AbstractGaussianContribution.java

  1. /* Copyright 2002-2022 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 <T extends CalculusFieldElement<T>> void init(final FieldSpacecraftState<T> initialState, final FieldAbsoluteDate<T> target) {
  198.         // Initialize the numerical force model
  199.         contribution.init(initialState, target);
  200.     }

  202.     /** {@inheritDoc} */
  203.     @Override
  204.     public List<ParameterDriver> getParametersDrivers() {

  206.         // Parameter drivers
  207.         final List<ParameterDriver> drivers = new ArrayList<>();

  209.         // Loop on drivers (without central attraction coefficient driver)
  210.         for (final ParameterDriver driver : getParametersDriversWithoutMu()) {
  211.             drivers.add(driver);
  212.         }

  214.         // We put central attraction coefficient driver at the end of the array
  215.         drivers.add(gmParameterDriver);
  216.         return drivers;

  218.     }

  220.     /**
  221.      * Get the drivers for force model parameters except the one for the central
  222.      * attraction coefficient.
  223.      * <p>
  224.      * The driver for central attraction coefficient is automatically added at the
  225.      * last element of the {@link ParameterDriver} array into
  226.      * {@link #getParametersDrivers()} method.
  227.      * </p>
  228.      * @return drivers for force model parameters
  229.      */
  230.     protected abstract List<ParameterDriver> getParametersDriversWithoutMu();

  232.     /** {@inheritDoc} */
  233.     @Override
  234.     public List<ShortPeriodTerms> initializeShortPeriodTerms(final AuxiliaryElements auxiliaryElements, final PropagationType type,
  235.             final double[] parameters) {

  237.         final List<ShortPeriodTerms> list = new ArrayList<ShortPeriodTerms>();
  238.         gaussianSPCoefs = new GaussianShortPeriodicCoefficients(coefficientsKeyPrefix, JMAX, INTERPOLATION_POINTS,
  239.                 new TimeSpanMap<Slot>(new Slot(JMAX, INTERPOLATION_POINTS)));
  240.         list.add(gaussianSPCoefs);
  241.         return list;

  243.     }

  245.     /** {@inheritDoc} */
  246.     @Override
  247.     public <T extends CalculusFieldElement<T>> List<FieldShortPeriodTerms<T>> initializeShortPeriodTerms(
  248.             final FieldAuxiliaryElements<T> auxiliaryElements, final PropagationType type, final T[] parameters) {

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

  252.         final FieldGaussianShortPeriodicCoefficients<T> fgspc = new FieldGaussianShortPeriodicCoefficients<>(
  253.                 coefficientsKeyPrefix, JMAX, INTERPOLATION_POINTS,
  254.                 new FieldTimeSpanMap<>(new FieldSlot<>(JMAX, INTERPOLATION_POINTS), field));
  255.         gaussianFieldSPCoefs.put(field, fgspc);
  256.         return Collections.singletonList(fgspc);
  257.     }

  259.     /**
  260.      * Performs initialization at each integration step for the current force model.
  261.      * <p>
  262.      * This method aims at being called before mean elements rates computation.
  263.      * </p>
  264.      * @param auxiliaryElements auxiliary elements related to the current orbit
  265.      * @param parameters        parameters values of the force model parameters
  266.      * @return new force model context
  267.      */
  268.     private AbstractGaussianContributionContext initializeStep(final AuxiliaryElements auxiliaryElements,
  269.             final double[] parameters) {
  270.         return new AbstractGaussianContributionContext(auxiliaryElements, parameters);
  271.     }

  273.     /**
  274.      * Performs initialization at each integration step for the current force model.
  275.      * <p>
  276.      * This method aims at being called before mean elements rates computation.
  277.      * </p>
  278.      * @param <T>               type of the elements
  279.      * @param auxiliaryElements auxiliary elements related to the current orbit
  280.      * @param parameters        parameters values of the force model parameters
  281.      * @return new force model context
  282.      */
  283.     private <T extends CalculusFieldElement<T>> FieldAbstractGaussianContributionContext<T> initializeStep(
  284.             final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters) {
  285.         return new FieldAbstractGaussianContributionContext<>(auxiliaryElements, parameters);
  286.     }

  288.     /** {@inheritDoc} */
  289.     @Override
  290.     public double[] getMeanElementRate(final SpacecraftState state, final AuxiliaryElements auxiliaryElements,
  291.             final double[] parameters) {

  293.         // Container for attributes
  294.         final AbstractGaussianContributionContext context = initializeStep(auxiliaryElements, parameters);
  295.         double[] meanElementRate = new double[6];
  296.         // Computes the limits for the integral
  297.         final double[] ll = getLLimits(state, auxiliaryElements);
  298.         // Computes integrated mean element rates if Llow < Lhigh
  299.         if (ll[0] < ll[1]) {
  300.             meanElementRate = getMeanElementRate(state, integrator, ll[0], ll[1], context, parameters);
  301.             if (isDirty) {
  302.                 boolean next = true;
  303.                 for (int i = 0; i < MAX_ORDER_RANK && next; i++) {
  304.                     final double[] meanRates = getMeanElementRate(state, new GaussQuadrature(GAUSS_ORDER[i]), ll[0],
  305.                             ll[1], context, parameters);
  306.                     if (getRatesDiff(meanElementRate, meanRates, context) < threshold) {
  307.                         integrator = new GaussQuadrature(GAUSS_ORDER[i]);
  308.                         next = false;
  309.                     }
  310.                 }
  311.                 isDirty = false;
  312.             }
  313.         }
  314.         return meanElementRate;
  315.     }

  317.     /** {@inheritDoc} */
  318.     @Override
  319.     public <T extends CalculusFieldElement<T>> T[] getMeanElementRate(final FieldSpacecraftState<T> state,
  320.             final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters) {

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

  326.         T[] meanElementRate = MathArrays.buildArray(field, 6);
  327.         // Computes the limits for the integral
  328.         final T[] ll = getLLimits(state, auxiliaryElements);
  329.         // Computes integrated mean element rates if Llow < Lhigh
  330.         if (ll[0].getReal() < ll[1].getReal()) {
  331.             meanElementRate = getMeanElementRate(state, integrator, ll[0], ll[1], context, parameters);
  332.             if (isDirty) {
  333.                 boolean next = true;
  334.                 for (int i = 0; i < MAX_ORDER_RANK && next; i++) {
  335.                     final T[] meanRates = getMeanElementRate(state, new GaussQuadrature(GAUSS_ORDER[i]), ll[0], ll[1],
  336.                             context, parameters);
  337.                     if (getRatesDiff(meanElementRate, meanRates, context).getReal() < threshold) {
  338.                         integrator = new GaussQuadrature(GAUSS_ORDER[i]);
  339.                         next = false;
  340.                     }
  341.                 }
  342.                 isDirty = false;
  343.             }
  344.         }

  346.         return meanElementRate;
  347.     }

  349.     /**
  350.      * Compute the limits in L, the true longitude, for integration.
  351.      *
  352.      * @param state             current state information: date, kinematics,
  353.      *                          attitude
  354.      * @param auxiliaryElements auxiliary elements related to the current orbit
  355.      * @return the integration limits in L
  356.      */
  357.     protected abstract double[] getLLimits(SpacecraftState state, AuxiliaryElements auxiliaryElements);

  359.     /**
  360.      * Compute the limits in L, the true longitude, for integration.
  361.      *
  362.      * @param <T>               type of the elements
  363.      * @param state             current state information: date, kinematics,
  364.      *                          attitude
  365.      * @param auxiliaryElements auxiliary elements related to the current orbit
  366.      * @return the integration limits in L
  367.      */
  368.     protected abstract <T extends CalculusFieldElement<T>> T[] getLLimits(FieldSpacecraftState<T> state,
  369.             FieldAuxiliaryElements<T> auxiliaryElements);

  371.     /**
  372.      * Computes the mean equinoctial elements rates da<sub>i</sub> / dt.
  373.      *
  374.      * @param state      current state
  375.      * @param gauss      Gauss quadrature
  376.      * @param low        lower bound of the integral interval
  377.      * @param high       upper bound of the integral interval
  378.      * @param context    container for attributes
  379.      * @param parameters values of the force model parameters
  380.      * @return the mean element rates
  381.      */
  382.     protected double[] getMeanElementRate(final SpacecraftState state, final GaussQuadrature gauss, final double low,
  383.             final double high, final AbstractGaussianContributionContext context, final double[] parameters) {

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

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

  390.         // Constant multiplier for integral
  391.         final double coef = 1. / (2. * FastMath.PI * auxiliaryElements.getB());
  392.         // Corrects mean element rates
  393.         for (int i = 0; i < 6; i++) {
  394.             meanElementRate[i] *= coef;
  395.         }
  396.         return meanElementRate;
  397.     }

  399.     /**
  400.      * Computes the mean equinoctial elements rates da<sub>i</sub> / dt.
  401.      *
  402.      * @param <T>        type of the elements
  403.      * @param state      current state
  404.      * @param gauss      Gauss quadrature
  405.      * @param low        lower bound of the integral interval
  406.      * @param high       upper bound of the integral interval
  407.      * @param context    container for attributes
  408.      * @param parameters values of the force model parameters
  409.      * @return the mean element rates
  410.      */
  411.     protected <T extends CalculusFieldElement<T>> T[] getMeanElementRate(final FieldSpacecraftState<T> state,
  412.             final GaussQuadrature gauss, final T low, final T high,
  413.             final FieldAbstractGaussianContributionContext<T> context, final T[] parameters) {

  415.         // Field
  416.         final Field<T> field = context.getA().getField();

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

  421.         final T[] meanElementRate = gauss.integrate(new FieldIntegrableFunction<>(state, true, 0, parameters, field),
  422.                 low, high, field);
  423.         // Constant multiplier for integral
  424.         final T coef = auxiliaryElements.getB().multiply(low.getPi()).multiply(2.).reciprocal();
  425.         // Corrects mean element rates
  426.         for (int i = 0; i < 6; i++) {
  427.             meanElementRate[i] = meanElementRate[i].multiply(coef);
  428.         }
  429.         return meanElementRate;
  430.     }

  432.     /**
  433.      * Estimates the weighted magnitude of the difference between 2 sets of
  434.      * equinoctial elements rates.
  435.      *
  436.      * @param meanRef reference rates
  437.      * @param meanCur current rates
  438.      * @param context container for attributes
  439.      * @return estimated magnitude of weighted differences
  440.      */
  441.     private double getRatesDiff(final double[] meanRef, final double[] meanCur,
  442.             final AbstractGaussianContributionContext context) {

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

  447.         double maxDiff = FastMath.abs(meanRef[0] - meanCur[0]) / auxiliaryElements.getSma();
  448.         // Corrects mean element rates
  449.         for (int i = 1; i < meanRef.length; i++) {
  450.             maxDiff = FastMath.max(maxDiff, FastMath.abs(meanRef[i] - meanCur[i]));
  451.         }
  452.         return maxDiff;
  453.     }

  455.     /**
  456.      * Estimates the weighted magnitude of the difference between 2 sets of
  457.      * equinoctial elements rates.
  458.      *
  459.      * @param <T>     type of the elements
  460.      * @param meanRef reference rates
  461.      * @param meanCur current rates
  462.      * @param context container for attributes
  463.      * @return estimated magnitude of weighted differences
  464.      */
  465.     private <T extends CalculusFieldElement<T>> T getRatesDiff(final T[] meanRef, final T[] meanCur,
  466.             final FieldAbstractGaussianContributionContext<T> context) {

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

  471.         T maxDiff = FastMath.abs(meanRef[0].subtract(meanCur[0])).divide(auxiliaryElements.getSma());
  472.         ;
  473.         // Corrects mean element rates
  474.         for (int i = 1; i < meanRef.length; i++) {
  475.             maxDiff = FastMath.max(maxDiff, FastMath.abs(meanRef[i].subtract(meanCur[i])));
  476.         }
  477.         return maxDiff;
  478.     }

  480.     /** {@inheritDoc} */
  481.     @Override
  482.     public void registerAttitudeProvider(final AttitudeProvider provider) {
  483.         this.attitudeProvider = provider;
  484.     }

  486.     /** {@inheritDoc} */
  487.     @Override
  488.     public void updateShortPeriodTerms(final double[] parameters, final SpacecraftState... meanStates) {

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

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

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

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

  502.             // Generate the Cij and Sij coefficients
  503.             final FourierCjSjCoefficients fourierCjSj = new FourierCjSjCoefficients(meanState, JMAX, auxiliaryElements,
  504.                     parameters);

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

  509.             gaussianSPCoefs.computeCoefficients(meanState, slot, fourierCjSj, uijvij, context.getMeanMotion(),
  510.                     auxiliaryElements.getSma());

  512.         }

  514.     }

  516.     /** {@inheritDoc} */
  517.     @Override
  518.     @SuppressWarnings("unchecked")
  519.     public <T extends CalculusFieldElement<T>> void updateShortPeriodTerms(final T[] parameters,
  520.             final FieldSpacecraftState<T>... meanStates) {

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

  525.         final FieldGaussianShortPeriodicCoefficients<T> fgspc = (FieldGaussianShortPeriodicCoefficients<T>) gaussianFieldSPCoefs
  526.                 .get(field);
  527.         final FieldSlot<T> slot = fgspc.createSlot(meanStates);
  528.         for (final FieldSpacecraftState<T> meanState : meanStates) {

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

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

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

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

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

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

  550.         }

  552.     }

  554.     /**
  555.      * Compute the auxiliary quantities ρ<sub>j</sub> and σ<sub>j</sub>.
  556.      * <p>
  557.      * The expressions used are equations 2.5.3-(4) from the Danielson paper. <br/>
  558.      * ρ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>C<sub>j</sub>(k, h) <br/>
  559.      * σ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>S<sub>j</sub>(k, h) <br/>
  560.      * </p>
  561.      * @param date              current date
  562.      * @param auxiliaryElements auxiliary elements related to the current orbit
  563.      * @return computed coefficients
  564.      */
  565.     private double[][] computeRhoSigmaCoefficients(final AbsoluteDate date, final AuxiliaryElements auxiliaryElements) {
  566.         final double[][] currentRhoSigmaj = new double[2][3 * JMAX + 1];
  567.         final CjSjCoefficient cjsjKH = new CjSjCoefficient(auxiliaryElements.getK(), auxiliaryElements.getH());
  568.         final double b = 1. / (1 + auxiliaryElements.getB());

  570.         // (-b)<sup>j</sup>
  571.         double mbtj = 1;

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

  575.             // Compute current rho and sigma;
  576.             mbtj *= -b;
  577.             final double coef = (1 + j * auxiliaryElements.getB()) * mbtj;
  578.             currentRhoSigmaj[0][j] = coef * cjsjKH.getCj(j);
  579.             currentRhoSigmaj[1][j] = coef * cjsjKH.getSj(j);
  580.         }
  581.         return currentRhoSigmaj;
  582.     }

  584.     /**
  585.      * Compute the auxiliary quantities ρ<sub>j</sub> and σ<sub>j</sub>.
  586.      * <p>
  587.      * The expressions used are equations 2.5.3-(4) from the Danielson paper. <br/>
  588.      * ρ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>C<sub>j</sub>(k, h) <br/>
  589.      * σ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>S<sub>j</sub>(k, h) <br/>
  590.      * </p>
  591.      * @param <T>     type of the elements
  592.      * @param date    current date
  593.      * @param context container for attributes
  594.      * @param field   field used by default
  595.      * @return computed coefficients
  596.      */
  597.     private <T extends CalculusFieldElement<T>> T[][] computeRhoSigmaCoefficients(final FieldAbsoluteDate<T> date,
  598.             final FieldAbstractGaussianContributionContext<T> context, final Field<T> field) {
  599.         // zero
  600.         final T zero = field.getZero();

  602.         final FieldAuxiliaryElements<T> auxiliaryElements = context.getFieldAuxiliaryElements();
  603.         final T[][] currentRhoSigmaj = MathArrays.buildArray(field, 2, 3 * JMAX + 1);
  604.         final FieldCjSjCoefficient<T> cjsjKH = new FieldCjSjCoefficient<>(auxiliaryElements.getK(),
  605.                 auxiliaryElements.getH(), field);
  606.         final T b = auxiliaryElements.getB().add(1.).reciprocal();

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

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

  613.             // Compute current rho and sigma;
  614.             mbtj = mbtj.multiply(b.negate());
  615.             final T coef = mbtj.multiply(auxiliaryElements.getB().multiply(j).add(1.));
  616.             currentRhoSigmaj[0][j] = coef.multiply(cjsjKH.getCj(j));
  617.             currentRhoSigmaj[1][j] = coef.multiply(cjsjKH.getSj(j));
  618.         }
  619.         return currentRhoSigmaj;
  620.     }

  622.     /**
  623.      * Internal class for numerical quadrature.
  624.      * <p>
  625.      * This class is a rewrite of {@link IntegrableFunction} for field elements
  626.      * </p>
  627.      */
  628.     protected class FieldIntegrableFunction<T extends CalculusFieldElement<T>>
  629.             implements CalculusFieldUnivariateVectorFunction<T> {

  631.         /** Current state. */
  632.         private final FieldSpacecraftState<T> state;

  634.         /**
  635.          * Signal that this class is used to compute the values required by the mean
  636.          * element variations or by the short periodic element variations.
  637.          */
  638.         private final boolean meanMode;

  640.         /**
  641.          * The j index.
  642.          * <p>
  643.          * Used only for short periodic variation. Ignored for mean elements variation.
  644.          * </p>
  645.          */
  646.         private final int j;

  648.         /** Container for attributes. */
  649.         private final FieldAbstractGaussianContributionContext<T> context;

  651.         /** Auxiliary Elements. */
  652.         private final FieldAuxiliaryElements<T> auxiliaryElements;

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

  657.         /**
  658.          * Build a new instance with a new field.
  659.          * @param state      current state information: date, kinematics, attitude
  660.          * @param meanMode   if true return the value associated to the mean elements
  661.          *                   variation, if false return the values associated to the
  662.          *                   short periodic elements variation
  663.          * @param j          the j index. used only for short periodic variation.
  664.          *                   Ignored for mean elements variation.
  665.          * @param parameters values of the force model parameters
  666.          * @param field      field utilized by default
  667.          */
  668.         public FieldIntegrableFunction(final FieldSpacecraftState<T> state, final boolean meanMode, final int j,
  669.                 final T[] parameters, final Field<T> field) {

  671.             this.meanMode = meanMode;
  672.             this.j = j;
  673.             this.parameters = parameters.clone();
  674.             this.auxiliaryElements = new FieldAuxiliaryElements<>(state.getOrbit(), I);
  675.             this.context = new FieldAbstractGaussianContributionContext<>(auxiliaryElements, this.parameters);
  676.             // remove derivatives from state
  677.             final T[] stateVector = MathArrays.buildArray(field, 6);
  678.             OrbitType.EQUINOCTIAL.mapOrbitToArray(state.getOrbit(), PositionAngle.TRUE, stateVector, null);
  679.             final FieldOrbit<T> fixedOrbit = OrbitType.EQUINOCTIAL.mapArrayToOrbit(stateVector, null,
  680.                     PositionAngle.TRUE, state.getDate(), context.getMu(), state.getFrame());
  681.             this.state = new FieldSpacecraftState<>(fixedOrbit, state.getAttitude(), state.getMass());
  682.         }

  684.         /** {@inheritDoc} */
  685.         @Override
  686.         public T[] value(final T x) {

  688.             // Parameters for array building
  689.             final Field<T> field = auxiliaryElements.getDate().getField();
  690.             final int dimension = 6;

  692.             // Compute the time difference from the true longitude difference
  693.             final T shiftedLm = trueToMean(x);
  694.             final T dLm = shiftedLm.subtract(auxiliaryElements.getLM());
  695.             final T dt = dLm.divide(context.getMeanMotion());

  697.             final FieldSinCos<T> scL = FastMath.sinCos(x);
  698.             final T cosL = scL.cos();
  699.             final T sinL = scL.sin();
  700.             final T roa  = auxiliaryElements.getB().multiply(auxiliaryElements.getB()).divide(auxiliaryElements.getH().multiply(sinL).add(auxiliaryElements.getK().multiply(cosL)).add(1.));
  701.             final T roa2 = roa.multiply(roa);
  702.             final T r = auxiliaryElements.getSma().multiply(roa);
  703.             final T X = r.multiply(cosL);
  704.             final T Y = r.multiply(sinL);
  705.             final T naob = context.getMeanMotion().multiply(auxiliaryElements.getSma())
  706.                     .divide(auxiliaryElements.getB());
  707.             final T Xdot = naob.multiply(auxiliaryElements.getH().add(sinL)).negate();
  708.             final T Ydot = naob.multiply(auxiliaryElements.getK().add(cosL));
  709.             final FieldVector3D<T> vel = new FieldVector3D<>(Xdot, auxiliaryElements.getVectorF(), Ydot,
  710.                     auxiliaryElements.getVectorG());

  712.             // Compute acceleration
  713.             FieldVector3D<T> acc = FieldVector3D.getZero(field);

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

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

  724.             // Get the corresponding attitude
  725.             final FieldAttitude<T> recomposedAttitude = attitudeProvider.getAttitude(recomposedOrbit,
  726.                     recomposedOrbit.getDate(), recomposedOrbit.getFrame());

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

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

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

  749.             // Compute mean elements rates
  750.             T[] val = null;
  751.             if (meanMode) {
  752.                 val = MathArrays.buildArray(field, dimension);
  753.                 for (int i = 0; i < 6; i++) {
  754.                     // da<sub>i</sub>/dt
  755.                     val[i] = deriv[i].multiply(roa2);
  756.                 }
  757.             } else {
  758.                 val = MathArrays.buildArray(field, dimension * 2);
  759.                 //Compute cos(j*L) and sin(j*L);
  760.                 final FieldSinCos<T> scjL = FastMath.sinCos(x.multiply(j));
  761.                 final T cosjL = j == 1 ? cosL : scjL.cos();
  762.                 final T sinjL = j == 1 ? sinL : scjL.sin();

  764.                 for (int i = 0; i < 6; i++) {
  765.                     // da<sub>i</sub>/dv * cos(jL)
  766.                     val[i] = deriv[i].multiply(cosjL);
  767.                     // da<sub>i</sub>/dv * sin(jL)
  768.                     val[i + 6] = deriv[i].multiply(sinjL);
  769.                 }
  770.             }

  772.             return val;
  773.         }

  775.         /**
  776.          * Converts true longitude to mean longitude.
  777.          * @param x True longitude
  778.          * @return Eccentric longitude
  779.          */
  780.         private T trueToMean(final T x) {
  781.             return eccentricToMean(trueToEccentric(x));
  782.         }

  784.         /**
  785.          * Converts true longitude to eccentric longitude.
  786.          * @param lv True longitude
  787.          * @return Eccentric longitude
  788.          */
  789.         private T trueToEccentric (final T lv) {
  790.             final FieldSinCos<T> sclV = FastMath.sinCos(lv);
  791.             final T cosLv   = sclV.cos();
  792.             final T sinLv   = sclV.sin();
  793.             final T num     = auxiliaryElements.getH().multiply(cosLv).subtract(auxiliaryElements.getK().multiply(sinLv));
  794.             final T den     = auxiliaryElements.getB().add(auxiliaryElements.getK().multiply(cosLv)).add(auxiliaryElements.getH().multiply(sinLv)).add(1.);
  795.             return FastMath.atan(num.divide(den)).multiply(2.).add(lv);
  796.         }

  798.         /**
  799.          * Converts eccentric longitude to mean longitude.
  800.          * @param le Eccentric longitude
  801.          * @return Mean longitude
  802.          */
  803.         private T eccentricToMean (final T le) {
  804.             final FieldSinCos<T> scle = FastMath.sinCos(le);
  805.             return le.subtract(auxiliaryElements.getK().multiply(scle.sin())).add(auxiliaryElements.getH().multiply(scle.cos()));
  806.         }

  808.         /**
  809.          * Compute δa/δv.
  810.          * @param vel satellite velocity
  811.          * @return δa/δv
  812.          */
  813.         private FieldVector3D<T> getAoV(final FieldVector3D<T> vel) {
  814.             return new FieldVector3D<>(context.getTon2a(), vel);
  815.         }

  817.         /**
  818.          * Compute δh/δv.
  819.          * @param X    satellite position component along f, equinoctial reference frame
  820.          *             1st vector
  821.          * @param Y    satellite position component along g, equinoctial reference frame
  822.          *             2nd vector
  823.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  824.          *             1st vector
  825.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  826.          *             2nd vector
  827.          * @return δh/δv
  828.          */
  829.         private FieldVector3D<T> getHoV(final T X, final T Y, final T Xdot, final T Ydot) {
  830.             final T kf = (Xdot.multiply(Y).multiply(2.).subtract(X.multiply(Ydot))).multiply(context.getOoMU());
  831.             final T kg = X.multiply(Xdot).multiply(context.getOoMU());
  832.             final T kw = auxiliaryElements.getK().multiply(
  833.                     auxiliaryElements.getQ().multiply(Y).multiply(I).subtract(auxiliaryElements.getP().multiply(X)))
  834.                     .multiply(context.getOOAB());
  835.             return new FieldVector3D<>(kf, auxiliaryElements.getVectorF(), kg.negate(), auxiliaryElements.getVectorG(),
  836.                     kw, auxiliaryElements.getVectorW());
  837.         }

  839.         /**
  840.          * Compute δk/δv.
  841.          * @param X    satellite position component along f, equinoctial reference frame
  842.          *             1st vector
  843.          * @param Y    satellite position component along g, equinoctial reference frame
  844.          *             2nd vector
  845.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  846.          *             1st vector
  847.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  848.          *             2nd vector
  849.          * @return δk/δv
  850.          */
  851.         private FieldVector3D<T> getKoV(final T X, final T Y, final T Xdot, final T Ydot) {
  852.             final T kf = Y.multiply(Ydot).multiply(context.getOoMU());
  853.             final T kg = (X.multiply(Ydot).multiply(2.).subtract(Xdot.multiply(Y))).multiply(context.getOoMU());
  854.             final T kw = auxiliaryElements.getH().multiply(
  855.                     auxiliaryElements.getQ().multiply(Y).multiply(I).subtract(auxiliaryElements.getP().multiply(X)))
  856.                     .multiply(context.getOOAB());
  857.             return new FieldVector3D<>(kf.negate(), auxiliaryElements.getVectorF(), kg, auxiliaryElements.getVectorG(),
  858.                     kw.negate(), auxiliaryElements.getVectorW());
  859.         }

  861.         /**
  862.          * Compute δp/δv.
  863.          * @param Y satellite position component along g, equinoctial reference frame
  864.          *          2nd vector
  865.          * @return δp/δv
  866.          */
  867.         private FieldVector3D<T> getPoV(final T Y) {
  868.             return new FieldVector3D<>(context.getCo2AB().multiply(Y), auxiliaryElements.getVectorW());
  869.         }

  871.         /**
  872.          * Compute δq/δv.
  873.          * @param X satellite position component along f, equinoctial reference frame
  874.          *          1st vector
  875.          * @return δq/δv
  876.          */
  877.         private FieldVector3D<T> getQoV(final T X) {
  878.             return new FieldVector3D<>(context.getCo2AB().multiply(X).multiply(I), auxiliaryElements.getVectorW());
  879.         }

  881.         /**
  882.          * Compute δλ/δv.
  883.          * @param X    satellite position component along f, equinoctial reference frame
  884.          *             1st vector
  885.          * @param Y    satellite position component along g, equinoctial reference frame
  886.          *             2nd vector
  887.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  888.          *             1st vector
  889.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  890.          *             2nd vector
  891.          * @return δλ/δv
  892.          */
  893.         private FieldVector3D<T> getLoV(final T X, final T Y, final T Xdot, final T Ydot) {
  894.             final FieldVector3D<T> pos = new FieldVector3D<>(X, auxiliaryElements.getVectorF(), Y,
  895.                     auxiliaryElements.getVectorG());
  896.             final FieldVector3D<T> v2 = new FieldVector3D<>(auxiliaryElements.getK(), getHoV(X, Y, Xdot, Ydot),
  897.                     auxiliaryElements.getH().negate(), getKoV(X, Y, Xdot, Ydot));
  898.             return new FieldVector3D<>(context.getOOA().multiply(-2.), pos, context.getOoBpo(), v2,
  899.                     context.getOOA().multiply(auxiliaryElements.getQ().multiply(Y).multiply(I)
  900.                             .subtract(auxiliaryElements.getP().multiply(X))),
  901.                     auxiliaryElements.getVectorW());
  902.         }

  904.     }

  906.     /** Internal class for numerical quadrature. */
  907.     protected class IntegrableFunction implements UnivariateVectorFunction {

  909.         /** Current state. */
  910.         private final SpacecraftState state;

  912.         /**
  913.          * Signal that this class is used to compute the values required by the mean
  914.          * element variations or by the short periodic element variations.
  915.          */
  916.         private final boolean meanMode;

  918.         /**
  919.          * The j index.
  920.          * <p>
  921.          * Used only for short periodic variation. Ignored for mean elements variation.
  922.          * </p>
  923.          */
  924.         private final int j;

  926.         /** Container for attributes. */
  927.         private final AbstractGaussianContributionContext context;

  929.         /** Auxiliary Elements. */
  930.         private final AuxiliaryElements auxiliaryElements;

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

  935.         /**
  936.          * Build a new instance.
  937.          * @param state      current state information: date, kinematics, attitude
  938.          * @param meanMode   if true return the value associated to the mean elements
  939.          *                   variation, if false return the values associated to the
  940.          *                   short periodic elements variation
  941.          * @param j          the j index. used only for short periodic variation.
  942.          *                   Ignored for mean elements variation.
  943.          * @param parameters values of the force model parameters
  944.          */
  945.         IntegrableFunction(final SpacecraftState state, final boolean meanMode, final int j,
  946.                 final double[] parameters) {

  948.             this.meanMode = meanMode;
  949.             this.j = j;
  950.             this.parameters = parameters.clone();
  951.             this.auxiliaryElements = new AuxiliaryElements(state.getOrbit(), I);
  952.             this.context = new AbstractGaussianContributionContext(auxiliaryElements, this.parameters);
  953.             // remove derivatives from state
  954.             final double[] stateVector = new double[6];
  955.             OrbitType.EQUINOCTIAL.mapOrbitToArray(state.getOrbit(), PositionAngle.TRUE, stateVector, null);
  956.             final Orbit fixedOrbit = OrbitType.EQUINOCTIAL.mapArrayToOrbit(stateVector, null, PositionAngle.TRUE,
  957.                     state.getDate(), context.getMu(), state.getFrame());
  958.             this.state = new SpacecraftState(fixedOrbit, state.getAttitude(), state.getMass());
  959.         }

  961.         /** {@inheritDoc} */
  962.         @Override
  963.         public double[] value(final double x) {

  965.             // Compute the time difference from the true longitude difference
  966.             final double shiftedLm = trueToMean(x);
  967.             final double dLm = shiftedLm - auxiliaryElements.getLM();
  968.             final double dt = dLm / context.getMeanMotion();

  970.             final SinCos scL  = FastMath.sinCos(x);
  971.             final double cosL = scL.cos();
  972.             final double sinL = scL.sin();
  973.             final double roa  = auxiliaryElements.getB() * auxiliaryElements.getB() / (1. + auxiliaryElements.getH() * sinL + auxiliaryElements.getK() * cosL);
  974.             final double roa2 = roa * roa;
  975.             final double r = auxiliaryElements.getSma() * roa;
  976.             final double X = r * cosL;
  977.             final double Y = r * sinL;
  978.             final double naob = context.getMeanMotion() * auxiliaryElements.getSma() / auxiliaryElements.getB();
  979.             final double Xdot = -naob * (auxiliaryElements.getH() + sinL);
  980.             final double Ydot = naob * (auxiliaryElements.getK() + cosL);
  981.             final Vector3D vel = new Vector3D(Xdot, auxiliaryElements.getVectorF(), Ydot,
  982.                     auxiliaryElements.getVectorG());

  984.             // Compute acceleration
  985.             Vector3D acc = Vector3D.ZERO;

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

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

  995.             // Get the corresponding attitude
  996.             final Attitude recomposedAttitude = attitudeProvider.getAttitude(recomposedOrbit, recomposedOrbit.getDate(),
  997.                     recomposedOrbit.getFrame());

  999.             // create shifted SpacecraftState with attitude at specified time
  1000.             final SpacecraftState shiftedState = new SpacecraftState(recomposedOrbit, recomposedAttitude,
  1001.                     state.getMass());

  1003.             acc = contribution.acceleration(shiftedState, parameters);

  1005.             // Compute the derivatives of the elements by the speed
  1006.             final double[] deriv = new double[6];
  1007.             // da/dv
  1008.             deriv[0] = getAoV(vel).dotProduct(acc);
  1009.             // dex/dv
  1010.             deriv[1] = getKoV(X, Y, Xdot, Ydot).dotProduct(acc);
  1011.             // dey/dv
  1012.             deriv[2] = getHoV(X, Y, Xdot, Ydot).dotProduct(acc);
  1013.             // dhx/dv
  1014.             deriv[3] = getQoV(X).dotProduct(acc);
  1015.             // dhy/dv
  1016.             deriv[4] = getPoV(Y).dotProduct(acc);
  1017.             // dλ/dv
  1018.             deriv[5] = getLoV(X, Y, Xdot, Ydot).dotProduct(acc);

  1020.             // Compute mean elements rates
  1021.             double[] val = null;
  1022.             if (meanMode) {
  1023.                 val = new double[6];
  1024.                 for (int i = 0; i < 6; i++) {
  1025.                     // da<sub>i</sub>/dt
  1026.                     val[i] = roa2 * deriv[i];
  1027.                 }
  1028.             } else {
  1029.                 val = new double[12];
  1030.                 //Compute cos(j*L) and sin(j*L);
  1031.                 final SinCos scjL  = FastMath.sinCos(j * x);
  1032.                 final double cosjL = j == 1 ? cosL : scjL.cos();
  1033.                 final double sinjL = j == 1 ? sinL : scjL.sin();

  1035.                 for (int i = 0; i < 6; i++) {
  1036.                     // da<sub>i</sub>/dv * cos(jL)
  1037.                     val[i] = cosjL * deriv[i];
  1038.                     // da<sub>i</sub>/dv * sin(jL)
  1039.                     val[i + 6] = sinjL * deriv[i];
  1040.                 }
  1041.             }
  1042.             return val;
  1043.         }

  1045.         /**
  1046.          * Converts true longitude to eccentric longitude.
  1047.          * @param lv True longitude
  1048.          * @return Eccentric longitude
  1049.          */
  1050.         private double trueToEccentric (final double lv) {
  1051.             final SinCos scLv    = FastMath.sinCos(lv);
  1052.             final double num     = auxiliaryElements.getH() * scLv.cos() - auxiliaryElements.getK() * scLv.sin();
  1053.             final double den     = auxiliaryElements.getB() + 1. + auxiliaryElements.getK() * scLv.cos() + auxiliaryElements.getH() * scLv.sin();
  1054.             return lv + 2. * FastMath.atan(num / den);
  1055.         }

  1057.         /**
  1058.          * Converts eccentric longitude to mean longitude.
  1059.          * @param le Eccentric longitude
  1060.          * @return Mean longitude
  1061.          */
  1062.         private double eccentricToMean (final double le) {
  1063.             final SinCos scLe = FastMath.sinCos(le);
  1064.             return le - auxiliaryElements.getK() * scLe.sin() + auxiliaryElements.getH() * scLe.cos();
  1065.         }

  1067.         /**
  1068.          * Converts true longitude to mean longitude.
  1069.          * @param lv True longitude
  1070.          * @return Eccentric longitude
  1071.          */
  1072.         private double trueToMean(final double lv) {
  1073.             return eccentricToMean(trueToEccentric(lv));
  1074.         }

  1076.         /**
  1077.          * Compute δa/δv.
  1078.          * @param vel satellite velocity
  1079.          * @return δa/δv
  1080.          */
  1081.         private Vector3D getAoV(final Vector3D vel) {
  1082.             return new Vector3D(context.getTon2a(), vel);
  1083.         }

  1085.         /**
  1086.          * Compute δh/δv.
  1087.          * @param X    satellite position component along f, equinoctial reference frame
  1088.          *             1st vector
  1089.          * @param Y    satellite position component along g, equinoctial reference frame
  1090.          *             2nd vector
  1091.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1092.          *             1st vector
  1093.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1094.          *             2nd vector
  1095.          * @return δh/δv
  1096.          */
  1097.         private Vector3D getHoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1098.             final double kf = (2. * Xdot * Y - X * Ydot) * context.getOoMU();
  1099.             final double kg = X * Xdot * context.getOoMU();
  1100.             final double kw = auxiliaryElements.getK() *
  1101.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOAB();
  1102.             return new Vector3D(kf, auxiliaryElements.getVectorF(), -kg, auxiliaryElements.getVectorG(), kw,
  1103.                     auxiliaryElements.getVectorW());
  1104.         }

  1106.         /**
  1107.          * Compute δk/δv.
  1108.          * @param X    satellite position component along f, equinoctial reference frame
  1109.          *             1st vector
  1110.          * @param Y    satellite position component along g, equinoctial reference frame
  1111.          *             2nd vector
  1112.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1113.          *             1st vector
  1114.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1115.          *             2nd vector
  1116.          * @return δk/δv
  1117.          */
  1118.         private Vector3D getKoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1119.             final double kf = Y * Ydot * context.getOoMU();
  1120.             final double kg = (2. * X * Ydot - Xdot * Y) * context.getOoMU();
  1121.             final double kw = auxiliaryElements.getH() *
  1122.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOAB();
  1123.             return new Vector3D(-kf, auxiliaryElements.getVectorF(), kg, auxiliaryElements.getVectorG(), -kw,
  1124.                     auxiliaryElements.getVectorW());
  1125.         }

  1127.         /**
  1128.          * Compute δp/δv.
  1129.          * @param Y satellite position component along g, equinoctial reference frame
  1130.          *          2nd vector
  1131.          * @return δp/δv
  1132.          */
  1133.         private Vector3D getPoV(final double Y) {
  1134.             return new Vector3D(context.getCo2AB() * Y, auxiliaryElements.getVectorW());
  1135.         }

  1137.         /**
  1138.          * Compute δq/δv.
  1139.          * @param X satellite position component along f, equinoctial reference frame
  1140.          *          1st vector
  1141.          * @return δq/δv
  1142.          */
  1143.         private Vector3D getQoV(final double X) {
  1144.             return new Vector3D(I * context.getCo2AB() * X, auxiliaryElements.getVectorW());
  1145.         }

  1147.         /**
  1148.          * Compute δλ/δv.
  1149.          * @param X    satellite position component along f, equinoctial reference frame
  1150.          *             1st vector
  1151.          * @param Y    satellite position component along g, equinoctial reference frame
  1152.          *             2nd vector
  1153.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1154.          *             1st vector
  1155.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1156.          *             2nd vector
  1157.          * @return δλ/δv
  1158.          */
  1159.         private Vector3D getLoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1160.             final Vector3D pos = new Vector3D(X, auxiliaryElements.getVectorF(), Y, auxiliaryElements.getVectorG());
  1161.             final Vector3D v2 = new Vector3D(auxiliaryElements.getK(), getHoV(X, Y, Xdot, Ydot),
  1162.                     -auxiliaryElements.getH(), getKoV(X, Y, Xdot, Ydot));
  1163.             return new Vector3D(-2. * context.getOOA(), pos, context.getOoBpo(), v2,
  1164.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOA(),
  1165.                     auxiliaryElements.getVectorW());
  1166.         }

  1168.     }

  1170.     /**
  1171.      * Class used to {@link #integrate(UnivariateVectorFunction, double, double)
  1172.      * integrate} a {@link org.hipparchus.analysis.UnivariateVectorFunction
  1173.      * function} of the orbital elements using the Gaussian quadrature rule to get
  1174.      * the acceleration.
  1175.      */
  1176.     protected static class GaussQuadrature {

  1178.         // Points and weights for the available quadrature orders

  1180.         /** Points for quadrature of order 12. */
  1181.         private static final double[] P_12 = { -0.98156063424671910000, -0.90411725637047490000,
  1182.             -0.76990267419430470000, -0.58731795428661740000, -0.36783149899818024000, -0.12523340851146890000,
  1183.             0.12523340851146890000, 0.36783149899818024000, 0.58731795428661740000, 0.76990267419430470000,
  1184.             0.90411725637047490000, 0.98156063424671910000 };

  1186.         /** Weights for quadrature of order 12. */
  1187.         private static final double[] W_12 = { 0.04717533638651220000, 0.10693932599531830000, 0.16007832854334633000,
  1188.             0.20316742672306584000, 0.23349253653835478000, 0.24914704581340286000, 0.24914704581340286000,
  1189.             0.23349253653835478000, 0.20316742672306584000, 0.16007832854334633000, 0.10693932599531830000,
  1190.             0.04717533638651220000 };

  1192.         /** Points for quadrature of order 16. */
  1193.         private static final double[] P_16 = { -0.98940093499164990000, -0.94457502307323260000,
  1194.             -0.86563120238783160000, -0.75540440835500310000, -0.61787624440264380000, -0.45801677765722737000,
  1195.             -0.28160355077925890000, -0.09501250983763745000, 0.09501250983763745000, 0.28160355077925890000,
  1196.             0.45801677765722737000, 0.61787624440264380000, 0.75540440835500310000, 0.86563120238783160000,
  1197.             0.94457502307323260000, 0.98940093499164990000 };

  1199.         /** Weights for quadrature of order 16. */
  1200.         private static final double[] W_16 = { 0.02715245941175405800, 0.06225352393864777000, 0.09515851168249283000,
  1201.             0.12462897125553388000, 0.14959598881657685000, 0.16915651939500256000, 0.18260341504492360000,
  1202.             0.18945061045506847000, 0.18945061045506847000, 0.18260341504492360000, 0.16915651939500256000,
  1203.             0.14959598881657685000, 0.12462897125553388000, 0.09515851168249283000, 0.06225352393864777000,
  1204.             0.02715245941175405800 };

  1206.         /** Points for quadrature of order 20. */
  1207.         private static final double[] P_20 = { -0.99312859918509490000, -0.96397192727791390000,
  1208.             -0.91223442825132600000, -0.83911697182221890000, -0.74633190646015080000, -0.63605368072651510000,
  1209.             -0.51086700195082700000, -0.37370608871541955000, -0.22778585114164507000, -0.07652652113349734000,
  1210.             0.07652652113349734000, 0.22778585114164507000, 0.37370608871541955000, 0.51086700195082700000,
  1211.             0.63605368072651510000, 0.74633190646015080000, 0.83911697182221890000, 0.91223442825132600000,
  1212.             0.96397192727791390000, 0.99312859918509490000 };

  1214.         /** Weights for quadrature of order 20. */
  1215.         private static final double[] W_20 = { 0.01761400713915226400, 0.04060142980038684000, 0.06267204833410904000,
  1216.             0.08327674157670477000, 0.10193011981724048000, 0.11819453196151844000, 0.13168863844917678000,
  1217.             0.14209610931838212000, 0.14917298647260380000, 0.15275338713072600000, 0.15275338713072600000,
  1218.             0.14917298647260380000, 0.14209610931838212000, 0.13168863844917678000, 0.11819453196151844000,
  1219.             0.10193011981724048000, 0.08327674157670477000, 0.06267204833410904000, 0.04060142980038684000,
  1220.             0.01761400713915226400 };

  1222.         /** Points for quadrature of order 24. */
  1223.         private static final double[] P_24 = { -0.99518721999702130000, -0.97472855597130950000,
  1224.             -0.93827455200273270000, -0.88641552700440100000, -0.82000198597390300000, -0.74012419157855440000,
  1225.             -0.64809365193697550000, -0.54542147138883950000, -0.43379350762604520000, -0.31504267969616340000,
  1226.             -0.19111886747361634000, -0.06405689286260563000, 0.06405689286260563000, 0.19111886747361634000,
  1227.             0.31504267969616340000, 0.43379350762604520000, 0.54542147138883950000, 0.64809365193697550000,
  1228.             0.74012419157855440000, 0.82000198597390300000, 0.88641552700440100000, 0.93827455200273270000,
  1229.             0.97472855597130950000, 0.99518721999702130000 };

  1231.         /** Weights for quadrature of order 24. */
  1232.         private static final double[] W_24 = { 0.01234122979998733500, 0.02853138862893380600, 0.04427743881741981000,
  1233.             0.05929858491543691500, 0.07334648141108027000, 0.08619016153195320000, 0.09761865210411391000,
  1234.             0.10744427011596558000, 0.11550566805372553000, 0.12167047292780335000, 0.12583745634682825000,
  1235.             0.12793819534675221000, 0.12793819534675221000, 0.12583745634682825000, 0.12167047292780335000,
  1236.             0.11550566805372553000, 0.10744427011596558000, 0.09761865210411391000, 0.08619016153195320000,
  1237.             0.07334648141108027000, 0.05929858491543691500, 0.04427743881741981000, 0.02853138862893380600,
  1238.             0.01234122979998733500 };

  1240.         /** Points for quadrature of order 32. */
  1241.         private static final double[] P_32 = { -0.99726386184948160000, -0.98561151154526840000,
  1242.             -0.96476225558750640000, -0.93490607593773970000, -0.89632115576605220000, -0.84936761373256990000,
  1243.             -0.79448379596794250000, -0.73218211874028970000, -0.66304426693021520000, -0.58771575724076230000,
  1244.             -0.50689990893222950000, -0.42135127613063540000, -0.33186860228212767000, -0.23928736225213710000,
  1245.             -0.14447196158279646000, -0.04830766568773831000, 0.04830766568773831000, 0.14447196158279646000,
  1246.             0.23928736225213710000, 0.33186860228212767000, 0.42135127613063540000, 0.50689990893222950000,
  1247.             0.58771575724076230000, 0.66304426693021520000, 0.73218211874028970000, 0.79448379596794250000,
  1248.             0.84936761373256990000, 0.89632115576605220000, 0.93490607593773970000, 0.96476225558750640000,
  1249.             0.98561151154526840000, 0.99726386184948160000 };

  1251.         /** Weights for quadrature of order 32. */
  1252.         private static final double[] W_32 = { 0.00701861000947013600, 0.01627439473090571200, 0.02539206530926214200,
  1253.             0.03427386291302141000, 0.04283589802222658600, 0.05099805926237621600, 0.05868409347853559000,
  1254.             0.06582222277636193000, 0.07234579410884862000, 0.07819389578707042000, 0.08331192422694673000,
  1255.             0.08765209300440380000, 0.09117387869576390000, 0.09384439908080441000, 0.09563872007927487000,
  1256.             0.09654008851472784000, 0.09654008851472784000, 0.09563872007927487000, 0.09384439908080441000,
  1257.             0.09117387869576390000, 0.08765209300440380000, 0.08331192422694673000, 0.07819389578707042000,
  1258.             0.07234579410884862000, 0.06582222277636193000, 0.05868409347853559000, 0.05099805926237621600,
  1259.             0.04283589802222658600, 0.03427386291302141000, 0.02539206530926214200, 0.01627439473090571200,
  1260.             0.00701861000947013600 };

  1262.         /** Points for quadrature of order 40. */
  1263.         private static final double[] P_40 = { -0.99823770971055930000, -0.99072623869945710000,
  1264.             -0.97725994998377420000, -0.95791681921379170000, -0.93281280827867660000, -0.90209880696887420000,
  1265.             -0.86595950321225960000, -0.82461223083331170000, -0.77830565142651940000, -0.72731825518992710000,
  1266.             -0.67195668461417960000, -0.61255388966798030000, -0.54946712509512820000, -0.48307580168617870000,
  1267.             -0.41377920437160500000, -0.34199409082575850000, -0.26815218500725370000, -0.19269758070137110000,
  1268.             -0.11608407067525522000, -0.03877241750605081600, 0.03877241750605081600, 0.11608407067525522000,
  1269.             0.19269758070137110000, 0.26815218500725370000, 0.34199409082575850000, 0.41377920437160500000,
  1270.             0.48307580168617870000, 0.54946712509512820000, 0.61255388966798030000, 0.67195668461417960000,
  1271.             0.72731825518992710000, 0.77830565142651940000, 0.82461223083331170000, 0.86595950321225960000,
  1272.             0.90209880696887420000, 0.93281280827867660000, 0.95791681921379170000, 0.97725994998377420000,
  1273.             0.99072623869945710000, 0.99823770971055930000 };

  1275.         /** Weights for quadrature of order 40. */
  1276.         private static final double[] W_40 = { 0.00452127709853309800, 0.01049828453115270400, 0.01642105838190797300,
  1277.             0.02224584919416689000, 0.02793700698002338000, 0.03346019528254786500, 0.03878216797447199000,
  1278.             0.04387090818567333000, 0.04869580763507221000, 0.05322784698393679000, 0.05743976909939157000,
  1279.             0.06130624249292891000, 0.06480401345660108000, 0.06791204581523394000, 0.07061164739128681000,
  1280.             0.07288658239580408000, 0.07472316905796833000, 0.07611036190062619000, 0.07703981816424793000,
  1281.             0.07750594797842482000, 0.07750594797842482000, 0.07703981816424793000, 0.07611036190062619000,
  1282.             0.07472316905796833000, 0.07288658239580408000, 0.07061164739128681000, 0.06791204581523394000,
  1283.             0.06480401345660108000, 0.06130624249292891000, 0.05743976909939157000, 0.05322784698393679000,
  1284.             0.04869580763507221000, 0.04387090818567333000, 0.03878216797447199000, 0.03346019528254786500,
  1285.             0.02793700698002338000, 0.02224584919416689000, 0.01642105838190797300, 0.01049828453115270400,
  1286.             0.00452127709853309800 };

  1288.         /** Points for quadrature of order 48. */
  1289.         private static final double[] P_48 = { -0.99877100725242610000, -0.99353017226635080000,
  1290.             -0.98412458372282700000, -0.97059159254624720000, -0.95298770316043080000, -0.93138669070655440000,
  1291.             -0.90587913671556960000, -0.87657202027424800000, -0.84358826162439350000, -0.80706620402944250000,
  1292.             -0.76715903251574020000, -0.72403413092381470000, -0.67787237963266400000, -0.62886739677651370000,
  1293.             -0.57722472608397270000, -0.52316097472223300000, -0.46690290475095840000, -0.40868648199071680000,
  1294.             -0.34875588629216070000, -0.28736248735545555000, -0.22476379039468908000, -0.16122235606889174000,
  1295.             -0.09700469920946270000, -0.03238017096286937000, 0.03238017096286937000, 0.09700469920946270000,
  1296.             0.16122235606889174000, 0.22476379039468908000, 0.28736248735545555000, 0.34875588629216070000,
  1297.             0.40868648199071680000, 0.46690290475095840000, 0.52316097472223300000, 0.57722472608397270000,
  1298.             0.62886739677651370000, 0.67787237963266400000, 0.72403413092381470000, 0.76715903251574020000,
  1299.             0.80706620402944250000, 0.84358826162439350000, 0.87657202027424800000, 0.90587913671556960000,
  1300.             0.93138669070655440000, 0.95298770316043080000, 0.97059159254624720000, 0.98412458372282700000,
  1301.             0.99353017226635080000, 0.99877100725242610000 };

  1303.         /** Weights for quadrature of order 48. */
  1304.         private static final double[] W_48 = { 0.00315334605230596250, 0.00732755390127620800, 0.01147723457923446900,
  1305.             0.01557931572294386600, 0.01961616045735556700, 0.02357076083932435600, 0.02742650970835688000,
  1306.             0.03116722783279807000, 0.03477722256477045000, 0.03824135106583080600, 0.04154508294346483000,
  1307.             0.04467456085669424000, 0.04761665849249054000, 0.05035903555385448000, 0.05289018948519365000,
  1308.             0.05519950369998416500, 0.05727729210040315000, 0.05911483969839566000, 0.06070443916589384000,
  1309.             0.06203942315989268000, 0.06311419228625403000, 0.06392423858464817000, 0.06446616443595010000,
  1310.             0.06473769681268386000, 0.06473769681268386000, 0.06446616443595010000, 0.06392423858464817000,
  1311.             0.06311419228625403000, 0.06203942315989268000, 0.06070443916589384000, 0.05911483969839566000,
  1312.             0.05727729210040315000, 0.05519950369998416500, 0.05289018948519365000, 0.05035903555385448000,
  1313.             0.04761665849249054000, 0.04467456085669424000, 0.04154508294346483000, 0.03824135106583080600,
  1314.             0.03477722256477045000, 0.03116722783279807000, 0.02742650970835688000, 0.02357076083932435600,
  1315.             0.01961616045735556700, 0.01557931572294386600, 0.01147723457923446900, 0.00732755390127620800,
  1316.             0.00315334605230596250 };

  1318.         /** Node points. */
  1319.         private final double[] nodePoints;

  1321.         /** Node weights. */
  1322.         private final double[] nodeWeights;

  1324.         /** Number of points. */
  1325.         private final int numberOfPoints;

  1327.         /**
  1328.          * Creates a Gauss integrator of the given order.
  1329.          *
  1330.          * @param numberOfPoints Order of the integration rule.
  1331.          */
  1332.         GaussQuadrature(final int numberOfPoints) {

  1334.             this.numberOfPoints = numberOfPoints;

  1336.             switch (numberOfPoints) {
  1337.                 case 12:
  1338.                     this.nodePoints = P_12.clone();
  1339.                     this.nodeWeights = W_12.clone();
  1340.                     break;
  1341.                 case 16:
  1342.                     this.nodePoints = P_16.clone();
  1343.                     this.nodeWeights = W_16.clone();
  1344.                     break;
  1345.                 case 20:
  1346.                     this.nodePoints = P_20.clone();
  1347.                     this.nodeWeights = W_20.clone();
  1348.                     break;
  1349.                 case 24:
  1350.                     this.nodePoints = P_24.clone();
  1351.                     this.nodeWeights = W_24.clone();
  1352.                     break;
  1353.                 case 32:
  1354.                     this.nodePoints = P_32.clone();
  1355.                     this.nodeWeights = W_32.clone();
  1356.                     break;
  1357.                 case 40:
  1358.                     this.nodePoints = P_40.clone();
  1359.                     this.nodeWeights = W_40.clone();
  1360.                     break;
  1361.                 case 48:
  1362.                 default:
  1363.                     this.nodePoints = P_48.clone();
  1364.                     this.nodeWeights = W_48.clone();
  1365.                     break;
  1366.             }

  1368.         }

  1370.         /**
  1371.          * Integrates a given function on the given interval.
  1372.          *
  1373.          * @param f          Function to integrate.
  1374.          * @param lowerBound Lower bound of the integration interval.
  1375.          * @param upperBound Upper bound of the integration interval.
  1376.          * @return the integral of the weighted function.
  1377.          */
  1378.         public double[] integrate(final UnivariateVectorFunction f, final double lowerBound, final double upperBound) {

  1380.             final double[] adaptedPoints = nodePoints.clone();
  1381.             final double[] adaptedWeights = nodeWeights.clone();
  1382.             transform(adaptedPoints, adaptedWeights, lowerBound, upperBound);
  1383.             return basicIntegrate(f, adaptedPoints, adaptedWeights);
  1384.         }

  1386.         /**
  1387.          * Integrates a given function on the given interval.
  1388.          *
  1389.          * @param <T>        the type of the field elements
  1390.          * @param f          Function to integrate.
  1391.          * @param lowerBound Lower bound of the integration interval.
  1392.          * @param upperBound Upper bound of the integration interval.
  1393.          * @param field      field utilized by default
  1394.          * @return the integral of the weighted function.
  1395.          */
  1396.         public <T extends CalculusFieldElement<T>> T[] integrate(final CalculusFieldUnivariateVectorFunction<T> f,
  1397.                 final T lowerBound, final T upperBound, final Field<T> field) {

  1399.             final T zero = field.getZero();

  1401.             final T[] adaptedPoints = MathArrays.buildArray(field, numberOfPoints);
  1402.             final T[] adaptedWeights = MathArrays.buildArray(field, numberOfPoints);

  1404.             for (int i = 0; i < numberOfPoints; i++) {
  1405.                 adaptedPoints[i] = zero.add(nodePoints[i]);
  1406.                 adaptedWeights[i] = zero.add(nodeWeights[i]);
  1407.             }

  1409.             transform(adaptedPoints, adaptedWeights, lowerBound, upperBound);
  1410.             return basicIntegrate(f, adaptedPoints, adaptedWeights, field);
  1411.         }

  1413.         /**
  1414.          * Performs a change of variable so that the integration can be performed on an
  1415.          * arbitrary interval {@code [a, b]}.
  1416.          * <p>
  1417.          * It is assumed that the natural interval is {@code [-1, 1]}.
  1418.          * </p>
  1419.          *
  1420.          * @param points  Points to adapt to the new interval.
  1421.          * @param weights Weights to adapt to the new interval.
  1422.          * @param a       Lower bound of the integration interval.
  1423.          * @param b       Lower bound of the integration interval.
  1424.          */
  1425.         private void transform(final double[] points, final double[] weights, final double a, final double b) {
  1426.             // Scaling
  1427.             final double scale = (b - a) / 2;
  1428.             final double shift = a + scale;
  1429.             for (int i = 0; i < points.length; i++) {
  1430.                 points[i] = points[i] * scale + shift;
  1431.                 weights[i] *= scale;
  1432.             }
  1433.         }

  1435.         /**
  1436.          * Performs a change of variable so that the integration can be performed on an
  1437.          * arbitrary interval {@code [a, b]}.
  1438.          * <p>
  1439.          * It is assumed that the natural interval is {@code [-1, 1]}.
  1440.          * </p>
  1441.          * @param <T>     the type of the field elements
  1442.          * @param points  Points to adapt to the new interval.
  1443.          * @param weights Weights to adapt to the new interval.
  1444.          * @param a       Lower bound of the integration interval.
  1445.          * @param b       Lower bound of the integration interval
  1446.          */
  1447.         private <T extends CalculusFieldElement<T>> void transform(final T[] points, final T[] weights, final T a,
  1448.                 final T b) {
  1449.             // Scaling
  1450.             final T scale = (b.subtract(a)).divide(2.);
  1451.             final T shift = a.add(scale);
  1452.             for (int i = 0; i < points.length; i++) {
  1453.                 points[i] = scale.multiply(points[i]).add(shift);
  1454.                 weights[i] = scale.multiply(weights[i]);
  1455.             }
  1456.         }

  1458.         /**
  1459.          * Returns an estimate of the integral of {@code f(x) * w(x)}, where {@code w}
  1460.          * is a weight function that depends on the actual flavor of the Gauss
  1461.          * integration scheme.
  1462.          *
  1463.          * @param f       Function to integrate.
  1464.          * @param points  Nodes.
  1465.          * @param weights Nodes weights.
  1466.          * @return the integral of the weighted function.
  1467.          */
  1468.         private double[] basicIntegrate(final UnivariateVectorFunction f, final double[] points,
  1469.                 final double[] weights) {
  1470.             double x = points[0];
  1471.             double w = weights[0];
  1472.             double[] v = f.value(x);
  1473.             final double[] y = new double[v.length];
  1474.             for (int j = 0; j < v.length; j++) {
  1475.                 y[j] = w * v[j];
  1476.             }
  1477.             final double[] t = y.clone();
  1478.             final double[] c = new double[v.length];
  1479.             final double[] s = t.clone();
  1480.             for (int i = 1; i < points.length; i++) {
  1481.                 x = points[i];
  1482.                 w = weights[i];
  1483.                 v = f.value(x);
  1484.                 for (int j = 0; j < v.length; j++) {
  1485.                     y[j] = w * v[j] - c[j];
  1486.                     t[j] = s[j] + y[j];
  1487.                     c[j] = (t[j] - s[j]) - y[j];
  1488.                     s[j] = t[j];
  1489.                 }
  1490.             }
  1491.             return s;
  1492.         }

  1494.         /**
  1495.          * Returns an estimate of the integral of {@code f(x) * w(x)}, where {@code w}
  1496.          * is a weight function that depends on the actual flavor of the Gauss
  1497.          * integration scheme.
  1498.          *
  1499.          * @param <T>     the type of the field elements.
  1500.          * @param f       Function to integrate.
  1501.          * @param points  Nodes.
  1502.          * @param weights Nodes weight
  1503.          * @param field   field utilized by default
  1504.          * @return the integral of the weighted function.
  1505.          */
  1506.         private <T extends CalculusFieldElement<T>> T[] basicIntegrate(final CalculusFieldUnivariateVectorFunction<T> f,
  1507.                 final T[] points, final T[] weights, final Field<T> field) {

  1509.             T x = points[0];
  1510.             T w = weights[0];
  1511.             T[] v = f.value(x);

  1513.             final T[] y = MathArrays.buildArray(field, v.length);
  1514.             for (int j = 0; j < v.length; j++) {
  1515.                 y[j] = v[j].multiply(w);
  1516.             }
  1517.             final T[] t = y.clone();
  1518.             final T[] c = MathArrays.buildArray(field, v.length);
  1519.             ;
  1520.             final T[] s = t.clone();
  1521.             for (int i = 1; i < points.length; i++) {
  1522.                 x = points[i];
  1523.                 w = weights[i];
  1524.                 v = f.value(x);
  1525.                 for (int j = 0; j < v.length; j++) {
  1526.                     y[j] = v[j].multiply(w).subtract(c[j]);
  1527.                     t[j] = y[j].add(s[j]);
  1528.                     c[j] = (t[j].subtract(s[j])).subtract(y[j]);
  1529.                     s[j] = t[j];
  1530.                 }
  1531.             }
  1532.             return s;
  1533.         }

  1535.     }

  1537.     /**
  1538.      * Compute the C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup>
  1539.      * coefficients.
  1540.      * <p>
  1541.      * Those coefficients are given in Danielson paper by expression 4.4-(6)
  1542.      * </p>
  1543.      * @author Petre Bazavan
  1544.      * @author Lucian Barbulescu
  1545.      */
  1546.     protected class FourierCjSjCoefficients {

  1548.         /** Maximum possible value for j. */
  1549.         private final int jMax;

  1551.         /**
  1552.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1553.          * <p>
  1554.          * the index i corresponds to the following elements: <br/>
  1555.          * - 0 for a <br>
  1556.          * - 1 for k <br>
  1557.          * - 2 for h <br>
  1558.          * - 3 for q <br>
  1559.          * - 4 for p <br>
  1560.          * - 5 for λ <br>
  1561.          * </p>
  1562.          */
  1563.         private final double[][] cCoef;

  1565.         /**
  1566.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1567.          * <p>
  1568.          * the index i corresponds to the following elements: <br/>
  1569.          * - 0 for a <br>
  1570.          * - 1 for k <br>
  1571.          * - 2 for h <br>
  1572.          * - 3 for q <br>
  1573.          * - 4 for p <br>
  1574.          * - 5 for λ <br>
  1575.          * </p>
  1576.          */
  1577.         private final double[][] sCoef;

  1579.         /**
  1580.          * Standard constructor.
  1581.          * @param state             the current state
  1582.          * @param jMax              maximum value for j
  1583.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1584.          * @param parameters        values of the force model parameters
  1585.          */
  1586.         FourierCjSjCoefficients(final SpacecraftState state, final int jMax, final AuxiliaryElements auxiliaryElements,
  1587.                 final double[] parameters) {

  1589.             // Initialise the fields
  1590.             this.jMax = jMax;

  1592.             // Allocate the arrays
  1593.             final int rows = jMax + 1;
  1594.             cCoef = new double[rows][6];
  1595.             sCoef = new double[rows][6];

  1597.             // Compute the coefficients
  1598.             computeCoefficients(state, auxiliaryElements, parameters);
  1599.         }

  1601.         /**
  1602.          * Compute the Fourrier coefficients.
  1603.          * <p>
  1604.          * Only the C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup> coefficients
  1605.          * need to be computed as D<sub>i</sub><sup>m</sup> is always 0.
  1606.          * </p>
  1607.          * @param state             the current state
  1608.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1609.          * @param parameters        values of the force model parameters
  1610.          */
  1611.         private void computeCoefficients(final SpacecraftState state, final AuxiliaryElements auxiliaryElements,
  1612.                 final double[] parameters) {

  1614.             // Computes the limits for the integral
  1615.             final double[] ll = getLLimits(state, auxiliaryElements);
  1616.             // Computes integrated mean element rates if Llow < Lhigh
  1617.             if (ll[0] < ll[1]) {
  1618.                 // Compute 1 / PI
  1619.                 final double ooPI = 1 / FastMath.PI;

  1621.                 // loop through all values of j
  1622.                 for (int j = 0; j <= jMax; j++) {
  1623.                     final double[] curentCoefficients = integrator
  1624.                             .integrate(new IntegrableFunction(state, false, j, parameters), ll[0], ll[1]);

  1626.                     // divide by PI and set the values for the coefficients
  1627.                     for (int i = 0; i < 6; i++) {
  1628.                         cCoef[j][i] = ooPI * curentCoefficients[i];
  1629.                         sCoef[j][i] = ooPI * curentCoefficients[i + 6];
  1630.                     }
  1631.                 }
  1632.             }
  1633.         }

  1635.         /**
  1636.          * Get the coefficient C<sub>i</sub><sup>j</sup>.
  1637.          * @param i i index - corresponds to the required variation
  1638.          * @param j j index
  1639.          * @return the coefficient C<sub>i</sub><sup>j</sup>
  1640.          */
  1641.         public double getCij(final int i, final int j) {
  1642.             return cCoef[j][i];
  1643.         }

  1645.         /**
  1646.          * Get the coefficient S<sub>i</sub><sup>j</sup>.
  1647.          * @param i i index - corresponds to the required variation
  1648.          * @param j j index
  1649.          * @return the coefficient S<sub>i</sub><sup>j</sup>
  1650.          */
  1651.         public double getSij(final int i, final int j) {
  1652.             return sCoef[j][i];
  1653.         }
  1654.     }

  1656.     /**
  1657.      * Compute the C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup>
  1658.      * coefficients with field elements.
  1659.      * <p>
  1660.      * Those coefficients are given in Danielson paper by expression 4.4-(6)
  1661.      * </p>
  1662.      * @author Petre Bazavan
  1663.      * @author Lucian Barbulescu
  1664.      */
  1665.     protected class FieldFourierCjSjCoefficients<T extends CalculusFieldElement<T>> {

  1667.         /** Maximum possible value for j. */
  1668.         private final int jMax;

  1670.         /**
  1671.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1672.          * <p>
  1673.          * the index i corresponds to the following elements: <br/>
  1674.          * - 0 for a <br>
  1675.          * - 1 for k <br>
  1676.          * - 2 for h <br>
  1677.          * - 3 for q <br>
  1678.          * - 4 for p <br>
  1679.          * - 5 for λ <br>
  1680.          * </p>
  1681.          */
  1682.         private final T[][] cCoef;

  1684.         /**
  1685.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1686.          * <p>
  1687.          * the index i corresponds to the following elements: <br/>
  1688.          * - 0 for a <br>
  1689.          * - 1 for k <br>
  1690.          * - 2 for h <br>
  1691.          * - 3 for q <br>
  1692.          * - 4 for p <br>
  1693.          * - 5 for λ <br>
  1694.          * </p>
  1695.          */
  1696.         private final T[][] sCoef;

  1698.         /**
  1699.          * Standard constructor.
  1700.          * @param state             the current state
  1701.          * @param jMax              maximum value for j
  1702.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1703.          * @param parameters        values of the force model parameters
  1704.          * @param field             field used by default
  1705.          */
  1706.         FieldFourierCjSjCoefficients(final FieldSpacecraftState<T> state, final int jMax,
  1707.                 final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters, final Field<T> field) {
  1708.             // Initialise the fields
  1709.             this.jMax = jMax;

  1711.             // Allocate the arrays
  1712.             final int rows = jMax + 1;
  1713.             cCoef = MathArrays.buildArray(field, rows, 6);
  1714.             sCoef = MathArrays.buildArray(field, rows, 6);

  1716.             // Compute the coefficients
  1717.             computeCoefficients(state, auxiliaryElements, parameters, field);
  1718.         }

  1720.         /**
  1721.          * Compute the Fourrier coefficients.
  1722.          * <p>
  1723.          * Only the C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup> coefficients
  1724.          * need to be computed as D<sub>i</sub><sup>m</sup> is always 0.
  1725.          * </p>
  1726.          * @param state             the current state
  1727.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1728.          * @param parameters        values of the force model parameters
  1729.          * @param field             field used by default
  1730.          */
  1731.         private void computeCoefficients(final FieldSpacecraftState<T> state,
  1732.                 final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters, final Field<T> field) {
  1733.             // Zero
  1734.             final T zero = field.getZero();
  1735.             // Computes the limits for the integral
  1736.             final T[] ll = getLLimits(state, auxiliaryElements);
  1737.             // Computes integrated mean element rates if Llow < Lhigh
  1738.             if (ll[0].getReal() < ll[1].getReal()) {
  1739.                 // Compute 1 / PI
  1740.                 final T ooPI = zero.getPi().reciprocal();

  1742.                 // loop through all values of j
  1743.                 for (int j = 0; j <= jMax; j++) {
  1744.                     final T[] curentCoefficients = integrator.integrate(
  1745.                             new FieldIntegrableFunction<>(state, false, j, parameters, field), ll[0], ll[1], field);

  1747.                     // divide by PI and set the values for the coefficients
  1748.                     for (int i = 0; i < 6; i++) {
  1749.                         cCoef[j][i] = curentCoefficients[i].multiply(ooPI);
  1750.                         sCoef[j][i] = curentCoefficients[i + 6].multiply(ooPI);
  1751.                     }
  1752.                 }
  1753.             }
  1754.         }

  1756.         /**
  1757.          * Get the coefficient C<sub>i</sub><sup>j</sup>.
  1758.          * @param i i index - corresponds to the required variation
  1759.          * @param j j index
  1760.          * @return the coefficient C<sub>i</sub><sup>j</sup>
  1761.          */
  1762.         public T getCij(final int i, final int j) {
  1763.             return cCoef[j][i];
  1764.         }

  1766.         /**
  1767.          * Get the coefficient S<sub>i</sub><sup>j</sup>.
  1768.          * @param i i index - corresponds to the required variation
  1769.          * @param j j index
  1770.          * @return the coefficient S<sub>i</sub><sup>j</sup>
  1771.          */
  1772.         public T getSij(final int i, final int j) {
  1773.             return sCoef[j][i];
  1774.         }
  1775.     }

  1777.     /**
  1778.      * This class handles the short periodic coefficients described in Danielson
  1779.      * 2.5.3-26.
  1780.      *
  1781.      * <p>
  1782.      * The value of M is 0. Also, since the values of the Fourier coefficient
  1783.      * D<sub>i</sub><sup>m</sup> is 0 then the values of the coefficients
  1784.      * D<sub>i</sub><sup>m</sup> for m &gt; 2 are also 0.
  1785.      * </p>
  1786.      * @author Petre Bazavan
  1787.      * @author Lucian Barbulescu
  1788.      *
  1789.      */
  1790.     protected static class GaussianShortPeriodicCoefficients implements ShortPeriodTerms {

  1792.         /** Maximum value for j index. */
  1793.         private final int jMax;

  1795.         /** Number of points used in the interpolation process. */
  1796.         private final int interpolationPoints;

  1798.         /** Prefix for coefficients keys. */
  1799.         private final String coefficientsKeyPrefix;

  1801.         /** All coefficients slots. */
  1802.         private final transient TimeSpanMap<Slot> slots;

  1804.         /**
  1805.          * Constructor.
  1806.          * @param coefficientsKeyPrefix prefix for coefficients keys
  1807.          * @param jMax                  maximum value for j index
  1808.          * @param interpolationPoints   number of points used in the interpolation
  1809.          *                              process
  1810.          * @param slots                 all coefficients slots
  1811.          */
  1812.         GaussianShortPeriodicCoefficients(final String coefficientsKeyPrefix, final int jMax,
  1813.                 final int interpolationPoints, final TimeSpanMap<Slot> slots) {
  1814.             // Initialize fields
  1815.             this.jMax = jMax;
  1816.             this.interpolationPoints = interpolationPoints;
  1817.             this.coefficientsKeyPrefix = coefficientsKeyPrefix;
  1818.             this.slots = slots;
  1819.         }

  1821.         /**
  1822.          * Get the slot valid for some date.
  1823.          * @param meanStates mean states defining the slot
  1824.          * @return slot valid at the specified date
  1825.          */
  1826.         public Slot createSlot(final SpacecraftState... meanStates) {
  1827.             final Slot slot = new Slot(jMax, interpolationPoints);
  1828.             final AbsoluteDate first = meanStates[0].getDate();
  1829.             final AbsoluteDate last = meanStates[meanStates.length - 1].getDate();
  1830.             final int compare = first.compareTo(last);
  1831.             if (compare < 0) {
  1832.                 slots.addValidAfter(slot, first, false);
  1833.             } else if (compare > 0) {
  1834.                 slots.addValidBefore(slot, first, false);
  1835.             } else {
  1836.                 // single date, valid for all time
  1837.                 slots.addValidAfter(slot, AbsoluteDate.PAST_INFINITY, false);
  1838.             }
  1839.             return slot;
  1840.         }

  1842.         /**
  1843.          * Compute the short periodic coefficients.
  1844.          *
  1845.          * @param state       current state information: date, kinematics, attitude
  1846.          * @param slot        coefficients slot
  1847.          * @param fourierCjSj Fourier coefficients
  1848.          * @param uijvij      U and V coefficients
  1849.          * @param n           Keplerian mean motion
  1850.          * @param a           semi major axis
  1851.          */
  1852.         private void computeCoefficients(final SpacecraftState state, final Slot slot,
  1853.                 final FourierCjSjCoefficients fourierCjSj, final UijVijCoefficients uijvij, final double n,
  1854.                 final double a) {

  1856.             // get the current date
  1857.             final AbsoluteDate date = state.getDate();

  1859.             // compute the k₂⁰ coefficient
  1860.             final double k20 = computeK20(jMax, uijvij.currentRhoSigmaj);

  1862.             // 1. / n
  1863.             final double oon = 1. / n;
  1864.             // 3. / (2 * a * n)
  1865.             final double to2an = 1.5 * oon / a;
  1866.             // 3. / (4 * a * n)
  1867.             final double to4an = to2an / 2;

  1869.             // Compute the coefficients for each element
  1870.             final int size = jMax + 1;
  1871.             final double[] di1 = new double[6];
  1872.             final double[] di2 = new double[6];
  1873.             final double[][] currentCij = new double[size][6];
  1874.             final double[][] currentSij = new double[size][6];
  1875.             for (int i = 0; i < 6; i++) {

  1877.                 // compute D<sub>i</sub>¹ and D<sub>i</sub>² (all others are 0)
  1878.                 di1[i] = -oon * fourierCjSj.getCij(i, 0);
  1879.                 if (i == 5) {
  1880.                     di1[i] += to2an * uijvij.getU1(0, 0);
  1881.                 }
  1882.                 di2[i] = 0.;
  1883.                 if (i == 5) {
  1884.                     di2[i] += -to4an * fourierCjSj.getCij(0, 0);
  1885.                 }

  1887.                 // the C<sub>i</sub>⁰ is computed based on all others
  1888.                 currentCij[0][i] = -di2[i] * k20;

  1890.                 for (int j = 1; j <= jMax; j++) {
  1891.                     // compute the current C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup>
  1892.                     currentCij[j][i] = oon * uijvij.getU1(j, i);
  1893.                     if (i == 5) {
  1894.                         currentCij[j][i] += -to2an * uijvij.getU2(j);
  1895.                     }
  1896.                     currentSij[j][i] = oon * uijvij.getV1(j, i);
  1897.                     if (i == 5) {
  1898.                         currentSij[j][i] += -to2an * uijvij.getV2(j);
  1899.                     }

  1901.                     // add the computed coefficients to C<sub>i</sub>⁰
  1902.                     currentCij[0][i] += -(currentCij[j][i] * uijvij.currentRhoSigmaj[0][j] +
  1903.                             currentSij[j][i] * uijvij.currentRhoSigmaj[1][j]);
  1904.                 }

  1906.             }

  1908.             // add the values to the interpolators
  1909.             slot.cij[0].addGridPoint(date, currentCij[0]);
  1910.             slot.dij[1].addGridPoint(date, di1);
  1911.             slot.dij[2].addGridPoint(date, di2);
  1912.             for (int j = 1; j <= jMax; j++) {
  1913.                 slot.cij[j].addGridPoint(date, currentCij[j]);
  1914.                 slot.sij[j].addGridPoint(date, currentSij[j]);
  1915.             }

  1917.         }

  1919.         /**
  1920.          * Compute the coefficient k₂⁰ by using the equation 2.5.3-(9a) from Danielson.
  1921.          * <p>
  1922.          * After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:<br>
  1923.          * k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  1924.          * ρ<sub>k</sub>²)]
  1925.          * </p>
  1926.          * @param kMax             max value fot k index
  1927.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  1928.          *                         σ<sub>j</sub> coefficients
  1929.          * @return the coefficient k₂⁰
  1930.          */
  1931.         private double computeK20(final int kMax, final double[][] currentRhoSigmaj) {
  1932.             double k20 = 0.;

  1934.             for (int kIndex = 1; kIndex <= kMax; kIndex++) {
  1935.                 // After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:
  1936.                 // k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  1937.                 // ρ<sub>k</sub>²)]
  1938.                 double currentTerm = currentRhoSigmaj[1][kIndex] * currentRhoSigmaj[1][kIndex] +
  1939.                         currentRhoSigmaj[0][kIndex] * currentRhoSigmaj[0][kIndex];

  1941.                 // multiply by 2 / k²
  1942.                 currentTerm *= 2. / (kIndex * kIndex);

  1944.                 // add the term to the result
  1945.                 k20 += currentTerm;
  1946.             }

  1948.             return k20;
  1949.         }

  1951.         /** {@inheritDoc} */
  1952.         @Override
  1953.         public double[] value(final Orbit meanOrbit) {

  1955.             // select the coefficients slot
  1956.             final Slot slot = slots.get(meanOrbit.getDate());

  1958.             // Get the True longitude L
  1959.             final double L = meanOrbit.getLv();

  1961.             // Compute the center (l - λ)
  1962.             final double center = L - meanOrbit.getLM();
  1963.             // Compute (l - λ)²
  1964.             final double center2 = center * center;

  1966.             // Initialize short periodic variations
  1967.             final double[] shortPeriodicVariation = slot.cij[0].value(meanOrbit.getDate());
  1968.             final double[] d1 = slot.dij[1].value(meanOrbit.getDate());
  1969.             final double[] d2 = slot.dij[2].value(meanOrbit.getDate());
  1970.             for (int i = 0; i < 6; i++) {
  1971.                 shortPeriodicVariation[i] += center * d1[i] + center2 * d2[i];
  1972.             }

  1974.             for (int j = 1; j <= JMAX; j++) {
  1975.                 final double[] c = slot.cij[j].value(meanOrbit.getDate());
  1976.                 final double[] s = slot.sij[j].value(meanOrbit.getDate());
  1977.                 final SinCos sc  = FastMath.sinCos(j * L);
  1978.                 final double cos = sc.cos();
  1979.                 final double sin = sc.sin();
  1980.                 for (int i = 0; i < 6; i++) {
  1981.                     // add corresponding term to the short periodic variation
  1982.                     shortPeriodicVariation[i] += c[i] * cos;
  1983.                     shortPeriodicVariation[i] += s[i] * sin;
  1984.                 }
  1985.             }

  1987.             return shortPeriodicVariation;

  1989.         }

  1991.         /** {@inheritDoc} */
  1992.         public String getCoefficientsKeyPrefix() {
  1993.             return coefficientsKeyPrefix;
  1994.         }

  1996.         /**
  1997.          * {@inheritDoc}
  1998.          * <p>
  1999.          * For Gaussian forces, there are JMAX cj coefficients, JMAX sj coefficients and
  2000.          * 3 dj coefficients. As JMAX = 12, this sums up to 27 coefficients. The j index
  2001.          * is the integer multiplier for the true longitude argument in the cj and sj
  2002.          * coefficients and to the degree in the polynomial dj coefficients.
  2003.          * </p>
  2004.          */
  2005.         @Override
  2006.         public Map<String, double[]> getCoefficients(final AbsoluteDate date, final Set<String> selected) {

  2008.             // select the coefficients slot
  2009.             final Slot slot = slots.get(date);

  2011.             final Map<String, double[]> coefficients = new HashMap<String, double[]>(2 * JMAX + 3);
  2012.             storeIfSelected(coefficients, selected, slot.cij[0].value(date), "d", 0);
  2013.             storeIfSelected(coefficients, selected, slot.dij[1].value(date), "d", 1);
  2014.             storeIfSelected(coefficients, selected, slot.dij[2].value(date), "d", 2);
  2015.             for (int j = 1; j <= JMAX; j++) {
  2016.                 storeIfSelected(coefficients, selected, slot.cij[j].value(date), "c", j);
  2017.                 storeIfSelected(coefficients, selected, slot.sij[j].value(date), "s", j);
  2018.             }

  2020.             return coefficients;

  2022.         }

  2024.         /**
  2025.          * Put a coefficient in a map if selected.
  2026.          * @param map      map to populate
  2027.          * @param selected set of coefficients that should be put in the map (empty set
  2028.          *                 means all coefficients are selected)
  2029.          * @param value    coefficient value
  2030.          * @param id       coefficient identifier
  2031.          * @param indices  list of coefficient indices
  2032.          */
  2033.         private void storeIfSelected(final Map<String, double[]> map, final Set<String> selected, final double[] value,
  2034.                 final String id, final int... indices) {
  2035.             final StringBuilder keyBuilder = new StringBuilder(getCoefficientsKeyPrefix());
  2036.             keyBuilder.append(id);
  2037.             for (int index : indices) {
  2038.                 keyBuilder.append('[').append(index).append(']');
  2039.             }
  2040.             final String key = keyBuilder.toString();
  2041.             if (selected.isEmpty() || selected.contains(key)) {
  2042.                 map.put(key, value);
  2043.             }
  2044.         }

  2046.     }

  2048.     /**
  2049.      * This class handles the short periodic coefficients described in Danielson
  2050.      * 2.5.3-26.
  2051.      *
  2052.      * <p>
  2053.      * The value of M is 0. Also, since the values of the Fourier coefficient
  2054.      * D<sub>i</sub><sup>m</sup> is 0 then the values of the coefficients
  2055.      * D<sub>i</sub><sup>m</sup> for m &gt; 2 are also 0.
  2056.      * </p>
  2057.      * @author Petre Bazavan
  2058.      * @author Lucian Barbulescu
  2059.      *
  2060.      */
  2061.     protected static class FieldGaussianShortPeriodicCoefficients<T extends CalculusFieldElement<T>>
  2062.             implements FieldShortPeriodTerms<T> {

  2064.         /** Maximum value for j index. */
  2065.         private final int jMax;

  2067.         /** Number of points used in the interpolation process. */
  2068.         private final int interpolationPoints;

  2070.         /** Prefix for coefficients keys. */
  2071.         private final String coefficientsKeyPrefix;

  2073.         /** All coefficients slots. */
  2074.         private final transient FieldTimeSpanMap<FieldSlot<T>, T> slots;

  2076.         /**
  2077.          * Constructor.
  2078.          * @param coefficientsKeyPrefix prefix for coefficients keys
  2079.          * @param jMax                  maximum value for j index
  2080.          * @param interpolationPoints   number of points used in the interpolation
  2081.          *                              process
  2082.          * @param slots                 all coefficients slots
  2083.          */
  2084.         FieldGaussianShortPeriodicCoefficients(final String coefficientsKeyPrefix, final int jMax,
  2085.                 final int interpolationPoints, final FieldTimeSpanMap<FieldSlot<T>, T> slots) {
  2086.             // Initialize fields
  2087.             this.jMax = jMax;
  2088.             this.interpolationPoints = interpolationPoints;
  2089.             this.coefficientsKeyPrefix = coefficientsKeyPrefix;
  2090.             this.slots = slots;
  2091.         }

  2093.         /**
  2094.          * Get the slot valid for some date.
  2095.          * @param meanStates mean states defining the slot
  2096.          * @return slot valid at the specified date
  2097.          */
  2098.         @SuppressWarnings("unchecked")
  2099.         public FieldSlot<T> createSlot(final FieldSpacecraftState<T>... meanStates) {
  2100.             final FieldSlot<T> slot = new FieldSlot<>(jMax, interpolationPoints);
  2101.             final FieldAbsoluteDate<T> first = meanStates[0].getDate();
  2102.             final FieldAbsoluteDate<T> last = meanStates[meanStates.length - 1].getDate();
  2103.             if (first.compareTo(last) <= 0) {
  2104.                 slots.addValidAfter(slot, first);
  2105.             } else {
  2106.                 slots.addValidBefore(slot, first);
  2107.             }
  2108.             return slot;
  2109.         }

  2111.         /**
  2112.          * Compute the short periodic coefficients.
  2113.          *
  2114.          * @param state       current state information: date, kinematics, attitude
  2115.          * @param slot        coefficients slot
  2116.          * @param fourierCjSj Fourier coefficients
  2117.          * @param uijvij      U and V coefficients
  2118.          * @param n           Keplerian mean motion
  2119.          * @param a           semi major axis
  2120.          * @param field       field used by default
  2121.          */
  2122.         private void computeCoefficients(final FieldSpacecraftState<T> state, final FieldSlot<T> slot,
  2123.                 final FieldFourierCjSjCoefficients<T> fourierCjSj, final FieldUijVijCoefficients<T> uijvij, final T n,
  2124.                 final T a, final Field<T> field) {

  2126.             // Zero
  2127.             final T zero = field.getZero();

  2129.             // get the current date
  2130.             final FieldAbsoluteDate<T> date = state.getDate();

  2132.             // compute the k₂⁰ coefficient
  2133.             final T k20 = computeK20(jMax, uijvij.currentRhoSigmaj, field);

  2135.             // 1. / n
  2136.             final T oon = n.reciprocal();
  2137.             // 3. / (2 * a * n)
  2138.             final T to2an = oon.multiply(1.5).divide(a);
  2139.             // 3. / (4 * a * n)
  2140.             final T to4an = to2an.divide(2.);

  2142.             // Compute the coefficients for each element
  2143.             final int size = jMax + 1;
  2144.             final T[] di1 = MathArrays.buildArray(field, 6);
  2145.             final T[] di2 = MathArrays.buildArray(field, 6);
  2146.             final T[][] currentCij = MathArrays.buildArray(field, size, 6);
  2147.             final T[][] currentSij = MathArrays.buildArray(field, size, 6);
  2148.             for (int i = 0; i < 6; i++) {

  2150.                 // compute D<sub>i</sub>¹ and D<sub>i</sub>² (all others are 0)
  2151.                 di1[i] = oon.negate().multiply(fourierCjSj.getCij(i, 0));
  2152.                 if (i == 5) {
  2153.                     di1[i] = di1[i].add(to2an.multiply(uijvij.getU1(0, 0)));
  2154.                 }
  2155.                 di2[i] = zero;
  2156.                 if (i == 5) {
  2157.                     di2[i] = di2[i].add(to4an.negate().multiply(fourierCjSj.getCij(0, 0)));
  2158.                 }

  2160.                 // the C<sub>i</sub>⁰ is computed based on all others
  2161.                 currentCij[0][i] = di2[i].negate().multiply(k20);

  2163.                 for (int j = 1; j <= jMax; j++) {
  2164.                     // compute the current C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup>
  2165.                     currentCij[j][i] = oon.multiply(uijvij.getU1(j, i));
  2166.                     if (i == 5) {
  2167.                         currentCij[j][i] = currentCij[j][i].add(to2an.negate().multiply(uijvij.getU2(j)));
  2168.                     }
  2169.                     currentSij[j][i] = oon.multiply(uijvij.getV1(j, i));
  2170.                     if (i == 5) {
  2171.                         currentSij[j][i] = currentSij[j][i].add(to2an.negate().multiply(uijvij.getV2(j)));
  2172.                     }

  2174.                     // add the computed coefficients to C<sub>i</sub>⁰
  2175.                     currentCij[0][i] = currentCij[0][i].add(currentCij[j][i].multiply(uijvij.currentRhoSigmaj[0][j])
  2176.                             .add(currentSij[j][i].multiply(uijvij.currentRhoSigmaj[1][j])).negate());
  2177.                 }

  2179.             }

  2181.             // add the values to the interpolators
  2182.             slot.cij[0].addGridPoint(date, currentCij[0]);
  2183.             slot.dij[1].addGridPoint(date, di1);
  2184.             slot.dij[2].addGridPoint(date, di2);
  2185.             for (int j = 1; j <= jMax; j++) {
  2186.                 slot.cij[j].addGridPoint(date, currentCij[j]);
  2187.                 slot.sij[j].addGridPoint(date, currentSij[j]);
  2188.             }

  2190.         }

  2192.         /**
  2193.          * Compute the coefficient k₂⁰ by using the equation 2.5.3-(9a) from Danielson.
  2194.          * <p>
  2195.          * After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:<br>
  2196.          * k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  2197.          * ρ<sub>k</sub>²)]
  2198.          * </p>
  2199.          * @param kMax             max value fot k index
  2200.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2201.          *                         σ<sub>j</sub> coefficients
  2202.          * @param field            field used by default
  2203.          * @return the coefficient k₂⁰
  2204.          */
  2205.         private T computeK20(final int kMax, final T[][] currentRhoSigmaj, final Field<T> field) {
  2206.             final T zero = field.getZero();
  2207.             T k20 = zero;

  2209.             for (int kIndex = 1; kIndex <= kMax; kIndex++) {
  2210.                 // After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:
  2211.                 // k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  2212.                 // ρ<sub>k</sub>²)]
  2213.                 T currentTerm = currentRhoSigmaj[1][kIndex].multiply(currentRhoSigmaj[1][kIndex])
  2214.                         .add(currentRhoSigmaj[0][kIndex].multiply(currentRhoSigmaj[0][kIndex]));

  2216.                 // multiply by 2 / k²
  2217.                 currentTerm = currentTerm.multiply(2. / (kIndex * kIndex));

  2219.                 // add the term to the result
  2220.                 k20 = k20.add(currentTerm);
  2221.             }

  2223.             return k20;
  2224.         }

  2226.         /** {@inheritDoc} */
  2227.         @Override
  2228.         public T[] value(final FieldOrbit<T> meanOrbit) {

  2230.             // select the coefficients slot
  2231.             final FieldSlot<T> slot = slots.get(meanOrbit.getDate());

  2233.             // Get the True longitude L
  2234.             final T L = meanOrbit.getLv();

  2236.             // Compute the center (l - λ)
  2237.             final T center = L.subtract(meanOrbit.getLM());
  2238.             // Compute (l - λ)²
  2239.             final T center2 = center.multiply(center);

  2241.             // Initialize short periodic variations
  2242.             final T[] shortPeriodicVariation = slot.cij[0].value(meanOrbit.getDate());
  2243.             final T[] d1 = slot.dij[1].value(meanOrbit.getDate());
  2244.             final T[] d2 = slot.dij[2].value(meanOrbit.getDate());
  2245.             for (int i = 0; i < 6; i++) {
  2246.                 shortPeriodicVariation[i] = shortPeriodicVariation[i]
  2247.                         .add(center.multiply(d1[i]).add(center2.multiply(d2[i])));
  2248.             }

  2250.             for (int j = 1; j <= JMAX; j++) {
  2251.                 final T[] c = slot.cij[j].value(meanOrbit.getDate());
  2252.                 final T[] s = slot.sij[j].value(meanOrbit.getDate());
  2253.                 final FieldSinCos<T> sc = FastMath.sinCos(L.multiply(j));
  2254.                 final T cos = sc.cos();
  2255.                 final T sin = sc.sin();
  2256.                 for (int i = 0; i < 6; i++) {
  2257.                     // add corresponding term to the short periodic variation
  2258.                     shortPeriodicVariation[i] = shortPeriodicVariation[i].add(c[i].multiply(cos));
  2259.                     shortPeriodicVariation[i] = shortPeriodicVariation[i].add(s[i].multiply(sin));
  2260.                 }
  2261.             }

  2263.             return shortPeriodicVariation;

  2265.         }

  2267.         /** {@inheritDoc} */
  2268.         public String getCoefficientsKeyPrefix() {
  2269.             return coefficientsKeyPrefix;
  2270.         }

  2272.         /**
  2273.          * {@inheritDoc}
  2274.          * <p>
  2275.          * For Gaussian forces, there are JMAX cj coefficients, JMAX sj coefficients and
  2276.          * 3 dj coefficients. As JMAX = 12, this sums up to 27 coefficients. The j index
  2277.          * is the integer multiplier for the true longitude argument in the cj and sj
  2278.          * coefficients and to the degree in the polynomial dj coefficients.
  2279.          * </p>
  2280.          */
  2281.         @Override
  2282.         public Map<String, T[]> getCoefficients(final FieldAbsoluteDate<T> date, final Set<String> selected) {

  2284.             // select the coefficients slot
  2285.             final FieldSlot<T> slot = slots.get(date);

  2287.             final Map<String, T[]> coefficients = new HashMap<String, T[]>(2 * JMAX + 3);
  2288.             storeIfSelected(coefficients, selected, slot.cij[0].value(date), "d", 0);
  2289.             storeIfSelected(coefficients, selected, slot.dij[1].value(date), "d", 1);
  2290.             storeIfSelected(coefficients, selected, slot.dij[2].value(date), "d", 2);
  2291.             for (int j = 1; j <= JMAX; j++) {
  2292.                 storeIfSelected(coefficients, selected, slot.cij[j].value(date), "c", j);
  2293.                 storeIfSelected(coefficients, selected, slot.sij[j].value(date), "s", j);
  2294.             }

  2296.             return coefficients;

  2298.         }

  2300.         /**
  2301.          * Put a coefficient in a map if selected.
  2302.          * @param map      map to populate
  2303.          * @param selected set of coefficients that should be put in the map (empty set
  2304.          *                 means all coefficients are selected)
  2305.          * @param value    coefficient value
  2306.          * @param id       coefficient identifier
  2307.          * @param indices  list of coefficient indices
  2308.          */
  2309.         private void storeIfSelected(final Map<String, T[]> map, final Set<String> selected, final T[] value,
  2310.                 final String id, final int... indices) {
  2311.             final StringBuilder keyBuilder = new StringBuilder(getCoefficientsKeyPrefix());
  2312.             keyBuilder.append(id);
  2313.             for (int index : indices) {
  2314.                 keyBuilder.append('[').append(index).append(']');
  2315.             }
  2316.             final String key = keyBuilder.toString();
  2317.             if (selected.isEmpty() || selected.contains(key)) {
  2318.                 map.put(key, value);
  2319.             }
  2320.         }

  2322.     }

  2324.     /**
  2325.      * The U<sub>i</sub><sup>j</sup> and V<sub>i</sub><sup>j</sup> coefficients
  2326.      * described by equations 2.5.3-(21) and 2.5.3-(22) from Danielson.
  2327.      * <p>
  2328.      * The index i takes only the values 1 and 2<br>
  2329.      * For U only the index 0 for j is used.
  2330.      * </p>
  2331.      *
  2332.      * @author Petre Bazavan
  2333.      * @author Lucian Barbulescu
  2334.      */
  2335.     protected static class UijVijCoefficients {

  2337.         /**
  2338.          * The U₁<sup>j</sup> coefficients.
  2339.          * <p>
  2340.          * The first index identifies the Fourier coefficients used<br>
  2341.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2342.          * S<sub>i</sub><sup>j</sup><br>
  2343.          * The only exception is when j = 0 when only the coefficient for fourier index
  2344.          * = 1 (i == 0) is needed.<br>
  2345.          * Also, for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are
  2346.          * computed, because are required to compute the coefficients U₂<sup>j</sup>
  2347.          * </p>
  2348.          */
  2349.         private final double[][] u1ij;

  2351.         /**
  2352.          * The V₁<sup>j</sup> coefficients.
  2353.          * <p>
  2354.          * The first index identifies the Fourier coefficients used<br>
  2355.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2356.          * S<sub>i</sub><sup>j</sup><br>
  2357.          * for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are computed,
  2358.          * because are required to compute the coefficients V₂<sup>j</sup>
  2359.          * </p>
  2360.          */
  2361.         private final double[][] v1ij;

  2363.         /**
  2364.          * The U₂<sup>j</sup> coefficients.
  2365.          * <p>
  2366.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2367.          * they are the only ones required.
  2368.          * </p>
  2369.          */
  2370.         private final double[] u2ij;

  2372.         /**
  2373.          * The V₂<sup>j</sup> coefficients.
  2374.          * <p>
  2375.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2376.          * they are the only ones required.
  2377.          * </p>
  2378.          */
  2379.         private final double[] v2ij;

  2381.         /**
  2382.          * The current computed values for the ρ<sub>j</sub> and σ<sub>j</sub>
  2383.          * coefficients.
  2384.          */
  2385.         private final double[][] currentRhoSigmaj;

  2387.         /**
  2388.          * The C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup> Fourier
  2389.          * coefficients.
  2390.          */
  2391.         private final FourierCjSjCoefficients fourierCjSj;

  2393.         /** The maximum value for j index. */
  2394.         private final int jMax;

  2396.         /**
  2397.          * Constructor.
  2398.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2399.          *                         σ<sub>j</sub> coefficients
  2400.          * @param fourierCjSj      the fourier coefficients C<sub>i</sub><sup>j</sup>
  2401.          *                         and the S<sub>i</sub><sup>j</sup>
  2402.          * @param jMax             maximum value for j index
  2403.          */
  2404.         UijVijCoefficients(final double[][] currentRhoSigmaj, final FourierCjSjCoefficients fourierCjSj,
  2405.                 final int jMax) {
  2406.             this.currentRhoSigmaj = currentRhoSigmaj;
  2407.             this.fourierCjSj = fourierCjSj;
  2408.             this.jMax = jMax;

  2410.             // initialize the internal arrays.
  2411.             this.u1ij = new double[6][2 * jMax + 1];
  2412.             this.v1ij = new double[6][2 * jMax + 1];
  2413.             this.u2ij = new double[jMax + 1];
  2414.             this.v2ij = new double[jMax + 1];

  2416.             // compute the coefficients
  2417.             computeU1V1Coefficients();
  2418.             computeU2V2Coefficients();
  2419.         }

  2421.         /** Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients. */
  2422.         private void computeU1V1Coefficients() {
  2423.             // generate the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients
  2424.             // for j >= 1
  2425.             // also the U₁⁰ for Fourier index = 1 (i == 0) coefficient will be computed
  2426.             u1ij[0][0] = 0;
  2427.             for (int j = 1; j <= jMax; j++) {
  2428.                 // compute 1 / j
  2429.                 final double ooj = 1. / j;

  2431.                 for (int i = 0; i < 6; i++) {
  2432.                     // j is aready between 1 and J
  2433.                     u1ij[i][j] = fourierCjSj.getSij(i, j);
  2434.                     v1ij[i][j] = fourierCjSj.getCij(i, j);

  2436.                     // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2437.                     if (j > 1) {
  2438.                         // k starts with 1 because j-J is less than or equal to 0
  2439.                         for (int kIndex = 1; kIndex <= j - 1; kIndex++) {
  2440.                             // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2441.                             // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2442.                             u1ij[i][j] += fourierCjSj.getCij(i, j - kIndex) * currentRhoSigmaj[1][kIndex] +
  2443.                                     fourierCjSj.getSij(i, j - kIndex) * currentRhoSigmaj[0][kIndex];

  2445.                             // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2446.                             // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2447.                             v1ij[i][j] += fourierCjSj.getCij(i, j - kIndex) * currentRhoSigmaj[0][kIndex] -
  2448.                                     fourierCjSj.getSij(i, j - kIndex) * currentRhoSigmaj[1][kIndex];
  2449.                         }
  2450.                     }

  2452.                     // since j must be between 1 and J-1 and is already between 1 and J
  2453.                     // the following sum is skiped only for j = jMax
  2454.                     if (j != jMax) {
  2455.                         for (int kIndex = 1; kIndex <= jMax - j; kIndex++) {
  2456.                             // -C<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub> +
  2457.                             // S<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub>
  2458.                             u1ij[i][j] += -fourierCjSj.getCij(i, j + kIndex) * currentRhoSigmaj[1][kIndex] +
  2459.                                     fourierCjSj.getSij(i, j + kIndex) * currentRhoSigmaj[0][kIndex];

  2461.                             // C<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub> +
  2462.                             // S<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub>
  2463.                             v1ij[i][j] += fourierCjSj.getCij(i, j + kIndex) * currentRhoSigmaj[0][kIndex] +
  2464.                                     fourierCjSj.getSij(i, j + kIndex) * currentRhoSigmaj[1][kIndex];
  2465.                         }
  2466.                     }

  2468.                     for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2469.                         // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2470.                         // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2471.                         u1ij[i][j] += -fourierCjSj.getCij(i, kIndex) * currentRhoSigmaj[1][j + kIndex] -
  2472.                                 fourierCjSj.getSij(i, kIndex) * currentRhoSigmaj[0][j + kIndex];

  2474.                         // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2475.                         // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2476.                         v1ij[i][j] += fourierCjSj.getCij(i, kIndex) * currentRhoSigmaj[0][j + kIndex] +
  2477.                                 fourierCjSj.getSij(i, kIndex) * currentRhoSigmaj[1][j + kIndex];
  2478.                     }

  2480.                     // divide by 1 / j
  2481.                     u1ij[i][j] *= -ooj;
  2482.                     v1ij[i][j] *= ooj;

  2484.                     // if index = 1 (i == 0) add the computed terms to U₁⁰
  2485.                     if (i == 0) {
  2486.                         // - (U₁<sup>j</sup> * ρ<sub>j</sub> + V₁<sup>j</sup> * σ<sub>j</sub>
  2487.                         u1ij[0][0] += -u1ij[0][j] * currentRhoSigmaj[0][j] - v1ij[0][j] * currentRhoSigmaj[1][j];
  2488.                     }
  2489.                 }
  2490.             }

  2492.             // Terms with j > jMax are required only when computing the coefficients
  2493.             // U₂<sup>j</sup> and V₂<sup>j</sup>
  2494.             // and those coefficients are only required for Fourier index = 1 (i == 0).
  2495.             for (int j = jMax + 1; j <= 2 * jMax; j++) {
  2496.                 // compute 1 / j
  2497.                 final double ooj = 1. / j;
  2498.                 // the value of i is 0
  2499.                 u1ij[0][j] = 0.;
  2500.                 v1ij[0][j] = 0.;

  2502.                 // k starts from j-J as it is always greater than or equal to 1
  2503.                 for (int kIndex = j - jMax; kIndex <= j - 1; kIndex++) {
  2504.                     // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2505.                     // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2506.                     u1ij[0][j] += fourierCjSj.getCij(0, j - kIndex) * currentRhoSigmaj[1][kIndex] +
  2507.                             fourierCjSj.getSij(0, j - kIndex) * currentRhoSigmaj[0][kIndex];

  2509.                     // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2510.                     // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2511.                     v1ij[0][j] += fourierCjSj.getCij(0, j - kIndex) * currentRhoSigmaj[0][kIndex] -
  2512.                             fourierCjSj.getSij(0, j - kIndex) * currentRhoSigmaj[1][kIndex];
  2513.                 }
  2514.                 for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2515.                     // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2516.                     // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2517.                     u1ij[0][j] += -fourierCjSj.getCij(0, kIndex) * currentRhoSigmaj[1][j + kIndex] -
  2518.                             fourierCjSj.getSij(0, kIndex) * currentRhoSigmaj[0][j + kIndex];

  2520.                     // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2521.                     // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2522.                     v1ij[0][j] += fourierCjSj.getCij(0, kIndex) * currentRhoSigmaj[0][j + kIndex] +
  2523.                             fourierCjSj.getSij(0, kIndex) * currentRhoSigmaj[1][j + kIndex];
  2524.                 }

  2526.                 // divide by 1 / j
  2527.                 u1ij[0][j] *= -ooj;
  2528.                 v1ij[0][j] *= ooj;
  2529.             }
  2530.         }

  2532.         /**
  2533.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2534.          * <p>
  2535.          * Only the coefficients for Fourier index = 1 (i == 0) are required.
  2536.          * </p>
  2537.          */
  2538.         private void computeU2V2Coefficients() {
  2539.             for (int j = 1; j <= jMax; j++) {
  2540.                 // compute 1 / j
  2541.                 final double ooj = 1. / j;

  2543.                 // only the values for i == 0 are computed
  2544.                 u2ij[j] = v1ij[0][j];
  2545.                 v2ij[j] = u1ij[0][j];

  2547.                 // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2548.                 if (j > 1) {
  2549.                     for (int l = 1; l <= j - 1; l++) {
  2550.                         // U₁<sup>j-l</sup> * σ<sub>l</sub> +
  2551.                         // V₁<sup>j-l</sup> * ρ<sub>l</sub>
  2552.                         u2ij[j] += u1ij[0][j - l] * currentRhoSigmaj[1][l] + v1ij[0][j - l] * currentRhoSigmaj[0][l];

  2554.                         // U₁<sup>j-l</sup> * ρ<sub>l</sub> -
  2555.                         // V₁<sup>j-l</sup> * σ<sub>l</sub>
  2556.                         v2ij[j] += u1ij[0][j - l] * currentRhoSigmaj[0][l] - v1ij[0][j - l] * currentRhoSigmaj[1][l];
  2557.                     }
  2558.                 }

  2560.                 for (int l = 1; l <= jMax; l++) {
  2561.                     // -U₁<sup>j+l</sup> * σ<sub>l</sub> +
  2562.                     // U₁<sup>l</sup> * σ<sub>j+l</sub> +
  2563.                     // V₁<sup>j+l</sup> * ρ<sub>l</sub> -
  2564.                     // V₁<sup>l</sup> * ρ<sub>j+l</sub>
  2565.                     u2ij[j] += -u1ij[0][j + l] * currentRhoSigmaj[1][l] + u1ij[0][l] * currentRhoSigmaj[1][j + l] +
  2566.                             v1ij[0][j + l] * currentRhoSigmaj[0][l] - v1ij[0][l] * currentRhoSigmaj[0][j + l];

  2568.                     // U₁<sup>j+l</sup> * ρ<sub>l</sub> +
  2569.                     // U₁<sup>l</sup> * ρ<sub>j+l</sub> +
  2570.                     // V₁<sup>j+l</sup> * σ<sub>l</sub> +
  2571.                     // V₁<sup>l</sup> * σ<sub>j+l</sub>
  2572.                     u2ij[j] += u1ij[0][j + l] * currentRhoSigmaj[0][l] + u1ij[0][l] * currentRhoSigmaj[0][j + l] +
  2573.                             v1ij[0][j + l] * currentRhoSigmaj[1][l] + v1ij[0][l] * currentRhoSigmaj[1][j + l];
  2574.                 }

  2576.                 // divide by 1 / j
  2577.                 u2ij[j] *= -ooj;
  2578.                 v2ij[j] *= ooj;
  2579.             }
  2580.         }

  2582.         /**
  2583.          * Get the coefficient U₁<sup>j</sup> for Fourier index i.
  2584.          *
  2585.          * @param j j index
  2586.          * @param i Fourier index (starts at 0)
  2587.          * @return the coefficient U₁<sup>j</sup> for the given Fourier index i
  2588.          */
  2589.         public double getU1(final int j, final int i) {
  2590.             return u1ij[i][j];
  2591.         }

  2593.         /**
  2594.          * Get the coefficient V₁<sup>j</sup> for Fourier index i.
  2595.          *
  2596.          * @param j j index
  2597.          * @param i Fourier index (starts at 0)
  2598.          * @return the coefficient V₁<sup>j</sup> for the given Fourier index i
  2599.          */
  2600.         public double getV1(final int j, final int i) {
  2601.             return v1ij[i][j];
  2602.         }

  2604.         /**
  2605.          * Get the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2606.          *
  2607.          * @param j j index
  2608.          * @return the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2609.          */
  2610.         public double getU2(final int j) {
  2611.             return u2ij[j];
  2612.         }

  2614.         /**
  2615.          * Get the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2616.          *
  2617.          * @param j j index
  2618.          * @return the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2619.          */
  2620.         public double getV2(final int j) {
  2621.             return v2ij[j];
  2622.         }
  2623.     }

  2625.     /**
  2626.      * The U<sub>i</sub><sup>j</sup> and V<sub>i</sub><sup>j</sup> coefficients
  2627.      * described by equations 2.5.3-(21) and 2.5.3-(22) from Danielson.
  2628.      * <p>
  2629.      * The index i takes only the values 1 and 2<br>
  2630.      * For U only the index 0 for j is used.
  2631.      * </p>
  2632.      *
  2633.      * @author Petre Bazavan
  2634.      * @author Lucian Barbulescu
  2635.      */
  2636.     protected static class FieldUijVijCoefficients<T extends CalculusFieldElement<T>> {

  2638.         /**
  2639.          * The U₁<sup>j</sup> coefficients.
  2640.          * <p>
  2641.          * The first index identifies the Fourier coefficients used<br>
  2642.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2643.          * S<sub>i</sub><sup>j</sup><br>
  2644.          * The only exception is when j = 0 when only the coefficient for fourier index
  2645.          * = 1 (i == 0) is needed.<br>
  2646.          * Also, for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are
  2647.          * computed, because are required to compute the coefficients U₂<sup>j</sup>
  2648.          * </p>
  2649.          */
  2650.         private final T[][] u1ij;

  2652.         /**
  2653.          * The V₁<sup>j</sup> coefficients.
  2654.          * <p>
  2655.          * The first index identifies the Fourier coefficients used<br>
  2656.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2657.          * S<sub>i</sub><sup>j</sup><br>
  2658.          * for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are computed,
  2659.          * because are required to compute the coefficients V₂<sup>j</sup>
  2660.          * </p>
  2661.          */
  2662.         private final T[][] v1ij;

  2664.         /**
  2665.          * The U₂<sup>j</sup> coefficients.
  2666.          * <p>
  2667.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2668.          * they are the only ones required.
  2669.          * </p>
  2670.          */
  2671.         private final T[] u2ij;

  2673.         /**
  2674.          * The V₂<sup>j</sup> coefficients.
  2675.          * <p>
  2676.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2677.          * they are the only ones required.
  2678.          * </p>
  2679.          */
  2680.         private final T[] v2ij;

  2682.         /**
  2683.          * The current computed values for the ρ<sub>j</sub> and σ<sub>j</sub>
  2684.          * coefficients.
  2685.          */
  2686.         private final T[][] currentRhoSigmaj;

  2688.         /**
  2689.          * The C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup> Fourier
  2690.          * coefficients.
  2691.          */
  2692.         private final FieldFourierCjSjCoefficients<T> fourierCjSj;

  2694.         /** The maximum value for j index. */
  2695.         private final int jMax;

  2697.         /**
  2698.          * Constructor.
  2699.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2700.          *                         σ<sub>j</sub> coefficients
  2701.          * @param fourierCjSj      the fourier coefficients C<sub>i</sub><sup>j</sup>
  2702.          *                         and the S<sub>i</sub><sup>j</sup>
  2703.          * @param jMax             maximum value for j index
  2704.          * @param field            field used by default
  2705.          */
  2706.         FieldUijVijCoefficients(final T[][] currentRhoSigmaj, final FieldFourierCjSjCoefficients<T> fourierCjSj,
  2707.                 final int jMax, final Field<T> field) {
  2708.             this.currentRhoSigmaj = currentRhoSigmaj;
  2709.             this.fourierCjSj = fourierCjSj;
  2710.             this.jMax = jMax;

  2712.             // initialize the internal arrays.
  2713.             this.u1ij = MathArrays.buildArray(field, 6, 2 * jMax + 1);
  2714.             this.v1ij = MathArrays.buildArray(field, 6, 2 * jMax + 1);
  2715.             this.u2ij = MathArrays.buildArray(field, jMax + 1);
  2716.             this.v2ij = MathArrays.buildArray(field, jMax + 1);

  2718.             // compute the coefficients
  2719.             computeU1V1Coefficients(field);
  2720.             computeU2V2Coefficients(field);
  2721.         }

  2723.         /**
  2724.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2725.          * @param field field used by default
  2726.          */
  2727.         private void computeU1V1Coefficients(final Field<T> field) {
  2728.             // Zero
  2729.             final T zero = field.getZero();

  2731.             // generate the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients
  2732.             // for j >= 1
  2733.             // also the U₁⁰ for Fourier index = 1 (i == 0) coefficient will be computed
  2734.             u1ij[0][0] = zero;
  2735.             for (int j = 1; j <= jMax; j++) {
  2736.                 // compute 1 / j
  2737.                 final double ooj = 1. / j;

  2739.                 for (int i = 0; i < 6; i++) {
  2740.                     // j is aready between 1 and J
  2741.                     u1ij[i][j] = fourierCjSj.getSij(i, j);
  2742.                     v1ij[i][j] = fourierCjSj.getCij(i, j);

  2744.                     // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2745.                     if (j > 1) {
  2746.                         // k starts with 1 because j-J is less than or equal to 0
  2747.                         for (int kIndex = 1; kIndex <= j - 1; kIndex++) {
  2748.                             // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2749.                             // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2750.                             u1ij[i][j] = u1ij[i][j]
  2751.                                     .add(fourierCjSj.getCij(i, j - kIndex).multiply(currentRhoSigmaj[1][kIndex]).add(
  2752.                                             fourierCjSj.getSij(i, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2754.                             // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2755.                             // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2756.                             v1ij[i][j] = v1ij[i][j].add(
  2757.                                     fourierCjSj.getCij(i, j - kIndex).multiply(currentRhoSigmaj[0][kIndex]).subtract(
  2758.                                             fourierCjSj.getSij(i, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2759.                         }
  2760.                     }

  2762.                     // since j must be between 1 and J-1 and is already between 1 and J
  2763.                     // the following sum is skiped only for j = jMax
  2764.                     if (j != jMax) {
  2765.                         for (int kIndex = 1; kIndex <= jMax - j; kIndex++) {
  2766.                             // -C<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub> +
  2767.                             // S<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub>
  2768.                             u1ij[i][j] = u1ij[i][j].add(fourierCjSj.getCij(i, j + kIndex).negate()
  2769.                                     .multiply(currentRhoSigmaj[1][kIndex])
  2770.                                     .add(fourierCjSj.getSij(i, j + kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2772.                             // C<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub> +
  2773.                             // S<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub>
  2774.                             v1ij[i][j] = v1ij[i][j]
  2775.                                     .add(fourierCjSj.getCij(i, j + kIndex).multiply(currentRhoSigmaj[0][kIndex]).add(
  2776.                                             fourierCjSj.getSij(i, j + kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2777.                         }
  2778.                     }

  2780.                     for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2781.                         // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2782.                         // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2783.                         u1ij[i][j] = u1ij[i][j].add(fourierCjSj.getCij(i, kIndex).negate()
  2784.                                 .multiply(currentRhoSigmaj[1][j + kIndex])
  2785.                                 .subtract(fourierCjSj.getSij(i, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])));

  2787.                         // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2788.                         // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2789.                         v1ij[i][j] = v1ij[i][j]
  2790.                                 .add(fourierCjSj.getCij(i, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])
  2791.                                         .add(fourierCjSj.getSij(i, kIndex).multiply(currentRhoSigmaj[1][j + kIndex])));
  2792.                     }

  2794.                     // divide by 1 / j
  2795.                     u1ij[i][j] = u1ij[i][j].multiply(-ooj);
  2796.                     v1ij[i][j] = v1ij[i][j].multiply(ooj);

  2798.                     // if index = 1 (i == 0) add the computed terms to U₁⁰
  2799.                     if (i == 0) {
  2800.                         // - (U₁<sup>j</sup> * ρ<sub>j</sub> + V₁<sup>j</sup> * σ<sub>j</sub>
  2801.                         u1ij[0][0] = u1ij[0][0].add(u1ij[0][j].negate().multiply(currentRhoSigmaj[0][j])
  2802.                                 .subtract(v1ij[0][j].multiply(currentRhoSigmaj[1][j])));
  2803.                     }
  2804.                 }
  2805.             }

  2807.             // Terms with j > jMax are required only when computing the coefficients
  2808.             // U₂<sup>j</sup> and V₂<sup>j</sup>
  2809.             // and those coefficients are only required for Fourier index = 1 (i == 0).
  2810.             for (int j = jMax + 1; j <= 2 * jMax; j++) {
  2811.                 // compute 1 / j
  2812.                 final double ooj = 1. / j;
  2813.                 // the value of i is 0
  2814.                 u1ij[0][j] = zero;
  2815.                 v1ij[0][j] = zero;

  2817.                 // k starts from j-J as it is always greater than or equal to 1
  2818.                 for (int kIndex = j - jMax; kIndex <= j - 1; kIndex++) {
  2819.                     // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2820.                     // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2821.                     u1ij[0][j] = u1ij[0][j].add(fourierCjSj.getCij(0, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])
  2822.                             .add(fourierCjSj.getSij(0, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2824.                     // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2825.                     // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2826.                     v1ij[0][j] = v1ij[0][j].add(fourierCjSj.getCij(0, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])
  2827.                             .subtract(fourierCjSj.getSij(0, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2828.                 }
  2829.                 for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2830.                     // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2831.                     // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2832.                     u1ij[0][j] = u1ij[0][j]
  2833.                             .add(fourierCjSj.getCij(0, kIndex).negate().multiply(currentRhoSigmaj[1][j + kIndex])
  2834.                                     .subtract(fourierCjSj.getSij(0, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])));

  2836.                     // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2837.                     // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2838.                     v1ij[0][j] = v1ij[0][j].add(fourierCjSj.getCij(0, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])
  2839.                             .add(fourierCjSj.getSij(0, kIndex).multiply(currentRhoSigmaj[1][j + kIndex])));
  2840.                 }

  2842.                 // divide by 1 / j
  2843.                 u1ij[0][j] = u1ij[0][j].multiply(-ooj);
  2844.                 v1ij[0][j] = v1ij[0][j].multiply(ooj);
  2845.             }
  2846.         }

  2848.         /**
  2849.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2850.          * <p>
  2851.          * Only the coefficients for Fourier index = 1 (i == 0) are required.
  2852.          * </p>
  2853.          * @param field field used by default
  2854.          */
  2855.         private void computeU2V2Coefficients(final Field<T> field) {
  2856.             for (int j = 1; j <= jMax; j++) {
  2857.                 // compute 1 / j
  2858.                 final double ooj = 1. / j;

  2860.                 // only the values for i == 0 are computed
  2861.                 u2ij[j] = v1ij[0][j];
  2862.                 v2ij[j] = u1ij[0][j];

  2864.                 // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2865.                 if (j > 1) {
  2866.                     for (int l = 1; l <= j - 1; l++) {
  2867.                         // U₁<sup>j-l</sup> * σ<sub>l</sub> +
  2868.                         // V₁<sup>j-l</sup> * ρ<sub>l</sub>
  2869.                         u2ij[j] = u2ij[j].add(u1ij[0][j - l].multiply(currentRhoSigmaj[1][l])
  2870.                                 .add(v1ij[0][j - l].multiply(currentRhoSigmaj[0][l])));

  2872.                         // U₁<sup>j-l</sup> * ρ<sub>l</sub> -
  2873.                         // V₁<sup>j-l</sup> * σ<sub>l</sub>
  2874.                         v2ij[j] = v2ij[j].add(u1ij[0][j - l].multiply(currentRhoSigmaj[0][l])
  2875.                                 .subtract(v1ij[0][j - l].multiply(currentRhoSigmaj[1][l])));
  2876.                     }
  2877.                 }

  2879.                 for (int l = 1; l <= jMax; l++) {
  2880.                     // -U₁<sup>j+l</sup> * σ<sub>l</sub> +
  2881.                     // U₁<sup>l</sup> * σ<sub>j+l</sub> +
  2882.                     // V₁<sup>j+l</sup> * ρ<sub>l</sub> -
  2883.                     // V₁<sup>l</sup> * ρ<sub>j+l</sub>
  2884.                     u2ij[j] = u2ij[j].add(u1ij[0][j + l].negate().multiply(currentRhoSigmaj[1][l])
  2885.                             .add(u1ij[0][l].multiply(currentRhoSigmaj[1][j + l]))
  2886.                             .add(v1ij[0][j + l].multiply(currentRhoSigmaj[0][l]))
  2887.                             .subtract(v1ij[0][l].multiply(currentRhoSigmaj[0][j + l])));

  2889.                     // U₁<sup>j+l</sup> * ρ<sub>l</sub> +
  2890.                     // U₁<sup>l</sup> * ρ<sub>j+l</sub> +
  2891.                     // V₁<sup>j+l</sup> * σ<sub>l</sub> +
  2892.                     // V₁<sup>l</sup> * σ<sub>j+l</sub>
  2893.                     u2ij[j] = u2ij[j].add(u1ij[0][j + l].multiply(currentRhoSigmaj[0][l])
  2894.                             .add(u1ij[0][l].multiply(currentRhoSigmaj[0][j + l]))
  2895.                             .add(v1ij[0][j + l].multiply(currentRhoSigmaj[1][l]))
  2896.                             .add(v1ij[0][l].multiply(currentRhoSigmaj[1][j + l])));
  2897.                 }

  2899.                 // divide by 1 / j
  2900.                 u2ij[j] = u2ij[j].multiply(-ooj);
  2901.                 v2ij[j] = v2ij[j].multiply(ooj);
  2902.             }
  2903.         }

  2905.         /**
  2906.          * Get the coefficient U₁<sup>j</sup> for Fourier index i.
  2907.          *
  2908.          * @param j j index
  2909.          * @param i Fourier index (starts at 0)
  2910.          * @return the coefficient U₁<sup>j</sup> for the given Fourier index i
  2911.          */
  2912.         public T getU1(final int j, final int i) {
  2913.             return u1ij[i][j];
  2914.         }

  2916.         /**
  2917.          * Get the coefficient V₁<sup>j</sup> for Fourier index i.
  2918.          *
  2919.          * @param j j index
  2920.          * @param i Fourier index (starts at 0)
  2921.          * @return the coefficient V₁<sup>j</sup> for the given Fourier index i
  2922.          */
  2923.         public T getV1(final int j, final int i) {
  2924.             return v1ij[i][j];
  2925.         }

  2927.         /**
  2928.          * Get the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2929.          *
  2930.          * @param j j index
  2931.          * @return the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2932.          */
  2933.         public T getU2(final int j) {
  2934.             return u2ij[j];
  2935.         }

  2937.         /**
  2938.          * Get the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2939.          *
  2940.          * @param j j index
  2941.          * @return the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2942.          */
  2943.         public T getV2(final int j) {
  2944.             return v2ij[j];
  2945.         }
  2946.     }

  2948.     /** Coefficients valid for one time slot. */
  2949.     protected static class Slot {

  2951.         /**
  2952.          * The coefficients D<sub>i</sub><sup>j</sup>.
  2953.          * <p>
  2954.          * Only for j = 1 and j = 2 the coefficients are not 0. <br>
  2955.          * i corresponds to the equinoctial element, as follows: - i=0 for a <br/>
  2956.          * - i=1 for k <br/>
  2957.          * - i=2 for h <br/>
  2958.          * - i=3 for q <br/>
  2959.          * - i=4 for p <br/>
  2960.          * - i=5 for λ <br/>
  2961.          * </p>
  2962.          */
  2963.         private final ShortPeriodicsInterpolatedCoefficient[] dij;

  2965.         /**
  2966.          * The coefficients C<sub>i</sub><sup>j</sup>.
  2967.          * <p>
  2968.          * The index order is cij[j][i] <br/>
  2969.          * i corresponds to the equinoctial element, as follows: <br/>
  2970.          * - i=0 for a <br/>
  2971.          * - i=1 for k <br/>
  2972.          * - i=2 for h <br/>
  2973.          * - i=3 for q <br/>
  2974.          * - i=4 for p <br/>
  2975.          * - i=5 for λ <br/>
  2976.          * </p>
  2977.          */
  2978.         private final ShortPeriodicsInterpolatedCoefficient[] cij;

  2980.         /**
  2981.          * The coefficients S<sub>i</sub><sup>j</sup>.
  2982.          * <p>
  2983.          * The index order is sij[j][i] <br/>
  2984.          * i corresponds to the equinoctial element, as follows: <br/>
  2985.          * - i=0 for a <br/>
  2986.          * - i=1 for k <br/>
  2987.          * - i=2 for h <br/>
  2988.          * - i=3 for q <br/>
  2989.          * - i=4 for p <br/>
  2990.          * - i=5 for λ <br/>
  2991.          * </p>
  2992.          */
  2993.         private final ShortPeriodicsInterpolatedCoefficient[] sij;

  2995.         /**
  2996.          * Simple constructor.
  2997.          * @param jMax                maximum value for j index
  2998.          * @param interpolationPoints number of points used in the interpolation process
  2999.          */
  3000.         Slot(final int jMax, final int interpolationPoints) {

  3002.             dij = new ShortPeriodicsInterpolatedCoefficient[3];
  3003.             cij = new ShortPeriodicsInterpolatedCoefficient[jMax + 1];
  3004.             sij = new ShortPeriodicsInterpolatedCoefficient[jMax + 1];

  3006.             // Initialize the C<sub>i</sub><sup>j</sup>, S<sub>i</sub><sup>j</sup> and
  3007.             // D<sub>i</sub><sup>j</sup> coefficients
  3008.             for (int j = 0; j <= jMax; j++) {
  3009.                 cij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3010.                 if (j > 0) {
  3011.                     sij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3012.                 }
  3013.                 // Initialize only the non-zero D<sub>i</sub><sup>j</sup> coefficients
  3014.                 if (j == 1 || j == 2) {
  3015.                     dij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3016.                 }
  3017.             }

  3019.         }

  3021.     }

  3023.     /** Coefficients valid for one time slot. */
  3024.     protected static class FieldSlot<T extends CalculusFieldElement<T>> {

  3026.         /**
  3027.          * The coefficients D<sub>i</sub><sup>j</sup>.
  3028.          * <p>
  3029.          * Only for j = 1 and j = 2 the coefficients are not 0. <br>
  3030.          * i corresponds to the equinoctial element, as follows: - i=0 for a <br/>
  3031.          * - i=1 for k <br/>
  3032.          * - i=2 for h <br/>
  3033.          * - i=3 for q <br/>
  3034.          * - i=4 for p <br/>
  3035.          * - i=5 for λ <br/>
  3036.          * </p>
  3037.          */
  3038.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] dij;

  3040.         /**
  3041.          * The coefficients C<sub>i</sub><sup>j</sup>.
  3042.          * <p>
  3043.          * The index order is cij[j][i] <br/>
  3044.          * i corresponds to the equinoctial element, as follows: <br/>
  3045.          * - i=0 for a <br/>
  3046.          * - i=1 for k <br/>
  3047.          * - i=2 for h <br/>
  3048.          * - i=3 for q <br/>
  3049.          * - i=4 for p <br/>
  3050.          * - i=5 for λ <br/>
  3051.          * </p>
  3052.          */
  3053.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] cij;

  3055.         /**
  3056.          * The coefficients S<sub>i</sub><sup>j</sup>.
  3057.          * <p>
  3058.          * The index order is sij[j][i] <br/>
  3059.          * i corresponds to the equinoctial element, as follows: <br/>
  3060.          * - i=0 for a <br/>
  3061.          * - i=1 for k <br/>
  3062.          * - i=2 for h <br/>
  3063.          * - i=3 for q <br/>
  3064.          * - i=4 for p <br/>
  3065.          * - i=5 for λ <br/>
  3066.          * </p>
  3067.          */
  3068.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] sij;

  3070.         /**
  3071.          * Simple constructor.
  3072.          * @param jMax                maximum value for j index
  3073.          * @param interpolationPoints number of points used in the interpolation process
  3074.          */
  3075.         @SuppressWarnings("unchecked")
  3076.         FieldSlot(final int jMax, final int interpolationPoints) {

  3078.             dij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3079.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, 3);
  3080.             cij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3081.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, jMax + 1);
  3082.             sij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3083.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, jMax + 1);

  3085.             // Initialize the C<sub>i</sub><sup>j</sup>, S<sub>i</sub><sup>j</sup> and
  3086.             // D<sub>i</sub><sup>j</sup> coefficients
  3087.             for (int j = 0; j <= jMax; j++) {
  3088.                 cij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3089.                 if (j > 0) {
  3090.                     sij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3091.                 }
  3092.                 // Initialize only the non-zero D<sub>i</sub><sup>j</sup> coefficients
  3093.                 if (j == 1 || j == 2) {
  3094.                     dij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3095.                 }
  3096.             }

  3098.         }

  3100.     }

  3102. }
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