval is {@code [-1, 1]}.
  1411.          * </p>
  1412.          *
  1413.          * @param points  Points to adapt to the new interval.
  1414.          * @param weights Weights to adapt to the new interval.
  1415.          * @param a       Lower bound of the integration interval.
  1416.          * @param b       Lower bound of the integration interval.
  1417.          */
  1418.         private void transform(final double[] points, final double[] weights, final double a, final double b) {
  1419.             // Scaling
  1420.             final double scale = (b - a) / 2;
  1421.             final double shift = a + scale;
  1422.             for (int i = 0; i < points.length; i++) {
  1423.                 points[i] = points[i] * scale + shift;
  1424.                 weights[i] *= scale;
  1425.             }
  1426.         }

  1428.         /**
  1429.          * Performs a change of variable so that the integration can be performed on an
  1430.          * arbitrary interval {@code [a, b]}.
  1431.          * <p>
  1432.          * It is assumed that the natural interval is {@code [-1, 1]}.
  1433.          * </p>
  1434.          * @param <T>     the type of the field elements
  1435.          * @param points  Points to adapt to the new interval.
  1436.          * @param weights Weights to adapt to the new interval.
  1437.          * @param a       Lower bound of the integration interval.
  1438.          * @param b       Lower bound of the integration interval
  1439.          */
  1440.         private <T extends CalculusFieldElement<T>> void transform(final T[] points, final T[] weights, final T a,
  1441.                 final T b) {
  1442.             // Scaling
  1443.             final T scale = (b.subtract(a)).divide(2.);
  1444.             final T shift = a.add(scale);
  1445.             for (int i = 0; i < points.length; i++) {
  1446.                 points[i] = scale.multiply(points[i]).add(shift);
  1447.                 weights[i] = scale.multiply(weights[i]);
  1448.             }
  1449.         }

  1451.         /**
  1452.          * Returns an estimate of the integral of {@code f(x) * w(x)}, where {@code w}
  1453.          * is a weight function that depends on the actual flavor of the Gauss
  1454.          * integration scheme.
  1455.          *
  1456.          * @param f       Function to integrate.
  1457.          * @param points  Nodes.
  1458.          * @param weights Nodes weights.
  1459.          * @return the integral of the weighted function.
  1460.          */
  1461.         private double[] basicIntegrate(final UnivariateVectorFunction f, final double[] points,
  1462.                 final double[] weights) {
  1463.             double x = points[0];
  1464.             double w = weights[0];
  1465.             double[] v = f.value(x);
  1466.             final double[] y = new double[v.length];
  1467.             for (int j = 0; j < v.length; j++) {
  1468.                 y[j] = w * v[j];
  1469.             }
  1470.             final double[] t = y.clone();
  1471.             final double[] c = new double[v.length];
  1472.             final double[] s = t.clone();
  1473.             for (int i = 1; i < points.length; i++) {
  1474.                 x = points[i];
  1475.                 w = weights[i];
  1476.                 v = f.value(x);
  1477.                 for (int j = 0; j < v.length; j++) {
  1478.                     y[j] = w * v[j] - c[j];
  1479.                     t[j] = s[j] + y[j];
  1480.                     c[j] = (t[j] - s[j]) - y[j];
  1481.                     s[j] = t[j];
  1482.                 }
  1483.             }
  1484.             return s;
  1485.         }

  1487.         /**
  1488.          * Returns an estimate of the integral of {@code f(x) * w(x)}, where {@code w}
  1489.          * is a weight function that depends on the actual flavor of the Gauss
  1490.          * integration scheme.
  1491.          *
  1492.          * @param <T>     the type of the field elements.
  1493.          * @param f       Function to integrate.
  1494.          * @param points  Nodes.
  1495.          * @param weights Nodes weight
  1496.          * @param field   field utilized by default
  1497.          * @return the integral of the weighted function.
  1498.          */
  1499.         private <T extends CalculusFieldElement<T>> T[] basicIntegrate(final CalculusFieldUnivariateVectorFunction<T> f,
  1500.                 final T[] points, final T[] weights, final Field<T> field) {

  1502.             T x = points[0];
  1503.             T w = weights[0];
  1504.             T[] v = f.value(x);

  1506.             final T[] y = MathArrays.buildArray(field, v.length);
  1507.             for (int j = 0; j < v.length; j++) {
  1508.                 y[j] = v[j].multiply(w);
  1509.             }
  1510.             final T[] t = y.clone();
  1511.             final T[] c = MathArrays.buildArray(field, v.length);
  1512.             ;
  1513.             final T[] s = t.clone();
  1514.             for (int i = 1; i < points.length; i++) {
  1515.                 x = points[i];
  1516.                 w = weights[i];
  1517.                 v = f.value(x);
  1518.                 for (int j = 0; j < v.length; j++) {
  1519.                     y[j] = v[j].multiply(w).subtract(c[j]);
  1520.                     t[j] = y[j].add(s[j]);
  1521.                     c[j] = (t[j].subtract(s[j])).subtract(y[j]);
  1522.                     s[j] = t[j];
  1523.                 }
  1524.             }
  1525.             return s;
  1526.         }

  1528.     }

  1530.     /**
  1531.      * Compute the C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup>
  1532.      * coefficients.
  1533.      * <p>
  1534.      * Those coefficients are given in Danielson paper by expression 4.4-(6)
  1535.      * </p>
  1536.      * @author Petre Bazavan
  1537.      * @author Lucian Barbulescu
  1538.      */
  1539.     protected class FourierCjSjCoefficients {

  1541.         /** Maximum possible value for j. */
  1542.         private final int jMax;

  1544.         /**
  1545.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1546.          * <p>
  1547.          * the index i corresponds to the following elements: <br/>
  1548.          * - 0 for a <br>
  1549.          * - 1 for k <br>
  1550.          * - 2 for h <br>
  1551.          * - 3 for q <br>
  1552.          * - 4 for p <br>
  1553.          * - 5 for λ <br>
  1554.          * </p>
  1555.          */
  1556.         private final double[][] cCoef;

  1558.         /**
  1559.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1560.          * <p>
  1561.          * the index i corresponds to the following elements: <br/>
  1562.          * - 0 for a <br>
  1563.          * - 1 for k <br>
  1564.          * - 2 for h <br>
  1565.          * - 3 for q <br>
  1566.          * - 4 for p <br>
  1567.          * - 5 for λ <br>
  1568.          * </p>
  1569.          */
  1570.         private final double[][] sCoef;

  1572.         /**
  1573.          * Standard constructor.
  1574.          * @param state             the current state
  1575.          * @param jMax              maximum value for j
  1576.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1577.          * @param parameters        values of the force model parameters
  1578.          */
  1579.         FourierCjSjCoefficients(final SpacecraftState state, final int jMax, final AuxiliaryElements auxiliaryElements,
  1580.                 final double[] parameters) {

  1582.             // Initialise the fields
  1583.             this.jMax = jMax;

  1585.             // Allocate the arrays
  1586.             final int rows = jMax + 1;
  1587.             cCoef = new double[rows][6];
  1588.             sCoef = new double[rows][6];

  1590.             // Compute the coefficients
  1591.             computeCoefficients(state, auxiliaryElements, parameters);
  1592.         }

  1594.         /**
  1595.          * Compute the Fourrier coefficients.
  1596.          * <p>
  1597.          * Only the C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup> coefficients
  1598.          * need to be computed as D<sub>i</sub><sup>m</sup> is always 0.
  1599.          * </p>
  1600.          * @param state             the current state
  1601.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1602.          * @param parameters        values of the force model parameters
  1603.          */
  1604.         private void computeCoefficients(final SpacecraftState state, final AuxiliaryElements auxiliaryElements,
  1605.                 final double[] parameters) {

  1607.             // Computes the limits for the integral
  1608.             final double[] ll = getLLimits(state, auxiliaryElements);
  1609.             // Computes integrated mean element rates if Llow < Lhigh
  1610.             if (ll[0] < ll[1]) {
  1611.                 // Compute 1 / PI
  1612.                 final double ooPI = 1 / FastMath.PI;

  1614.                 // loop through all values of j
  1615.                 for (int j = 0; j <= jMax; j++) {
  1616.                     final double[] curentCoefficients = integrator
  1617.                             .integrate(new IntegrableFunction(state, false, j, parameters), ll[0], ll[1]);

  1619.                     // divide by PI and set the values for the coefficients
  1620.                     for (int i = 0; i < 6; i++) {
  1621.                         cCoef[j][i] = ooPI * curentCoefficients[i];
  1622.                         sCoef[j][i] = ooPI * curentCoefficients[i + 6];
  1623.                     }
  1624.                 }
  1625.             }
  1626.         }

  1628.         /**
  1629.          * Get the coefficient C<sub>i</sub><sup>j</sup>.
  1630.          * @param i i index - corresponds to the required variation
  1631.          * @param j j index
  1632.          * @return the coefficient C<sub>i</sub><sup>j</sup>
  1633.          */
  1634.         public double getCij(final int i, final int j) {
  1635.             return cCoef[j][i];
  1636.         }

  1638.         /**
  1639.          * Get the coefficient S<sub>i</sub><sup>j</sup>.
  1640.          * @param i i index - corresponds to the required variation
  1641.          * @param j j index
  1642.          * @return the coefficient S<sub>i</sub><sup>j</sup>
  1643.          */
  1644.         public double getSij(final int i, final int j) {
  1645.             return sCoef[j][i];
  1646.         }
  1647.     }

  1649.     /**
  1650.      * Compute the C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup>
  1651.      * coefficients with field elements.
  1652.      * <p>
  1653.      * Those coefficients are given in Danielson paper by expression 4.4-(6)
  1654.      * </p>
  1655.      * @author Petre Bazavan
  1656.      * @author Lucian Barbulescu
  1657.      */
  1658.     protected class FieldFourierCjSjCoefficients<T extends CalculusFieldElement<T>> {

  1660.         /** Maximum possible value for j. */
  1661.         private final int jMax;

  1663.         /**
  1664.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1665.          * <p>
  1666.          * the index i corresponds to the following elements: <br/>
  1667.          * - 0 for a <br>
  1668.          * - 1 for k <br>
  1669.          * - 2 for h <br>
  1670.          * - 3 for q <br>
  1671.          * - 4 for p <br>
  1672.          * - 5 for λ <br>
  1673.          * </p>
  1674.          */
  1675.         private final T[][] cCoef;

  1677.         /**
  1678.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1679.          * <p>
  1680.          * the index i corresponds to the following elements: <br/>
  1681.          * - 0 for a <br>
  1682.          * - 1 for k <br>
  1683.          * - 2 for h <br>
  1684.          * - 3 for q <br>
  1685.          * - 4 for p <br>
  1686.          * - 5 for λ <br>
  1687.          * </p>
  1688.          */
  1689.         private final T[][] sCoef;

  1691.         /**
  1692.          * Standard constructor.
  1693.          * @param state             the current state
  1694.          * @param jMax              maximum value for j
  1695.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1696.          * @param parameters        values of the force model parameters
  1697.          * @param field             field used by default
  1698.          */
  1699.         FieldFourierCjSjCoefficients(final FieldSpacecraftState<T> state, final int jMax,
  1700.                 final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters, final Field<T> field) {
  1701.             // Initialise the fields
  1702.             this.jMax = jMax;

  1704.             // Allocate the arrays
  1705.             final int rows = jMax + 1;
  1706.             cCoef = MathArrays.buildArray(field, rows, 6);
  1707.             sCoef = MathArrays.buildArray(field, rows, 6);

  1709.             // Compute the coefficients
  1710.             computeCoefficients(state, auxiliaryElements, parameters, field);
  1711.         }

  1713.         /**
  1714.          * Compute the Fourrier coefficients.
  1715.          * <p>
  1716.          * Only the C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup> coefficients
  1717.          * need to be computed as D<sub>i</sub><sup>m</sup> is always 0.
  1718.          * </p>
  1719.          * @param state             the current state
  1720.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1721.          * @param parameters        values of the force model parameters
  1722.          * @param field             field used by default
  1723.          */
  1724.         private void computeCoefficients(final FieldSpacecraftState<T> state,
  1725.                 final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters, final Field<T> field) {
  1726.             // Zero
  1727.             final T zero = field.getZero();
  1728.             // Computes the limits for the integral
  1729.             final T[] ll = getLLimits(state, auxiliaryElements);
  1730.             // Computes integrated mean element rates if Llow < Lhigh
  1731.             if (ll[0].getReal() < ll[1].getReal()) {
  1732.                 // Compute 1 / PI
  1733.                 final T ooPI = zero.getPi().reciprocal();

  1735.                 // loop through all values of j
  1736.                 for (int j = 0; j <= jMax; j++) {
  1737.                     final T[] curentCoefficients = integrator.integrate(
  1738.                             new FieldIntegrableFunction<>(state, false, j, parameters, field), ll[0], ll[1], field);

  1740.                     // divide by PI and set the values for the coefficients
  1741.                     for (int i = 0; i < 6; i++) {
  1742.                         cCoef[j][i] = curentCoefficients[i].multiply(ooPI);
  1743.                         sCoef[j][i] = curentCoefficients[i + 6].multiply(ooPI);
  1744.                     }
  1745.                 }
  1746.             }
  1747.         }

  1749.         /**
  1750.          * Get the coefficient C<sub>i</sub><sup>j</sup>.
  1751.          * @param i i index - corresponds to the required variation
  1752.          * @param j j index
  1753.          * @return the coefficient C<sub>i</sub><sup>j</sup>
  1754.          */
  1755.         public T getCij(final int i, final int j) {
  1756.             return cCoef[j][i];
  1757.         }

  1759.         /**
  1760.          * Get the coefficient S<sub>i</sub><sup>j</sup>.
  1761.          * @param i i index - corresponds to the required variation
  1762.          * @param j j index
  1763.          * @return the coefficient S<sub>i</sub><sup>j</sup>
  1764.          */
  1765.         public T getSij(final int i, final int j) {
  1766.             return sCoef[j][i];
  1767.         }
  1768.     }

  1770.     /**
  1771.      * This class handles the short periodic coefficients described in Danielson
  1772.      * 2.5.3-26.
  1773.      *
  1774.      * <p>
  1775.      * The value of M is 0. Also, since the values of the Fourier coefficient
  1776.      * D<sub>i</sub><sup>m</sup> is 0 then the values of the coefficients
  1777.      * D<sub>i</sub><sup>m</sup> for m &gt; 2 are also 0.
  1778.      * </p>
  1779.      * @author Petre Bazavan
  1780.      * @author Lucian Barbulescu
  1781.      *
  1782.      */
  1783.     protected static class GaussianShortPeriodicCoefficients implements ShortPeriodTerms {

  1785.         /** Maximum value for j index. */
  1786.         private final int jMax;

  1788.         /** Number of points used in the interpolation process. */
  1789.         private final int interpolationPoints;

  1791.         /** Prefix for coefficients keys. */
  1792.         private final String coefficientsKeyPrefix;

  1794.         /** All coefficients slots. */
  1795.         private final transient TimeSpanMap<Slot> slots;

  1797.         /**
  1798.          * Constructor.
  1799.          * @param coefficientsKeyPrefix prefix for coefficients keys
  1800.          * @param jMax                  maximum value for j index
  1801.          * @param interpolationPoints   number of points used in the interpolation
  1802.          *                              process
  1803.          * @param slots                 all coefficients slots
  1804.          */
  1805.         GaussianShortPeriodicCoefficients(final String coefficientsKeyPrefix, final int jMax,
  1806.                 final int interpolationPoints, final TimeSpanMap<Slot> slots) {
  1807.             // Initialize fields
  1808.             this.jMax = jMax;
  1809.             this.interpolationPoints = interpolationPoints;
  1810.             this.coefficientsKeyPrefix = coefficientsKeyPrefix;
  1811.             this.slots = slots;
  1812.         }

  1814.         /**
  1815.          * Get the slot valid for some date.
  1816.          * @param meanStates mean states defining the slot
  1817.          * @return slot valid at the specified date
  1818.          */
  1819.         public Slot createSlot(final SpacecraftState... meanStates) {
  1820.             final Slot slot = new Slot(jMax, interpolationPoints);
  1821.             final AbsoluteDate first = meanStates[0].getDate();
  1822.             final AbsoluteDate last = meanStates[meanStates.length - 1].getDate();
  1823.             final int compare = first.compareTo(last);
  1824.             if (compare < 0) {
  1825.                 slots.addValidAfter(slot, first);
  1826.             } else if (compare > 0) {
  1827.                 slots.addValidBefore(slot, first);
  1828.             } else {
  1829.                 // single date, valid for all time
  1830.                 slots.addValidAfter(slot, AbsoluteDate.PAST_INFINITY);
  1831.             }
  1832.             return slot;
  1833.         }

  1835.         /**
  1836.          * Compute the short periodic coefficients.
  1837.          *
  1838.          * @param state       current state information: date, kinematics, attitude
  1839.          * @param slot        coefficients slot
  1840.          * @param fourierCjSj Fourier coefficients
  1841.          * @param uijvij      U and V coefficients
  1842.          * @param n           Keplerian mean motion
  1843.          * @param a           semi major axis
  1844.          */
  1845.         private void computeCoefficients(final SpacecraftState state, final Slot slot,
  1846.                 final FourierCjSjCoefficients fourierCjSj, final UijVijCoefficients uijvij, final double n,
  1847.                 final double a) {

  1849.             // get the current date
  1850.             final AbsoluteDate date = state.getDate();

  1852.             // compute the k₂⁰ coefficient
  1853.             final double k20 = computeK20(jMax, uijvij.currentRhoSigmaj);

  1855.             // 1. / n
  1856.             final double oon = 1. / n;
  1857.             // 3. / (2 * a * n)
  1858.             final double to2an = 1.5 * oon / a;
  1859.             // 3. / (4 * a * n)
  1860.             final double to4an = to2an / 2;

  1862.             // Compute the coefficients for each element
  1863.             final int size = jMax + 1;
  1864.             final double[] di1 = new double[6];
  1865.             final double[] di2 = new double[6];
  1866.             final double[][] currentCij = new double[size][6];
  1867.             final double[][] currentSij = new double[size][6];
  1868.             for (int i = 0; i < 6; i++) {

  1870.                 // compute D<sub>i</sub>¹ and D<sub>i</sub>² (all others are 0)
  1871.                 di1[i] = -oon * fourierCjSj.getCij(i, 0);
  1872.                 if (i == 5) {
  1873.                     di1[i] += to2an * uijvij.getU1(0, 0);
  1874.                 }
  1875.                 di2[i] = 0.;
  1876.                 if (i == 5) {
  1877.                     di2[i] += -to4an * fourierCjSj.getCij(0, 0);
  1878.                 }

  1880.                 // the C<sub>i</sub>⁰ is computed based on all others
  1881.                 currentCij[0][i] = -di2[i] * k20;

  1883.                 for (int j = 1; j <= jMax; j++) {
  1884.                     // compute the current C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup>
  1885.                     currentCij[j][i] = oon * uijvij.getU1(j, i);
  1886.                     if (i == 5) {
  1887.                         currentCij[j][i] += -to2an * uijvij.getU2(j);
  1888.                     }
  1889.                     currentSij[j][i] = oon * uijvij.getV1(j, i);
  1890.                     if (i == 5) {
  1891.                         currentSij[j][i] += -to2an * uijvij.getV2(j);
  1892.                     }

  1894.                     // add the computed coefficients to C<sub>i</sub>⁰
  1895.                     currentCij[0][i] += -(currentCij[j][i] * uijvij.currentRhoSigmaj[0][j] +
  1896.                             currentSij[j][i] * uijvij.currentRhoSigmaj[1][j]);
  1897.                 }

  1899.             }

  1901.             // add the values to the interpolators
  1902.             slot.cij[0].addGridPoint(date, currentCij[0]);
  1903.             slot.dij[1].addGridPoint(date, di1);
  1904.             slot.dij[2].addGridPoint(date, di2);
  1905.             for (int j = 1; j <= jMax; j++) {
  1906.                 slot.cij[j].addGridPoint(date, currentCij[j]);
  1907.                 slot.sij[j].addGridPoint(date, currentSij[j]);
  1908.             }

  1910.         }

  1912.         /**
  1913.          * Compute the coefficient k₂⁰ by using the equation 2.5.3-(9a) from Danielson.
  1914.          * <p>
  1915.          * After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:<br>
  1916.          * k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  1917.          * ρ<sub>k</sub>²)]
  1918.          * </p>
  1919.          * @param kMax             max value fot k index
  1920.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  1921.          *                         σ<sub>j</sub> coefficients
  1922.          * @return the coefficient k₂⁰
  1923.          */
  1924.         private double computeK20(final int kMax, final double[][] currentRhoSigmaj) {
  1925.             double k20 = 0.;

  1927.             for (int kIndex = 1; kIndex <= kMax; kIndex++) {
  1928.                 // After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:
  1929.                 // k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  1930.                 // ρ<sub>k</sub>²)]
  1931.                 double currentTerm = currentRhoSigmaj[1][kIndex] * currentRhoSigmaj[1][kIndex] +
  1932.                         currentRhoSigmaj[0][kIndex] * currentRhoSigmaj[0][kIndex];

  1934.                 // multiply by 2 / k²
  1935.                 currentTerm *= 2. / (kIndex * kIndex);

  1937.                 // add the term to the result
  1938.                 k20 += currentTerm;
  1939.             }

  1941.             return k20;
  1942.         }

  1944.         /** {@inheritDoc} */
  1945.         @Override
  1946.         public double[] value(final Orbit meanOrbit) {

  1948.             // select the coefficients slot
  1949.             final Slot slot = slots.get(meanOrbit.getDate());

  1951.             // Get the True longitude L
  1952.             final double L = meanOrbit.getLv();

  1954.             // Compute the center (l - λ)
  1955.             final double center = L - meanOrbit.getLM();
  1956.             // Compute (l - λ)²
  1957.             final double center2 = center * center;

  1959.             // Initialize short periodic variations
  1960.             final double[] shortPeriodicVariation = slot.cij[0].value(meanOrbit.getDate());
  1961.             final double[] d1 = slot.dij[1].value(meanOrbit.getDate());
  1962.             final double[] d2 = slot.dij[2].value(meanOrbit.getDate());
  1963.             for (int i = 0; i < 6; i++) {
  1964.                 shortPeriodicVariation[i] += center * d1[i] + center2 * d2[i];
  1965.             }

  1967.             for (int j = 1; j <= JMAX; j++) {
  1968.                 final double[] c = slot.cij[j].value(meanOrbit.getDate());
  1969.                 final double[] s = slot.sij[j].value(meanOrbit.getDate());
  1970.                 final SinCos sc  = FastMath.sinCos(j * L);
  1971.                 final double cos = sc.cos();
  1972.                 final double sin = sc.sin();
  1973.                 for (int i = 0; i < 6; i++) {
  1974.                     // add corresponding term to the short periodic variation
  1975.                     shortPeriodicVariation[i] += c[i] * cos;
  1976.                     shortPeriodicVariation[i] += s[i] * sin;
  1977.                 }
  1978.             }

  1980.             return shortPeriodicVariation;

  1982.         }

  1984.         /** {@inheritDoc} */
  1985.         public String getCoefficientsKeyPrefix() {
  1986.             return coefficientsKeyPrefix;
  1987.         }

  1989.         /**
  1990.          * {@inheritDoc}
  1991.          * <p>
  1992.          * For Gaussian forces, there are JMAX cj coefficients, JMAX sj coefficients and
  1993.          * 3 dj coefficients. As JMAX = 12, this sums up to 27 coefficients. The j index
  1994.          * is the integer multiplier for the true longitude argument in the cj and sj
  1995.          * coefficients and to the degree in the polynomial dj coefficients.
  1996.          * </p>
  1997.          */
  1998.         @Override
  1999.         public Map<String, double[]> getCoefficients(final AbsoluteDate date, final Set<String> selected) {

  2001.             // select the coefficients slot
  2002.             final Slot slot = slots.get(date);

  2004.             final Map<String, double[]> coefficients = new HashMap<String, double[]>(2 * JMAX + 3);
  2005.             storeIfSelected(coefficients, selected, slot.cij[0].value(date), "d", 0);
  2006.             storeIfSelected(coefficients, selected, slot.dij[1].value(date), "d", 1);
  2007.             storeIfSelected(coefficients, selected, slot.dij[2].value(date), "d", 2);
  2008.             for (int j = 1; j <= JMAX; j++) {
  2009.                 storeIfSelected(coefficients, selected, slot.cij[j].value(date), "c", j);
  2010.                 storeIfSelected(coefficients, selected, slot.sij[j].value(date), "s", j);
  2011.             }

  2013.             return coefficients;

  2015.         }

  2017.         /**
  2018.          * Put a coefficient in a map if selected.
  2019.          * @param map      map to populate
  2020.          * @param selected set of coefficients that should be put in the map (empty set
  2021.          *                 means all coefficients are selected)
  2022.          * @param value    coefficient value
  2023.          * @param id       coefficient identifier
  2024.          * @param indices  list of coefficient indices
  2025.          */
  2026.         private void storeIfSelected(final Map<String, double[]> map, final Set<String> selected, final double[] value,
  2027.                 final String id, final int... indices) {
  2028.             final StringBuilder keyBuilder = new StringBuilder(getCoefficientsKeyPrefix());
  2029.             keyBuilder.append(id);
  2030.             for (int index : indices) {
  2031.                 keyBuilder.append('[').append(index).append(']');
  2032.             }
  2033.             final String key = keyBuilder.toString();
  2034.             if (selected.isEmpty() || selected.contains(key)) {
  2035.                 map.put(key, value);
  2036.             }
  2037.         }

  2039.     }

  2041.     /**
  2042.      * This class handles the short periodic coefficients described in Danielson
  2043.      * 2.5.3-26.
  2044.      *
  2045.      * <p>
  2046.      * The value of M is 0. Also, since the values of the Fourier coefficient
  2047.      * D<sub>i</sub><sup>m</sup> is 0 then the values of the coefficients
  2048.      * D<sub>i</sub><sup>m</sup> for m &gt; 2 are also 0.
  2049.      * </p>
  2050.      * @author Petre Bazavan
  2051.      * @author Lucian Barbulescu
  2052.      *
  2053.      */
  2054.     protected static class FieldGaussianShortPeriodicCoefficients<T extends CalculusFieldElement<T>>
  2055.             implements FieldShortPeriodTerms<T> {

  2057.         /** Maximum value for j index. */
  2058.         private final int jMax;

  2060.         /** Number of points used in the interpolation process. */
  2061.         private final int interpolationPoints;

  2063.         /** Prefix for coefficients keys. */
  2064.         private final String coefficientsKeyPrefix;

  2066.         /** All coefficients slots. */
  2067.         private final transient FieldTimeSpanMap<FieldSlot<T>, T> slots;

  2069.         /**
  2070.          * Constructor.
  2071.          * @param coefficientsKeyPrefix prefix for coefficients keys
  2072.          * @param jMax                  maximum value for j index
  2073.          * @param interpolationPoints   number of points used in the interpolation
  2074.          *                              process
  2075.          * @param slots                 all coefficients slots
  2076.          */
  2077.         FieldGaussianShortPeriodicCoefficients(final String coefficientsKeyPrefix, final int jMax,
  2078.                 final int interpolationPoints, final FieldTimeSpanMap<FieldSlot<T>, T> slots) {
  2079.             // Initialize fields
  2080.             this.jMax = jMax;
  2081.             this.interpolationPoints = interpolationPoints;
  2082.             this.coefficientsKeyPrefix = coefficientsKeyPrefix;
  2083.             this.slots = slots;
  2084.         }

  2086.         /**
  2087.          * Get the slot valid for some date.
  2088.          * @param meanStates mean states defining the slot
  2089.          * @return slot valid at the specified date
  2090.          */
  2091.         @SuppressWarnings("unchecked")
  2092.         public FieldSlot<T> createSlot(final FieldSpacecraftState<T>... meanStates) {
  2093.             final FieldSlot<T> slot = new FieldSlot<>(jMax, interpolationPoints);
  2094.             final FieldAbsoluteDate<T> first = meanStates[0].getDate();
  2095.             final FieldAbsoluteDate<T> last = meanStates[meanStates.length - 1].getDate();
  2096.             if (first.compareTo(last) <= 0) {
  2097.                 slots.addValidAfter(slot, first);
  2098.             } else {
  2099.                 slots.addValidBefore(slot, first);
  2100.             }
  2101.             return slot;
  2102.         }

  2104.         /**
  2105.          * Compute the short periodic coefficients.
  2106.          *
  2107.          * @param state       current state information: date, kinematics, attitude
  2108.          * @param slot        coefficients slot
  2109.          * @param fourierCjSj Fourier coefficients
  2110.          * @param uijvij      U and V coefficients
  2111.          * @param n           Keplerian mean motion
  2112.          * @param a           semi major axis
  2113.          * @param field       field used by default
  2114.          */
  2115.         private void computeCoefficients(final FieldSpacecraftState<T> state, final FieldSlot<T> slot,
  2116.                 final FieldFourierCjSjCoefficients<T> fourierCjSj, final FieldUijVijCoefficients<T> uijvij, final T n,
  2117.                 final T a, final Field<T> field) {

  2119.             // Zero
  2120.             final T zero = field.getZero();

  2122.             // get the current date
  2123.             final FieldAbsoluteDate<T> date = state.getDate();

  2125.             // compute the k₂⁰ coefficient
  2126.             final T k20 = computeK20(jMax, uijvij.currentRhoSigmaj, field);

  2128.             // 1. / n
  2129.             final T oon = n.reciprocal();
  2130.             // 3. / (2 * a * n)
  2131.             final T to2an = oon.multiply(1.5).divide(a);
  2132.             // 3. / (4 * a * n)
  2133.             final T to4an = to2an.divide(2.);

  2135.             // Compute the coefficients for each element
  2136.             final int size = jMax + 1;
  2137.             final T[] di1 = MathArrays.buildArray(field, 6);
  2138.             final T[] di2 = MathArrays.buildArray(field, 6);
  2139.             final T[][] currentCij = MathArrays.buildArray(field, size, 6);
  2140.             final T[][] currentSij = MathArrays.buildArray(field, size, 6);
  2141.             for (int i = 0; i < 6; i++) {

  2143.                 // compute D<sub>i</sub>¹ and D<sub>i</sub>² (all others are 0)
  2144.                 di1[i] = oon.negate().multiply(fourierCjSj.getCij(i, 0));
  2145.                 if (i == 5) {
  2146.                     di1[i] = di1[i].add(to2an.multiply(uijvij.getU1(0, 0)));
  2147.                 }
  2148.                 di2[i] = zero;
  2149.                 if (i == 5) {
  2150.                     di2[i] = di2[i].add(to4an.negate().multiply(fourierCjSj.getCij(0, 0)));
  2151.                 }

  2153.                 // the C<sub>i</sub>⁰ is computed based on all others
  2154.                 currentCij[0][i] = di2[i].negate().multiply(k20);

  2156.                 for (int j = 1; j <= jMax; j++) {
  2157.                     // compute the current C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup>
  2158.                     currentCij[j][i] = oon.multiply(uijvij.getU1(j, i));
  2159.                     if (i == 5) {
  2160.                         currentCij[j][i] = currentCij[j][i].add(to2an.negate().multiply(uijvij.getU2(j)));
  2161.                     }
  2162.                     currentSij[j][i] = oon.multiply(uijvij.getV1(j, i));
  2163.                     if (i == 5) {
  2164.                         currentSij[j][i] = currentSij[j][i].add(to2an.negate().multiply(uijvij.getV2(j)));
  2165.                     }

  2167.                     // add the computed coefficients to C<sub>i</sub>⁰
  2168.                     currentCij[0][i] = currentCij[0][i].add(currentCij[j][i].multiply(uijvij.currentRhoSigmaj[0][j])
  2169.                             .add(currentSij[j][i].multiply(uijvij.currentRhoSigmaj[1][j])).negate());
  2170.                 }

  2172.             }

  2174.             // add the values to the interpolators
  2175.             slot.cij[0].addGridPoint(date, currentCij[0]);
  2176.             slot.dij[1].addGridPoint(date, di1);
  2177.             slot.dij[2].addGridPoint(date, di2);
  2178.             for (int j = 1; j <= jMax; j++) {
  2179.                 slot.cij[j].addGridPoint(date, currentCij[j]);
  2180.                 slot.sij[j].addGridPoint(date, currentSij[j]);
  2181.             }

  2183.         }

  2185.         /**
  2186.          * Compute the coefficient k₂⁰ by using the equation 2.5.3-(9a) from Danielson.
  2187.          * <p>
  2188.          * After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:<br>
  2189.          * k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  2190.          * ρ<sub>k</sub>²)]
  2191.          * </p>
  2192.          * @param kMax             max value fot k index
  2193.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2194.          *                         σ<sub>j</sub> coefficients
  2195.          * @param field            field used by default
  2196.          * @return the coefficient k₂⁰
  2197.          */
  2198.         private T computeK20(final int kMax, final T[][] currentRhoSigmaj, final Field<T> field) {
  2199.             final T zero = field.getZero();
  2200.             T k20 = zero;

  2202.             for (int kIndex = 1; kIndex <= kMax; kIndex++) {
  2203.                 // After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:
  2204.                 // k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  2205.                 // ρ<sub>k</sub>²)]
  2206.                 T currentTerm = currentRhoSigmaj[1][kIndex].multiply(currentRhoSigmaj[1][kIndex])
  2207.                         .add(currentRhoSigmaj[0][kIndex].multiply(currentRhoSigmaj[0][kIndex]));

  2209.                 // multiply by 2 / k²
  2210.                 currentTerm = currentTerm.multiply(2. / (kIndex * kIndex));

  2212.                 // add the term to the result
  2213.                 k20 = k20.add(currentTerm);
  2214.             }

  2216.             return k20;
  2217.         }

  2219.         /** {@inheritDoc} */
  2220.         @Override
  2221.         public T[] value(final FieldOrbit<T> meanOrbit) {

  2223.             // select the coefficients slot
  2224.             final FieldSlot<T> slot = slots.get(meanOrbit.getDate());

  2226.             // Get the True longitude L
  2227.             final T L = meanOrbit.getLv();

  2229.             // Compute the center (l - λ)
  2230.             final T center = L.subtract(meanOrbit.getLM());
  2231.             // Compute (l - λ)²
  2232.             final T center2 = center.multiply(center);

  2234.             // Initialize short periodic variations
  2235.             final T[] shortPeriodicVariation = slot.cij[0].value(meanOrbit.getDate());
  2236.             final T[] d1 = slot.dij[1].value(meanOrbit.getDate());
  2237.             final T[] d2 = slot.dij[2].value(meanOrbit.getDate());
  2238.             for (int i = 0; i < 6; i++) {
  2239.                 shortPeriodicVariation[i] = shortPeriodicVariation[i]
  2240.                         .add(center.multiply(d1[i]).add(center2.multiply(d2[i])));
  2241.             }

  2243.             for (int j = 1; j <= JMAX; j++) {
  2244.                 final T[] c = slot.cij[j].value(meanOrbit.getDate());
  2245.                 final T[] s = slot.sij[j].value(meanOrbit.getDate());
  2246.                 final FieldSinCos<T> sc = FastMath.sinCos(L.multiply(j));
  2247.                 final T cos = sc.cos();
  2248.                 final T sin = sc.sin();
  2249.                 for (int i = 0; i < 6; i++) {
  2250.                     // add corresponding term to the short periodic variation
  2251.                     shortPeriodicVariation[i] = shortPeriodicVariation[i].add(c[i].multiply(cos));
  2252.                     shortPeriodicVariation[i] = shortPeriodicVariation[i].add(s[i].multiply(sin));
  2253.                 }
  2254.             }

  2256.             return shortPeriodicVariation;

  2258.         }

  2260.         /** {@inheritDoc} */
  2261.         public String getCoefficientsKeyPrefix() {
  2262.             return coefficientsKeyPrefix;
  2263.         }

  2265.         /**
  2266.          * {@inheritDoc}
  2267.          * <p>
  2268.          * For Gaussian forces, there are JMAX cj coefficients, JMAX sj coefficients and
  2269.          * 3 dj coefficients. As JMAX = 12, this sums up to 27 coefficients. The j index
  2270.          * is the integer multiplier for the true longitude argument in the cj and sj
  2271.          * coefficients and to the degree in the polynomial dj coefficients.
  2272.          * </p>
  2273.          */
  2274.         @Override
  2275.         public Map<String, T[]> getCoefficients(final FieldAbsoluteDate<T> date, final Set<String> selected) {

  2277.             // select the coefficients slot
  2278.             final FieldSlot<T> slot = slots.get(date);

  2280.             final Map<String, T[]> coefficients = new HashMap<String, T[]>(2 * JMAX + 3);
  2281.             storeIfSelected(coefficients, selected, slot.cij[0].value(date), "d", 0);
  2282.             storeIfSelected(coefficients, selected, slot.dij[1].value(date), "d", 1);
  2283.             storeIfSelected(coefficients, selected, slot.dij[2].value(date), "d", 2);
  2284.             for (int j = 1; j <= JMAX; j++) {
  2285.                 storeIfSelected(coefficients, selected, slot.cij[j].value(date), "c", j);
  2286.                 storeIfSelected(coefficients, selected, slot.sij[j].value(date), "s", j);
  2287.             }

  2289.             return coefficients;

  2291.         }

  2293.         /**
  2294.          * Put a coefficient in a map if selected.
  2295.          * @param map      map to populate
  2296.          * @param selected set of coefficients that should be put in the map (empty set
  2297.          *                 means all coefficients are selected)
  2298.          * @param value    coefficient value
  2299.          * @param id       coefficient identifier
  2300.          * @param indices  list of coefficient indices
  2301.          */
  2302.         private void storeIfSelected(final Map<String, T[]> map, final Set<String> selected, final T[] value,
  2303.                 final String id, final int... indices) {
  2304.             final StringBuilder keyBuilder = new StringBuilder(getCoefficientsKeyPrefix());
  2305.             keyBuilder.append(id);
  2306.             for (int index : indices) {
  2307.                 keyBuilder.append('[').append(index).append(']');
  2308.             }
  2309.             final String key = keyBuilder.toString();
  2310.             if (selected.isEmpty() || selected.contains(key)) {
  2311.                 map.put(key, value);
  2312.             }
  2313.         }

  2315.     }

  2317.     /**
  2318.      * The U<sub>i</sub><sup>j</sup> and V<sub>i</sub><sup>j</sup> coefficients
  2319.      * described by equations 2.5.3-(21) and 2.5.3-(22) from Danielson.
  2320.      * <p>
  2321.      * The index i takes only the values 1 and 2<br>
  2322.      * For U only the index 0 for j is used.
  2323.      * </p>
  2324.      *
  2325.      * @author Petre Bazavan
  2326.      * @author Lucian Barbulescu
  2327.      */
  2328.     protected static class UijVijCoefficients {

  2330.         /**
  2331.          * The U₁<sup>j</sup> coefficients.
  2332.          * <p>
  2333.          * The first index identifies the Fourier coefficients used<br>
  2334.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2335.          * S<sub>i</sub><sup>j</sup><br>
  2336.          * The only exception is when j = 0 when only the coefficient for fourier index
  2337.          * = 1 (i == 0) is needed.<br>
  2338.          * Also, for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are
  2339.          * computed, because are required to compute the coefficients U₂<sup>j</sup>
  2340.          * </p>
  2341.          */
  2342.         private final double[][] u1ij;

  2344.         /**
  2345.          * The V₁<sup>j</sup> coefficients.
  2346.          * <p>
  2347.          * The first index identifies the Fourier coefficients used<br>
  2348.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2349.          * S<sub>i</sub><sup>j</sup><br>
  2350.          * for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are computed,
  2351.          * because are required to compute the coefficients V₂<sup>j</sup>
  2352.          * </p>
  2353.          */
  2354.         private final double[][] v1ij;

  2356.         /**
  2357.          * The U₂<sup>j</sup> coefficients.
  2358.          * <p>
  2359.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2360.          * they are the only ones required.
  2361.          * </p>
  2362.          */
  2363.         private final double[] u2ij;

  2365.         /**
  2366.          * The V₂<sup>j</sup> coefficients.
  2367.          * <p>
  2368.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2369.          * they are the only ones required.
  2370.          * </p>
  2371.          */
  2372.         private final double[] v2ij;

  2374.         /**
  2375.          * The current computed values for the ρ<sub>j</sub> and σ<sub>j</sub>
  2376.          * coefficients.
  2377.          */
  2378.         private final double[][] currentRhoSigmaj;

  2380.         /**
  2381.          * The C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup> Fourier
  2382.          * coefficients.
  2383.          */
  2384.         private final FourierCjSjCoefficients fourierCjSj;

  2386.         /** The maximum value for j index. */
  2387.         private final int jMax;

  2389.         /**
  2390.          * Constructor.
  2391.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2392.          *                         σ<sub>j</sub> coefficients
  2393.          * @param fourierCjSj      the fourier coefficients C<sub>i</sub><sup>j</sup>
  2394.          *                         and the S<sub>i</sub><sup>j</sup>
  2395.          * @param jMax             maximum value for j index
  2396.          */
  2397.         UijVijCoefficients(final double[][] currentRhoSigmaj, final FourierCjSjCoefficients fourierCjSj,
  2398.                 final int jMax) {
  2399.             this.currentRhoSigmaj = currentRhoSigmaj;
  2400.             this.fourierCjSj = fourierCjSj;
  2401.             this.jMax = jMax;

  2403.             // initialize the internal arrays.
  2404.             this.u1ij = new double[6][2 * jMax + 1];
  2405.             this.v1ij = new double[6][2 * jMax + 1];
  2406.             this.u2ij = new double[jMax + 1];
  2407.             this.v2ij = new double[jMax + 1];

  2409.             // compute the coefficients
  2410.             computeU1V1Coefficients();
  2411.             computeU2V2Coefficients();
  2412.         }

  2414.         /** Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients. */
  2415.         private void computeU1V1Coefficients() {
  2416.             // generate the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients
  2417.             // for j >= 1
  2418.             // also the U₁⁰ for Fourier index = 1 (i == 0) coefficient will be computed
  2419.             u1ij[0][0] = 0;
  2420.             for (int j = 1; j <= jMax; j++) {
  2421.                 // compute 1 / j
  2422.                 final double ooj = 1. / j;

  2424.                 for (int i = 0; i < 6; i++) {
  2425.                     // j is aready between 1 and J
  2426.                     u1ij[i][j] = fourierCjSj.getSij(i, j);
  2427.                     v1ij[i][j] = fourierCjSj.getCij(i, j);

  2429.                     // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2430.                     if (j > 1) {
  2431.                         // k starts with 1 because j-J is less than or equal to 0
  2432.                         for (int kIndex = 1; kIndex <= j - 1; kIndex++) {
  2433.                             // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2434.                             // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2435.                             u1ij[i][j] += fourierCjSj.getCij(i, j - kIndex) * currentRhoSigmaj[1][kIndex] +
  2436.                                     fourierCjSj.getSij(i, j - kIndex) * currentRhoSigmaj[0][kIndex];

  2438.                             // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2439.                             // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2440.                             v1ij[i][j] += fourierCjSj.getCij(i, j - kIndex) * currentRhoSigmaj[0][kIndex] -
  2441.                                     fourierCjSj.getSij(i, j - kIndex) * currentRhoSigmaj[1][kIndex];
  2442.                         }
  2443.                     }

  2445.                     // since j must be between 1 and J-1 and is already between 1 and J
  2446.                     // the following sum is skiped only for j = jMax
  2447.                     if (j != jMax) {
  2448.                         for (int kIndex = 1; kIndex <= jMax - j; kIndex++) {
  2449.                             // -C<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub> +
  2450.                             // S<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub>
  2451.                             u1ij[i][j] += -fourierCjSj.getCij(i, j + kIndex) * currentRhoSigmaj[1][kIndex] +
  2452.                                     fourierCjSj.getSij(i, j + kIndex) * currentRhoSigmaj[0][kIndex];

  2454.                             // C<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub> +
  2455.                             // S<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub>
  2456.                             v1ij[i][j] += fourierCjSj.getCij(i, j + kIndex) * currentRhoSigmaj[0][kIndex] +
  2457.                                     fourierCjSj.getSij(i, j + kIndex) * currentRhoSigmaj[1][kIndex];
  2458.                         }
  2459.                     }

  2461.                     for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2462.                         // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2463.                         // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2464.                         u1ij[i][j] += -fourierCjSj.getCij(i, kIndex) * currentRhoSigmaj[1][j + kIndex] -
  2465.                                 fourierCjSj.getSij(i, kIndex) * currentRhoSigmaj[0][j + kIndex];

  2467.                         // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2468.                         // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2469.                         v1ij[i][j] += fourierCjSj.getCij(i, kIndex) * currentRhoSigmaj[0][j + kIndex] +
  2470.                                 fourierCjSj.getSij(i, kIndex) * currentRhoSigmaj[1][j + kIndex];
  2471.                     }

  2473.                     // divide by 1 / j
  2474.                     u1ij[i][j] *= -ooj;
  2475.                     v1ij[i][j] *= ooj;

  2477.                     // if index = 1 (i == 0) add the computed terms to U₁⁰
  2478.                     if (i == 0) {
  2479.                         // - (U₁<sup>j</sup> * ρ<sub>j</sub> + V₁<sup>j</sup> * σ<sub>j</sub>
  2480.                         u1ij[0][0] += -u1ij[0][j] * currentRhoSigmaj[0][j] - v1ij[0][j] * currentRhoSigmaj[1][j];
  2481.                     }
  2482.                 }
  2483.             }

  2485.             // Terms with j > jMax are required only when computing the coefficients
  2486.             // U₂<sup>j</sup> and V₂<sup>j</sup>
  2487.             // and those coefficients are only required for Fourier index = 1 (i == 0).
  2488.             for (int j = jMax + 1; j <= 2 * jMax; j++) {
  2489.                 // compute 1 / j
  2490.                 final double ooj = 1. / j;
  2491.                 // the value of i is 0
  2492.                 u1ij[0][j] = 0.;
  2493.                 v1ij[0][j] = 0.;

  2495.                 // k starts from j-J as it is always greater than or equal to 1
  2496.                 for (int kIndex = j - jMax; kIndex <= j - 1; kIndex++) {
  2497.                     // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2498.                     // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2499.                     u1ij[0][j] += fourierCjSj.getCij(0, j - kIndex) * currentRhoSigmaj[1][kIndex] +
  2500.                             fourierCjSj.getSij(0, j - kIndex) * currentRhoSigmaj[0][kIndex];

  2502.                     // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2503.                     // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2504.                     v1ij[0][j] += fourierCjSj.getCij(0, j - kIndex) * currentRhoSigmaj[0][kIndex] -
  2505.                             fourierCjSj.getSij(0, j - kIndex) * currentRhoSigmaj[1][kIndex];
  2506.                 }
  2507.                 for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2508.                     // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2509.                     // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2510.                     u1ij[0][j] += -fourierCjSj.getCij(0, kIndex) * currentRhoSigmaj[1][j + kIndex] -
  2511.                             fourierCjSj.getSij(0, kIndex) * currentRhoSigmaj[0][j + kIndex];

  2513.                     // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2514.                     // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2515.                     v1ij[0][j] += fourierCjSj.getCij(0, kIndex) * currentRhoSigmaj[0][j + kIndex] +
  2516.                             fourierCjSj.getSij(0, kIndex) * currentRhoSigmaj[1][j + kIndex];
  2517.                 }

  2519.                 // divide by 1 / j
  2520.                 u1ij[0][j] *= -ooj;
  2521.                 v1ij[0][j] *= ooj;
  2522.             }
  2523.         }

  2525.         /**
  2526.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2527.          * <p>
  2528.          * Only the coefficients for Fourier index = 1 (i == 0) are required.
  2529.          * </p>
  2530.          */
  2531.         private void computeU2V2Coefficients() {
  2532.             for (int j = 1; j <= jMax; j++) {
  2533.                 // compute 1 / j
  2534.                 final double ooj = 1. / j;

  2536.                 // only the values for i == 0 are computed
  2537.                 u2ij[j] = v1ij[0][j];
  2538.                 v2ij[j] = u1ij[0][j];

  2540.                 // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2541.                 if (j > 1) {
  2542.                     for (int l = 1; l <= j - 1; l++) {
  2543.                         // U₁<sup>j-l</sup> * σ<sub>l</sub> +
  2544.                         // V₁<sup>j-l</sup> * ρ<sub>l</sub>
  2545.                         u2ij[j] += u1ij[0][j - l] * currentRhoSigmaj[1][l] + v1ij[0][j - l] * currentRhoSigmaj[0][l];

  2547.                         // U₁<sup>j-l</sup> * ρ<sub>l</sub> -
  2548.                         // V₁<sup>j-l</sup> * σ<sub>l</sub>
  2549.                         v2ij[j] += u1ij[0][j - l] * currentRhoSigmaj[0][l] - v1ij[0][j - l] * currentRhoSigmaj[1][l];
  2550.                     }
  2551.                 }

  2553.                 for (int l = 1; l <= jMax; l++) {
  2554.                     // -U₁<sup>j+l</sup> * σ<sub>l</sub> +
  2555.                     // U₁<sup>l</sup> * σ<sub>j+l</sub> +
  2556.                     // V₁<sup>j+l</sup> * ρ<sub>l</sub> -
  2557.                     // V₁<sup>l</sup> * ρ<sub>j+l</sub>
  2558.                     u2ij[j] += -u1ij[0][j + l] * currentRhoSigmaj[1][l] + u1ij[0][l] * currentRhoSigmaj[1][j + l] +
  2559.                             v1ij[0][j + l] * currentRhoSigmaj[0][l] - v1ij[0][l] * currentRhoSigmaj[0][j + l];

  2561.                     // U₁<sup>j+l</sup> * ρ<sub>l</sub> +
  2562.                     // U₁<sup>l</sup> * ρ<sub>j+l</sub> +
  2563.                     // V₁<sup>j+l</sup> * σ<sub>l</sub> +
  2564.                     // V₁<sup>l</sup> * σ<sub>j+l</sub>
  2565.                     u2ij[j] += u1ij[0][j + l] * currentRhoSigmaj[0][l] + u1ij[0][l] * currentRhoSigmaj[0][j + l] +
  2566.                             v1ij[0][j + l] * currentRhoSigmaj[1][l] + v1ij[0][l] * currentRhoSigmaj[1][j + l];
  2567.                 }

  2569.                 // divide by 1 / j
  2570.                 u2ij[j] *= -ooj;
  2571.                 v2ij[j] *= ooj;
  2572.             }
  2573.         }

  2575.         /**
  2576.          * Get the coefficient U₁<sup>j</sup> for Fourier index i.
  2577.          *
  2578.          * @param j j index
  2579.          * @param i Fourier index (starts at 0)
  2580.          * @return the coefficient U₁<sup>j</sup> for the given Fourier index i
  2581.          */
  2582.         public double getU1(final int j, final int i) {
  2583.             return u1ij[i][j];
  2584.         }

  2586.         /**
  2587.          * Get the coefficient V₁<sup>j</sup> for Fourier index i.
  2588.          *
  2589.          * @param j j index
  2590.          * @param i Fourier index (starts at 0)
  2591.          * @return the coefficient V₁<sup>j</sup> for the given Fourier index i
  2592.          */
  2593.         public double getV1(final int j, final int i) {
  2594.             return v1ij[i][j];
  2595.         }

  2597.         /**
  2598.          * Get the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2599.          *
  2600.          * @param j j index
  2601.          * @return the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2602.          */
  2603.         public double getU2(final int j) {
  2604.             return u2ij[j];
  2605.         }

  2607.         /**
  2608.          * Get the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2609.          *
  2610.          * @param j j index
  2611.          * @return the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2612.          */
  2613.         public double getV2(final int j) {
  2614.             return v2ij[j];
  2615.         }
  2616.     }

  2618.     /**
  2619.      * The U<sub>i</sub><sup>j</sup> and V<sub>i</sub><sup>j</sup> coefficients
  2620.      * described by equations 2.5.3-(21) and 2.5.3-(22) from Danielson.
  2621.      * <p>
  2622.      * The index i takes only the values 1 and 2<br>
  2623.      * For U only the index 0 for j is used.
  2624.      * </p>
  2625.      *
  2626.      * @author Petre Bazavan
  2627.      * @author Lucian Barbulescu
  2628.      */
  2629.     protected static class FieldUijVijCoefficients<T extends CalculusFieldElement<T>> {

  2631.         /**
  2632.          * The U₁<sup>j</sup> coefficients.
  2633.          * <p>
  2634.          * The first index identifies the Fourier coefficients used<br>
  2635.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2636.          * S<sub>i</sub><sup>j</sup><br>
  2637.          * The only exception is when j = 0 when only the coefficient for fourier index
  2638.          * = 1 (i == 0) is needed.<br>
  2639.          * Also, for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are
  2640.          * computed, because are required to compute the coefficients U₂<sup>j</sup>
  2641.          * </p>
  2642.          */
  2643.         private final T[][] u1ij;

  2645.         /**
  2646.          * The V₁<sup>j</sup> coefficients.
  2647.          * <p>
  2648.          * The first index identifies the Fourier coefficients used<br>
  2649.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2650.          * S<sub>i</sub><sup>j</sup><br>
  2651.          * for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are computed,
  2652.          * because are required to compute the coefficients V₂<sup>j</sup>
  2653.          * </p>
  2654.          */
  2655.         private final T[][] v1ij;

  2657.         /**
  2658.          * The U₂<sup>j</sup> coefficients.
  2659.          * <p>
  2660.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2661.          * they are the only ones required.
  2662.          * </p>
  2663.          */
  2664.         private final T[] u2ij;

  2666.         /**
  2667.          * The V₂<sup>j</sup> coefficients.
  2668.          * <p>
  2669.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2670.          * they are the only ones required.
  2671.          * </p>
  2672.          */
  2673.         private final T[] v2ij;

  2675.         /**
  2676.          * The current computed values for the ρ<sub>j</sub> and σ<sub>j</sub>
  2677.          * coefficients.
  2678.          */
  2679.         private final T[][] currentRhoSigmaj;

  2681.         /**
  2682.          * The C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup> Fourier
  2683.          * coefficients.
  2684.          */
  2685.         private final FieldFourierCjSjCoefficients<T> fourierCjSj;

  2687.         /** The maximum value for j index. */
  2688.         private final int jMax;

  2690.         /**
  2691.          * Constructor.
  2692.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2693.          *                         σ<sub>j</sub> coefficients
  2694.          * @param fourierCjSj      the fourier coefficients C<sub>i</sub><sup>j</sup>
  2695.          *                         and the S<sub>i</sub><sup>j</sup>
  2696.          * @param jMax             maximum value for j index
  2697.          * @param field            field used by default
  2698.          */
  2699.         FieldUijVijCoefficients(final T[][] currentRhoSigmaj, final FieldFourierCjSjCoefficients<T> fourierCjSj,
  2700.                 final int jMax, final Field<T> field) {
  2701.             this.currentRhoSigmaj = currentRhoSigmaj;
  2702.             this.fourierCjSj = fourierCjSj;
  2703.             this.jMax = jMax;

  2705.             // initialize the internal arrays.
  2706.             this.u1ij = MathArrays.buildArray(field, 6, 2 * jMax + 1);
  2707.             this.v1ij = MathArrays.buildArray(field, 6, 2 * jMax + 1);
  2708.             this.u2ij = MathArrays.buildArray(field, jMax + 1);
  2709.             this.v2ij = MathArrays.buildArray(field, jMax + 1);

  2711.             // compute the coefficients
  2712.             computeU1V1Coefficients(field);
  2713.             computeU2V2Coefficients(field);
  2714.         }

  2716.         /**
  2717.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2718.          * @param field field used by default
  2719.          */
  2720.         private void computeU1V1Coefficients(final Field<T> field) {
  2721.             // Zero
  2722.             final T zero = field.getZero();

  2724.             // generate the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients
  2725.             // for j >= 1
  2726.             // also the U₁⁰ for Fourier index = 1 (i == 0) coefficient will be computed
  2727.             u1ij[0][0] = zero;
  2728.             for (int j = 1; j <= jMax; j++) {
  2729.                 // compute 1 / j
  2730.                 final double ooj = 1. / j;

  2732.                 for (int i = 0; i < 6; i++) {
  2733.                     // j is aready between 1 and J
  2734.                     u1ij[i][j] = fourierCjSj.getSij(i, j);
  2735.                     v1ij[i][j] = fourierCjSj.getCij(i, j);

  2737.                     // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2738.                     if (j > 1) {
  2739.                         // k starts with 1 because j-J is less than or equal to 0
  2740.                         for (int kIndex = 1; kIndex <= j - 1; kIndex++) {
  2741.                             // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2742.                             // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2743.                             u1ij[i][j] = u1ij[i][j]
  2744.                                     .add(fourierCjSj.getCij(i, j - kIndex).multiply(currentRhoSigmaj[1][kIndex]).add(
  2745.                                             fourierCjSj.getSij(i, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2747.                             // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2748.                             // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2749.                             v1ij[i][j] = v1ij[i][j].add(
  2750.                                     fourierCjSj.getCij(i, j - kIndex).multiply(currentRhoSigmaj[0][kIndex]).subtract(
  2751.                                             fourierCjSj.getSij(i, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2752.                         }
  2753.                     }

  2755.                     // since j must be between 1 and J-1 and is already between 1 and J
  2756.                     // the following sum is skiped only for j = jMax
  2757.                     if (j != jMax) {
  2758.                         for (int kIndex = 1; kIndex <= jMax - j; kIndex++) {
  2759.                             // -C<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub> +
  2760.                             // S<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub>
  2761.                             u1ij[i][j] = u1ij[i][j].add(fourierCjSj.getCij(i, j + kIndex).negate()
  2762.                                     .multiply(currentRhoSigmaj[1][kIndex])
  2763.                                     .add(fourierCjSj.getSij(i, j + kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2765.                             // C<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub> +
  2766.                             // S<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub>
  2767.                             v1ij[i][j] = v1ij[i][j]
  2768.                                     .add(fourierCjSj.getCij(i, j + kIndex).multiply(currentRhoSigmaj[0][kIndex]).add(
  2769.                                             fourierCjSj.getSij(i, j + kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2770.                         }
  2771.                     }

  2773.                     for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2774.                         // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2775.                         // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2776.                         u1ij[i][j] = u1ij[i][j].add(fourierCjSj.getCij(i, kIndex).negate()
  2777.                                 .multiply(currentRhoSigmaj[1][j + kIndex])
  2778.                                 .subtract(fourierCjSj.getSij(i, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])));

  2780.                         // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2781.                         // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2782.                         v1ij[i][j] = v1ij[i][j]
  2783.                                 .add(fourierCjSj.getCij(i, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])
  2784.                                         .add(fourierCjSj.getSij(i, kIndex).multiply(currentRhoSigmaj[1][j + kIndex])));
  2785.                     }

  2787.                     // divide by 1 / j
  2788.                     u1ij[i][j] = u1ij[i][j].multiply(-ooj);
  2789.                     v1ij[i][j] = v1ij[i][j].multiply(ooj);

  2791.                     // if index = 1 (i == 0) add the computed terms to U₁⁰
  2792.                     if (i == 0) {
  2793.                         // - (U₁<sup>j</sup> * ρ<sub>j</sub> + V₁<sup>j</sup> * σ<sub>j</sub>
  2794.                         u1ij[0][0] = u1ij[0][0].add(u1ij[0][j].negate().multiply(currentRhoSigmaj[0][j])
  2795.                                 .subtract(v1ij[0][j].multiply(currentRhoSigmaj[1][j])));
  2796.                     }
  2797.                 }
  2798.             }

  2800.             // Terms with j > jMax are required only when computing the coefficients
  2801.             // U₂<sup>j</sup> and V₂<sup>j</sup>
  2802.             // and those coefficients are only required for Fourier index = 1 (i == 0).
  2803.             for (int j = jMax + 1; j <= 2 * jMax; j++) {
  2804.                 // compute 1 / j
  2805.                 final double ooj = 1. / j;
  2806.                 // the value of i is 0
  2807.                 u1ij[0][j] = zero;
  2808.                 v1ij[0][j] = zero;

  2810.                 // k starts from j-J as it is always greater than or equal to 1
  2811.                 for (int kIndex = j - jMax; kIndex <= j - 1; kIndex++) {
  2812.                     // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2813.                     // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2814.                     u1ij[0][j] = u1ij[0][j].add(fourierCjSj.getCij(0, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])
  2815.                             .add(fourierCjSj.getSij(0, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2817.                     // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2818.                     // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2819.                     v1ij[0][j] = v1ij[0][j].add(fourierCjSj.getCij(0, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])
  2820.                             .subtract(fourierCjSj.getSij(0, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2821.                 }
  2822.                 for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2823.                     // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2824.                     // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2825.                     u1ij[0][j] = u1ij[0][j]
  2826.                             .add(fourierCjSj.getCij(0, kIndex).negate().multiply(currentRhoSigmaj[1][j + kIndex])
  2827.                                     .subtract(fourierCjSj.getSij(0, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])));

  2829.                     // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2830.                     // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2831.                     v1ij[0][j] = v1ij[0][j].add(fourierCjSj.getCij(0, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])
  2832.                             .add(fourierCjSj.getSij(0, kIndex).multiply(currentRhoSigmaj[1][j + kIndex])));
  2833.                 }

  2835.                 // divide by 1 / j
  2836.                 u1ij[0][j] = u1ij[0][j].multiply(-ooj);
  2837.                 v1ij[0][j] = v1ij[0][j].multiply(ooj);
  2838.             }
  2839.         }

  2841.         /**
  2842.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2843.          * <p>
  2844.          * Only the coefficients for Fourier index = 1 (i == 0) are required.
  2845.          * </p>
  2846.          * @param field field used by default
  2847.          */
  2848.         private void computeU2V2Coefficients(final Field<T> field) {
  2849.             for (int j = 1; j <= jMax; j++) {
  2850.                 // compute 1 / j
  2851.                 final double ooj = 1. / j;

  2853.                 // only the values for i == 0 are computed
  2854.                 u2ij[j] = v1ij[0][j];
  2855.                 v2ij[j] = u1ij[0][j];

  2857.                 // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2858.                 if (j > 1) {
  2859.                     for (int l = 1; l <= j - 1; l++) {
  2860.                         // U₁<sup>j-l</sup> * σ<sub>l</sub> +
  2861.                         // V₁<sup>j-l</sup> * ρ<sub>l</sub>
  2862.                         u2ij[j] = u2ij[j].add(u1ij[0][j - l].multiply(currentRhoSigmaj[1][l])
  2863.                                 .add(v1ij[0][j - l].multiply(currentRhoSigmaj[0][l])));

  2865.                         // U₁<sup>j-l</sup> * ρ<sub>l</sub> -
  2866.                         // V₁<sup>j-l</sup> * σ<sub>l</sub>
  2867.                         v2ij[j] = v2ij[j].add(u1ij[0][j - l].multiply(currentRhoSigmaj[0][l])
  2868.                                 .subtract(v1ij[0][j - l].multiply(currentRhoSigmaj[1][l])));
  2869.                     }
  2870.                 }

  2872.                 for (int l = 1; l <= jMax; l++) {
  2873.                     // -U₁<sup>j+l</sup> * σ<sub>l</sub> +
  2874.                     // U₁<sup>l</sup> * σ<sub>j+l</sub> +
  2875.                     // V₁<sup>j+l</sup> * ρ<sub>l</sub> -
  2876.                     // V₁<sup>l</sup> * ρ<sub>j+l</sub>
  2877.                     u2ij[j] = u2ij[j].add(u1ij[0][j + l].negate().multiply(currentRhoSigmaj[1][l])
  2878.                             .add(u1ij[0][l].multiply(currentRhoSigmaj[1][j + l]))
  2879.                             .add(v1ij[0][j + l].multiply(currentRhoSigmaj[0][l]))
  2880.                             .subtract(v1ij[0][l].multiply(currentRhoSigmaj[0][j + l])));

  2882.                     // U₁<sup>j+l</sup> * ρ<sub>l</sub> +
  2883.                     // U₁<sup>l</sup> * ρ<sub>j+l</sub> +
  2884.                     // V₁<sup>j+l</sup> * σ<sub>l</sub> +
  2885.                     // V₁<sup>l</sup> * σ<sub>j+l</sub>
  2886.                     u2ij[j] = u2ij[j].add(u1ij[0][j + l].multiply(currentRhoSigmaj[0][l])
  2887.                             .add(u1ij[0][l].multiply(currentRhoSigmaj[0][j + l]))
  2888.                             .add(v1ij[0][j + l].multiply(currentRhoSigmaj[1][l]))
  2889.                             .add(v1ij[0][l].multiply(currentRhoSigmaj[1][j + l])));
  2890.                 }

  2892.                 // divide by 1 / j
  2893.                 u2ij[j] = u2ij[j].multiply(-ooj);
  2894.                 v2ij[j] = v2ij[j].multiply(ooj);
  2895.             }
  2896.         }

  2898.         /**
  2899.          * Get the coefficient U₁<sup>j</sup> for Fourier index i.
  2900.          *
  2901.          * @param j j index
  2902.          * @param i Fourier index (starts at 0)
  2903.          * @return the coefficient U₁<sup>j</sup> for the given Fourier index i
  2904.          */
  2905.         public T getU1(final int j, final int i) {
  2906.             return u1ij[i][j];
  2907.         }

  2909.         /**
  2910.          * Get the coefficient V₁<sup>j</sup> for Fourier index i.
  2911.          *
  2912.          * @param j j index
  2913.          * @param i Fourier index (starts at 0)
  2914.          * @return the coefficient V₁<sup>j</sup> for the given Fourier index i
  2915.          */
  2916.         public T getV1(final int j, final int i) {
  2917.             return v1ij[i][j];
  2918.         }

  2920.         /**
  2921.          * Get the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2922.          *
  2923.          * @param j j index
  2924.          * @return the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2925.          */
  2926.         public T getU2(final int j) {
  2927.             return u2ij[j];
  2928.         }

  2930.         /**
  2931.          * Get the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2932.          *
  2933.          * @param j j index
  2934.          * @return the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2935.          */
  2936.         public T getV2(final int j) {
  2937.             return v2ij[j];
  2938.         }
  2939.     }

  2941.     /** Coefficients valid for one time slot. */
  2942.     protected static class Slot {

  2944.         /**
  2945.          * The coefficients D<sub>i</sub><sup>j</sup>.
  2946.          * <p>
  2947.          * Only for j = 1 and j = 2 the coefficients are not 0. <br>
  2948.          * i corresponds to the equinoctial element, as follows: - i=0 for a <br/>
  2949.          * - i=1 for k <br/>
  2950.          * - i=2 for h <br/>
  2951.          * - i=3 for q <br/>
  2952.          * - i=4 for p <br/>
  2953.          * - i=5 for λ <br/>
  2954.          * </p>
  2955.          */
  2956.         private final ShortPeriodicsInterpolatedCoefficient[] dij;

  2958.         /**
  2959.          * The coefficients C<sub>i</sub><sup>j</sup>.
  2960.          * <p>
  2961.          * The index order is cij[j][i] <br/>
  2962.          * i corresponds to the equinoctial element, as follows: <br/>
  2963.          * - i=0 for a <br/>
  2964.          * - i=1 for k <br/>
  2965.          * - i=2 for h <br/>
  2966.          * - i=3 for q <br/>
  2967.          * - i=4 for p <br/>
  2968.          * - i=5 for λ <br/>
  2969.          * </p>
  2970.          */
  2971.         private final ShortPeriodicsInterpolatedCoefficient[] cij;

  2973.         /**
  2974.          * The coefficients S<sub>i</sub><sup>j</sup>.
  2975.          * <p>
  2976.          * The index order is sij[j][i] <br/>
  2977.          * i corresponds to the equinoctial element, as follows: <br/>
  2978.          * - i=0 for a <br/>
  2979.          * - i=1 for k <br/>
  2980.          * - i=2 for h <br/>
  2981.          * - i=3 for q <br/>
  2982.          * - i=4 for p <br/>
  2983.          * - i=5 for λ <br/>
  2984.          * </p>
  2985.          */
  2986.         private final ShortPeriodicsInterpolatedCoefficient[] sij;

  2988.         /**
  2989.          * Simple constructor.
  2990.          * @param jMax                maximum value for j index
  2991.          * @param interpolationPoints number of points used in the interpolation process
  2992.          */
  2993.         Slot(final int jMax, final int interpolationPoints) {

  2995.             dij = new ShortPeriodicsInterpolatedCoefficient[3];
  2996.             cij = new ShortPeriodicsInterpolatedCoefficient[jMax + 1];
  2997.             sij = new ShortPeriodicsInterpolatedCoefficient[jMax + 1];

  2999.             // Initialize the C<sub>i</sub><sup>j</sup>, S<sub>i</sub><sup>j</sup> and
  3000.             // D<sub>i</sub><sup>j</sup> coefficients
  3001.             for (int j = 0; j <= jMax; j++) {
  3002.                 cij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3003.                 if (j > 0) {
  3004.                     sij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3005.                 }
  3006.                 // Initialize only the non-zero D<sub>i</sub><sup>j</sup> coefficients
  3007.                 if (j == 1 || j == 2) {
  3008.                     dij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3009.                 }
  3010.             }

  3012.         }

  3014.     }

  3016.     /** Coefficients valid for one time slot. */
  3017.     protected static class FieldSlot<T extends CalculusFieldElement<T>> {

  3019.         /**
  3020.          * The coefficients D<sub>i</sub><sup>j</sup>.
  3021.          * <p>
  3022.          * Only for j = 1 and j = 2 the coefficients are not 0. <br>
  3023.          * i corresponds to the equinoctial element, as follows: - i=0 for a <br/>
  3024.          * - i=1 for k <br/>
  3025.          * - i=2 for h <br/>
  3026.          * - i=3 for q <br/>
  3027.          * - i=4 for p <br/>
  3028.          * - i=5 for λ <br/>
  3029.          * </p>
  3030.          */
  3031.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] dij;

  3033.         /**
  3034.          * The coefficients C<sub>i</sub><sup>j</sup>.
  3035.          * <p>
  3036.          * The index order is cij[j][i] <br/>
  3037.          * i corresponds to the equinoctial element, as follows: <br/>
  3038.          * - i=0 for a <br/>
  3039.          * - i=1 for k <br/>
  3040.          * - i=2 for h <br/>
  3041.          * - i=3 for q <br/>
  3042.          * - i=4 for p <br/>
  3043.          * - i=5 for λ <br/>
  3044.          * </p>
  3045.          */
  3046.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] cij;

  3048.         /**
  3049.          * The coefficients S<sub>i</sub><sup>j</sup>.
  3050.          * <p>
  3051.          * The index order is sij[j][i] <br/>
  3052.          * i corresponds to the equinoctial element, as follows: <br/>
  3053.          * - i=0 for a <br/>
  3054.          * - i=1 for k <br/>
  3055.          * - i=2 for h <br/>
  3056.          * - i=3 for q <br/>
  3057.          * - i=4 for p <br/>
  3058.          * - i=5 for λ <br/>
  3059.          * </p>
  3060.          */
  3061.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] sij;

  3063.         /**
  3064.          * Simple constructor.
  3065.          * @param jMax                maximum value for j index
  3066.          * @param interpolationPoints number of points used in the interpolation process
  3067.          */
  3068.         @SuppressWarnings("unchecked")
  3069.         FieldSlot(final int jMax, final int interpolationPoints) {

  3071.             dij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3072.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, 3);
  3073.             cij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3074.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, jMax + 1);
  3075.             sij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3076.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, jMax + 1);

  3078.             // Initialize the C<sub>i</sub><sup>j</sup>, S<sub>i</sub><sup>j</sup> and
  3079.             // D<sub>i</sub><sup>j</sup> coefficients
  3080.             for (int j = 0; j <= jMax; j++) {
  3081.                 cij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3082.                 if (j > 0) {
  3083.                     sij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3084.                 }
  3085.                 // Initialize only the non-zero D<sub>i</sub><sup>j</sup> coefficients
  3086.                 if (j == 1 || j == 2) {
  3087.                     dij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3088.                 }
  3089.             }

  3091.         }

  3093.     }

  3095. }
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