AbstractGaussianContribution.java

  1. /* Copyright 2002-2023 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.PositionAngleType;
  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.      *                          only 1 value for each parameterDriver
  267.      * @return new force model context
  268.      */
  269.     private AbstractGaussianContributionContext initializeStep(final AuxiliaryElements auxiliaryElements,
  270.             final double[] parameters) {
  271.         return new AbstractGaussianContributionContext(auxiliaryElements, parameters);
  272.     }

  274.     /**
  275.      * Performs initialization at each integration step for the current force model.
  276.      * <p>
  277.      * This method aims at being called before mean elements rates computation.
  278.      * </p>
  279.      * @param <T>               type of the elements
  280.      * @param auxiliaryElements auxiliary elements related to the current orbit
  281.      * @param parameters        parameters values of the force model parameters
  282.      *                          (only 1 values for each parameters corresponding
  283.      *                          to state date) by getting the parameters for a specific date.
  284.      * @return new force model context
  285.      */
  286.     private <T extends CalculusFieldElement<T>> FieldAbstractGaussianContributionContext<T> initializeStep(
  287.             final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters) {
  288.         return new FieldAbstractGaussianContributionContext<>(auxiliaryElements, parameters);
  289.     }

  291.     /** {@inheritDoc} */
  292.     @Override
  293.     public double[] getMeanElementRate(final SpacecraftState state, final AuxiliaryElements auxiliaryElements,
  294.             final double[] parameters) {

  296.         // Container for attributes

  298.         final AbstractGaussianContributionContext context = initializeStep(auxiliaryElements, parameters);
  299.         double[] meanElementRate = new double[6];
  300.         // Computes the limits for the integral
  301.         final double[] ll = getLLimits(state, auxiliaryElements);
  302.         // Computes integrated mean element rates if Llow < Lhigh
  303.         if (ll[0] < ll[1]) {
  304.             meanElementRate = getMeanElementRate(state, integrator, ll[0], ll[1], context, parameters);
  305.             if (isDirty) {
  306.                 boolean next = true;
  307.                 for (int i = 0; i < MAX_ORDER_RANK && next; i++) {
  308.                     final double[] meanRates = getMeanElementRate(state, new GaussQuadrature(GAUSS_ORDER[i]), ll[0],
  309.                             ll[1], context, parameters);
  310.                     if (getRatesDiff(meanElementRate, meanRates, context) < threshold) {
  311.                         integrator = new GaussQuadrature(GAUSS_ORDER[i]);
  312.                         next = false;
  313.                     }
  314.                 }
  315.                 isDirty = false;
  316.             }
  317.         }
  318.         return meanElementRate;
  319.     }

  321.     /** {@inheritDoc} */
  322.     @Override
  323.     public <T extends CalculusFieldElement<T>> T[] getMeanElementRate(final FieldSpacecraftState<T> state,
  324.             final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters) {

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

  330.         T[] meanElementRate = MathArrays.buildArray(field, 6);
  331.         // Computes the limits for the integral
  332.         final T[] ll = getLLimits(state, auxiliaryElements);
  333.         // Computes integrated mean element rates if Llow < Lhigh
  334.         if (ll[0].getReal() < ll[1].getReal()) {
  335.             meanElementRate = getMeanElementRate(state, integrator, ll[0], ll[1], context, parameters);
  336.             if (isDirty) {
  337.                 boolean next = true;
  338.                 for (int i = 0; i < MAX_ORDER_RANK && next; i++) {
  339.                     final T[] meanRates = getMeanElementRate(state, new GaussQuadrature(GAUSS_ORDER[i]), ll[0], ll[1],
  340.                             context, parameters);
  341.                     if (getRatesDiff(meanElementRate, meanRates, context).getReal() < threshold) {
  342.                         integrator = new GaussQuadrature(GAUSS_ORDER[i]);
  343.                         next = false;
  344.                     }
  345.                 }
  346.                 isDirty = false;
  347.             }
  348.         }

  350.         return meanElementRate;
  351.     }

  353.     /**
  354.      * Compute the limits in L, the true longitude, for integration.
  355.      *
  356.      * @param state             current state information: date, kinematics,
  357.      *                          attitude
  358.      * @param auxiliaryElements auxiliary elements related to the current orbit
  359.      * @return the integration limits in L
  360.      */
  361.     protected abstract double[] getLLimits(SpacecraftState state, AuxiliaryElements auxiliaryElements);

  363.     /**
  364.      * Compute the limits in L, the true longitude, for integration.
  365.      *
  366.      * @param <T>               type of the elements
  367.      * @param state             current state information: date, kinematics,
  368.      *                          attitude
  369.      * @param auxiliaryElements auxiliary elements related to the current orbit
  370.      * @return the integration limits in L
  371.      */
  372.     protected abstract <T extends CalculusFieldElement<T>> T[] getLLimits(FieldSpacecraftState<T> state,
  373.             FieldAuxiliaryElements<T> auxiliaryElements);

  375.     /**
  376.      * Computes the mean equinoctial elements rates da<sub>i</sub> / dt.
  377.      *
  378.      * @param state      current state
  379.      * @param gauss      Gauss quadrature
  380.      * @param low        lower bound of the integral interval
  381.      * @param high       upper bound of the integral interval
  382.      * @param context    container for attributes
  383.      * @param parameters values of the force model parameters
  384.      * at state date (1 values for each parameters)
  385.      * @return the mean element rates
  386.      */
  387.     protected double[] getMeanElementRate(final SpacecraftState state, final GaussQuadrature gauss, final double low,
  388.             final double high, final AbstractGaussianContributionContext context, final double[] parameters) {

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

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

  395.         // Constant multiplier for integral
  396.         final double coef = 1. / (2. * FastMath.PI * auxiliaryElements.getB());
  397.         // Corrects mean element rates
  398.         for (int i = 0; i < 6; i++) {
  399.             meanElementRate[i] *= coef;
  400.         }
  401.         return meanElementRate;
  402.     }

  404.     /**
  405.      * Computes the mean equinoctial elements rates da<sub>i</sub> / dt.
  406.      *
  407.      * @param <T>        type of the elements
  408.      * @param state      current state
  409.      * @param gauss      Gauss quadrature
  410.      * @param low        lower bound of the integral interval
  411.      * @param high       upper bound of the integral interval
  412.      * @param context    container for attributes
  413.      * @param parameters values of the force model parameters(1 values for each parameters)
  414.      * @return the mean element rates
  415.      */
  416.     protected <T extends CalculusFieldElement<T>> T[] getMeanElementRate(final FieldSpacecraftState<T> state,
  417.             final GaussQuadrature gauss, final T low, final T high,
  418.             final FieldAbstractGaussianContributionContext<T> context, final T[] parameters) {

  420.         // Field
  421.         final Field<T> field = context.getA().getField();

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

  426.         final T[] meanElementRate = gauss.integrate(new FieldIntegrableFunction<>(state, true, 0, parameters, field),
  427.                 low, high, field);
  428.         // Constant multiplier for integral
  429.         final T coef = auxiliaryElements.getB().multiply(low.getPi()).multiply(2.).reciprocal();
  430.         // Corrects mean element rates
  431.         for (int i = 0; i < 6; i++) {
  432.             meanElementRate[i] = meanElementRate[i].multiply(coef);
  433.         }
  434.         return meanElementRate;
  435.     }

  437.     /**
  438.      * Estimates the weighted magnitude of the difference between 2 sets of
  439.      * equinoctial elements rates.
  440.      *
  441.      * @param meanRef reference rates
  442.      * @param meanCur current rates
  443.      * @param context container for attributes
  444.      * @return estimated magnitude of weighted differences
  445.      */
  446.     private double getRatesDiff(final double[] meanRef, final double[] meanCur,
  447.             final AbstractGaussianContributionContext context) {

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

  452.         double maxDiff = FastMath.abs(meanRef[0] - meanCur[0]) / auxiliaryElements.getSma();
  453.         // Corrects mean element rates
  454.         for (int i = 1; i < meanRef.length; i++) {
  455.             maxDiff = FastMath.max(maxDiff, FastMath.abs(meanRef[i] - meanCur[i]));
  456.         }
  457.         return maxDiff;
  458.     }

  460.     /**
  461.      * Estimates the weighted magnitude of the difference between 2 sets of
  462.      * equinoctial elements rates.
  463.      *
  464.      * @param <T>     type of the elements
  465.      * @param meanRef reference rates
  466.      * @param meanCur current rates
  467.      * @param context container for attributes
  468.      * @return estimated magnitude of weighted differences
  469.      */
  470.     private <T extends CalculusFieldElement<T>> T getRatesDiff(final T[] meanRef, final T[] meanCur,
  471.             final FieldAbstractGaussianContributionContext<T> context) {

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

  476.         T maxDiff = FastMath.abs(meanRef[0].subtract(meanCur[0])).divide(auxiliaryElements.getSma());
  477.         ;
  478.         // Corrects mean element rates
  479.         for (int i = 1; i < meanRef.length; i++) {
  480.             maxDiff = FastMath.max(maxDiff, FastMath.abs(meanRef[i].subtract(meanCur[i])));
  481.         }
  482.         return maxDiff;
  483.     }

  485.     /** {@inheritDoc} */
  486.     @Override
  487.     public void registerAttitudeProvider(final AttitudeProvider provider) {
  488.         this.attitudeProvider = provider;
  489.     }

  491.     /** {@inheritDoc} */
  492.     @Override
  493.     public void updateShortPeriodTerms(final double[] parameters, final SpacecraftState... meanStates) {

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

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

  501.             // Container of attributes
  502.             // Extract the proper parameters valid for the corresponding meanState date from the input array
  503.             final double[] extractedParameters = this.extractParameters(parameters, auxiliaryElements.getDate());
  504.             final AbstractGaussianContributionContext context = initializeStep(auxiliaryElements, extractedParameters);

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

  509.             // Generate the Cij and Sij coefficients
  510.             final FourierCjSjCoefficients fourierCjSj = new FourierCjSjCoefficients(meanState, JMAX, auxiliaryElements,
  511.                                                                                     extractedParameters);

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

  516.             gaussianSPCoefs.computeCoefficients(meanState, slot, fourierCjSj, uijvij, context.getMeanMotion(),
  517.                     auxiliaryElements.getSma());

  519.         }

  521.     }

  523.     /** {@inheritDoc} */
  524.     @Override
  525.     @SuppressWarnings("unchecked")
  526.     public <T extends CalculusFieldElement<T>> void updateShortPeriodTerms(final T[] parameters,
  527.             final FieldSpacecraftState<T>... meanStates) {

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

  532.         final FieldGaussianShortPeriodicCoefficients<T> fgspc = (FieldGaussianShortPeriodicCoefficients<T>) gaussianFieldSPCoefs
  533.                 .get(field);
  534.         final FieldSlot<T> slot = fgspc.createSlot(meanStates);
  535.         for (final FieldSpacecraftState<T> meanState : meanStates) {

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

  540.             // Container of attributes
  541.             // Extract the proper parameters valid for the corresponding meanState date from the input array
  542.             final T[] extractedParameters = this.extractParameters(parameters, auxiliaryElements.getDate());
  543.             final FieldAbstractGaussianContributionContext<T> context = initializeStep(auxiliaryElements, extractedParameters);

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

  548.             // Generate the Cij and Sij coefficients
  549.             final FieldFourierCjSjCoefficients<T> fourierCjSj = new FieldFourierCjSjCoefficients<>(meanState, JMAX,
  550.                     auxiliaryElements, extractedParameters, field);

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

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

  559.         }

  561.     }

  563.     /**
  564.      * Compute the auxiliary quantities ρ<sub>j</sub> and σ<sub>j</sub>.
  565.      * <p>
  566.      * The expressions used are equations 2.5.3-(4) from the Danielson paper. <br/>
  567.      * ρ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>C<sub>j</sub>(k, h) <br/>
  568.      * σ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>S<sub>j</sub>(k, h) <br/>
  569.      * </p>
  570.      * @param date              current date
  571.      * @param auxiliaryElements auxiliary elements related to the current orbit
  572.      * @return computed coefficients
  573.      */
  574.     private double[][] computeRhoSigmaCoefficients(final AbsoluteDate date, final AuxiliaryElements auxiliaryElements) {
  575.         final double[][] currentRhoSigmaj = new double[2][3 * JMAX + 1];
  576.         final CjSjCoefficient cjsjKH = new CjSjCoefficient(auxiliaryElements.getK(), auxiliaryElements.getH());
  577.         final double b = 1. / (1 + auxiliaryElements.getB());

  579.         // (-b)<sup>j</sup>
  580.         double mbtj = 1;

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

  584.             // Compute current rho and sigma;
  585.             mbtj *= -b;
  586.             final double coef = (1 + j * auxiliaryElements.getB()) * mbtj;
  587.             currentRhoSigmaj[0][j] = coef * cjsjKH.getCj(j);
  588.             currentRhoSigmaj[1][j] = coef * cjsjKH.getSj(j);
  589.         }
  590.         return currentRhoSigmaj;
  591.     }

  593.     /**
  594.      * Compute the auxiliary quantities ρ<sub>j</sub> and σ<sub>j</sub>.
  595.      * <p>
  596.      * The expressions used are equations 2.5.3-(4) from the Danielson paper. <br/>
  597.      * ρ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>C<sub>j</sub>(k, h) <br/>
  598.      * σ<sub>j</sub> = (1+jB)(-b)<sup>j</sup>S<sub>j</sub>(k, h) <br/>
  599.      * </p>
  600.      * @param <T>     type of the elements
  601.      * @param date    current date
  602.      * @param context container for attributes
  603.      * @param field   field used by default
  604.      * @return computed coefficients
  605.      */
  606.     private <T extends CalculusFieldElement<T>> T[][] computeRhoSigmaCoefficients(final FieldAbsoluteDate<T> date,
  607.             final FieldAbstractGaussianContributionContext<T> context, final Field<T> field) {
  608.         // zero
  609.         final T zero = field.getZero();

  611.         final FieldAuxiliaryElements<T> auxiliaryElements = context.getFieldAuxiliaryElements();
  612.         final T[][] currentRhoSigmaj = MathArrays.buildArray(field, 2, 3 * JMAX + 1);
  613.         final FieldCjSjCoefficient<T> cjsjKH = new FieldCjSjCoefficient<>(auxiliaryElements.getK(),
  614.                 auxiliaryElements.getH(), field);
  615.         final T b = auxiliaryElements.getB().add(1.).reciprocal();

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

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

  622.             // Compute current rho and sigma;
  623.             mbtj = mbtj.multiply(b.negate());
  624.             final T coef = mbtj.multiply(auxiliaryElements.getB().multiply(j).add(1.));
  625.             currentRhoSigmaj[0][j] = coef.multiply(cjsjKH.getCj(j));
  626.             currentRhoSigmaj[1][j] = coef.multiply(cjsjKH.getSj(j));
  627.         }
  628.         return currentRhoSigmaj;
  629.     }

  631.     /**
  632.      * Internal class for numerical quadrature.
  633.      * <p>
  634.      * This class is a rewrite of {@link IntegrableFunction} for field elements
  635.      * </p>
  636.      * @param <T> type of the field elements
  637.      */
  638.     protected class FieldIntegrableFunction<T extends CalculusFieldElement<T>>
  639.             implements CalculusFieldUnivariateVectorFunction<T> {

  641.         /** Current state. */
  642.         private final FieldSpacecraftState<T> state;

  644.         /**
  645.          * Signal that this class is used to compute the values required by the mean
  646.          * element variations or by the short periodic element variations.
  647.          */
  648.         private final boolean meanMode;

  650.         /**
  651.          * The j index.
  652.          * <p>
  653.          * Used only for short periodic variation. Ignored for mean elements variation.
  654.          * </p>
  655.          */
  656.         private final int j;

  658.         /** Container for attributes. */
  659.         private final FieldAbstractGaussianContributionContext<T> context;

  661.         /** Auxiliary Elements. */
  662.         private final FieldAuxiliaryElements<T> auxiliaryElements;

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

  667.         /**
  668.          * Build a new instance with a new field.
  669.          * @param state      current state information: date, kinematics, attitude
  670.          * @param meanMode   if true return the value associated to the mean elements
  671.          *                   variation, if false return the values associated to the
  672.          *                   short periodic elements variation
  673.          * @param j          the j index. used only for short periodic variation.
  674.          *                   Ignored for mean elements variation.
  675.          * @param parameters values of the force model parameters (only 1 values
  676.          *                   for each parameters corresponding to state date) obtained by
  677.          *                   calling the extract parameter method {@link #extractParameters(double[], AbsoluteDate)}
  678.          *                   to selected the right value for state date or by getting the parameters for a specific date
  679.          * @param field      field utilized by default
  680.          */
  681.         public FieldIntegrableFunction(final FieldSpacecraftState<T> state, final boolean meanMode, final int j,
  682.                 final T[] parameters, final Field<T> field) {

  684.             this.meanMode = meanMode;
  685.             this.j = j;
  686.             this.parameters = parameters.clone();
  687.             this.auxiliaryElements = new FieldAuxiliaryElements<>(state.getOrbit(), I);
  688.             this.context = new FieldAbstractGaussianContributionContext<>(auxiliaryElements, this.parameters);
  689.             // remove derivatives from state
  690.             final T[] stateVector = MathArrays.buildArray(field, 6);
  691.             OrbitType.EQUINOCTIAL.mapOrbitToArray(state.getOrbit(), PositionAngleType.TRUE, stateVector, null);
  692.             final FieldOrbit<T> fixedOrbit = OrbitType.EQUINOCTIAL.mapArrayToOrbit(stateVector, null,
  693.                     PositionAngleType.TRUE, state.getDate(), context.getMu(), state.getFrame());
  694.             this.state = new FieldSpacecraftState<>(fixedOrbit, state.getAttitude(), state.getMass());
  695.         }

  697.         /** {@inheritDoc} */
  698.         @Override
  699.         public T[] value(final T x) {

  701.             // Parameters for array building
  702.             final Field<T> field = auxiliaryElements.getDate().getField();
  703.             final int dimension = 6;

  705.             // Compute the time difference from the true longitude difference
  706.             final T shiftedLm = trueToMean(x);
  707.             final T dLm = shiftedLm.subtract(auxiliaryElements.getLM());
  708.             final T dt = dLm.divide(context.getMeanMotion());

  710.             final FieldSinCos<T> scL = FastMath.sinCos(x);
  711.             final T cosL = scL.cos();
  712.             final T sinL = scL.sin();
  713.             final T roa  = auxiliaryElements.getB().multiply(auxiliaryElements.getB()).divide(auxiliaryElements.getH().multiply(sinL).add(auxiliaryElements.getK().multiply(cosL)).add(1.));
  714.             final T roa2 = roa.multiply(roa);
  715.             final T r = auxiliaryElements.getSma().multiply(roa);
  716.             final T X = r.multiply(cosL);
  717.             final T Y = r.multiply(sinL);
  718.             final T naob = context.getMeanMotion().multiply(auxiliaryElements.getSma())
  719.                     .divide(auxiliaryElements.getB());
  720.             final T Xdot = naob.multiply(auxiliaryElements.getH().add(sinL)).negate();
  721.             final T Ydot = naob.multiply(auxiliaryElements.getK().add(cosL));
  722.             final FieldVector3D<T> vel = new FieldVector3D<>(Xdot, auxiliaryElements.getVectorF(), Ydot,
  723.                     auxiliaryElements.getVectorG());

  725.             // Compute acceleration
  726.             FieldVector3D<T> acc = FieldVector3D.getZero(field);

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

  731.             // Recompose an orbit with time held fixed to be compliant with DSST theory
  732.             final FieldOrbit<T> recomposedOrbit = new FieldEquinoctialOrbit<>(shiftedOrbit.getA(),
  733.                     shiftedOrbit.getEquinoctialEx(), shiftedOrbit.getEquinoctialEy(), shiftedOrbit.getHx(),
  734.                     shiftedOrbit.getHy(), shiftedOrbit.getLv(), PositionAngleType.TRUE, shiftedOrbit.getFrame(),
  735.                     state.getDate(), context.getMu());

  737.             // Get the corresponding attitude
  738.             final FieldAttitude<T> recomposedAttitude = attitudeProvider.getAttitude(recomposedOrbit,
  739.                     recomposedOrbit.getDate(), recomposedOrbit.getFrame());

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

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

  747.             // Compute the derivatives of the elements by the speed
  748.             final T[] deriv = MathArrays.buildArray(field, dimension);
  749.             // da/dv
  750.             deriv[0] = getAoV(vel).dotProduct(acc);
  751.             // dex/dv
  752.             deriv[1] = getKoV(X, Y, Xdot, Ydot).dotProduct(acc);
  753.             // dey/dv
  754.             deriv[2] = getHoV(X, Y, Xdot, Ydot).dotProduct(acc);
  755.             // dhx/dv
  756.             deriv[3] = getQoV(X).dotProduct(acc);
  757.             // dhy/dv
  758.             deriv[4] = getPoV(Y).dotProduct(acc);
  759.             // dλ/dv
  760.             deriv[5] = getLoV(X, Y, Xdot, Ydot).dotProduct(acc);

  762.             // Compute mean elements rates
  763.             T[] val = null;
  764.             if (meanMode) {
  765.                 val = MathArrays.buildArray(field, dimension);
  766.                 for (int i = 0; i < 6; i++) {
  767.                     // da<sub>i</sub>/dt
  768.                     val[i] = deriv[i].multiply(roa2);
  769.                 }
  770.             } else {
  771.                 val = MathArrays.buildArray(field, dimension * 2);
  772.                 //Compute cos(j*L) and sin(j*L);
  773.                 final FieldSinCos<T> scjL = FastMath.sinCos(x.multiply(j));
  774.                 final T cosjL = j == 1 ? cosL : scjL.cos();
  775.                 final T sinjL = j == 1 ? sinL : scjL.sin();

  777.                 for (int i = 0; i < 6; i++) {
  778.                     // da<sub>i</sub>/dv * cos(jL)
  779.                     val[i] = deriv[i].multiply(cosjL);
  780.                     // da<sub>i</sub>/dv * sin(jL)
  781.                     val[i + 6] = deriv[i].multiply(sinjL);
  782.                 }
  783.             }

  785.             return val;
  786.         }

  788.         /**
  789.          * Converts true longitude to mean longitude.
  790.          * @param x True longitude
  791.          * @return Eccentric longitude
  792.          */
  793.         private T trueToMean(final T x) {
  794.             return eccentricToMean(trueToEccentric(x));
  795.         }

  797.         /**
  798.          * Converts true longitude to eccentric longitude.
  799.          * @param lv True longitude
  800.          * @return Eccentric longitude
  801.          */
  802.         private T trueToEccentric (final T lv) {
  803.             final FieldSinCos<T> sclV = FastMath.sinCos(lv);
  804.             final T cosLv   = sclV.cos();
  805.             final T sinLv   = sclV.sin();
  806.             final T num     = auxiliaryElements.getH().multiply(cosLv).subtract(auxiliaryElements.getK().multiply(sinLv));
  807.             final T den     = auxiliaryElements.getB().add(auxiliaryElements.getK().multiply(cosLv)).add(auxiliaryElements.getH().multiply(sinLv)).add(1.);
  808.             return FastMath.atan(num.divide(den)).multiply(2.).add(lv);
  809.         }

  811.         /**
  812.          * Converts eccentric longitude to mean longitude.
  813.          * @param le Eccentric longitude
  814.          * @return Mean longitude
  815.          */
  816.         private T eccentricToMean (final T le) {
  817.             final FieldSinCos<T> scle = FastMath.sinCos(le);
  818.             return le.subtract(auxiliaryElements.getK().multiply(scle.sin())).add(auxiliaryElements.getH().multiply(scle.cos()));
  819.         }

  821.         /**
  822.          * Compute δa/δv.
  823.          * @param vel satellite velocity
  824.          * @return δa/δv
  825.          */
  826.         private FieldVector3D<T> getAoV(final FieldVector3D<T> vel) {
  827.             return new FieldVector3D<>(context.getTon2a(), vel);
  828.         }

  830.         /**
  831.          * Compute δh/δv.
  832.          * @param X    satellite position component along f, equinoctial reference frame
  833.          *             1st vector
  834.          * @param Y    satellite position component along g, equinoctial reference frame
  835.          *             2nd vector
  836.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  837.          *             1st vector
  838.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  839.          *             2nd vector
  840.          * @return δh/δv
  841.          */
  842.         private FieldVector3D<T> getHoV(final T X, final T Y, final T Xdot, final T Ydot) {
  843.             final T kf = (Xdot.multiply(Y).multiply(2.).subtract(X.multiply(Ydot))).multiply(context.getOoMU());
  844.             final T kg = X.multiply(Xdot).multiply(context.getOoMU());
  845.             final T kw = auxiliaryElements.getK().multiply(
  846.                     auxiliaryElements.getQ().multiply(Y).multiply(I).subtract(auxiliaryElements.getP().multiply(X)))
  847.                     .multiply(context.getOOAB());
  848.             return new FieldVector3D<>(kf, auxiliaryElements.getVectorF(), kg.negate(), auxiliaryElements.getVectorG(),
  849.                     kw, auxiliaryElements.getVectorW());
  850.         }

  852.         /**
  853.          * Compute δk/δv.
  854.          * @param X    satellite position component along f, equinoctial reference frame
  855.          *             1st vector
  856.          * @param Y    satellite position component along g, equinoctial reference frame
  857.          *             2nd vector
  858.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  859.          *             1st vector
  860.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  861.          *             2nd vector
  862.          * @return δk/δv
  863.          */
  864.         private FieldVector3D<T> getKoV(final T X, final T Y, final T Xdot, final T Ydot) {
  865.             final T kf = Y.multiply(Ydot).multiply(context.getOoMU());
  866.             final T kg = (X.multiply(Ydot).multiply(2.).subtract(Xdot.multiply(Y))).multiply(context.getOoMU());
  867.             final T kw = auxiliaryElements.getH().multiply(
  868.                     auxiliaryElements.getQ().multiply(Y).multiply(I).subtract(auxiliaryElements.getP().multiply(X)))
  869.                     .multiply(context.getOOAB());
  870.             return new FieldVector3D<>(kf.negate(), auxiliaryElements.getVectorF(), kg, auxiliaryElements.getVectorG(),
  871.                     kw.negate(), auxiliaryElements.getVectorW());
  872.         }

  874.         /**
  875.          * Compute δp/δv.
  876.          * @param Y satellite position component along g, equinoctial reference frame
  877.          *          2nd vector
  878.          * @return δp/δv
  879.          */
  880.         private FieldVector3D<T> getPoV(final T Y) {
  881.             return new FieldVector3D<>(context.getCo2AB().multiply(Y), auxiliaryElements.getVectorW());
  882.         }

  884.         /**
  885.          * Compute δq/δv.
  886.          * @param X satellite position component along f, equinoctial reference frame
  887.          *          1st vector
  888.          * @return δq/δv
  889.          */
  890.         private FieldVector3D<T> getQoV(final T X) {
  891.             return new FieldVector3D<>(context.getCo2AB().multiply(X).multiply(I), auxiliaryElements.getVectorW());
  892.         }

  894.         /**
  895.          * Compute δλ/δv.
  896.          * @param X    satellite position component along f, equinoctial reference frame
  897.          *             1st vector
  898.          * @param Y    satellite position component along g, equinoctial reference frame
  899.          *             2nd vector
  900.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  901.          *             1st vector
  902.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  903.          *             2nd vector
  904.          * @return δλ/δv
  905.          */
  906.         private FieldVector3D<T> getLoV(final T X, final T Y, final T Xdot, final T Ydot) {
  907.             final FieldVector3D<T> pos = new FieldVector3D<>(X, auxiliaryElements.getVectorF(), Y,
  908.                     auxiliaryElements.getVectorG());
  909.             final FieldVector3D<T> v2 = new FieldVector3D<>(auxiliaryElements.getK(), getHoV(X, Y, Xdot, Ydot),
  910.                     auxiliaryElements.getH().negate(), getKoV(X, Y, Xdot, Ydot));
  911.             return new FieldVector3D<>(context.getOOA().multiply(-2.), pos, context.getOoBpo(), v2,
  912.                     context.getOOA().multiply(auxiliaryElements.getQ().multiply(Y).multiply(I)
  913.                             .subtract(auxiliaryElements.getP().multiply(X))),
  914.                     auxiliaryElements.getVectorW());
  915.         }

  917.     }

  919.     /** Internal class for numerical quadrature. */
  920.     protected class IntegrableFunction implements UnivariateVectorFunction {

  922.         /** Current state. */
  923.         private final SpacecraftState state;

  925.         /**
  926.          * Signal that this class is used to compute the values required by the mean
  927.          * element variations or by the short periodic element variations.
  928.          */
  929.         private final boolean meanMode;

  931.         /**
  932.          * The j index.
  933.          * <p>
  934.          * Used only for short periodic variation. Ignored for mean elements variation.
  935.          * </p>
  936.          */
  937.         private final int j;

  939.         /** Container for attributes. */
  940.         private final AbstractGaussianContributionContext context;

  942.         /** Auxiliary Elements. */
  943.         private final AuxiliaryElements auxiliaryElements;

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

  948.         /**
  949.          * Build a new instance.
  950.          * @param state      current state information: date, kinematics, attitude
  951.          * @param meanMode   if true return the value associated to the mean elements
  952.          *                   variation, if false return the values associated to the
  953.          *                   short periodic elements variation
  954.          * @param j          the j index. used only for short periodic variation.
  955.          *                   Ignored for mean elements variation.
  956.          * @param parameters list of the estimated values for each driver at state date of the force model parameters
  957.          *                   only 1 value for each parameter
  958.          */
  959.         IntegrableFunction(final SpacecraftState state, final boolean meanMode, final int j,
  960.                 final double[] parameters) {

  962.             this.meanMode = meanMode;
  963.             this.j = j;
  964.             this.parameters = parameters.clone();
  965.             this.auxiliaryElements = new AuxiliaryElements(state.getOrbit(), I);
  966.             this.context = new AbstractGaussianContributionContext(auxiliaryElements, this.parameters);
  967.             // remove derivatives from state
  968.             final double[] stateVector = new double[6];
  969.             OrbitType.EQUINOCTIAL.mapOrbitToArray(state.getOrbit(), PositionAngleType.TRUE, stateVector, null);
  970.             final Orbit fixedOrbit = OrbitType.EQUINOCTIAL.mapArrayToOrbit(stateVector, null, PositionAngleType.TRUE,
  971.                     state.getDate(), context.getMu(), state.getFrame());
  972.             this.state = new SpacecraftState(fixedOrbit, state.getAttitude(), state.getMass());
  973.         }

  975.         /** {@inheritDoc} */
  976.         @Override
  977.         public double[] value(final double x) {

  979.             // Compute the time difference from the true longitude difference
  980.             final double shiftedLm = trueToMean(x);
  981.             final double dLm = shiftedLm - auxiliaryElements.getLM();
  982.             final double dt = dLm / context.getMeanMotion();

  984.             final SinCos scL  = FastMath.sinCos(x);
  985.             final double cosL = scL.cos();
  986.             final double sinL = scL.sin();
  987.             final double roa  = auxiliaryElements.getB() * auxiliaryElements.getB() / (1. + auxiliaryElements.getH() * sinL + auxiliaryElements.getK() * cosL);
  988.             final double roa2 = roa * roa;
  989.             final double r = auxiliaryElements.getSma() * roa;
  990.             final double X = r * cosL;
  991.             final double Y = r * sinL;
  992.             final double naob = context.getMeanMotion() * auxiliaryElements.getSma() / auxiliaryElements.getB();
  993.             final double Xdot = -naob * (auxiliaryElements.getH() + sinL);
  994.             final double Ydot = naob * (auxiliaryElements.getK() + cosL);
  995.             final Vector3D vel = new Vector3D(Xdot, auxiliaryElements.getVectorF(), Ydot,
  996.                     auxiliaryElements.getVectorG());

  998.             // Compute acceleration
  999.             Vector3D acc = Vector3D.ZERO;

  1001.             // shift the orbit to dt
  1002.             final Orbit shiftedOrbit = state.getOrbit().shiftedBy(dt);

  1004.             // Recompose an orbit with time held fixed to be compliant with DSST theory
  1005.             final Orbit recomposedOrbit = new EquinoctialOrbit(shiftedOrbit.getA(), shiftedOrbit.getEquinoctialEx(),
  1006.                     shiftedOrbit.getEquinoctialEy(), shiftedOrbit.getHx(), shiftedOrbit.getHy(), shiftedOrbit.getLv(),
  1007.                     PositionAngleType.TRUE, shiftedOrbit.getFrame(), state.getDate(), context.getMu());

  1009.             // Get the corresponding attitude
  1010.             final Attitude recomposedAttitude = attitudeProvider.getAttitude(recomposedOrbit, recomposedOrbit.getDate(),
  1011.                     recomposedOrbit.getFrame());

  1013.             // create shifted SpacecraftState with attitude at specified time
  1014.             final SpacecraftState shiftedState = new SpacecraftState(recomposedOrbit, recomposedAttitude,
  1015.                     state.getMass());

  1017.             // here parameters is a list of all span values of each parameter driver
  1018.             acc = contribution.acceleration(shiftedState, parameters);

  1020.             // Compute the derivatives of the elements by the speed
  1021.             final double[] deriv = new double[6];
  1022.             // da/dv
  1023.             deriv[0] = getAoV(vel).dotProduct(acc);
  1024.             // dex/dv
  1025.             deriv[1] = getKoV(X, Y, Xdot, Ydot).dotProduct(acc);
  1026.             // dey/dv
  1027.             deriv[2] = getHoV(X, Y, Xdot, Ydot).dotProduct(acc);
  1028.             // dhx/dv
  1029.             deriv[3] = getQoV(X).dotProduct(acc);
  1030.             // dhy/dv
  1031.             deriv[4] = getPoV(Y).dotProduct(acc);
  1032.             // dλ/dv
  1033.             deriv[5] = getLoV(X, Y, Xdot, Ydot).dotProduct(acc);

  1035.             // Compute mean elements rates
  1036.             double[] val = null;
  1037.             if (meanMode) {
  1038.                 val = new double[6];
  1039.                 for (int i = 0; i < 6; i++) {
  1040.                     // da<sub>i</sub>/dt
  1041.                     val[i] = roa2 * deriv[i];
  1042.                 }
  1043.             } else {
  1044.                 val = new double[12];
  1045.                 //Compute cos(j*L) and sin(j*L);
  1046.                 final SinCos scjL  = FastMath.sinCos(j * x);
  1047.                 final double cosjL = j == 1 ? cosL : scjL.cos();
  1048.                 final double sinjL = j == 1 ? sinL : scjL.sin();

  1050.                 for (int i = 0; i < 6; i++) {
  1051.                     // da<sub>i</sub>/dv * cos(jL)
  1052.                     val[i] = cosjL * deriv[i];
  1053.                     // da<sub>i</sub>/dv * sin(jL)
  1054.                     val[i + 6] = sinjL * deriv[i];
  1055.                 }
  1056.             }
  1057.             return val;
  1058.         }

  1060.         /**
  1061.          * Converts true longitude to eccentric longitude.
  1062.          * @param lv True longitude
  1063.          * @return Eccentric longitude
  1064.          */
  1065.         private double trueToEccentric (final double lv) {
  1066.             final SinCos scLv    = FastMath.sinCos(lv);
  1067.             final double num     = auxiliaryElements.getH() * scLv.cos() - auxiliaryElements.getK() * scLv.sin();
  1068.             final double den     = auxiliaryElements.getB() + 1. + auxiliaryElements.getK() * scLv.cos() + auxiliaryElements.getH() * scLv.sin();
  1069.             return lv + 2. * FastMath.atan(num / den);
  1070.         }

  1072.         /**
  1073.          * Converts eccentric longitude to mean longitude.
  1074.          * @param le Eccentric longitude
  1075.          * @return Mean longitude
  1076.          */
  1077.         private double eccentricToMean (final double le) {
  1078.             final SinCos scLe = FastMath.sinCos(le);
  1079.             return le - auxiliaryElements.getK() * scLe.sin() + auxiliaryElements.getH() * scLe.cos();
  1080.         }

  1082.         /**
  1083.          * Converts true longitude to mean longitude.
  1084.          * @param lv True longitude
  1085.          * @return Eccentric longitude
  1086.          */
  1087.         private double trueToMean(final double lv) {
  1088.             return eccentricToMean(trueToEccentric(lv));
  1089.         }

  1091.         /**
  1092.          * Compute δa/δv.
  1093.          * @param vel satellite velocity
  1094.          * @return δa/δv
  1095.          */
  1096.         private Vector3D getAoV(final Vector3D vel) {
  1097.             return new Vector3D(context.getTon2a(), vel);
  1098.         }

  1100.         /**
  1101.          * Compute δh/δv.
  1102.          * @param X    satellite position component along f, equinoctial reference frame
  1103.          *             1st vector
  1104.          * @param Y    satellite position component along g, equinoctial reference frame
  1105.          *             2nd vector
  1106.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1107.          *             1st vector
  1108.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1109.          *             2nd vector
  1110.          * @return δh/δv
  1111.          */
  1112.         private Vector3D getHoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1113.             final double kf = (2. * Xdot * Y - X * Ydot) * context.getOoMU();
  1114.             final double kg = X * Xdot * context.getOoMU();
  1115.             final double kw = auxiliaryElements.getK() *
  1116.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOAB();
  1117.             return new Vector3D(kf, auxiliaryElements.getVectorF(), -kg, auxiliaryElements.getVectorG(), kw,
  1118.                     auxiliaryElements.getVectorW());
  1119.         }

  1121.         /**
  1122.          * Compute δk/δv.
  1123.          * @param X    satellite position component along f, equinoctial reference frame
  1124.          *             1st vector
  1125.          * @param Y    satellite position component along g, equinoctial reference frame
  1126.          *             2nd vector
  1127.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1128.          *             1st vector
  1129.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1130.          *             2nd vector
  1131.          * @return δk/δv
  1132.          */
  1133.         private Vector3D getKoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1134.             final double kf = Y * Ydot * context.getOoMU();
  1135.             final double kg = (2. * X * Ydot - Xdot * Y) * context.getOoMU();
  1136.             final double kw = auxiliaryElements.getH() *
  1137.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOAB();
  1138.             return new Vector3D(-kf, auxiliaryElements.getVectorF(), kg, auxiliaryElements.getVectorG(), -kw,
  1139.                     auxiliaryElements.getVectorW());
  1140.         }

  1142.         /**
  1143.          * Compute δp/δv.
  1144.          * @param Y satellite position component along g, equinoctial reference frame
  1145.          *          2nd vector
  1146.          * @return δp/δv
  1147.          */
  1148.         private Vector3D getPoV(final double Y) {
  1149.             return new Vector3D(context.getCo2AB() * Y, auxiliaryElements.getVectorW());
  1150.         }

  1152.         /**
  1153.          * Compute δq/δv.
  1154.          * @param X satellite position component along f, equinoctial reference frame
  1155.          *          1st vector
  1156.          * @return δq/δv
  1157.          */
  1158.         private Vector3D getQoV(final double X) {
  1159.             return new Vector3D(I * context.getCo2AB() * X, auxiliaryElements.getVectorW());
  1160.         }

  1162.         /**
  1163.          * Compute δλ/δv.
  1164.          * @param X    satellite position component along f, equinoctial reference frame
  1165.          *             1st vector
  1166.          * @param Y    satellite position component along g, equinoctial reference frame
  1167.          *             2nd vector
  1168.          * @param Xdot satellite velocity component along f, equinoctial reference frame
  1169.          *             1st vector
  1170.          * @param Ydot satellite velocity component along g, equinoctial reference frame
  1171.          *             2nd vector
  1172.          * @return δλ/δv
  1173.          */
  1174.         private Vector3D getLoV(final double X, final double Y, final double Xdot, final double Ydot) {
  1175.             final Vector3D pos = new Vector3D(X, auxiliaryElements.getVectorF(), Y, auxiliaryElements.getVectorG());
  1176.             final Vector3D v2 = new Vector3D(auxiliaryElements.getK(), getHoV(X, Y, Xdot, Ydot),
  1177.                     -auxiliaryElements.getH(), getKoV(X, Y, Xdot, Ydot));
  1178.             return new Vector3D(-2. * context.getOOA(), pos, context.getOoBpo(), v2,
  1179.                     (I * auxiliaryElements.getQ() * Y - auxiliaryElements.getP() * X) * context.getOOA(),
  1180.                     auxiliaryElements.getVectorW());
  1181.         }

  1183.     }

  1185.     /**
  1186.      * Class used to {@link #integrate(UnivariateVectorFunction, double, double)
  1187.      * integrate} a {@link org.hipparchus.analysis.UnivariateVectorFunction
  1188.      * function} of the orbital elements using the Gaussian quadrature rule to get
  1189.      * the acceleration.
  1190.      */
  1191.     protected static class GaussQuadrature {

  1193.         // Points and weights for the available quadrature orders

  1195.         /** Points for quadrature of order 12. */
  1196.         private static final double[] P_12 = { -0.98156063424671910000, -0.90411725637047490000,
  1197.             -0.76990267419430470000, -0.58731795428661740000, -0.36783149899818024000, -0.12523340851146890000,
  1198.             0.12523340851146890000, 0.36783149899818024000, 0.58731795428661740000, 0.76990267419430470000,
  1199.             0.90411725637047490000, 0.98156063424671910000 };

  1201.         /** Weights for quadrature of order 12. */
  1202.         private static final double[] W_12 = { 0.04717533638651220000, 0.10693932599531830000, 0.16007832854334633000,
  1203.             0.20316742672306584000, 0.23349253653835478000, 0.24914704581340286000, 0.24914704581340286000,
  1204.             0.23349253653835478000, 0.20316742672306584000, 0.16007832854334633000, 0.10693932599531830000,
  1205.             0.04717533638651220000 };

  1207.         /** Points for quadrature of order 16. */
  1208.         private static final double[] P_16 = { -0.98940093499164990000, -0.94457502307323260000,
  1209.             -0.86563120238783160000, -0.75540440835500310000, -0.61787624440264380000, -0.45801677765722737000,
  1210.             -0.28160355077925890000, -0.09501250983763745000, 0.09501250983763745000, 0.28160355077925890000,
  1211.             0.45801677765722737000, 0.61787624440264380000, 0.75540440835500310000, 0.86563120238783160000,
  1212.             0.94457502307323260000, 0.98940093499164990000 };

  1214.         /** Weights for quadrature of order 16. */
  1215.         private static final double[] W_16 = { 0.02715245941175405800, 0.06225352393864777000, 0.09515851168249283000,
  1216.             0.12462897125553388000, 0.14959598881657685000, 0.16915651939500256000, 0.18260341504492360000,
  1217.             0.18945061045506847000, 0.18945061045506847000, 0.18260341504492360000, 0.16915651939500256000,
  1218.             0.14959598881657685000, 0.12462897125553388000, 0.09515851168249283000, 0.06225352393864777000,
  1219.             0.02715245941175405800 };

  1221.         /** Points for quadrature of order 20. */
  1222.         private static final double[] P_20 = { -0.99312859918509490000, -0.96397192727791390000,
  1223.             -0.91223442825132600000, -0.83911697182221890000, -0.74633190646015080000, -0.63605368072651510000,
  1224.             -0.51086700195082700000, -0.37370608871541955000, -0.22778585114164507000, -0.07652652113349734000,
  1225.             0.07652652113349734000, 0.22778585114164507000, 0.37370608871541955000, 0.51086700195082700000,
  1226.             0.63605368072651510000, 0.74633190646015080000, 0.83911697182221890000, 0.91223442825132600000,
  1227.             0.96397192727791390000, 0.99312859918509490000 };

  1229.         /** Weights for quadrature of order 20. */
  1230.         private static final double[] W_20 = { 0.01761400713915226400, 0.04060142980038684000, 0.06267204833410904000,
  1231.             0.08327674157670477000, 0.10193011981724048000, 0.11819453196151844000, 0.13168863844917678000,
  1232.             0.14209610931838212000, 0.14917298647260380000, 0.15275338713072600000, 0.15275338713072600000,
  1233.             0.14917298647260380000, 0.14209610931838212000, 0.13168863844917678000, 0.11819453196151844000,
  1234.             0.10193011981724048000, 0.08327674157670477000, 0.06267204833410904000, 0.04060142980038684000,
  1235.             0.01761400713915226400 };

  1237.         /** Points for quadrature of order 24. */
  1238.         private static final double[] P_24 = { -0.99518721999702130000, -0.97472855597130950000,
  1239.             -0.93827455200273270000, -0.88641552700440100000, -0.82000198597390300000, -0.74012419157855440000,
  1240.             -0.64809365193697550000, -0.54542147138883950000, -0.43379350762604520000, -0.31504267969616340000,
  1241.             -0.19111886747361634000, -0.06405689286260563000, 0.06405689286260563000, 0.19111886747361634000,
  1242.             0.31504267969616340000, 0.43379350762604520000, 0.54542147138883950000, 0.64809365193697550000,
  1243.             0.74012419157855440000, 0.82000198597390300000, 0.88641552700440100000, 0.93827455200273270000,
  1244.             0.97472855597130950000, 0.99518721999702130000 };

  1246.         /** Weights for quadrature of order 24. */
  1247.         private static final double[] W_24 = { 0.01234122979998733500, 0.02853138862893380600, 0.04427743881741981000,
  1248.             0.05929858491543691500, 0.07334648141108027000, 0.08619016153195320000, 0.09761865210411391000,
  1249.             0.10744427011596558000, 0.11550566805372553000, 0.12167047292780335000, 0.12583745634682825000,
  1250.             0.12793819534675221000, 0.12793819534675221000, 0.12583745634682825000, 0.12167047292780335000,
  1251.             0.11550566805372553000, 0.10744427011596558000, 0.09761865210411391000, 0.08619016153195320000,
  1252.             0.07334648141108027000, 0.05929858491543691500, 0.04427743881741981000, 0.02853138862893380600,
  1253.             0.01234122979998733500 };

  1255.         /** Points for quadrature of order 32. */
  1256.         private static final double[] P_32 = { -0.99726386184948160000, -0.98561151154526840000,
  1257.             -0.96476225558750640000, -0.93490607593773970000, -0.89632115576605220000, -0.84936761373256990000,
  1258.             -0.79448379596794250000, -0.73218211874028970000, -0.66304426693021520000, -0.58771575724076230000,
  1259.             -0.50689990893222950000, -0.42135127613063540000, -0.33186860228212767000, -0.23928736225213710000,
  1260.             -0.14447196158279646000, -0.04830766568773831000, 0.04830766568773831000, 0.14447196158279646000,
  1261.             0.23928736225213710000, 0.33186860228212767000, 0.42135127613063540000, 0.50689990893222950000,
  1262.             0.58771575724076230000, 0.66304426693021520000, 0.73218211874028970000, 0.79448379596794250000,
  1263.             0.84936761373256990000, 0.89632115576605220000, 0.93490607593773970000, 0.96476225558750640000,
  1264.             0.98561151154526840000, 0.99726386184948160000 };

  1266.         /** Weights for quadrature of order 32. */
  1267.         private static final double[] W_32 = { 0.00701861000947013600, 0.01627439473090571200, 0.02539206530926214200,
  1268.             0.03427386291302141000, 0.04283589802222658600, 0.05099805926237621600, 0.05868409347853559000,
  1269.             0.06582222277636193000, 0.07234579410884862000, 0.07819389578707042000, 0.08331192422694673000,
  1270.             0.08765209300440380000, 0.09117387869576390000, 0.09384439908080441000, 0.09563872007927487000,
  1271.             0.09654008851472784000, 0.09654008851472784000, 0.09563872007927487000, 0.09384439908080441000,
  1272.             0.09117387869576390000, 0.08765209300440380000, 0.08331192422694673000, 0.07819389578707042000,
  1273.             0.07234579410884862000, 0.06582222277636193000, 0.05868409347853559000, 0.05099805926237621600,
  1274.             0.04283589802222658600, 0.03427386291302141000, 0.02539206530926214200, 0.01627439473090571200,
  1275.             0.00701861000947013600 };

  1277.         /** Points for quadrature of order 40. */
  1278.         private static final double[] P_40 = { -0.99823770971055930000, -0.99072623869945710000,
  1279.             -0.97725994998377420000, -0.95791681921379170000, -0.93281280827867660000, -0.90209880696887420000,
  1280.             -0.86595950321225960000, -0.82461223083331170000, -0.77830565142651940000, -0.72731825518992710000,
  1281.             -0.67195668461417960000, -0.61255388966798030000, -0.54946712509512820000, -0.48307580168617870000,
  1282.             -0.41377920437160500000, -0.34199409082575850000, -0.26815218500725370000, -0.19269758070137110000,
  1283.             -0.11608407067525522000, -0.03877241750605081600, 0.03877241750605081600, 0.11608407067525522000,
  1284.             0.19269758070137110000, 0.26815218500725370000, 0.34199409082575850000, 0.41377920437160500000,
  1285.             0.48307580168617870000, 0.54946712509512820000, 0.61255388966798030000, 0.67195668461417960000,
  1286.             0.72731825518992710000, 0.77830565142651940000, 0.82461223083331170000, 0.86595950321225960000,
  1287.             0.90209880696887420000, 0.93281280827867660000, 0.95791681921379170000, 0.97725994998377420000,
  1288.             0.99072623869945710000, 0.99823770971055930000 };

  1290.         /** Weights for quadrature of order 40. */
  1291.         private static final double[] W_40 = { 0.00452127709853309800, 0.01049828453115270400, 0.01642105838190797300,
  1292.             0.02224584919416689000, 0.02793700698002338000, 0.03346019528254786500, 0.03878216797447199000,
  1293.             0.04387090818567333000, 0.04869580763507221000, 0.05322784698393679000, 0.05743976909939157000,
  1294.             0.06130624249292891000, 0.06480401345660108000, 0.06791204581523394000, 0.07061164739128681000,
  1295.             0.07288658239580408000, 0.07472316905796833000, 0.07611036190062619000, 0.07703981816424793000,
  1296.             0.07750594797842482000, 0.07750594797842482000, 0.07703981816424793000, 0.07611036190062619000,
  1297.             0.07472316905796833000, 0.07288658239580408000, 0.07061164739128681000, 0.06791204581523394000,
  1298.             0.06480401345660108000, 0.06130624249292891000, 0.05743976909939157000, 0.05322784698393679000,
  1299.             0.04869580763507221000, 0.04387090818567333000, 0.03878216797447199000, 0.03346019528254786500,
  1300.             0.02793700698002338000, 0.02224584919416689000, 0.01642105838190797300, 0.01049828453115270400,
  1301.             0.00452127709853309800 };

  1303.         /** Points for quadrature of order 48. */
  1304.         private static final double[] P_48 = { -0.99877100725242610000, -0.99353017226635080000,
  1305.             -0.98412458372282700000, -0.97059159254624720000, -0.95298770316043080000, -0.93138669070655440000,
  1306.             -0.90587913671556960000, -0.87657202027424800000, -0.84358826162439350000, -0.80706620402944250000,
  1307.             -0.76715903251574020000, -0.72403413092381470000, -0.67787237963266400000, -0.62886739677651370000,
  1308.             -0.57722472608397270000, -0.52316097472223300000, -0.46690290475095840000, -0.40868648199071680000,
  1309.             -0.34875588629216070000, -0.28736248735545555000, -0.22476379039468908000, -0.16122235606889174000,
  1310.             -0.09700469920946270000, -0.03238017096286937000, 0.03238017096286937000, 0.09700469920946270000,
  1311.             0.16122235606889174000, 0.22476379039468908000, 0.28736248735545555000, 0.34875588629216070000,
  1312.             0.40868648199071680000, 0.46690290475095840000, 0.52316097472223300000, 0.57722472608397270000,
  1313.             0.62886739677651370000, 0.67787237963266400000, 0.72403413092381470000, 0.76715903251574020000,
  1314.             0.80706620402944250000, 0.84358826162439350000, 0.87657202027424800000, 0.90587913671556960000,
  1315.             0.93138669070655440000, 0.95298770316043080000, 0.97059159254624720000, 0.98412458372282700000,
  1316.             0.99353017226635080000, 0.99877100725242610000 };

  1318.         /** Weights for quadrature of order 48. */
  1319.         private static final double[] W_48 = { 0.00315334605230596250, 0.00732755390127620800, 0.01147723457923446900,
  1320.             0.01557931572294386600, 0.01961616045735556700, 0.02357076083932435600, 0.02742650970835688000,
  1321.             0.03116722783279807000, 0.03477722256477045000, 0.03824135106583080600, 0.04154508294346483000,
  1322.             0.04467456085669424000, 0.04761665849249054000, 0.05035903555385448000, 0.05289018948519365000,
  1323.             0.05519950369998416500, 0.05727729210040315000, 0.05911483969839566000, 0.06070443916589384000,
  1324.             0.06203942315989268000, 0.06311419228625403000, 0.06392423858464817000, 0.06446616443595010000,
  1325.             0.06473769681268386000, 0.06473769681268386000, 0.06446616443595010000, 0.06392423858464817000,
  1326.             0.06311419228625403000, 0.06203942315989268000, 0.06070443916589384000, 0.05911483969839566000,
  1327.             0.05727729210040315000, 0.05519950369998416500, 0.05289018948519365000, 0.05035903555385448000,
  1328.             0.04761665849249054000, 0.04467456085669424000, 0.04154508294346483000, 0.03824135106583080600,
  1329.             0.03477722256477045000, 0.03116722783279807000, 0.02742650970835688000, 0.02357076083932435600,
  1330.             0.01961616045735556700, 0.01557931572294386600, 0.01147723457923446900, 0.00732755390127620800,
  1331.             0.00315334605230596250 };

  1333.         /** Node points. */
  1334.         private final double[] nodePoints;

  1336.         /** Node weights. */
  1337.         private final double[] nodeWeights;

  1339.         /** Number of points. */
  1340.         private final int numberOfPoints;

  1342.         /**
  1343.          * Creates a Gauss integrator of the given order.
  1344.          *
  1345.          * @param numberOfPoints Order of the integration rule.
  1346.          */
  1347.         GaussQuadrature(final int numberOfPoints) {

  1349.             this.numberOfPoints = numberOfPoints;

  1351.             switch (numberOfPoints) {
  1352.                 case 12:
  1353.                     this.nodePoints = P_12.clone();
  1354.                     this.nodeWeights = W_12.clone();
  1355.                     break;
  1356.                 case 16:
  1357.                     this.nodePoints = P_16.clone();
  1358.                     this.nodeWeights = W_16.clone();
  1359.                     break;
  1360.                 case 20:
  1361.                     this.nodePoints = P_20.clone();
  1362.                     this.nodeWeights = W_20.clone();
  1363.                     break;
  1364.                 case 24:
  1365.                     this.nodePoints = P_24.clone();
  1366.                     this.nodeWeights = W_24.clone();
  1367.                     break;
  1368.                 case 32:
  1369.                     this.nodePoints = P_32.clone();
  1370.                     this.nodeWeights = W_32.clone();
  1371.                     break;
  1372.                 case 40:
  1373.                     this.nodePoints = P_40.clone();
  1374.                     this.nodeWeights = W_40.clone();
  1375.                     break;
  1376.                 case 48:
  1377.                 default:
  1378.                     this.nodePoints = P_48.clone();
  1379.                     this.nodeWeights = W_48.clone();
  1380.                     break;
  1381.             }

  1383.         }

  1385.         /**
  1386.          * Integrates a given function on the given interval.
  1387.          *
  1388.          * @param f          Function to integrate.
  1389.          * @param lowerBound Lower bound of the integration interval.
  1390.          * @param upperBound Upper bound of the integration interval.
  1391.          * @return the integral of the weighted function.
  1392.          */
  1393.         public double[] integrate(final UnivariateVectorFunction f, final double lowerBound, final double upperBound) {

  1395.             final double[] adaptedPoints = nodePoints.clone();
  1396.             final double[] adaptedWeights = nodeWeights.clone();
  1397.             transform(adaptedPoints, adaptedWeights, lowerBound, upperBound);
  1398.             return basicIntegrate(f, adaptedPoints, adaptedWeights);
  1399.         }

  1401.         /**
  1402.          * Integrates a given function on the given interval.
  1403.          *
  1404.          * @param <T>        the type of the field elements
  1405.          * @param f          Function to integrate.
  1406.          * @param lowerBound Lower bound of the integration interval.
  1407.          * @param upperBound Upper bound of the integration interval.
  1408.          * @param field      field utilized by default
  1409.          * @return the integral of the weighted function.
  1410.          */
  1411.         public <T extends CalculusFieldElement<T>> T[] integrate(final CalculusFieldUnivariateVectorFunction<T> f,
  1412.                 final T lowerBound, final T upperBound, final Field<T> field) {

  1414.             final T zero = field.getZero();

  1416.             final T[] adaptedPoints = MathArrays.buildArray(field, numberOfPoints);
  1417.             final T[] adaptedWeights = MathArrays.buildArray(field, numberOfPoints);

  1419.             for (int i = 0; i < numberOfPoints; i++) {
  1420.                 adaptedPoints[i] = zero.add(nodePoints[i]);
  1421.                 adaptedWeights[i] = zero.add(nodeWeights[i]);
  1422.             }

  1424.             transform(adaptedPoints, adaptedWeights, lowerBound, upperBound);
  1425.             return basicIntegrate(f, adaptedPoints, adaptedWeights, field);
  1426.         }

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

  1450.         /**
  1451.          * Performs a change of variable so that the integration can be performed on an
  1452.          * arbitrary interval {@code [a, b]}.
  1453.          * <p>
  1454.          * It is assumed that the natural interval is {@code [-1, 1]}.
  1455.          * </p>
  1456.          * @param <T>     the type of the field elements
  1457.          * @param points  Points to adapt to the new interval.
  1458.          * @param weights Weights to adapt to the new interval.
  1459.          * @param a       Lower bound of the integration interval.
  1460.          * @param b       Lower bound of the integration interval
  1461.          */
  1462.         private <T extends CalculusFieldElement<T>> void transform(final T[] points, final T[] weights, final T a,
  1463.                 final T b) {
  1464.             // Scaling
  1465.             final T scale = (b.subtract(a)).divide(2.);
  1466.             final T shift = a.add(scale);
  1467.             for (int i = 0; i < points.length; i++) {
  1468.                 points[i] = scale.multiply(points[i]).add(shift);
  1469.                 weights[i] = scale.multiply(weights[i]);
  1470.             }
  1471.         }

  1473.         /**
  1474.          * Returns an estimate of the integral of {@code f(x) * w(x)}, where {@code w}
  1475.          * is a weight function that depends on the actual flavor of the Gauss
  1476.          * integration scheme.
  1477.          *
  1478.          * @param f       Function to integrate.
  1479.          * @param points  Nodes.
  1480.          * @param weights Nodes weights.
  1481.          * @return the integral of the weighted function.
  1482.          */
  1483.         private double[] basicIntegrate(final UnivariateVectorFunction f, final double[] points,
  1484.                 final double[] weights) {
  1485.             double x = points[0];
  1486.             double w = weights[0];
  1487.             double[] v = f.value(x);
  1488.             final double[] y = new double[v.length];
  1489.             for (int j = 0; j < v.length; j++) {
  1490.                 y[j] = w * v[j];
  1491.             }
  1492.             final double[] t = y.clone();
  1493.             final double[] c = new double[v.length];
  1494.             final double[] s = t.clone();
  1495.             for (int i = 1; i < points.length; i++) {
  1496.                 x = points[i];
  1497.                 w = weights[i];
  1498.                 v = f.value(x);
  1499.                 for (int j = 0; j < v.length; j++) {
  1500.                     y[j] = w * v[j] - c[j];
  1501.                     t[j] = s[j] + y[j];
  1502.                     c[j] = (t[j] - s[j]) - y[j];
  1503.                     s[j] = t[j];
  1504.                 }
  1505.             }
  1506.             return s;
  1507.         }

  1509.         /**
  1510.          * Returns an estimate of the integral of {@code f(x) * w(x)}, where {@code w}
  1511.          * is a weight function that depends on the actual flavor of the Gauss
  1512.          * integration scheme.
  1513.          *
  1514.          * @param <T>     the type of the field elements.
  1515.          * @param f       Function to integrate.
  1516.          * @param points  Nodes.
  1517.          * @param weights Nodes weight
  1518.          * @param field   field utilized by default
  1519.          * @return the integral of the weighted function.
  1520.          */
  1521.         private <T extends CalculusFieldElement<T>> T[] basicIntegrate(final CalculusFieldUnivariateVectorFunction<T> f,
  1522.                 final T[] points, final T[] weights, final Field<T> field) {

  1524.             T x = points[0];
  1525.             T w = weights[0];
  1526.             T[] v = f.value(x);

  1528.             final T[] y = MathArrays.buildArray(field, v.length);
  1529.             for (int j = 0; j < v.length; j++) {
  1530.                 y[j] = v[j].multiply(w);
  1531.             }
  1532.             final T[] t = y.clone();
  1533.             final T[] c = MathArrays.buildArray(field, v.length);
  1534.             ;
  1535.             final T[] s = t.clone();
  1536.             for (int i = 1; i < points.length; i++) {
  1537.                 x = points[i];
  1538.                 w = weights[i];
  1539.                 v = f.value(x);
  1540.                 for (int j = 0; j < v.length; j++) {
  1541.                     y[j] = v[j].multiply(w).subtract(c[j]);
  1542.                     t[j] = y[j].add(s[j]);
  1543.                     c[j] = (t[j].subtract(s[j])).subtract(y[j]);
  1544.                     s[j] = t[j];
  1545.                 }
  1546.             }
  1547.             return s;
  1548.         }

  1550.     }

  1552.     /**
  1553.      * Compute the C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup>
  1554.      * coefficients.
  1555.      * <p>
  1556.      * Those coefficients are given in Danielson paper by expression 4.4-(6)
  1557.      * </p>
  1558.      * @author Petre Bazavan
  1559.      * @author Lucian Barbulescu
  1560.      */
  1561.     protected class FourierCjSjCoefficients {

  1563.         /** Maximum possible value for j. */
  1564.         private final int jMax;

  1566.         /**
  1567.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1568.          * <p>
  1569.          * the index i corresponds to the following elements: <br/>
  1570.          * - 0 for a <br>
  1571.          * - 1 for k <br>
  1572.          * - 2 for h <br>
  1573.          * - 3 for q <br>
  1574.          * - 4 for p <br>
  1575.          * - 5 for λ <br>
  1576.          * </p>
  1577.          */
  1578.         private final double[][] cCoef;

  1580.         /**
  1581.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1582.          * <p>
  1583.          * the index i corresponds to the following elements: <br/>
  1584.          * - 0 for a <br>
  1585.          * - 1 for k <br>
  1586.          * - 2 for h <br>
  1587.          * - 3 for q <br>
  1588.          * - 4 for p <br>
  1589.          * - 5 for λ <br>
  1590.          * </p>
  1591.          */
  1592.         private final double[][] sCoef;

  1594.         /**
  1595.          * Standard constructor.
  1596.          * @param state             the current state
  1597.          * @param jMax              maximum value for j
  1598.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1599.          * @param parameters        list of parameter values at state date for each driver
  1600.          * of the force model parameters (1 value per parameter)
  1601.          */
  1602.         FourierCjSjCoefficients(final SpacecraftState state, final int jMax, final AuxiliaryElements auxiliaryElements,
  1603.                 final double[] parameters) {

  1605.             // Initialise the fields
  1606.             this.jMax = jMax;

  1608.             // Allocate the arrays
  1609.             final int rows = jMax + 1;
  1610.             cCoef = new double[rows][6];
  1611.             sCoef = new double[rows][6];

  1613.             // Compute the coefficients
  1614.             computeCoefficients(state, auxiliaryElements, parameters);
  1615.         }

  1617.         /**
  1618.          * Compute the Fourrier coefficients.
  1619.          * <p>
  1620.          * Only the C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup> coefficients
  1621.          * need to be computed as D<sub>i</sub><sup>m</sup> is always 0.
  1622.          * </p>
  1623.          * @param state             the current state
  1624.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1625.          * @param parameters        list of parameter values at state date for each driver
  1626.          * of the force model parameters (1 value per parameter)
  1627.          */
  1628.         private void computeCoefficients(final SpacecraftState state, final AuxiliaryElements auxiliaryElements,
  1629.                 final double[] parameters) {

  1631.             // Computes the limits for the integral
  1632.             final double[] ll = getLLimits(state, auxiliaryElements);
  1633.             // Computes integrated mean element rates if Llow < Lhigh
  1634.             if (ll[0] < ll[1]) {
  1635.                 // Compute 1 / PI
  1636.                 final double ooPI = 1 / FastMath.PI;

  1638.                 // loop through all values of j
  1639.                 for (int j = 0; j <= jMax; j++) {
  1640.                     final double[] curentCoefficients = integrator
  1641.                             .integrate(new IntegrableFunction(state, false, j, parameters), ll[0], ll[1]);

  1643.                     // divide by PI and set the values for the coefficients
  1644.                     for (int i = 0; i < 6; i++) {
  1645.                         cCoef[j][i] = ooPI * curentCoefficients[i];
  1646.                         sCoef[j][i] = ooPI * curentCoefficients[i + 6];
  1647.                     }
  1648.                 }
  1649.             }
  1650.         }

  1652.         /**
  1653.          * Get the coefficient C<sub>i</sub><sup>j</sup>.
  1654.          * @param i i index - corresponds to the required variation
  1655.          * @param j j index
  1656.          * @return the coefficient C<sub>i</sub><sup>j</sup>
  1657.          */
  1658.         public double getCij(final int i, final int j) {
  1659.             return cCoef[j][i];
  1660.         }

  1662.         /**
  1663.          * Get the coefficient S<sub>i</sub><sup>j</sup>.
  1664.          * @param i i index - corresponds to the required variation
  1665.          * @param j j index
  1666.          * @return the coefficient S<sub>i</sub><sup>j</sup>
  1667.          */
  1668.         public double getSij(final int i, final int j) {
  1669.             return sCoef[j][i];
  1670.         }
  1671.     }

  1673.     /**
  1674.      * Compute the C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup>
  1675.      * coefficients with field elements.
  1676.      * <p>
  1677.      * Those coefficients are given in Danielson paper by expression 4.4-(6)
  1678.      * </p>
  1679.      * @author Petre Bazavan
  1680.      * @author Lucian Barbulescu
  1681.      * @param <T> type of the field elements
  1682.      */
  1683.     protected class FieldFourierCjSjCoefficients<T extends CalculusFieldElement<T>> {

  1685.         /** Maximum possible value for j. */
  1686.         private final int jMax;

  1688.         /**
  1689.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1690.          * <p>
  1691.          * the index i corresponds to the following elements: <br/>
  1692.          * - 0 for a <br>
  1693.          * - 1 for k <br>
  1694.          * - 2 for h <br>
  1695.          * - 3 for q <br>
  1696.          * - 4 for p <br>
  1697.          * - 5 for λ <br>
  1698.          * </p>
  1699.          */
  1700.         private final T[][] cCoef;

  1702.         /**
  1703.          * The C<sub>i</sub><sup>j</sup> coefficients.
  1704.          * <p>
  1705.          * the index i corresponds to the following elements: <br/>
  1706.          * - 0 for a <br>
  1707.          * - 1 for k <br>
  1708.          * - 2 for h <br>
  1709.          * - 3 for q <br>
  1710.          * - 4 for p <br>
  1711.          * - 5 for λ <br>
  1712.          * </p>
  1713.          */
  1714.         private final T[][] sCoef;

  1716.         /**
  1717.          * Standard constructor.
  1718.          * @param state             the current state
  1719.          * @param jMax              maximum value for j
  1720.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1721.          * @param parameters        values of the force model parameters
  1722.          * @param field             field used by default
  1723.          */
  1724.         FieldFourierCjSjCoefficients(final FieldSpacecraftState<T> state, final int jMax,
  1725.                 final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters, final Field<T> field) {
  1726.             // Initialise the fields
  1727.             this.jMax = jMax;

  1729.             // Allocate the arrays
  1730.             final int rows = jMax + 1;
  1731.             cCoef = MathArrays.buildArray(field, rows, 6);
  1732.             sCoef = MathArrays.buildArray(field, rows, 6);

  1734.             // Compute the coefficients
  1735.             computeCoefficients(state, auxiliaryElements, parameters, field);
  1736.         }

  1738.         /**
  1739.          * Compute the Fourrier coefficients.
  1740.          * <p>
  1741.          * Only the C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup> coefficients
  1742.          * need to be computed as D<sub>i</sub><sup>m</sup> is always 0.
  1743.          * </p>
  1744.          * @param state             the current state
  1745.          * @param auxiliaryElements auxiliary elements related to the current orbit
  1746.          * @param parameters        values of the force model parameters
  1747.          * @param field             field used by default
  1748.          */
  1749.         private void computeCoefficients(final FieldSpacecraftState<T> state,
  1750.                 final FieldAuxiliaryElements<T> auxiliaryElements, final T[] parameters, final Field<T> field) {
  1751.             // Zero
  1752.             final T zero = field.getZero();
  1753.             // Computes the limits for the integral
  1754.             final T[] ll = getLLimits(state, auxiliaryElements);
  1755.             // Computes integrated mean element rates if Llow < Lhigh
  1756.             if (ll[0].getReal() < ll[1].getReal()) {
  1757.                 // Compute 1 / PI
  1758.                 final T ooPI = zero.getPi().reciprocal();

  1760.                 // loop through all values of j
  1761.                 for (int j = 0; j <= jMax; j++) {
  1762.                     final T[] curentCoefficients = integrator.integrate(
  1763.                             new FieldIntegrableFunction<>(state, false, j, parameters, field), ll[0], ll[1], field);

  1765.                     // divide by PI and set the values for the coefficients
  1766.                     for (int i = 0; i < 6; i++) {
  1767.                         cCoef[j][i] = curentCoefficients[i].multiply(ooPI);
  1768.                         sCoef[j][i] = curentCoefficients[i + 6].multiply(ooPI);
  1769.                     }
  1770.                 }
  1771.             }
  1772.         }

  1774.         /**
  1775.          * Get the coefficient C<sub>i</sub><sup>j</sup>.
  1776.          * @param i i index - corresponds to the required variation
  1777.          * @param j j index
  1778.          * @return the coefficient C<sub>i</sub><sup>j</sup>
  1779.          */
  1780.         public T getCij(final int i, final int j) {
  1781.             return cCoef[j][i];
  1782.         }

  1784.         /**
  1785.          * Get the coefficient S<sub>i</sub><sup>j</sup>.
  1786.          * @param i i index - corresponds to the required variation
  1787.          * @param j j index
  1788.          * @return the coefficient S<sub>i</sub><sup>j</sup>
  1789.          */
  1790.         public T getSij(final int i, final int j) {
  1791.             return sCoef[j][i];
  1792.         }
  1793.     }

  1795.     /**
  1796.      * This class handles the short periodic coefficients described in Danielson
  1797.      * 2.5.3-26.
  1798.      *
  1799.      * <p>
  1800.      * The value of M is 0. Also, since the values of the Fourier coefficient
  1801.      * D<sub>i</sub><sup>m</sup> is 0 then the values of the coefficients
  1802.      * D<sub>i</sub><sup>m</sup> for m &gt; 2 are also 0.
  1803.      * </p>
  1804.      * @author Petre Bazavan
  1805.      * @author Lucian Barbulescu
  1806.      *
  1807.      */
  1808.     protected static class GaussianShortPeriodicCoefficients implements ShortPeriodTerms {

  1810.         /** Maximum value for j index. */
  1811.         private final int jMax;

  1813.         /** Number of points used in the interpolation process. */
  1814.         private final int interpolationPoints;

  1816.         /** Prefix for coefficients keys. */
  1817.         private final String coefficientsKeyPrefix;

  1819.         /** All coefficients slots. */
  1820.         private final transient TimeSpanMap<Slot> slots;

  1822.         /**
  1823.          * Constructor.
  1824.          * @param coefficientsKeyPrefix prefix for coefficients keys
  1825.          * @param jMax                  maximum value for j index
  1826.          * @param interpolationPoints   number of points used in the interpolation
  1827.          *                              process
  1828.          * @param slots                 all coefficients slots
  1829.          */
  1830.         GaussianShortPeriodicCoefficients(final String coefficientsKeyPrefix, final int jMax,
  1831.                 final int interpolationPoints, final TimeSpanMap<Slot> slots) {
  1832.             // Initialize fields
  1833.             this.jMax = jMax;
  1834.             this.interpolationPoints = interpolationPoints;
  1835.             this.coefficientsKeyPrefix = coefficientsKeyPrefix;
  1836.             this.slots = slots;
  1837.         }

  1839.         /**
  1840.          * Get the slot valid for some date.
  1841.          * @param meanStates mean states defining the slot
  1842.          * @return slot valid at the specified date
  1843.          */
  1844.         public Slot createSlot(final SpacecraftState... meanStates) {
  1845.             final Slot slot = new Slot(jMax, interpolationPoints);
  1846.             final AbsoluteDate first = meanStates[0].getDate();
  1847.             final AbsoluteDate last = meanStates[meanStates.length - 1].getDate();
  1848.             final int compare = first.compareTo(last);
  1849.             if (compare < 0) {
  1850.                 slots.addValidAfter(slot, first, false);
  1851.             } else if (compare > 0) {
  1852.                 slots.addValidBefore(slot, first, false);
  1853.             } else {
  1854.                 // single date, valid for all time
  1855.                 slots.addValidAfter(slot, AbsoluteDate.PAST_INFINITY, false);
  1856.             }
  1857.             return slot;
  1858.         }

  1860.         /**
  1861.          * Compute the short periodic coefficients.
  1862.          *
  1863.          * @param state       current state information: date, kinematics, attitude
  1864.          * @param slot        coefficients slot
  1865.          * @param fourierCjSj Fourier coefficients
  1866.          * @param uijvij      U and V coefficients
  1867.          * @param n           Keplerian mean motion
  1868.          * @param a           semi major axis
  1869.          */
  1870.         private void computeCoefficients(final SpacecraftState state, final Slot slot,
  1871.                 final FourierCjSjCoefficients fourierCjSj, final UijVijCoefficients uijvij, final double n,
  1872.                 final double a) {

  1874.             // get the current date
  1875.             final AbsoluteDate date = state.getDate();

  1877.             // compute the k₂⁰ coefficient
  1878.             final double k20 = computeK20(jMax, uijvij.currentRhoSigmaj);

  1880.             // 1. / n
  1881.             final double oon = 1. / n;
  1882.             // 3. / (2 * a * n)
  1883.             final double to2an = 1.5 * oon / a;
  1884.             // 3. / (4 * a * n)
  1885.             final double to4an = to2an / 2;

  1887.             // Compute the coefficients for each element
  1888.             final int size = jMax + 1;
  1889.             final double[] di1 = new double[6];
  1890.             final double[] di2 = new double[6];
  1891.             final double[][] currentCij = new double[size][6];
  1892.             final double[][] currentSij = new double[size][6];
  1893.             for (int i = 0; i < 6; i++) {

  1895.                 // compute D<sub>i</sub>¹ and D<sub>i</sub>² (all others are 0)
  1896.                 di1[i] = -oon * fourierCjSj.getCij(i, 0);
  1897.                 if (i == 5) {
  1898.                     di1[i] += to2an * uijvij.getU1(0, 0);
  1899.                 }
  1900.                 di2[i] = 0.;
  1901.                 if (i == 5) {
  1902.                     di2[i] += -to4an * fourierCjSj.getCij(0, 0);
  1903.                 }

  1905.                 // the C<sub>i</sub>⁰ is computed based on all others
  1906.                 currentCij[0][i] = -di2[i] * k20;

  1908.                 for (int j = 1; j <= jMax; j++) {
  1909.                     // compute the current C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup>
  1910.                     currentCij[j][i] = oon * uijvij.getU1(j, i);
  1911.                     if (i == 5) {
  1912.                         currentCij[j][i] += -to2an * uijvij.getU2(j);
  1913.                     }
  1914.                     currentSij[j][i] = oon * uijvij.getV1(j, i);
  1915.                     if (i == 5) {
  1916.                         currentSij[j][i] += -to2an * uijvij.getV2(j);
  1917.                     }

  1919.                     // add the computed coefficients to C<sub>i</sub>⁰
  1920.                     currentCij[0][i] += -(currentCij[j][i] * uijvij.currentRhoSigmaj[0][j] +
  1921.                             currentSij[j][i] * uijvij.currentRhoSigmaj[1][j]);
  1922.                 }

  1924.             }

  1926.             // add the values to the interpolators
  1927.             slot.cij[0].addGridPoint(date, currentCij[0]);
  1928.             slot.dij[1].addGridPoint(date, di1);
  1929.             slot.dij[2].addGridPoint(date, di2);
  1930.             for (int j = 1; j <= jMax; j++) {
  1931.                 slot.cij[j].addGridPoint(date, currentCij[j]);
  1932.                 slot.sij[j].addGridPoint(date, currentSij[j]);
  1933.             }

  1935.         }

  1937.         /**
  1938.          * Compute the coefficient k₂⁰ by using the equation 2.5.3-(9a) from Danielson.
  1939.          * <p>
  1940.          * After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:<br>
  1941.          * k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  1942.          * ρ<sub>k</sub>²)]
  1943.          * </p>
  1944.          * @param kMax             max value fot k index
  1945.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  1946.          *                         σ<sub>j</sub> coefficients
  1947.          * @return the coefficient k₂⁰
  1948.          */
  1949.         private double computeK20(final int kMax, final double[][] currentRhoSigmaj) {
  1950.             double k20 = 0.;

  1952.             for (int kIndex = 1; kIndex <= kMax; kIndex++) {
  1953.                 // After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:
  1954.                 // k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  1955.                 // ρ<sub>k</sub>²)]
  1956.                 double currentTerm = currentRhoSigmaj[1][kIndex] * currentRhoSigmaj[1][kIndex] +
  1957.                         currentRhoSigmaj[0][kIndex] * currentRhoSigmaj[0][kIndex];

  1959.                 // multiply by 2 / k²
  1960.                 currentTerm *= 2. / (kIndex * kIndex);

  1962.                 // add the term to the result
  1963.                 k20 += currentTerm;
  1964.             }

  1966.             return k20;
  1967.         }

  1969.         /** {@inheritDoc} */
  1970.         @Override
  1971.         public double[] value(final Orbit meanOrbit) {

  1973.             // select the coefficients slot
  1974.             final Slot slot = slots.get(meanOrbit.getDate());

  1976.             // Get the True longitude L
  1977.             final double L = meanOrbit.getLv();

  1979.             // Compute the center (l - λ)
  1980.             final double center = L - meanOrbit.getLM();
  1981.             // Compute (l - λ)²
  1982.             final double center2 = center * center;

  1984.             // Initialize short periodic variations
  1985.             final double[] shortPeriodicVariation = slot.cij[0].value(meanOrbit.getDate());
  1986.             final double[] d1 = slot.dij[1].value(meanOrbit.getDate());
  1987.             final double[] d2 = slot.dij[2].value(meanOrbit.getDate());
  1988.             for (int i = 0; i < 6; i++) {
  1989.                 shortPeriodicVariation[i] += center * d1[i] + center2 * d2[i];
  1990.             }

  1992.             for (int j = 1; j <= JMAX; j++) {
  1993.                 final double[] c = slot.cij[j].value(meanOrbit.getDate());
  1994.                 final double[] s = slot.sij[j].value(meanOrbit.getDate());
  1995.                 final SinCos sc  = FastMath.sinCos(j * L);
  1996.                 final double cos = sc.cos();
  1997.                 final double sin = sc.sin();
  1998.                 for (int i = 0; i < 6; i++) {
  1999.                     // add corresponding term to the short periodic variation
  2000.                     shortPeriodicVariation[i] += c[i] * cos;
  2001.                     shortPeriodicVariation[i] += s[i] * sin;
  2002.                 }
  2003.             }

  2005.             return shortPeriodicVariation;

  2007.         }

  2009.         /** {@inheritDoc} */
  2010.         public String getCoefficientsKeyPrefix() {
  2011.             return coefficientsKeyPrefix;
  2012.         }

  2014.         /**
  2015.          * {@inheritDoc}
  2016.          * <p>
  2017.          * For Gaussian forces, there are JMAX cj coefficients, JMAX sj coefficients and
  2018.          * 3 dj coefficients. As JMAX = 12, this sums up to 27 coefficients. The j index
  2019.          * is the integer multiplier for the true longitude argument in the cj and sj
  2020.          * coefficients and to the degree in the polynomial dj coefficients.
  2021.          * </p>
  2022.          */
  2023.         @Override
  2024.         public Map<String, double[]> getCoefficients(final AbsoluteDate date, final Set<String> selected) {

  2026.             // select the coefficients slot
  2027.             final Slot slot = slots.get(date);

  2029.             final Map<String, double[]> coefficients = new HashMap<String, double[]>(2 * JMAX + 3);
  2030.             storeIfSelected(coefficients, selected, slot.cij[0].value(date), "d", 0);
  2031.             storeIfSelected(coefficients, selected, slot.dij[1].value(date), "d", 1);
  2032.             storeIfSelected(coefficients, selected, slot.dij[2].value(date), "d", 2);
  2033.             for (int j = 1; j <= JMAX; j++) {
  2034.                 storeIfSelected(coefficients, selected, slot.cij[j].value(date), "c", j);
  2035.                 storeIfSelected(coefficients, selected, slot.sij[j].value(date), "s", j);
  2036.             }

  2038.             return coefficients;

  2040.         }

  2042.         /**
  2043.          * Put a coefficient in a map if selected.
  2044.          * @param map      map to populate
  2045.          * @param selected set of coefficients that should be put in the map (empty set
  2046.          *                 means all coefficients are selected)
  2047.          * @param value    coefficient value
  2048.          * @param id       coefficient identifier
  2049.          * @param indices  list of coefficient indices
  2050.          */
  2051.         private void storeIfSelected(final Map<String, double[]> map, final Set<String> selected, final double[] value,
  2052.                 final String id, final int... indices) {
  2053.             final StringBuilder keyBuilder = new StringBuilder(getCoefficientsKeyPrefix());
  2054.             keyBuilder.append(id);
  2055.             for (int index : indices) {
  2056.                 keyBuilder.append('[').append(index).append(']');
  2057.             }
  2058.             final String key = keyBuilder.toString();
  2059.             if (selected.isEmpty() || selected.contains(key)) {
  2060.                 map.put(key, value);
  2061.             }
  2062.         }

  2064.     }

  2066.     /**
  2067.      * This class handles the short periodic coefficients described in Danielson
  2068.      * 2.5.3-26.
  2069.      *
  2070.      * <p>
  2071.      * The value of M is 0. Also, since the values of the Fourier coefficient
  2072.      * D<sub>i</sub><sup>m</sup> is 0 then the values of the coefficients
  2073.      * D<sub>i</sub><sup>m</sup> for m &gt; 2 are also 0.
  2074.      * </p>
  2075.      * @author Petre Bazavan
  2076.      * @author Lucian Barbulescu
  2077.      * @param <T> type of the field elements
  2078.      */
  2079.     protected static class FieldGaussianShortPeriodicCoefficients<T extends CalculusFieldElement<T>>
  2080.             implements FieldShortPeriodTerms<T> {

  2082.         /** Maximum value for j index. */
  2083.         private final int jMax;

  2085.         /** Number of points used in the interpolation process. */
  2086.         private final int interpolationPoints;

  2088.         /** Prefix for coefficients keys. */
  2089.         private final String coefficientsKeyPrefix;

  2091.         /** All coefficients slots. */
  2092.         private final transient FieldTimeSpanMap<FieldSlot<T>, T> slots;

  2094.         /**
  2095.          * Constructor.
  2096.          * @param coefficientsKeyPrefix prefix for coefficients keys
  2097.          * @param jMax                  maximum value for j index
  2098.          * @param interpolationPoints   number of points used in the interpolation
  2099.          *                              process
  2100.          * @param slots                 all coefficients slots
  2101.          */
  2102.         FieldGaussianShortPeriodicCoefficients(final String coefficientsKeyPrefix, final int jMax,
  2103.                 final int interpolationPoints, final FieldTimeSpanMap<FieldSlot<T>, T> slots) {
  2104.             // Initialize fields
  2105.             this.jMax = jMax;
  2106.             this.interpolationPoints = interpolationPoints;
  2107.             this.coefficientsKeyPrefix = coefficientsKeyPrefix;
  2108.             this.slots = slots;
  2109.         }

  2111.         /**
  2112.          * Get the slot valid for some date.
  2113.          * @param meanStates mean states defining the slot
  2114.          * @return slot valid at the specified date
  2115.          */
  2116.         @SuppressWarnings("unchecked")
  2117.         public FieldSlot<T> createSlot(final FieldSpacecraftState<T>... meanStates) {
  2118.             final FieldSlot<T> slot = new FieldSlot<>(jMax, interpolationPoints);
  2119.             final FieldAbsoluteDate<T> first = meanStates[0].getDate();
  2120.             final FieldAbsoluteDate<T> last = meanStates[meanStates.length - 1].getDate();
  2121.             if (first.compareTo(last) <= 0) {
  2122.                 slots.addValidAfter(slot, first);
  2123.             } else {
  2124.                 slots.addValidBefore(slot, first);
  2125.             }
  2126.             return slot;
  2127.         }

  2129.         /**
  2130.          * Compute the short periodic coefficients.
  2131.          *
  2132.          * @param state       current state information: date, kinematics, attitude
  2133.          * @param slot        coefficients slot
  2134.          * @param fourierCjSj Fourier coefficients
  2135.          * @param uijvij      U and V coefficients
  2136.          * @param n           Keplerian mean motion
  2137.          * @param a           semi major axis
  2138.          * @param field       field used by default
  2139.          */
  2140.         private void computeCoefficients(final FieldSpacecraftState<T> state, final FieldSlot<T> slot,
  2141.                 final FieldFourierCjSjCoefficients<T> fourierCjSj, final FieldUijVijCoefficients<T> uijvij, final T n,
  2142.                 final T a, final Field<T> field) {

  2144.             // Zero
  2145.             final T zero = field.getZero();

  2147.             // get the current date
  2148.             final FieldAbsoluteDate<T> date = state.getDate();

  2150.             // compute the k₂⁰ coefficient
  2151.             final T k20 = computeK20(jMax, uijvij.currentRhoSigmaj, field);

  2153.             // 1. / n
  2154.             final T oon = n.reciprocal();
  2155.             // 3. / (2 * a * n)
  2156.             final T to2an = oon.multiply(1.5).divide(a);
  2157.             // 3. / (4 * a * n)
  2158.             final T to4an = to2an.divide(2.);

  2160.             // Compute the coefficients for each element
  2161.             final int size = jMax + 1;
  2162.             final T[] di1 = MathArrays.buildArray(field, 6);
  2163.             final T[] di2 = MathArrays.buildArray(field, 6);
  2164.             final T[][] currentCij = MathArrays.buildArray(field, size, 6);
  2165.             final T[][] currentSij = MathArrays.buildArray(field, size, 6);
  2166.             for (int i = 0; i < 6; i++) {

  2168.                 // compute D<sub>i</sub>¹ and D<sub>i</sub>² (all others are 0)
  2169.                 di1[i] = oon.negate().multiply(fourierCjSj.getCij(i, 0));
  2170.                 if (i == 5) {
  2171.                     di1[i] = di1[i].add(to2an.multiply(uijvij.getU1(0, 0)));
  2172.                 }
  2173.                 di2[i] = zero;
  2174.                 if (i == 5) {
  2175.                     di2[i] = di2[i].add(to4an.negate().multiply(fourierCjSj.getCij(0, 0)));
  2176.                 }

  2178.                 // the C<sub>i</sub>⁰ is computed based on all others
  2179.                 currentCij[0][i] = di2[i].negate().multiply(k20);

  2181.                 for (int j = 1; j <= jMax; j++) {
  2182.                     // compute the current C<sub>i</sub><sup>j</sup> and S<sub>i</sub><sup>j</sup>
  2183.                     currentCij[j][i] = oon.multiply(uijvij.getU1(j, i));
  2184.                     if (i == 5) {
  2185.                         currentCij[j][i] = currentCij[j][i].add(to2an.negate().multiply(uijvij.getU2(j)));
  2186.                     }
  2187.                     currentSij[j][i] = oon.multiply(uijvij.getV1(j, i));
  2188.                     if (i == 5) {
  2189.                         currentSij[j][i] = currentSij[j][i].add(to2an.negate().multiply(uijvij.getV2(j)));
  2190.                     }

  2192.                     // add the computed coefficients to C<sub>i</sub>⁰
  2193.                     currentCij[0][i] = currentCij[0][i].add(currentCij[j][i].multiply(uijvij.currentRhoSigmaj[0][j])
  2194.                             .add(currentSij[j][i].multiply(uijvij.currentRhoSigmaj[1][j])).negate());
  2195.                 }

  2197.             }

  2199.             // add the values to the interpolators
  2200.             slot.cij[0].addGridPoint(date, currentCij[0]);
  2201.             slot.dij[1].addGridPoint(date, di1);
  2202.             slot.dij[2].addGridPoint(date, di2);
  2203.             for (int j = 1; j <= jMax; j++) {
  2204.                 slot.cij[j].addGridPoint(date, currentCij[j]);
  2205.                 slot.sij[j].addGridPoint(date, currentSij[j]);
  2206.             }

  2208.         }

  2210.         /**
  2211.          * Compute the coefficient k₂⁰ by using the equation 2.5.3-(9a) from Danielson.
  2212.          * <p>
  2213.          * After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:<br>
  2214.          * k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  2215.          * ρ<sub>k</sub>²)]
  2216.          * </p>
  2217.          * @param kMax             max value fot k index
  2218.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2219.          *                         σ<sub>j</sub> coefficients
  2220.          * @param field            field used by default
  2221.          * @return the coefficient k₂⁰
  2222.          */
  2223.         private T computeK20(final int kMax, final T[][] currentRhoSigmaj, final Field<T> field) {
  2224.             final T zero = field.getZero();
  2225.             T k20 = zero;

  2227.             for (int kIndex = 1; kIndex <= kMax; kIndex++) {
  2228.                 // After inserting 2.5.3-(8) into 2.5.3-(9a) the result becomes:
  2229.                 // k₂⁰ = &Sigma;<sub>k=1</sub><sup>kMax</sup>[(2 / k²) * (σ<sub>k</sub>² +
  2230.                 // ρ<sub>k</sub>²)]
  2231.                 T currentTerm = currentRhoSigmaj[1][kIndex].multiply(currentRhoSigmaj[1][kIndex])
  2232.                         .add(currentRhoSigmaj[0][kIndex].multiply(currentRhoSigmaj[0][kIndex]));

  2234.                 // multiply by 2 / k²
  2235.                 currentTerm = currentTerm.multiply(2. / (kIndex * kIndex));

  2237.                 // add the term to the result
  2238.                 k20 = k20.add(currentTerm);
  2239.             }

  2241.             return k20;
  2242.         }

  2244.         /** {@inheritDoc} */
  2245.         @Override
  2246.         public T[] value(final FieldOrbit<T> meanOrbit) {

  2248.             // select the coefficients slot
  2249.             final FieldSlot<T> slot = slots.get(meanOrbit.getDate());

  2251.             // Get the True longitude L
  2252.             final T L = meanOrbit.getLv();

  2254.             // Compute the center (l - λ)
  2255.             final T center = L.subtract(meanOrbit.getLM());
  2256.             // Compute (l - λ)²
  2257.             final T center2 = center.multiply(center);

  2259.             // Initialize short periodic variations
  2260.             final T[] shortPeriodicVariation = slot.cij[0].value(meanOrbit.getDate());
  2261.             final T[] d1 = slot.dij[1].value(meanOrbit.getDate());
  2262.             final T[] d2 = slot.dij[2].value(meanOrbit.getDate());
  2263.             for (int i = 0; i < 6; i++) {
  2264.                 shortPeriodicVariation[i] = shortPeriodicVariation[i]
  2265.                         .add(center.multiply(d1[i]).add(center2.multiply(d2[i])));
  2266.             }

  2268.             for (int j = 1; j <= JMAX; j++) {
  2269.                 final T[] c = slot.cij[j].value(meanOrbit.getDate());
  2270.                 final T[] s = slot.sij[j].value(meanOrbit.getDate());
  2271.                 final FieldSinCos<T> sc = FastMath.sinCos(L.multiply(j));
  2272.                 final T cos = sc.cos();
  2273.                 final T sin = sc.sin();
  2274.                 for (int i = 0; i < 6; i++) {
  2275.                     // add corresponding term to the short periodic variation
  2276.                     shortPeriodicVariation[i] = shortPeriodicVariation[i].add(c[i].multiply(cos));
  2277.                     shortPeriodicVariation[i] = shortPeriodicVariation[i].add(s[i].multiply(sin));
  2278.                 }
  2279.             }

  2281.             return shortPeriodicVariation;

  2283.         }

  2285.         /** {@inheritDoc} */
  2286.         public String getCoefficientsKeyPrefix() {
  2287.             return coefficientsKeyPrefix;
  2288.         }

  2290.         /**
  2291.          * {@inheritDoc}
  2292.          * <p>
  2293.          * For Gaussian forces, there are JMAX cj coefficients, JMAX sj coefficients and
  2294.          * 3 dj coefficients. As JMAX = 12, this sums up to 27 coefficients. The j index
  2295.          * is the integer multiplier for the true longitude argument in the cj and sj
  2296.          * coefficients and to the degree in the polynomial dj coefficients.
  2297.          * </p>
  2298.          */
  2299.         @Override
  2300.         public Map<String, T[]> getCoefficients(final FieldAbsoluteDate<T> date, final Set<String> selected) {

  2302.             // select the coefficients slot
  2303.             final FieldSlot<T> slot = slots.get(date);

  2305.             final Map<String, T[]> coefficients = new HashMap<String, T[]>(2 * JMAX + 3);
  2306.             storeIfSelected(coefficients, selected, slot.cij[0].value(date), "d", 0);
  2307.             storeIfSelected(coefficients, selected, slot.dij[1].value(date), "d", 1);
  2308.             storeIfSelected(coefficients, selected, slot.dij[2].value(date), "d", 2);
  2309.             for (int j = 1; j <= JMAX; j++) {
  2310.                 storeIfSelected(coefficients, selected, slot.cij[j].value(date), "c", j);
  2311.                 storeIfSelected(coefficients, selected, slot.sij[j].value(date), "s", j);
  2312.             }

  2314.             return coefficients;

  2316.         }

  2318.         /**
  2319.          * Put a coefficient in a map if selected.
  2320.          * @param map      map to populate
  2321.          * @param selected set of coefficients that should be put in the map (empty set
  2322.          *                 means all coefficients are selected)
  2323.          * @param value    coefficient value
  2324.          * @param id       coefficient identifier
  2325.          * @param indices  list of coefficient indices
  2326.          */
  2327.         private void storeIfSelected(final Map<String, T[]> map, final Set<String> selected, final T[] value,
  2328.                 final String id, final int... indices) {
  2329.             final StringBuilder keyBuilder = new StringBuilder(getCoefficientsKeyPrefix());
  2330.             keyBuilder.append(id);
  2331.             for (int index : indices) {
  2332.                 keyBuilder.append('[').append(index).append(']');
  2333.             }
  2334.             final String key = keyBuilder.toString();
  2335.             if (selected.isEmpty() || selected.contains(key)) {
  2336.                 map.put(key, value);
  2337.             }
  2338.         }

  2340.     }

  2342.     /**
  2343.      * The U<sub>i</sub><sup>j</sup> and V<sub>i</sub><sup>j</sup> coefficients
  2344.      * described by equations 2.5.3-(21) and 2.5.3-(22) from Danielson.
  2345.      * <p>
  2346.      * The index i takes only the values 1 and 2<br>
  2347.      * For U only the index 0 for j is used.
  2348.      * </p>
  2349.      *
  2350.      * @author Petre Bazavan
  2351.      * @author Lucian Barbulescu
  2352.      */
  2353.     protected static class UijVijCoefficients {

  2355.         /**
  2356.          * The U₁<sup>j</sup> coefficients.
  2357.          * <p>
  2358.          * The first index identifies the Fourier coefficients used<br>
  2359.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2360.          * S<sub>i</sub><sup>j</sup><br>
  2361.          * The only exception is when j = 0 when only the coefficient for fourier index
  2362.          * = 1 (i == 0) is needed.<br>
  2363.          * Also, for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are
  2364.          * computed, because are required to compute the coefficients U₂<sup>j</sup>
  2365.          * </p>
  2366.          */
  2367.         private final double[][] u1ij;

  2369.         /**
  2370.          * The V₁<sup>j</sup> coefficients.
  2371.          * <p>
  2372.          * The first index identifies the Fourier coefficients used<br>
  2373.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2374.          * S<sub>i</sub><sup>j</sup><br>
  2375.          * for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are computed,
  2376.          * because are required to compute the coefficients V₂<sup>j</sup>
  2377.          * </p>
  2378.          */
  2379.         private final double[][] v1ij;

  2381.         /**
  2382.          * The U₂<sup>j</sup> coefficients.
  2383.          * <p>
  2384.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2385.          * they are the only ones required.
  2386.          * </p>
  2387.          */
  2388.         private final double[] u2ij;

  2390.         /**
  2391.          * The V₂<sup>j</sup> coefficients.
  2392.          * <p>
  2393.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2394.          * they are the only ones required.
  2395.          * </p>
  2396.          */
  2397.         private final double[] v2ij;

  2399.         /**
  2400.          * The current computed values for the ρ<sub>j</sub> and σ<sub>j</sub>
  2401.          * coefficients.
  2402.          */
  2403.         private final double[][] currentRhoSigmaj;

  2405.         /**
  2406.          * The C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup> Fourier
  2407.          * coefficients.
  2408.          */
  2409.         private final FourierCjSjCoefficients fourierCjSj;

  2411.         /** The maximum value for j index. */
  2412.         private final int jMax;

  2414.         /**
  2415.          * Constructor.
  2416.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2417.          *                         σ<sub>j</sub> coefficients
  2418.          * @param fourierCjSj      the fourier coefficients C<sub>i</sub><sup>j</sup>
  2419.          *                         and the S<sub>i</sub><sup>j</sup>
  2420.          * @param jMax             maximum value for j index
  2421.          */
  2422.         UijVijCoefficients(final double[][] currentRhoSigmaj, final FourierCjSjCoefficients fourierCjSj,
  2423.                 final int jMax) {
  2424.             this.currentRhoSigmaj = currentRhoSigmaj;
  2425.             this.fourierCjSj = fourierCjSj;
  2426.             this.jMax = jMax;

  2428.             // initialize the internal arrays.
  2429.             this.u1ij = new double[6][2 * jMax + 1];
  2430.             this.v1ij = new double[6][2 * jMax + 1];
  2431.             this.u2ij = new double[jMax + 1];
  2432.             this.v2ij = new double[jMax + 1];

  2434.             // compute the coefficients
  2435.             computeU1V1Coefficients();
  2436.             computeU2V2Coefficients();
  2437.         }

  2439.         /** Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients. */
  2440.         private void computeU1V1Coefficients() {
  2441.             // generate the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients
  2442.             // for j >= 1
  2443.             // also the U₁⁰ for Fourier index = 1 (i == 0) coefficient will be computed
  2444.             u1ij[0][0] = 0;
  2445.             for (int j = 1; j <= jMax; j++) {
  2446.                 // compute 1 / j
  2447.                 final double ooj = 1. / j;

  2449.                 for (int i = 0; i < 6; i++) {
  2450.                     // j is aready between 1 and J
  2451.                     u1ij[i][j] = fourierCjSj.getSij(i, j);
  2452.                     v1ij[i][j] = fourierCjSj.getCij(i, j);

  2454.                     // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2455.                     if (j > 1) {
  2456.                         // k starts with 1 because j-J is less than or equal to 0
  2457.                         for (int kIndex = 1; kIndex <= j - 1; kIndex++) {
  2458.                             // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2459.                             // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2460.                             u1ij[i][j] += fourierCjSj.getCij(i, j - kIndex) * currentRhoSigmaj[1][kIndex] +
  2461.                                     fourierCjSj.getSij(i, j - kIndex) * currentRhoSigmaj[0][kIndex];

  2463.                             // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2464.                             // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2465.                             v1ij[i][j] += fourierCjSj.getCij(i, j - kIndex) * currentRhoSigmaj[0][kIndex] -
  2466.                                     fourierCjSj.getSij(i, j - kIndex) * currentRhoSigmaj[1][kIndex];
  2467.                         }
  2468.                     }

  2470.                     // since j must be between 1 and J-1 and is already between 1 and J
  2471.                     // the following sum is skiped only for j = jMax
  2472.                     if (j != jMax) {
  2473.                         for (int kIndex = 1; kIndex <= jMax - j; kIndex++) {
  2474.                             // -C<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub> +
  2475.                             // S<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub>
  2476.                             u1ij[i][j] += -fourierCjSj.getCij(i, j + kIndex) * currentRhoSigmaj[1][kIndex] +
  2477.                                     fourierCjSj.getSij(i, j + kIndex) * currentRhoSigmaj[0][kIndex];

  2479.                             // C<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub> +
  2480.                             // S<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub>
  2481.                             v1ij[i][j] += fourierCjSj.getCij(i, j + kIndex) * currentRhoSigmaj[0][kIndex] +
  2482.                                     fourierCjSj.getSij(i, j + kIndex) * currentRhoSigmaj[1][kIndex];
  2483.                         }
  2484.                     }

  2486.                     for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2487.                         // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2488.                         // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2489.                         u1ij[i][j] += -fourierCjSj.getCij(i, kIndex) * currentRhoSigmaj[1][j + kIndex] -
  2490.                                 fourierCjSj.getSij(i, kIndex) * currentRhoSigmaj[0][j + kIndex];

  2492.                         // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2493.                         // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2494.                         v1ij[i][j] += fourierCjSj.getCij(i, kIndex) * currentRhoSigmaj[0][j + kIndex] +
  2495.                                 fourierCjSj.getSij(i, kIndex) * currentRhoSigmaj[1][j + kIndex];
  2496.                     }

  2498.                     // divide by 1 / j
  2499.                     u1ij[i][j] *= -ooj;
  2500.                     v1ij[i][j] *= ooj;

  2502.                     // if index = 1 (i == 0) add the computed terms to U₁⁰
  2503.                     if (i == 0) {
  2504.                         // - (U₁<sup>j</sup> * ρ<sub>j</sub> + V₁<sup>j</sup> * σ<sub>j</sub>
  2505.                         u1ij[0][0] += -u1ij[0][j] * currentRhoSigmaj[0][j] - v1ij[0][j] * currentRhoSigmaj[1][j];
  2506.                     }
  2507.                 }
  2508.             }

  2510.             // Terms with j > jMax are required only when computing the coefficients
  2511.             // U₂<sup>j</sup> and V₂<sup>j</sup>
  2512.             // and those coefficients are only required for Fourier index = 1 (i == 0).
  2513.             for (int j = jMax + 1; j <= 2 * jMax; j++) {
  2514.                 // compute 1 / j
  2515.                 final double ooj = 1. / j;
  2516.                 // the value of i is 0
  2517.                 u1ij[0][j] = 0.;
  2518.                 v1ij[0][j] = 0.;

  2520.                 // k starts from j-J as it is always greater than or equal to 1
  2521.                 for (int kIndex = j - jMax; kIndex <= j - 1; kIndex++) {
  2522.                     // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2523.                     // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2524.                     u1ij[0][j] += fourierCjSj.getCij(0, j - kIndex) * currentRhoSigmaj[1][kIndex] +
  2525.                             fourierCjSj.getSij(0, j - kIndex) * currentRhoSigmaj[0][kIndex];

  2527.                     // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2528.                     // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2529.                     v1ij[0][j] += fourierCjSj.getCij(0, j - kIndex) * currentRhoSigmaj[0][kIndex] -
  2530.                             fourierCjSj.getSij(0, j - kIndex) * currentRhoSigmaj[1][kIndex];
  2531.                 }
  2532.                 for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2533.                     // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2534.                     // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2535.                     u1ij[0][j] += -fourierCjSj.getCij(0, kIndex) * currentRhoSigmaj[1][j + kIndex] -
  2536.                             fourierCjSj.getSij(0, kIndex) * currentRhoSigmaj[0][j + kIndex];

  2538.                     // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2539.                     // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2540.                     v1ij[0][j] += fourierCjSj.getCij(0, kIndex) * currentRhoSigmaj[0][j + kIndex] +
  2541.                             fourierCjSj.getSij(0, kIndex) * currentRhoSigmaj[1][j + kIndex];
  2542.                 }

  2544.                 // divide by 1 / j
  2545.                 u1ij[0][j] *= -ooj;
  2546.                 v1ij[0][j] *= ooj;
  2547.             }
  2548.         }

  2550.         /**
  2551.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2552.          * <p>
  2553.          * Only the coefficients for Fourier index = 1 (i == 0) are required.
  2554.          * </p>
  2555.          */
  2556.         private void computeU2V2Coefficients() {
  2557.             for (int j = 1; j <= jMax; j++) {
  2558.                 // compute 1 / j
  2559.                 final double ooj = 1. / j;

  2561.                 // only the values for i == 0 are computed
  2562.                 u2ij[j] = v1ij[0][j];
  2563.                 v2ij[j] = u1ij[0][j];

  2565.                 // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2566.                 if (j > 1) {
  2567.                     for (int l = 1; l <= j - 1; l++) {
  2568.                         // U₁<sup>j-l</sup> * σ<sub>l</sub> +
  2569.                         // V₁<sup>j-l</sup> * ρ<sub>l</sub>
  2570.                         u2ij[j] += u1ij[0][j - l] * currentRhoSigmaj[1][l] + v1ij[0][j - l] * currentRhoSigmaj[0][l];

  2572.                         // U₁<sup>j-l</sup> * ρ<sub>l</sub> -
  2573.                         // V₁<sup>j-l</sup> * σ<sub>l</sub>
  2574.                         v2ij[j] += u1ij[0][j - l] * currentRhoSigmaj[0][l] - v1ij[0][j - l] * currentRhoSigmaj[1][l];
  2575.                     }
  2576.                 }

  2578.                 for (int l = 1; l <= jMax; l++) {
  2579.                     // -U₁<sup>j+l</sup> * σ<sub>l</sub> +
  2580.                     // U₁<sup>l</sup> * σ<sub>j+l</sub> +
  2581.                     // V₁<sup>j+l</sup> * ρ<sub>l</sub> -
  2582.                     // V₁<sup>l</sup> * ρ<sub>j+l</sub>
  2583.                     u2ij[j] += -u1ij[0][j + l] * currentRhoSigmaj[1][l] + u1ij[0][l] * currentRhoSigmaj[1][j + l] +
  2584.                             v1ij[0][j + l] * currentRhoSigmaj[0][l] - v1ij[0][l] * currentRhoSigmaj[0][j + l];

  2586.                     // U₁<sup>j+l</sup> * ρ<sub>l</sub> +
  2587.                     // U₁<sup>l</sup> * ρ<sub>j+l</sub> +
  2588.                     // V₁<sup>j+l</sup> * σ<sub>l</sub> +
  2589.                     // V₁<sup>l</sup> * σ<sub>j+l</sub>
  2590.                     u2ij[j] += u1ij[0][j + l] * currentRhoSigmaj[0][l] + u1ij[0][l] * currentRhoSigmaj[0][j + l] +
  2591.                             v1ij[0][j + l] * currentRhoSigmaj[1][l] + v1ij[0][l] * currentRhoSigmaj[1][j + l];
  2592.                 }

  2594.                 // divide by 1 / j
  2595.                 u2ij[j] *= -ooj;
  2596.                 v2ij[j] *= ooj;
  2597.             }
  2598.         }

  2600.         /**
  2601.          * Get the coefficient U₁<sup>j</sup> for Fourier index i.
  2602.          *
  2603.          * @param j j index
  2604.          * @param i Fourier index (starts at 0)
  2605.          * @return the coefficient U₁<sup>j</sup> for the given Fourier index i
  2606.          */
  2607.         public double getU1(final int j, final int i) {
  2608.             return u1ij[i][j];
  2609.         }

  2611.         /**
  2612.          * Get the coefficient V₁<sup>j</sup> for Fourier index i.
  2613.          *
  2614.          * @param j j index
  2615.          * @param i Fourier index (starts at 0)
  2616.          * @return the coefficient V₁<sup>j</sup> for the given Fourier index i
  2617.          */
  2618.         public double getV1(final int j, final int i) {
  2619.             return v1ij[i][j];
  2620.         }

  2622.         /**
  2623.          * Get the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2624.          *
  2625.          * @param j j index
  2626.          * @return the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2627.          */
  2628.         public double getU2(final int j) {
  2629.             return u2ij[j];
  2630.         }

  2632.         /**
  2633.          * Get the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2634.          *
  2635.          * @param j j index
  2636.          * @return the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2637.          */
  2638.         public double getV2(final int j) {
  2639.             return v2ij[j];
  2640.         }
  2641.     }

  2643.     /**
  2644.      * The U<sub>i</sub><sup>j</sup> and V<sub>i</sub><sup>j</sup> coefficients
  2645.      * described by equations 2.5.3-(21) and 2.5.3-(22) from Danielson.
  2646.      * <p>
  2647.      * The index i takes only the values 1 and 2<br>
  2648.      * For U only the index 0 for j is used.
  2649.      * </p>
  2650.      *
  2651.      * @author Petre Bazavan
  2652.      * @author Lucian Barbulescu
  2653.      * @param <T> type of the field elements
  2654.      */
  2655.     protected static class FieldUijVijCoefficients<T extends CalculusFieldElement<T>> {

  2657.         /**
  2658.          * The U₁<sup>j</sup> coefficients.
  2659.          * <p>
  2660.          * The first index identifies the Fourier coefficients used<br>
  2661.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2662.          * S<sub>i</sub><sup>j</sup><br>
  2663.          * The only exception is when j = 0 when only the coefficient for fourier index
  2664.          * = 1 (i == 0) is needed.<br>
  2665.          * Also, for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are
  2666.          * computed, because are required to compute the coefficients U₂<sup>j</sup>
  2667.          * </p>
  2668.          */
  2669.         private final T[][] u1ij;

  2671.         /**
  2672.          * The V₁<sup>j</sup> coefficients.
  2673.          * <p>
  2674.          * The first index identifies the Fourier coefficients used<br>
  2675.          * Those coefficients are computed for all Fourier C<sub>i</sub><sup>j</sup> and
  2676.          * S<sub>i</sub><sup>j</sup><br>
  2677.          * for fourier index = 1 (i == 0), the coefficients up to 2 * jMax are computed,
  2678.          * because are required to compute the coefficients V₂<sup>j</sup>
  2679.          * </p>
  2680.          */
  2681.         private final T[][] v1ij;

  2683.         /**
  2684.          * The U₂<sup>j</sup> coefficients.
  2685.          * <p>
  2686.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2687.          * they are the only ones required.
  2688.          * </p>
  2689.          */
  2690.         private final T[] u2ij;

  2692.         /**
  2693.          * The V₂<sup>j</sup> coefficients.
  2694.          * <p>
  2695.          * Only the coefficients that use the Fourier index = 1 (i == 0) are computed as
  2696.          * they are the only ones required.
  2697.          * </p>
  2698.          */
  2699.         private final T[] v2ij;

  2701.         /**
  2702.          * The current computed values for the ρ<sub>j</sub> and σ<sub>j</sub>
  2703.          * coefficients.
  2704.          */
  2705.         private final T[][] currentRhoSigmaj;

  2707.         /**
  2708.          * The C<sub>i</sub><sup>j</sup> and the S<sub>i</sub><sup>j</sup> Fourier
  2709.          * coefficients.
  2710.          */
  2711.         private final FieldFourierCjSjCoefficients<T> fourierCjSj;

  2713.         /** The maximum value for j index. */
  2714.         private final int jMax;

  2716.         /**
  2717.          * Constructor.
  2718.          * @param currentRhoSigmaj the current computed values for the ρ<sub>j</sub> and
  2719.          *                         σ<sub>j</sub> coefficients
  2720.          * @param fourierCjSj      the fourier coefficients C<sub>i</sub><sup>j</sup>
  2721.          *                         and the S<sub>i</sub><sup>j</sup>
  2722.          * @param jMax             maximum value for j index
  2723.          * @param field            field used by default
  2724.          */
  2725.         FieldUijVijCoefficients(final T[][] currentRhoSigmaj, final FieldFourierCjSjCoefficients<T> fourierCjSj,
  2726.                 final int jMax, final Field<T> field) {
  2727.             this.currentRhoSigmaj = currentRhoSigmaj;
  2728.             this.fourierCjSj = fourierCjSj;
  2729.             this.jMax = jMax;

  2731.             // initialize the internal arrays.
  2732.             this.u1ij = MathArrays.buildArray(field, 6, 2 * jMax + 1);
  2733.             this.v1ij = MathArrays.buildArray(field, 6, 2 * jMax + 1);
  2734.             this.u2ij = MathArrays.buildArray(field, jMax + 1);
  2735.             this.v2ij = MathArrays.buildArray(field, jMax + 1);

  2737.             // compute the coefficients
  2738.             computeU1V1Coefficients(field);
  2739.             computeU2V2Coefficients(field);
  2740.         }

  2742.         /**
  2743.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2744.          * @param field field used by default
  2745.          */
  2746.         private void computeU1V1Coefficients(final Field<T> field) {
  2747.             // Zero
  2748.             final T zero = field.getZero();

  2750.             // generate the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients
  2751.             // for j >= 1
  2752.             // also the U₁⁰ for Fourier index = 1 (i == 0) coefficient will be computed
  2753.             u1ij[0][0] = zero;
  2754.             for (int j = 1; j <= jMax; j++) {
  2755.                 // compute 1 / j
  2756.                 final double ooj = 1. / j;

  2758.                 for (int i = 0; i < 6; i++) {
  2759.                     // j is aready between 1 and J
  2760.                     u1ij[i][j] = fourierCjSj.getSij(i, j);
  2761.                     v1ij[i][j] = fourierCjSj.getCij(i, j);

  2763.                     // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2764.                     if (j > 1) {
  2765.                         // k starts with 1 because j-J is less than or equal to 0
  2766.                         for (int kIndex = 1; kIndex <= j - 1; kIndex++) {
  2767.                             // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2768.                             // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2769.                             u1ij[i][j] = u1ij[i][j]
  2770.                                     .add(fourierCjSj.getCij(i, j - kIndex).multiply(currentRhoSigmaj[1][kIndex]).add(
  2771.                                             fourierCjSj.getSij(i, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2773.                             // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2774.                             // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2775.                             v1ij[i][j] = v1ij[i][j].add(
  2776.                                     fourierCjSj.getCij(i, j - kIndex).multiply(currentRhoSigmaj[0][kIndex]).subtract(
  2777.                                             fourierCjSj.getSij(i, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2778.                         }
  2779.                     }

  2781.                     // since j must be between 1 and J-1 and is already between 1 and J
  2782.                     // the following sum is skiped only for j = jMax
  2783.                     if (j != jMax) {
  2784.                         for (int kIndex = 1; kIndex <= jMax - j; kIndex++) {
  2785.                             // -C<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub> +
  2786.                             // S<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub>
  2787.                             u1ij[i][j] = u1ij[i][j].add(fourierCjSj.getCij(i, j + kIndex).negate()
  2788.                                     .multiply(currentRhoSigmaj[1][kIndex])
  2789.                                     .add(fourierCjSj.getSij(i, j + kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2791.                             // C<sub>i</sub><sup>j+k</sup> * ρ<sub>k</sub> +
  2792.                             // S<sub>i</sub><sup>j+k</sup> * σ<sub>k</sub>
  2793.                             v1ij[i][j] = v1ij[i][j]
  2794.                                     .add(fourierCjSj.getCij(i, j + kIndex).multiply(currentRhoSigmaj[0][kIndex]).add(
  2795.                                             fourierCjSj.getSij(i, j + kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2796.                         }
  2797.                     }

  2799.                     for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2800.                         // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2801.                         // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2802.                         u1ij[i][j] = u1ij[i][j].add(fourierCjSj.getCij(i, kIndex).negate()
  2803.                                 .multiply(currentRhoSigmaj[1][j + kIndex])
  2804.                                 .subtract(fourierCjSj.getSij(i, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])));

  2806.                         // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2807.                         // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2808.                         v1ij[i][j] = v1ij[i][j]
  2809.                                 .add(fourierCjSj.getCij(i, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])
  2810.                                         .add(fourierCjSj.getSij(i, kIndex).multiply(currentRhoSigmaj[1][j + kIndex])));
  2811.                     }

  2813.                     // divide by 1 / j
  2814.                     u1ij[i][j] = u1ij[i][j].multiply(-ooj);
  2815.                     v1ij[i][j] = v1ij[i][j].multiply(ooj);

  2817.                     // if index = 1 (i == 0) add the computed terms to U₁⁰
  2818.                     if (i == 0) {
  2819.                         // - (U₁<sup>j</sup> * ρ<sub>j</sub> + V₁<sup>j</sup> * σ<sub>j</sub>
  2820.                         u1ij[0][0] = u1ij[0][0].add(u1ij[0][j].negate().multiply(currentRhoSigmaj[0][j])
  2821.                                 .subtract(v1ij[0][j].multiply(currentRhoSigmaj[1][j])));
  2822.                     }
  2823.                 }
  2824.             }

  2826.             // Terms with j > jMax are required only when computing the coefficients
  2827.             // U₂<sup>j</sup> and V₂<sup>j</sup>
  2828.             // and those coefficients are only required for Fourier index = 1 (i == 0).
  2829.             for (int j = jMax + 1; j <= 2 * jMax; j++) {
  2830.                 // compute 1 / j
  2831.                 final double ooj = 1. / j;
  2832.                 // the value of i is 0
  2833.                 u1ij[0][j] = zero;
  2834.                 v1ij[0][j] = zero;

  2836.                 // k starts from j-J as it is always greater than or equal to 1
  2837.                 for (int kIndex = j - jMax; kIndex <= j - 1; kIndex++) {
  2838.                     // C<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub> +
  2839.                     // S<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub>
  2840.                     u1ij[0][j] = u1ij[0][j].add(fourierCjSj.getCij(0, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])
  2841.                             .add(fourierCjSj.getSij(0, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])));

  2843.                     // C<sub>i</sub><sup>j-k</sup> * ρ<sub>k</sub> -
  2844.                     // S<sub>i</sub><sup>j-k</sup> * σ<sub>k</sub>
  2845.                     v1ij[0][j] = v1ij[0][j].add(fourierCjSj.getCij(0, j - kIndex).multiply(currentRhoSigmaj[0][kIndex])
  2846.                             .subtract(fourierCjSj.getSij(0, j - kIndex).multiply(currentRhoSigmaj[1][kIndex])));
  2847.                 }
  2848.                 for (int kIndex = 1; kIndex <= jMax; kIndex++) {
  2849.                     // C<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub> -
  2850.                     // S<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub>
  2851.                     u1ij[0][j] = u1ij[0][j]
  2852.                             .add(fourierCjSj.getCij(0, kIndex).negate().multiply(currentRhoSigmaj[1][j + kIndex])
  2853.                                     .subtract(fourierCjSj.getSij(0, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])));

  2855.                     // C<sub>i</sub><sup>k</sup> * ρ<sub>j+k</sub> +
  2856.                     // S<sub>i</sub><sup>k</sup> * σ<sub>j+k</sub>
  2857.                     v1ij[0][j] = v1ij[0][j].add(fourierCjSj.getCij(0, kIndex).multiply(currentRhoSigmaj[0][j + kIndex])
  2858.                             .add(fourierCjSj.getSij(0, kIndex).multiply(currentRhoSigmaj[1][j + kIndex])));
  2859.                 }

  2861.                 // divide by 1 / j
  2862.                 u1ij[0][j] = u1ij[0][j].multiply(-ooj);
  2863.                 v1ij[0][j] = v1ij[0][j].multiply(ooj);
  2864.             }
  2865.         }

  2867.         /**
  2868.          * Build the U₁<sup>j</sup> and V₁<sup>j</sup> coefficients.
  2869.          * <p>
  2870.          * Only the coefficients for Fourier index = 1 (i == 0) are required.
  2871.          * </p>
  2872.          * @param field field used by default
  2873.          */
  2874.         private void computeU2V2Coefficients(final Field<T> field) {
  2875.             for (int j = 1; j <= jMax; j++) {
  2876.                 // compute 1 / j
  2877.                 final double ooj = 1. / j;

  2879.                 // only the values for i == 0 are computed
  2880.                 u2ij[j] = v1ij[0][j];
  2881.                 v2ij[j] = u1ij[0][j];

  2883.                 // 1 - δ<sub>1j</sub> is 1 for all j > 1
  2884.                 if (j > 1) {
  2885.                     for (int l = 1; l <= j - 1; l++) {
  2886.                         // U₁<sup>j-l</sup> * σ<sub>l</sub> +
  2887.                         // V₁<sup>j-l</sup> * ρ<sub>l</sub>
  2888.                         u2ij[j] = u2ij[j].add(u1ij[0][j - l].multiply(currentRhoSigmaj[1][l])
  2889.                                 .add(v1ij[0][j - l].multiply(currentRhoSigmaj[0][l])));

  2891.                         // U₁<sup>j-l</sup> * ρ<sub>l</sub> -
  2892.                         // V₁<sup>j-l</sup> * σ<sub>l</sub>
  2893.                         v2ij[j] = v2ij[j].add(u1ij[0][j - l].multiply(currentRhoSigmaj[0][l])
  2894.                                 .subtract(v1ij[0][j - l].multiply(currentRhoSigmaj[1][l])));
  2895.                     }
  2896.                 }

  2898.                 for (int l = 1; l <= jMax; l++) {
  2899.                     // -U₁<sup>j+l</sup> * σ<sub>l</sub> +
  2900.                     // U₁<sup>l</sup> * σ<sub>j+l</sub> +
  2901.                     // V₁<sup>j+l</sup> * ρ<sub>l</sub> -
  2902.                     // V₁<sup>l</sup> * ρ<sub>j+l</sub>
  2903.                     u2ij[j] = u2ij[j].add(u1ij[0][j + l].negate().multiply(currentRhoSigmaj[1][l])
  2904.                             .add(u1ij[0][l].multiply(currentRhoSigmaj[1][j + l]))
  2905.                             .add(v1ij[0][j + l].multiply(currentRhoSigmaj[0][l]))
  2906.                             .subtract(v1ij[0][l].multiply(currentRhoSigmaj[0][j + l])));

  2908.                     // U₁<sup>j+l</sup> * ρ<sub>l</sub> +
  2909.                     // U₁<sup>l</sup> * ρ<sub>j+l</sub> +
  2910.                     // V₁<sup>j+l</sup> * σ<sub>l</sub> +
  2911.                     // V₁<sup>l</sup> * σ<sub>j+l</sub>
  2912.                     u2ij[j] = u2ij[j].add(u1ij[0][j + l].multiply(currentRhoSigmaj[0][l])
  2913.                             .add(u1ij[0][l].multiply(currentRhoSigmaj[0][j + l]))
  2914.                             .add(v1ij[0][j + l].multiply(currentRhoSigmaj[1][l]))
  2915.                             .add(v1ij[0][l].multiply(currentRhoSigmaj[1][j + l])));
  2916.                 }

  2918.                 // divide by 1 / j
  2919.                 u2ij[j] = u2ij[j].multiply(-ooj);
  2920.                 v2ij[j] = v2ij[j].multiply(ooj);
  2921.             }
  2922.         }

  2924.         /**
  2925.          * Get the coefficient U₁<sup>j</sup> for Fourier index i.
  2926.          *
  2927.          * @param j j index
  2928.          * @param i Fourier index (starts at 0)
  2929.          * @return the coefficient U₁<sup>j</sup> for the given Fourier index i
  2930.          */
  2931.         public T getU1(final int j, final int i) {
  2932.             return u1ij[i][j];
  2933.         }

  2935.         /**
  2936.          * Get the coefficient V₁<sup>j</sup> for Fourier index i.
  2937.          *
  2938.          * @param j j index
  2939.          * @param i Fourier index (starts at 0)
  2940.          * @return the coefficient V₁<sup>j</sup> for the given Fourier index i
  2941.          */
  2942.         public T getV1(final int j, final int i) {
  2943.             return v1ij[i][j];
  2944.         }

  2946.         /**
  2947.          * Get the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2948.          *
  2949.          * @param j j index
  2950.          * @return the coefficient U₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2951.          */
  2952.         public T getU2(final int j) {
  2953.             return u2ij[j];
  2954.         }

  2956.         /**
  2957.          * Get the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0).
  2958.          *
  2959.          * @param j j index
  2960.          * @return the coefficient V₂<sup>j</sup> for Fourier index = 1 (i == 0)
  2961.          */
  2962.         public T getV2(final int j) {
  2963.             return v2ij[j];
  2964.         }
  2965.     }

  2967.     /** Coefficients valid for one time slot. */
  2968.     protected static class Slot {

  2970.         /**
  2971.          * The coefficients D<sub>i</sub><sup>j</sup>.
  2972.          * <p>
  2973.          * Only for j = 1 and j = 2 the coefficients are not 0. <br>
  2974.          * i corresponds to the equinoctial element, as follows: - i=0 for a <br/>
  2975.          * - i=1 for k <br/>
  2976.          * - i=2 for h <br/>
  2977.          * - i=3 for q <br/>
  2978.          * - i=4 for p <br/>
  2979.          * - i=5 for λ <br/>
  2980.          * </p>
  2981.          */
  2982.         private final ShortPeriodicsInterpolatedCoefficient[] dij;

  2984.         /**
  2985.          * The coefficients C<sub>i</sub><sup>j</sup>.
  2986.          * <p>
  2987.          * The index order is cij[j][i] <br/>
  2988.          * i corresponds to the equinoctial element, as follows: <br/>
  2989.          * - i=0 for a <br/>
  2990.          * - i=1 for k <br/>
  2991.          * - i=2 for h <br/>
  2992.          * - i=3 for q <br/>
  2993.          * - i=4 for p <br/>
  2994.          * - i=5 for λ <br/>
  2995.          * </p>
  2996.          */
  2997.         private final ShortPeriodicsInterpolatedCoefficient[] cij;

  2999.         /**
  3000.          * The coefficients S<sub>i</sub><sup>j</sup>.
  3001.          * <p>
  3002.          * The index order is sij[j][i] <br/>
  3003.          * i corresponds to the equinoctial element, as follows: <br/>
  3004.          * - i=0 for a <br/>
  3005.          * - i=1 for k <br/>
  3006.          * - i=2 for h <br/>
  3007.          * - i=3 for q <br/>
  3008.          * - i=4 for p <br/>
  3009.          * - i=5 for λ <br/>
  3010.          * </p>
  3011.          */
  3012.         private final ShortPeriodicsInterpolatedCoefficient[] sij;

  3014.         /**
  3015.          * Simple constructor.
  3016.          * @param jMax                maximum value for j index
  3017.          * @param interpolationPoints number of points used in the interpolation process
  3018.          */
  3019.         Slot(final int jMax, final int interpolationPoints) {

  3021.             dij = new ShortPeriodicsInterpolatedCoefficient[3];
  3022.             cij = new ShortPeriodicsInterpolatedCoefficient[jMax + 1];
  3023.             sij = new ShortPeriodicsInterpolatedCoefficient[jMax + 1];

  3025.             // Initialize the C<sub>i</sub><sup>j</sup>, S<sub>i</sub><sup>j</sup> and
  3026.             // D<sub>i</sub><sup>j</sup> coefficients
  3027.             for (int j = 0; j <= jMax; j++) {
  3028.                 cij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3029.                 if (j > 0) {
  3030.                     sij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3031.                 }
  3032.                 // Initialize only the non-zero D<sub>i</sub><sup>j</sup> coefficients
  3033.                 if (j == 1 || j == 2) {
  3034.                     dij[j] = new ShortPeriodicsInterpolatedCoefficient(interpolationPoints);
  3035.                 }
  3036.             }

  3038.         }

  3040.     }

  3042.     /** Coefficients valid for one time slot.
  3043.      * @param <T> type of the field elements
  3044.      */
  3045.     protected static class FieldSlot<T extends CalculusFieldElement<T>> {

  3047.         /**
  3048.          * The coefficients D<sub>i</sub><sup>j</sup>.
  3049.          * <p>
  3050.          * Only for j = 1 and j = 2 the coefficients are not 0. <br>
  3051.          * i corresponds to the equinoctial element, as follows: - i=0 for a <br/>
  3052.          * - i=1 for k <br/>
  3053.          * - i=2 for h <br/>
  3054.          * - i=3 for q <br/>
  3055.          * - i=4 for p <br/>
  3056.          * - i=5 for λ <br/>
  3057.          * </p>
  3058.          */
  3059.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] dij;

  3061.         /**
  3062.          * The coefficients C<sub>i</sub><sup>j</sup>.
  3063.          * <p>
  3064.          * The index order is cij[j][i] <br/>
  3065.          * i corresponds to the equinoctial element, as follows: <br/>
  3066.          * - i=0 for a <br/>
  3067.          * - i=1 for k <br/>
  3068.          * - i=2 for h <br/>
  3069.          * - i=3 for q <br/>
  3070.          * - i=4 for p <br/>
  3071.          * - i=5 for λ <br/>
  3072.          * </p>
  3073.          */
  3074.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] cij;

  3076.         /**
  3077.          * The coefficients S<sub>i</sub><sup>j</sup>.
  3078.          * <p>
  3079.          * The index order is sij[j][i] <br/>
  3080.          * i corresponds to the equinoctial element, as follows: <br/>
  3081.          * - i=0 for a <br/>
  3082.          * - i=1 for k <br/>
  3083.          * - i=2 for h <br/>
  3084.          * - i=3 for q <br/>
  3085.          * - i=4 for p <br/>
  3086.          * - i=5 for λ <br/>
  3087.          * </p>
  3088.          */
  3089.         private final FieldShortPeriodicsInterpolatedCoefficient<T>[] sij;

  3091.         /**
  3092.          * Simple constructor.
  3093.          * @param jMax                maximum value for j index
  3094.          * @param interpolationPoints number of points used in the interpolation process
  3095.          */
  3096.         @SuppressWarnings("unchecked")
  3097.         FieldSlot(final int jMax, final int interpolationPoints) {

  3099.             dij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3100.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, 3);
  3101.             cij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3102.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, jMax + 1);
  3103.             sij = (FieldShortPeriodicsInterpolatedCoefficient<T>[]) Array
  3104.                     .newInstance(FieldShortPeriodicsInterpolatedCoefficient.class, jMax + 1);

  3106.             // Initialize the C<sub>i</sub><sup>j</sup>, S<sub>i</sub><sup>j</sup> and
  3107.             // D<sub>i</sub><sup>j</sup> coefficients
  3108.             for (int j = 0; j <= jMax; j++) {
  3109.                 cij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3110.                 if (j > 0) {
  3111.                     sij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3112.                 }
  3113.                 // Initialize only the non-zero D<sub>i</sub><sup>j</sup> coefficients
  3114.                 if (j == 1 || j == 2) {
  3115.                     dij[j] = new FieldShortPeriodicsInterpolatedCoefficient<>(interpolationPoints);
  3116.                 }
  3117.             }

  3119.         }

  3121.     }

  3123. }
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