NewcombOperators.java

  1. /* Copyright 2002-2025 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.utilities;

  19. import org.hipparchus.analysis.polynomials.PolynomialFunction;
  20. import org.hipparchus.util.FastMath;

  22. import java.util.ArrayList;
  23. import java.util.List;
  24. import java.util.Map;
  25. import java.util.SortedMap;
  26. import java.util.TreeMap;

  28. /**
  29.  * Implementation of the Modified Newcomb Operators.
  30.  *
  31.  *  <p> From equations 2.7.3 - (12)(13) of the Danielson paper, those operators
  32.  *  are defined as:
  33.  *
  34.  *  <p>
  35.  *  4(ρ + σ)Y<sub>ρ,σ</sub><sup>n,s</sup> = <br>
  36.  *     2(2s - n)Y<sub>ρ-1,σ</sub><sup>n,s+1</sup> + (s - n)Y<sub>ρ-2,σ</sub><sup>n,s+2</sup> <br>
  37.  *   - 2(2s + n)Y<sub>ρ,σ-1</sub><sup>n,s-1</sup> - (s+n)Y<sub>ρ,σ-2</sub><sup>n,s-2</sup> <br>
  38.  *   + 2(2ρ + 2σ + 2 + 3n)Y<sub>ρ-1,σ-1</sub><sup>n,s</sup>
  39.  *
  40.  *  <p> Initialization is given by : Y<sub>0,0</sub><sup>n,s</sup> = 1
  41.  *
  42.  *  <p> Internally, the Modified Newcomb Operators are stored as an array of
  43.  *  {@link PolynomialFunction} :
  44.  *
  45.  *  <p> Y<sub>ρ,σ</sub><sup>n,s</sup> = P<sub>k0</sub> + P<sub>k1</sub>n + ... +
  46.  *  P<sub>kj</sub>n<sup>j</sup>
  47.  *
  48.  * <p> where the P<sub>kj</sub> are given by
  49.  *
  50.  * <p> P<sub>kj</sub> = ∑<sub>j=0;ρ</sub> a<sub>j</sub>s<sup>j</sup>
  51.  *
  52.  * @author Romain Di Costanzo
  53.  * @author Pascal Parraud
  54.  */
  55. public class NewcombOperators {

  57.     /** Storage map. */
  58.     private static final Map<NewKey, Double> MAP = new TreeMap<>((k1, k2) -> {
  59.         if (k1.n == k2.n) {
  60.             if (k1.s == k2.s) {
  61.                 if (k1.rho == k2.rho) {
  62.                     return Integer.compare(k1.sigma, k2.sigma);
  63.                 } else if (k1.rho < k2.rho) {
  64.                     return -1;
  65.                 } else {
  66.                     return 1;
  67.                 }
  68.             } else if (k1.s < k2.s) {
  69.                 return -1;
  70.             } else {
  71.                 return 1;
  72.             }
  73.         } else if (k1.n < k2.n) {
  74.             return -1;
  75.         }
  76.         return 1;
  77.     });

  79.     /** Private constructor as class is a utility.
  80.      */
  81.     private NewcombOperators() {
  82.     }

  84.     /** Get the Newcomb operator evaluated at n, s, ρ, σ.
  85.      * <p>
  86.      * This method is guaranteed to be thread-safe
  87.      * </p>
  88.      *  @param rho ρ index
  89.      *  @param sigma σ index
  90.      *  @param n n index
  91.      *  @param s s index
  92.      *  @return Y<sub>ρ,σ</sub><sup>n,s</sup>
  93.      */
  94.     public static double getValue(final int rho, final int sigma, final int n, final int s) {

  96.         final NewKey key = new NewKey(n, s, rho, sigma);
  97.         synchronized (MAP) {
  98.             if (MAP.containsKey(key)) {
  99.                 return MAP.get(key);
  100.             }
  101.         }

  103.         // Get the Newcomb polynomials for the given rho and sigma
  104.         final List<PolynomialFunction> polynomials = PolynomialsGenerator.getPolynomials(rho, sigma);

  106.         // Compute the value from the list of polynomials for the given n and s
  107.         double nPower = 1.;
  108.         double value = 0.0;
  109.         for (final PolynomialFunction polynomial : polynomials) {
  110.             value += polynomial.value(s) * nPower;
  111.             nPower = n * nPower;
  112.         }
  113.         synchronized (MAP) {
  114.             MAP.put(key, value);
  115.         }

  117.         return value;

  119.     }

  121.     /** Generator for Newcomb polynomials. */
  122.     private static class PolynomialsGenerator {

  124.         /** Polynomials storage. */
  125.         private static final SortedMap<Couple, List<PolynomialFunction>> POLYNOMIALS =
  126.                 new TreeMap<>((c1, c2) -> {
  127.                     if (c1.rho == c2.rho) {
  128.                         if (c1.sigma < c2.sigma) {
  129.                             return -1;
  130.                         } else if (c1.sigma == c2.sigma) {
  131.                             return 0;
  132.                         }
  133.                     } else if (c1.rho < c2.rho) {
  134.                         return -1;
  135.                     }
  136.                     return 1;
  137.                 });

  139.         /** Private constructor as class is a utility.
  140.          */
  141.         private PolynomialsGenerator() {
  142.         }

  144.         /** Get the list of polynomials representing the Newcomb Operator for the (ρ,σ) couple.
  145.          * <p>
  146.          * This method is guaranteed to be thread-safe
  147.          * </p>
  148.          *  @param rho ρ value
  149.          *  @param sigma σ value
  150.          *  @return Polynomials representing the Newcomb Operator for the (ρ,σ) couple.
  151.          */
  152.         private static List<PolynomialFunction> getPolynomials(final int rho, final int sigma) {

  154.             final Couple couple = new Couple(rho, sigma);

  156.             synchronized (POLYNOMIALS) {

  158.                 if (POLYNOMIALS.isEmpty()) {
  159.                     // Initialize lists
  160.                     final List<PolynomialFunction> l00 = new ArrayList<>();
  161.                     final List<PolynomialFunction> l01 = new ArrayList<>();
  162.                     final List<PolynomialFunction> l10 = new ArrayList<>();
  163.                     final List<PolynomialFunction> l11 = new ArrayList<>();

  165.                     // Y(rho = 0, sigma = 0) = 1
  166.                     l00.add(new PolynomialFunction(new double[] {
  167.                         1.
  168.                     }));
  169.                     // Y(rho = 0, sigma = 1) =  -s - n/2
  170.                     l01.add(new PolynomialFunction(new double[] {
  171.                         0, -1.
  172.                     }));
  173.                     l01.add(new PolynomialFunction(new double[] {
  174.                         -0.5
  175.                     }));
  176.                     // Y(rho = 1, sigma = 0) =  s - n/2
  177.                     l10.add(new PolynomialFunction(new double[] {
  178.                         0, 1.
  179.                     }));
  180.                     l10.add(new PolynomialFunction(new double[] {
  181.                         -0.5
  182.                     }));
  183.                     // Y(rho = 1, sigma = 1) = 3/2 - s² + 5n/4 + n²/4
  184.                     l11.add(new PolynomialFunction(new double[] {
  185.                         1.5, 0., -1.
  186.                     }));
  187.                     l11.add(new PolynomialFunction(new double[] {
  188.                         1.25
  189.                     }));
  190.                     l11.add(new PolynomialFunction(new double[] {
  191.                         0.25
  192.                     }));

  194.                     // Initialize polynomials
  195.                     POLYNOMIALS.put(new Couple(0, 0), l00);
  196.                     POLYNOMIALS.put(new Couple(0, 1), l01);
  197.                     POLYNOMIALS.put(new Couple(1, 0), l10);
  198.                     POLYNOMIALS.put(new Couple(1, 1), l11);

  200.                 }

  202.                 // If order hasn't been computed yet, update the Newcomb polynomials
  203.                 if (!POLYNOMIALS.containsKey(couple)) {
  204.                     PolynomialsGenerator.computeFor(rho, sigma);
  205.                 }

  207.                 return POLYNOMIALS.get(couple);

  209.             }
  210.         }

  212.         /** Compute the Modified Newcomb Operators up to a given (ρ, σ) couple.
  213.          *  <p>
  214.          *  The recursive computation uses equation 2.7.3-(12) of the Danielson paper.
  215.          *  </p>
  216.          *  @param rho ρ value to reach
  217.          *  @param sigma σ value to reach
  218.          */
  219.         private static void computeFor(final int rho, final int sigma) {

  221.             // Initialize result :
  222.             List<PolynomialFunction> result = new ArrayList<>();

  224.             // Get the coefficient from the recurrence relation
  225.             final Map<Integer, List<PolynomialFunction>> map = generateRecurrenceCoefficients(rho, sigma);

  227.             // Compute (s - n) * Y[rho - 2, sigma][n, s + 2]
  228.             if (rho >= 2) {
  229.                 final List<PolynomialFunction> poly = map.get(0);
  230.                 final List<PolynomialFunction> list = getPolynomials(rho - 2, sigma);
  231.                 result = multiplyPolynomialList(poly, shiftList(list, 2));
  232.             }

  234.             // Compute 2(2rho + 2sigma + 2 + 3n) * Y[rho - 1, sigma - 1][n, s]
  235.             if (rho >= 1 && sigma >= 1) {
  236.                 final List<PolynomialFunction> poly = map.get(1);
  237.                 final List<PolynomialFunction> list = getPolynomials(rho - 1, sigma - 1);
  238.                 result = sumPolynomialList(result, multiplyPolynomialList(poly, list));
  239.             }

  241.             // Compute 2(2s - n) * Y[rho - 1, sigma][n, s + 1]
  242.             if (rho >= 1) {
  243.                 final List<PolynomialFunction> poly = map.get(2);
  244.                 final List<PolynomialFunction> list = getPolynomials(rho - 1, sigma);
  245.                 result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, 1)));
  246.             }

  248.             // Compute -(s + n) * Y[rho, sigma - 2][n, s - 2]
  249.             if (sigma >= 2) {
  250.                 final List<PolynomialFunction> poly = map.get(3);
  251.                 final List<PolynomialFunction> list = getPolynomials(rho, sigma - 2);
  252.                 result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, -2)));
  253.             }

  255.             // Compute -2(2s + n) * Y[rho, sigma - 1][n, s - 1]
  256.             if (sigma >= 1) {
  257.                 final List<PolynomialFunction> poly = map.get(4);
  258.                 final List<PolynomialFunction> list = getPolynomials(rho, sigma - 1);
  259.                 result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, -1)));
  260.             }

  262.             // Save polynomials for current (rho, sigma) couple
  263.             final Couple couple = new Couple(rho, sigma);
  264.             POLYNOMIALS.put(couple, result);
  265.         }

  267.         /** Multiply two lists of polynomials defined as the internal representation of the Newcomb Operator.
  268.          *  <p>
  269.          *  Let's call R<sub>s</sub>(n) the result returned by the method :
  270.          *  <pre>
  271.          *  R<sub>s</sub>(n) = (P<sub>s₀</sub> + P<sub>s₁</sub>n + ... + P<sub>s<sub>j</sub></sub>n<sup>j</sup>) *(Q<sub>s₀</sub> + Q<sub>s₁</sub>n + ... + Q<sub>s<sub>k</sub></sub>n<sup>k</sup>)
  272.          *  </pre>
  273.          *
  274.          *  where the P<sub>s<sub>j</sub></sub> and Q<sub>s<sub>k</sub></sub> are polynomials in s,
  275.          *  s being the index of the Y<sub>ρ,σ</sub><sup>n,s</sup> function
  276.          *
  277.          *  @param poly1 first list of polynomials
  278.          *  @param poly2 second list of polynomials
  279.          *  @return R<sub>s</sub>(n) as a list of {@link PolynomialFunction}
  280.          */
  281.         private static List<PolynomialFunction> multiplyPolynomialList(final List<PolynomialFunction> poly1,
  282.                                                                        final List<PolynomialFunction> poly2) {
  283.             // Initialize the result list of polynomial function
  284.             final List<PolynomialFunction> result = new ArrayList<>();
  285.             initializeListOfPolynomials(poly1.size() + poly2.size() - 1, result);

  287.             int i = 0;
  288.             // Iterate over first polynomial list
  289.             for (PolynomialFunction f1 : poly1) {
  290.                 // Iterate over second polynomial list
  291.                 for (int j = i; j < poly2.size() + i; j++) {
  292.                     final PolynomialFunction p2 = poly2.get(j - i);
  293.                     // Get previous polynomial for current 'n' order
  294.                     final PolynomialFunction previousP2 = result.get(j);
  295.                     // Replace the current order by summing the product of both of the polynomials
  296.                     result.set(j, previousP2.add(f1.multiply(p2)));
  297.                 }
  298.                 // shift polynomial order in 'n'
  299.                 i++;
  300.             }
  301.             return result;
  302.         }

  304.         /** Sum two lists of {@link PolynomialFunction}.
  305.          *  @param poly1 first list
  306.          *  @param poly2 second list
  307.          *  @return the summed list
  308.          */
  309.         private static List<PolynomialFunction> sumPolynomialList(final List<PolynomialFunction> poly1,
  310.                                                                   final List<PolynomialFunction> poly2) {
  311.             // identify the lowest degree polynomial
  312.             final int lowLength  = FastMath.min(poly1.size(), poly2.size());
  313.             final int highLength = FastMath.max(poly1.size(), poly2.size());
  314.             // Initialize the result list of polynomial function
  315.             final List<PolynomialFunction> result = new ArrayList<>();
  316.             initializeListOfPolynomials(highLength, result);

  318.             for (int i = 0; i < lowLength; i++) {
  319.                 // Add polynomials by increasing order of 'n'
  320.                 result.set(i, poly1.get(i).add(poly2.get(i)));
  321.             }
  322.             // Complete the list if lists are of different size:
  323.             for (int i = lowLength; i < highLength; i++) {
  324.                 if (poly1.size() < poly2.size()) {
  325.                     result.set(i, poly2.get(i));
  326.                 } else {
  327.                     result.set(i, poly1.get(i));
  328.                 }
  329.             }
  330.             return result;
  331.         }

  333.         /** Initialize an empty list of polynomials.
  334.          *  @param i order
  335.          *  @param result list into which polynomials should be added
  336.          */
  337.         private static void initializeListOfPolynomials(final int i,
  338.                                                         final List<PolynomialFunction> result) {
  339.             for (int k = 0; k < i; k++) {
  340.                 result.add(new PolynomialFunction(new double[] {0.}));
  341.             }
  342.         }

  344.         /** Shift a list of {@link PolynomialFunction}.
  345.          *
  346.          *  @param polynomialList list of {@link PolynomialFunction}
  347.          *  @param shift shift value
  348.          *  @return new list of shifted {@link PolynomialFunction}
  349.          */
  350.         private static List<PolynomialFunction> shiftList(final List<PolynomialFunction> polynomialList,
  351.                                                           final int shift) {
  352.             final List<PolynomialFunction> shiftedList = new ArrayList<>();
  353.             for (PolynomialFunction function : polynomialList) {
  354.                 shiftedList.add(new PolynomialFunction(shift(function.getCoefficients(), shift)));
  355.             }
  356.             return shiftedList;
  357.         }

  359.         /**
  360.          * Compute the coefficients of the polynomial \(P_s(x)\)
  361.          * whose values at point {@code x} will be the same as the those from the
  362.          * original polynomial \(P(x)\) when computed at {@code x + shift}.
  363.          * <p>
  364.          * More precisely, let \(\Delta = \) {@code shift} and let
  365.          * \(P_s(x) = P(x + \Delta)\).  The returned array
  366.          * consists of the coefficients of \(P_s\).  So if \(a_0, ..., a_{n-1}\)
  367.          * are the coefficients of \(P\), then the returned array
  368.          * \(b_0, ..., b_{n-1}\) satisfies the identity
  369.          * \(\sum_{i=0}^{n-1} b_i x^i = \sum_{i=0}^{n-1} a_i (x + \Delta)^i\) for all \(x\).
  370.          * </p>
  371.          * <p>
  372.          * This method is a modified version of the method with the same name
  373.          * in Hipparchus {@code PolynomialsUtils} class, simply changing
  374.          * computation of binomial coefficients so degrees higher than 66 can be used.
  375.          * </p>
  376.          *
  377.          * @param coefficients Coefficients of the original polynomial.
  378.          * @param shift Shift value.
  379.          * @return the coefficients \(b_i\) of the shifted
  380.          * polynomial.
  381.          */
  382.         public static double[] shift(final double[] coefficients,
  383.                                      final double shift) {
  384.             final int dp1 = coefficients.length;
  385.             final double[] newCoefficients = new double[dp1];

  387.             // Pascal triangle.
  388.             final double[][] coeff = new double[dp1][dp1];
  389.             coeff[0][0] = 1;
  390.             for (int i = 1; i < dp1; i++) {
  391.                 coeff[i][0] = 1;
  392.                 for (int j = 1; j < i; j++) {
  393.                     coeff[i][j] = coeff[i - 1][j - 1] + coeff[i - 1][j];
  394.                 }
  395.                 coeff[i][i] = 1;
  396.             }

  398.             // First polynomial coefficient.
  399.             double shiftI = 1;
  400.             for (double coefficient : coefficients) {
  401.                 newCoefficients[0] += coefficient * shiftI;
  402.                 shiftI *= shift;
  403.             }

  405.             // Superior order.
  406.             final int d = dp1 - 1;
  407.             for (int i = 0; i < d; i++) {
  408.                 double shiftJmI = 1;
  409.                 for (int j = i; j < d; j++) {
  410.                     newCoefficients[i + 1] += coeff[j + 1][j - i] * coefficients[j + 1] * shiftJmI;
  411.                     shiftJmI *= shift;
  412.                 }
  413.             }

  415.             return newCoefficients;
  416.         }

  418.         /** Generate recurrence coefficients.
  419.          *
  420.          *  @param rho ρ value
  421.          *  @param sigma σ value
  422.          *  @return recurrence coefficients
  423.          */
  424.         private static Map<Integer, List<PolynomialFunction>> generateRecurrenceCoefficients(final int rho, final int sigma) {
  425.             final double den   = 1. / (4. * (rho + sigma));
  426.             final double denx2 = 2. * den;
  427.             final double denx4 = 4. * den;
  428.             // Initialization :
  429.             final Map<Integer, List<PolynomialFunction>> list = new TreeMap<>();
  430.             final List<PolynomialFunction> poly0 = new ArrayList<>();
  431.             final List<PolynomialFunction> poly1 = new ArrayList<>();
  432.             final List<PolynomialFunction> poly2 = new ArrayList<>();
  433.             final List<PolynomialFunction> poly3 = new ArrayList<>();
  434.             final List<PolynomialFunction> poly4 = new ArrayList<>();
  435.             // (s - n)
  436.             poly0.add(new PolynomialFunction(new double[] {0., den}));
  437.             poly0.add(new PolynomialFunction(new double[] {-den}));
  438.             // 2(2 * rho + 2 * sigma + 2 + 3*n)
  439.             poly1.add(new PolynomialFunction(new double[] {1. + denx4}));
  440.             poly1.add(new PolynomialFunction(new double[] {denx2 + denx4}));
  441.             // 2(2s - n)
  442.             poly2.add(new PolynomialFunction(new double[] {0., denx4}));
  443.             poly2.add(new PolynomialFunction(new double[] {-denx2}));
  444.             // - (s + n)
  445.             poly3.add(new PolynomialFunction(new double[] {0., -den}));
  446.             poly3.add(new PolynomialFunction(new double[] {-den}));
  447.             // - 2(2s + n)
  448.             poly4.add(new PolynomialFunction(new double[] {0., -denx4}));
  449.             poly4.add(new PolynomialFunction(new double[] {-denx2}));

  451.             // Fill the map :
  452.             list.put(0, poly0);
  453.             list.put(1, poly1);
  454.             list.put(2, poly2);
  455.             list.put(3, poly3);
  456.             list.put(4, poly4);
  457.             return list;
  458.         }

  460.     }

  462.     /** Private class to define a couple of value. */
  463.     private static class Couple {

  465.         /** first couple value. */
  466.         private final int rho;

  468.         /** second couple value. */
  469.         private final int sigma;

  471.         /** Constructor.
  472.          * @param rho first couple value
  473.          * @param sigma second couple value
  474.          */
  475.         private Couple(final int rho, final int sigma) {
  476.             this.rho = rho;
  477.             this.sigma = sigma;
  478.         }

  480.     }

  482.     /** Newcomb operator's key. */
  483.     private static class NewKey {

  485.         /** n value. */
  486.         private final int n;

  488.         /** s value. */
  489.         private final int s;

  491.         /** ρ value. */
  492.         private final int rho;

  494.         /** σ value. */
  495.         private final int sigma;

  497.         /** Simpleconstructor.
  498.          * @param n n
  499.          * @param s s
  500.          * @param rho ρ
  501.          * @param sigma σ
  502.          */
  503.         NewKey(final int n, final int s, final int rho, final int sigma) {
  504.             this.n = n;
  505.             this.s = s;
  506.             this.rho = rho;
  507.             this.sigma = sigma;
  508.         }

  510.     }

  512. }
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