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    * Licensed to CS GROUP (CS) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
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    *
    *   http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
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   * See the License for the specific language governing permissions and
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  package org.orekit.forces.gravity;
  
  import org.hipparchus.analysis.differentiation.DerivativeStructure;
  import org.hipparchus.dfp.Dfp;
  import org.hipparchus.dfp.DfpField;
  
  import java.util.ArrayList;
  import java.util.HashMap;
  import java.util.List;
  import java.util.Map;
  
  /** Implementation of associated Legendre functions from defining formulas.
   * <p>
   * This implementation is for test purposes only! It is limited to low degrees
   * and order and is slow.
   * </p>
   * @author Luc Maisonobe
   */
  class AssociatedLegendreFunction {
  
      static final Map<Integer, List<Dfp[]>> LEGENDRE_POLYNOMIALS = new HashMap<Integer, List<Dfp[]>>();
      final int m;
      final Dfp[] polynomial;
      final Dfp normalization;
  
      private Dfp[] getLegendrePolynomial(int n, DfpField dfpField) {
  
          // get (or create) the list of polynomials for the specified field
          List<Dfp[]> list = LEGENDRE_POLYNOMIALS.get(dfpField.getRadixDigits());
          if (list == null) {
              list = new ArrayList<Dfp[]>();
              list.add(new Dfp[] {
                  dfpField.getOne()                     // P0(X) = 1
              });
              list.add(new Dfp[] {
                  dfpField.getZero(), dfpField.getOne() // P1(X) = 0 + 1 * X
              });
          }
  
          while (list.size() <= n) {
  
              // build polynomial Pk+1 using recursion formula
              // (k+1) P<sub>k+1</sub>(X) = (2k+1) X P<sub>k</sub>(X) - k P<sub>k-1</sub>(X)
              int   k       = list.size() - 1;
              Dfp   kDfp    = dfpField.newDfp(k);
              Dfp   kP1Dfp  = kDfp.add(dfpField.getOne());
              Dfp   ckDfp   = kP1Dfp.add(kDfp).divide(kP1Dfp);
              Dfp[] pk      = list.get(k);
              Dfp   ckM1Dfp = kDfp.divide(kP1Dfp).negate();
              Dfp[] pkM1    = list.get(k - 1);
  
              Dfp[] pkP1    = new Dfp[k + 2];
              pkP1[0] = ckM1Dfp.multiply(pkM1[0]);
              for (int i = 0; i < k; ++i) {
                  if ((k - i) % 2 == 1) {
                      pkP1[i + 1] = dfpField.getZero();
                  } else {
                      pkP1[i + 1] = ckDfp.multiply(pk[i]).add(ckM1Dfp.multiply(pkM1[i + 1]));
                  }
              }
              pkP1[k + 1] = ckDfp.multiply(pk[k]);
  
              list.add(pkP1);
  
          }
  
          // retrieve degree n polynomial
          return list.get(n);
  
      }
  
      private Dfp[] differentiateLegendrePolynomial(Dfp[] p, int m, DfpField dfpField) {
          Dfp[] dp = new Dfp[p.length - m];
          for (int i = 0; i < dp.length; ++i) {
              dp[i] = p[i + m];
              for (int j = m; j > 0; --j) {
                  dp[i] = dp[i].multiply(dfpField.newDfp(i + j));
              }
          }
          return dp;
      }
  
      public AssociatedLegendreFunction(boolean normalized, int n, int m, DfpField dfpField) {
 
         this.m = m;
 
         // store mth derivative of the degree n Legendre polynomial
         polynomial = differentiateLegendrePolynomial(getLegendrePolynomial(n, dfpField), m, dfpField);
 
         if (normalized) {
             // compute normalization coefficient
             Dfp c = dfpField.newDfp(((m == 0) ? 1.0 : 2.0) * (2 * n + 1));
             for (int k = 0; k < m; ++k) {
                 c = c.divide(dfpField.newDfp((n + 1 + k) * (n - k)));
             }
             this.normalization = c.sqrt();
         } else {
             this.normalization = dfpField.getOne();
         }
 
     }
 
     public DerivativeStructure value(DerivativeStructure t) {
         DerivativeStructure y1 = t.getField().getOne().multiply(polynomial[polynomial.length - 1].toDouble());
         for (int j = polynomial.length - 2; j >= 0; j--) {
             y1 = y1.multiply(t).add(polynomial[j].toDouble());
         }
         DerivativeStructure oneMinusT2 = t.getField().getOne().subtract(t.multiply(t));
         DerivativeStructure y2 = oneMinusT2.pow(m).sqrt();
         return y1.multiply(y2).multiply(normalization.toDouble());
     }
 
     public Dfp value(Dfp t) {
         Dfp y1 = polynomial[polynomial.length - 1];
         for (int j = polynomial.length - 2; j >= 0; j--) {
             y1 = y1.multiply(t).add(polynomial[j]);
         }
         Dfp oneMinusT2 = t.getField().getOne().subtract(t.multiply(t));
         Dfp y2 = t.getField().getOne();
         for (int j = 0; j < m; ++j) {
             y2 = y2.multiply(oneMinusT2);
         }
         y2 = y2.sqrt();
         return y1.multiply(y2).multiply(normalization);
     }
 
 }