NewcombOperators.java

  1. /* Copyright 2002-2022 CS GROUP
  2.  * Licensed to CS GROUP (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.propagation.semianalytical.dsst.utilities;

  19. import java.util.ArrayList;
  20. import java.util.List;
  21. import java.util.Map;
  22. import java.util.SortedMap;
  23. import java.util.TreeMap;

  25. import org.hipparchus.analysis.polynomials.PolynomialFunction;
  26. import org.hipparchus.util.FastMath;

  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.                     if (k1.sigma < k2.sigma) {
  63.                         return  -1;
  64.                     } else if (k1.sigma == k2.sigma) {
  65.                         return 0;
  66.                     } else {
  67.                         return 1;
  68.                     }
  69.                 } else if (k1.rho < k2.rho) {
  70.                     return -1;
  71.                 } else {
  72.                     return 1;
  73.                 }
  74.             } else if (k1.s < k2.s) {
  75.                 return -1;
  76.             } else {
  77.                 return 1;
  78.             }
  79.         } else if (k1.n < k2.n) {
  80.             return -1;
  81.         }
  82.         return 1;
  83.     });

  85.     /** Private constructor as class is a utility.
  86.      */
  87.     private NewcombOperators() {
  88.     }

  90.     /** Get the Newcomb operator evaluated at n, s, ρ, σ.
  91.      * <p>
  92.      * This method is guaranteed to be thread-safe
  93.      * </p>
  94.      *  @param rho ρ index
  95.      *  @param sigma σ index
  96.      *  @param n n index
  97.      *  @param s s index
  98.      *  @return Y<sub>ρ,σ</sub><sup>n,s</sup>
  99.      */
  100.     public static double getValue(final int rho, final int sigma, final int n, final int s) {

  102.         final NewKey key = new NewKey(n, s, rho, sigma);
  103.         synchronized (MAP) {
  104.             if (MAP.containsKey(key)) {
  105.                 return MAP.get(key);
  106.             }
  107.         }

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

  112.         // Compute the value from the list of polynomials for the given n and s
  113.         double nPower = 1.;
  114.         double value = 0.0;
  115.         for (final PolynomialFunction polynomial : polynomials) {
  116.             value += polynomial.value(s) * nPower;
  117.             nPower = n * nPower;
  118.         }
  119.         synchronized (MAP) {
  120.             MAP.put(key, value);
  121.         }

  123.         return value;

  125.     }

  127.     /** Generator for Newcomb polynomials. */
  128.     private static class PolynomialsGenerator {

  130.         /** Polynomials storage. */
  131.         private static final SortedMap<Couple, List<PolynomialFunction>> POLYNOMIALS =
  132.                 new TreeMap<>((c1, c2) -> {
  133.                     if (c1.rho == c2.rho) {
  134.                         if (c1.sigma < c2.sigma) {
  135.                             return -1;
  136.                         } else if (c1.sigma == c2.sigma) {
  137.                             return 0;
  138.                         }
  139.                     } else if (c1.rho < c2.rho) {
  140.                         return -1;
  141.                     }
  142.                     return 1;
  143.                 });

  145.         /** Private constructor as class is a utility.
  146.          */
  147.         private PolynomialsGenerator() {
  148.         }

  150.         /** Get the list of polynomials representing the Newcomb Operator for the (ρ,σ) couple.
  151.          * <p>
  152.          * This method is guaranteed to be thread-safe
  153.          * </p>
  154.          *  @param rho ρ value
  155.          *  @param sigma σ value
  156.          *  @return Polynomials representing the Newcomb Operator for the (ρ,σ) couple.
  157.          */
  158.         private static List<PolynomialFunction> getPolynomials(final int rho, final int sigma) {

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

  162.             synchronized (POLYNOMIALS) {

  164.                 if (POLYNOMIALS.isEmpty()) {
  165.                     // Initialize lists
  166.                     final List<PolynomialFunction> l00 = new ArrayList<PolynomialFunction>();
  167.                     final List<PolynomialFunction> l01 = new ArrayList<PolynomialFunction>();
  168.                     final List<PolynomialFunction> l10 = new ArrayList<PolynomialFunction>();
  169.                     final List<PolynomialFunction> l11 = new ArrayList<PolynomialFunction>();

  171.                     // Y(rho = 0, sigma = 0) = 1
  172.                     l00.add(new PolynomialFunction(new double[] {
  173.                         1.
  174.                     }));
  175.                     // Y(rho = 0, sigma = 1) =  -s - n/2
  176.                     l01.add(new PolynomialFunction(new double[] {
  177.                         0, -1.
  178.                     }));
  179.                     l01.add(new PolynomialFunction(new double[] {
  180.                         -0.5
  181.                     }));
  182.                     // Y(rho = 1, sigma = 0) =  s - n/2
  183.                     l10.add(new PolynomialFunction(new double[] {
  184.                         0, 1.
  185.                     }));
  186.                     l10.add(new PolynomialFunction(new double[] {
  187.                         -0.5
  188.                     }));
  189.                     // Y(rho = 1, sigma = 1) = 3/2 - s² + 5n/4 + n²/4
  190.                     l11.add(new PolynomialFunction(new double[] {
  191.                         1.5, 0., -1.
  192.                     }));
  193.                     l11.add(new PolynomialFunction(new double[] {
  194.                         1.25
  195.                     }));
  196.                     l11.add(new PolynomialFunction(new double[] {
  197.                         0.25
  198.                     }));

  200.                     // Initialize polynomials
  201.                     POLYNOMIALS.put(new Couple(0, 0), l00);
  202.                     POLYNOMIALS.put(new Couple(0, 1), l01);
  203.                     POLYNOMIALS.put(new Couple(1, 0), l10);
  204.                     POLYNOMIALS.put(new Couple(1, 1), l11);

  206.                 }

  208.                 // If order hasn't been computed yet, update the Newcomb polynomials
  209.                 if (!POLYNOMIALS.containsKey(couple)) {
  210.                     PolynomialsGenerator.computeFor(rho, sigma);
  211.                 }

  213.                 return POLYNOMIALS.get(couple);

  215.             }
  216.         }

  218.         /** Compute the Modified Newcomb Operators up to a given (ρ, σ) couple.
  219.          *  <p>
  220.          *  The recursive computation uses equation 2.7.3-(12) of the Danielson paper.
  221.          *  </p>
  222.          *  @param rho ρ value to reach
  223.          *  @param sigma σ value to reach
  224.          */
  225.         private static void computeFor(final int rho, final int sigma) {

  227.             // Initialize result :
  228.             List<PolynomialFunction> result = new ArrayList<PolynomialFunction>();

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

  233.             // Compute (s - n) * Y[rho - 2, sigma][n, s + 2]
  234.             if (rho >= 2) {
  235.                 final List<PolynomialFunction> poly = map.get(0);
  236.                 final List<PolynomialFunction> list = getPolynomials(rho - 2, sigma);
  237.                 result = multiplyPolynomialList(poly, shiftList(list, 2));
  238.             }

  240.             // Compute 2(2rho + 2sigma + 2 + 3n) * Y[rho - 1, sigma - 1][n, s]
  241.             if (rho >= 1 && sigma >= 1) {
  242.                 final List<PolynomialFunction> poly = map.get(1);
  243.                 final List<PolynomialFunction> list = getPolynomials(rho - 1, sigma - 1);
  244.                 result = sumPolynomialList(result, multiplyPolynomialList(poly, list));
  245.             }

  247.             // Compute 2(2s - n) * Y[rho - 1, sigma][n, s + 1]
  248.             if (rho >= 1) {
  249.                 final List<PolynomialFunction> poly = map.get(2);
  250.                 final List<PolynomialFunction> list = getPolynomials(rho - 1, sigma);
  251.                 result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, 1)));
  252.             }

  254.             // Compute -(s + n) * Y[rho, sigma - 2][n, s - 2]
  255.             if (sigma >= 2) {
  256.                 final List<PolynomialFunction> poly = map.get(3);
  257.                 final List<PolynomialFunction> list = getPolynomials(rho, sigma - 2);
  258.                 result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, -2)));
  259.             }

  261.             // Compute -2(2s + n) * Y[rho, sigma - 1][n, s - 1]
  262.             if (sigma >= 1) {
  263.                 final List<PolynomialFunction> poly = map.get(4);
  264.                 final List<PolynomialFunction> list = getPolynomials(rho, sigma - 1);
  265.                 result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, -1)));
  266.             }

  268.             // Save polynomials for current (rho, sigma) couple
  269.             final Couple couple = new Couple(rho, sigma);
  270.             POLYNOMIALS.put(couple, result);
  271.         }

  273.         /** Multiply two lists of polynomials defined as the internal representation of the Newcomb Operator.
  274.          *  <p>
  275.          *  Let's call R<sub>s</sub>(n) the result returned by the method :
  276.          *  <pre>
  277.          *  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>)
  278.          *  </pre>
  279.          *
  280.          *  where the P<sub>s<sub>j</sub></sub> and Q<sub>s<sub>k</sub></sub> are polynomials in s,
  281.          *  s being the index of the Y<sub>ρ,σ</sub><sup>n,s</sup> function
  282.          *
  283.          *  @param poly1 first list of polynomials
  284.          *  @param poly2 second list of polynomials
  285.          *  @return R<sub>s</sub>(n) as a list of {@link PolynomialFunction}
  286.          */
  287.         private static List<PolynomialFunction> multiplyPolynomialList(final List<PolynomialFunction> poly1,
  288.                                                                        final List<PolynomialFunction> poly2) {
  289.             // Initialize the result list of polynomial function
  290.             final List<PolynomialFunction> result = new ArrayList<PolynomialFunction>();
  291.             initializeListOfPolynomials(poly1.size() + poly2.size() - 1, result);

  293.             int i = 0;
  294.             // Iterate over first polynomial list
  295.             for (PolynomialFunction f1 : poly1) {
  296.                 // Iterate over second polynomial list
  297.                 for (int j = i; j < poly2.size() + i; j++) {
  298.                     final PolynomialFunction p2 = poly2.get(j - i);
  299.                     // Get previous polynomial for current 'n' order
  300.                     final PolynomialFunction previousP2 = result.get(j);
  301.                     // Replace the current order by summing the product of both of the polynomials
  302.                     result.set(j, previousP2.add(f1.multiply(p2)));
  303.                 }
  304.                 // shift polynomial order in 'n'
  305.                 i++;
  306.             }
  307.             return result;
  308.         }

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

  324.             for (int i = 0; i < lowLength; i++) {
  325.                 // Add polynomials by increasing order of 'n'
  326.                 result.set(i, poly1.get(i).add(poly2.get(i)));
  327.             }
  328.             // Complete the list if lists are of different size:
  329.             for (int i = lowLength; i < highLength; i++) {
  330.                 if (poly1.size() < poly2.size()) {
  331.                     result.set(i, poly2.get(i));
  332.                 } else {
  333.                     result.set(i, poly1.get(i));
  334.                 }
  335.             }
  336.             return result;
  337.         }

  339.         /** Initialize an empty list of polynomials.
  340.          *  @param i order
  341.          *  @param result list into which polynomials should be added
  342.          */
  343.         private static void initializeListOfPolynomials(final int i,
  344.                                                         final List<PolynomialFunction> result) {
  345.             for (int k = 0; k < i; k++) {
  346.                 result.add(new PolynomialFunction(new double[] {0.}));
  347.             }
  348.         }

  350.         /** Shift a list of {@link PolynomialFunction}.
  351.          *
  352.          *  @param polynomialList list of {@link PolynomialFunction}
  353.          *  @param shift shift value
  354.          *  @return new list of shifted {@link PolynomialFunction}
  355.          */
  356.         private static List<PolynomialFunction> shiftList(final List<PolynomialFunction> polynomialList,
  357.                                                           final int shift) {
  358.             final List<PolynomialFunction> shiftedList = new ArrayList<PolynomialFunction>();
  359.             for (PolynomialFunction function : polynomialList) {
  360.                 shiftedList.add(new PolynomialFunction(shift(function.getCoefficients(), shift)));
  361.             }
  362.             return shiftedList;
  363.         }

  365.         /**
  366.          * Compute the coefficients of the polynomial \(P_s(x)\)
  367.          * whose values at point {@code x} will be the same as the those from the
  368.          * original polynomial \(P(x)\) when computed at {@code x + shift}.
  369.          * <p>
  370.          * More precisely, let \(\Delta = \) {@code shift} and let
  371.          * \(P_s(x) = P(x + \Delta)\).  The returned array
  372.          * consists of the coefficients of \(P_s\).  So if \(a_0, ..., a_{n-1}\)
  373.          * are the coefficients of \(P\), then the returned array
  374.          * \(b_0, ..., b_{n-1}\) satisfies the identity
  375.          * \(\sum_{i=0}^{n-1} b_i x^i = \sum_{i=0}^{n-1} a_i (x + \Delta)^i\) for all \(x\).
  376.          * </p>
  377.          * <p>
  378.          * This method is a modified version of the method with the same name
  379.          * in Hipparchus {@code PolynomialsUtils} class, simply changing
  380.          * computation of binomial coefficients so degrees higher than 66 can be used.
  381.          * </p>
  382.          *
  383.          * @param coefficients Coefficients of the original polynomial.
  384.          * @param shift Shift value.
  385.          * @return the coefficients \(b_i\) of the shifted
  386.          * polynomial.
  387.          */
  388.         public static double[] shift(final double[] coefficients,
  389.                                      final double shift) {
  390.             final int dp1 = coefficients.length;
  391.             final double[] newCoefficients = new double[dp1];

  393.             // Pascal triangle.
  394.             final double[][] coeff = new double[dp1][dp1];
  395.             coeff[0][0] = 1;
  396.             for (int i = 1; i < dp1; i++) {
  397.                 coeff[i][0] = 1;
  398.                 for (int j = 1; j < i; j++) {
  399.                     coeff[i][j] = coeff[i - 1][j - 1] + coeff[i - 1][j];
  400.                 }
  401.                 coeff[i][i] = 1;
  402.             }

  404.             // First polynomial coefficient.
  405.             double shiftI = 1;
  406.             for (int i = 0; i < dp1; i++) {
  407.                 newCoefficients[0] += coefficients[i] * shiftI;
  408.                 shiftI *= shift;
  409.             }

  411.             // Superior order.
  412.             final int d = dp1 - 1;
  413.             for (int i = 0; i < d; i++) {
  414.                 double shiftJmI = 1;
  415.                 for (int j = i; j < d; j++) {
  416.                     newCoefficients[i + 1] += coeff[j + 1][j - i] * coefficients[j + 1] * shiftJmI;
  417.                     shiftJmI *= shift;
  418.                 }
  419.             }

  421.             return newCoefficients;
  422.         }

  424.         /** Generate recurrence coefficients.
  425.          *
  426.          *  @param rho ρ value
  427.          *  @param sigma σ value
  428.          *  @return recurrence coefficients
  429.          */
  430.         private static Map<Integer, List<PolynomialFunction>> generateRecurrenceCoefficients(final int rho, final int sigma) {
  431.             final double den   = 1. / (4. * (rho + sigma));
  432.             final double denx2 = 2. * den;
  433.             final double denx4 = 4. * den;
  434.             // Initialization :
  435.             final Map<Integer, List<PolynomialFunction>> list = new TreeMap<Integer, List<PolynomialFunction>>();
  436.             final List<PolynomialFunction> poly0 = new ArrayList<PolynomialFunction>();
  437.             final List<PolynomialFunction> poly1 = new ArrayList<PolynomialFunction>();
  438.             final List<PolynomialFunction> poly2 = new ArrayList<PolynomialFunction>();
  439.             final List<PolynomialFunction> poly3 = new ArrayList<PolynomialFunction>();
  440.             final List<PolynomialFunction> poly4 = new ArrayList<PolynomialFunction>();
  441.             // (s - n)
  442.             poly0.add(new PolynomialFunction(new double[] {0., den}));
  443.             poly0.add(new PolynomialFunction(new double[] {-den}));
  444.             // 2(2 * rho + 2 * sigma + 2 + 3*n)
  445.             poly1.add(new PolynomialFunction(new double[] {1. + denx4}));
  446.             poly1.add(new PolynomialFunction(new double[] {denx2 + denx4}));
  447.             // 2(2s - n)
  448.             poly2.add(new PolynomialFunction(new double[] {0., denx4}));
  449.             poly2.add(new PolynomialFunction(new double[] {-denx2}));
  450.             // - (s + n)
  451.             poly3.add(new PolynomialFunction(new double[] {0., -den}));
  452.             poly3.add(new PolynomialFunction(new double[] {-den}));
  453.             // - 2(2s + n)
  454.             poly4.add(new PolynomialFunction(new double[] {0., -denx4}));
  455.             poly4.add(new PolynomialFunction(new double[] {-denx2}));

  457.             // Fill the map :
  458.             list.put(0, poly0);
  459.             list.put(1, poly1);
  460.             list.put(2, poly2);
  461.             list.put(3, poly3);
  462.             list.put(4, poly4);
  463.             return list;
  464.         }

  466.     }

  468.     /** Private class to define a couple of value. */
  469.     private static class Couple {

  471.         /** first couple value. */
  472.         private final int rho;

  474.         /** second couple value. */
  475.         private final int sigma;

  477.         /** Constructor.
  478.          * @param rho first couple value
  479.          * @param sigma second couple value
  480.          */
  481.         private Couple(final int rho, final int sigma) {
  482.             this.rho = rho;
  483.             this.sigma = sigma;
  484.         }

  486.     }

  488.     /** Newcomb operator's key. */
  489.     private static class NewKey {

  491.         /** n value. */
  492.         private final int n;

  494.         /** s value. */
  495.         private final int s;

  497.         /** ρ value. */
  498.         private final int rho;

  500.         /** σ value. */
  501.         private final int sigma;

  503.         /** Simpleconstructor.
  504.          * @param n n
  505.          * @param s s
  506.          * @param rho ρ
  507.          * @param sigma σ
  508.          */
  509.         NewKey(final int n, final int s, final int rho, final int sigma) {
  510.             this.n = n;
  511.             this.s = s;
  512.             this.rho = rho;
  513.             this.sigma = sigma;
  514.         }

  516.     }

  518. }
Created with JaCoCo 0.8.8.202204050719