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1   /*
2    * Licensed to the Hipparchus project under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The Hipparchus project 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    *      https://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.hipparchus.special.elliptic.carlson;
18  
19  import org.hipparchus.CalculusFieldElement;
20  import org.hipparchus.complex.Complex;
21  import org.hipparchus.complex.FieldComplex;
22  import org.hipparchus.util.FastMath;
23  
24  /** Duplication algorithm for Carlson R<sub>J</sub> elliptic integral.
25   * @param <T> type of the field elements (really {@link Complex} or {@link FieldComplex})
26   * @since 2.0
27   */
28  class RjFieldDuplication<T extends CalculusFieldElement<T>> extends FieldDuplication<T> {
29  
30      /** Delta product. */
31      private T delta;
32  
33      /** sₘ iteration parameter. */
34      private T sM;
35  
36      /** Simple constructor.
37       * @param x first symmetric variable of the integral
38       * @param y second symmetric variable of the integral
39       * @param z third symmetric variable of the integral
40       * @param p fourth <em>not</em> symmetric variable of the integral
41       * @param delta precomputed value of (p-x)(p-y)(p-z)
42       */
43      RjFieldDuplication(final T x, final T y, final T z, final T p, final T delta) {
44          super(x, y, z, p);
45          this.delta = delta;
46      }
47  
48      /** {@inheritDoc} */
49      @Override
50      protected void initialMeanPoint(final T[] va) {
51          va[4] = va[0].add(va[1]).add(va[2]).add(va[3].multiply(2)).divide(5.0);
52      }
53  
54      /** {@inheritDoc} */
55      @Override
56      protected T convergenceCriterion(final T r, final T max) {
57          return max.divide(FastMath.sqrt(FastMath.sqrt(FastMath.sqrt(r.multiply(0.25)))));
58      }
59  
60      /** {@inheritDoc} */
61      @Override
62      protected void update(final int m, final T[] vaM, final T[] sqrtM, final  double fourM) {
63          final T dM =          sqrtM[3].add(sqrtM[0]).
64                       multiply(sqrtM[3].add(sqrtM[1])).
65                       multiply(sqrtM[3].add(sqrtM[2]));
66          if (m == 0) {
67              sM = dM.multiply(0.5);
68          } else {
69              // equation A.3 in Carlson[2000]
70              final T rM = sM.multiply(delta.divide(sM.multiply(sM).multiply(fourM)).add(1.0).sqrt().add(1.0));
71              sM = dM.multiply(rM).subtract(delta.divide(fourM * fourM)).
72                   divide(dM.add(rM.divide(fourM)).multiply(2));
73          }
74  
75          // equation 2.19 in Carlson[1995]
76          final T lambdaA = sqrtM[0].multiply(sqrtM[1]);
77          final T lambdaB = sqrtM[0].multiply(sqrtM[2]);
78          final T lambdaC = sqrtM[1].multiply(sqrtM[2]);
79  
80          // equations 2.19 and 2.20 in Carlson[1995]
81          vaM[0] = vaM[0].linearCombination(0.25, vaM[0], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // xₘ
82          vaM[1] = vaM[1].linearCombination(0.25, vaM[1], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // yₘ
83          vaM[2] = vaM[2].linearCombination(0.25, vaM[2], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // zₘ
84          vaM[3] = vaM[3].linearCombination(0.25, vaM[3], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // pₘ
85          vaM[4] = vaM[4].linearCombination(0.25, vaM[4], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // aₘ
86  
87      }
88  
89      /** {@inheritDoc} */
90      @Override
91      protected T evaluate(final T[] va0, final T aM, final  double fourM) {
92  
93          // compute symmetric differences
94          final T inv    = aM.multiply(fourM).reciprocal();
95          final T bigX   = va0[4].subtract(va0[0]).multiply(inv);
96          final T bigY   = va0[4].subtract(va0[1]).multiply(inv);
97          final T bigZ   = va0[4].subtract(va0[2]).multiply(inv);
98          final T bigP   = bigX.add(bigY).add(bigZ).multiply(-0.5);
99          final T bigP2  = bigP.multiply(bigP);
100 
101         // compute elementary symmetric functions (we already know e1 = 0 by construction)
102         final T xyz    = bigX.multiply(bigY).multiply(bigZ);
103         final T e2     = bigX.multiply(bigY.add(bigZ)).add(bigY.multiply(bigZ)).
104                                subtract(bigP.multiply(bigP).multiply(3));
105         final T e3     = xyz.add(bigP.multiply(2).multiply(e2.add(bigP2.multiply(2))));
106         final T e4     = xyz.multiply(2).add(bigP.multiply(e2.add(bigP2.multiply(3)))).multiply(bigP);
107         final T e5     = xyz.multiply(bigP2);
108 
109         final T e2e2   = e2.multiply(e2);
110         final T e2e3   = e2.multiply(e3);
111         final T e2e4   = e2.multiply(e4);
112         final T e2e5   = e2.multiply(e5);
113         final T e3e3   = e3.multiply(e3);
114         final T e3e4   = e3.multiply(e4);
115         final T e2e2e2 = e2e2.multiply(e2);
116         final T e2e2e3 = e2e2.multiply(e3);
117 
118         // evaluate integral using equation 19.36.1 in DLMF
119         // (which add more terms than equation 2.7 in Carlson[1995])
120         final T poly = e3e4.add(e2e5).multiply(RdRealDuplication.E3_E4_P_E2_E5).
121                        add(e2e2e3.multiply(RdRealDuplication.E2_E2_E3)).
122                        add(e2e4.multiply(RdRealDuplication.E2_E4)).
123                        add(e3e3.multiply(RdRealDuplication.E3_E3)).
124                        add(e2e2e2.multiply(RdRealDuplication.E2_E2_E2)).
125                        add(e5.multiply(RdRealDuplication.E5)).
126                        add(e2e3.multiply(RdRealDuplication.E2_E3)).
127                        add(e4.multiply(RdRealDuplication.E4)).
128                        add(e2e2.multiply(RdRealDuplication.E2_E2)).
129                        add(e3.multiply(RdRealDuplication.E3)).
130                        add(e2.multiply(RdRealDuplication.E2)).
131                        add(RdRealDuplication.CONSTANT).
132                        divide(RdRealDuplication.DENOMINATOR);
133         final T polyTerm = poly.divide(aM.multiply(FastMath.sqrt(aM)).multiply(fourM));
134 
135         // compute a single R_C term
136         final T rcTerm = new RcFieldDuplication<>(poly.getField().getOne(), delta.divide(sM.multiply(sM).multiply(fourM)).add(1)).
137                          integral().
138                          multiply(3).divide(sM);
139 
140         return polyTerm.add(rcTerm);
141 
142     }
143 
144 }