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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.analysis.integration.gauss;
18  
19  import org.hipparchus.CalculusFieldElement;
20  import org.hipparchus.Field;
21  import org.hipparchus.exception.MathIllegalArgumentException;
22  import org.hipparchus.util.MathArrays;
23  import org.hipparchus.util.Pair;
24  
25  /**
26   * Factory that creates Gauss-type quadrature rule using Laguerre polynomials.
27   *
28   * @see <a href="http://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature">Gauss-Laguerre quadrature (Wikipedia)</a>
29   * @param <T> Type of the number used to represent the points and weights of
30   * the quadrature rules.
31   * @since 2.0
32   */
33  public class FieldLaguerreRuleFactory<T extends CalculusFieldElement<T>> extends FieldAbstractRuleFactory<T> {
34  
35      /** Simple constructor
36       * @param field field to which rule coefficients belong
37       */
38      public FieldLaguerreRuleFactory(final Field<T> field) {
39          super(field);
40      }
41  
42      /** {@inheritDoc} */
43      @Override
44      public Pair<T[], T[]> computeRule(int numberOfPoints)
45          throws MathIllegalArgumentException {
46  
47          final Field<T> field = getField();
48  
49          // find nodes as roots of Laguerre polynomial
50          final Laguerre<T> p      =  new Laguerre<>(numberOfPoints);
51          final T[]      points = findRoots(numberOfPoints, p::ratio);
52  
53          // compute weights
54          final T[] weights = MathArrays.buildArray(field, numberOfPoints);
55          final int      n1         = numberOfPoints + 1;
56          final long     n1Squared  = n1 * (long) n1;
57          final Laguerre<T> laguerreN1 = new Laguerre<>(n1);
58          for (int i = 0; i < numberOfPoints; i++) {
59              final T y = laguerreN1.value(points[i]);
60              weights[i] = points[i].divide(y.multiply(y).multiply(n1Squared));
61          }
62  
63          return new Pair<>(points, weights);
64  
65      }
66  
67      /** Laguerre polynomial.
68       * @param <T> Type of the field elements.
69       */
70      private static class Laguerre<T extends CalculusFieldElement<T>> {
71  
72          /** Degree. */
73          private int degree;
74  
75          /** Simple constructor.
76           * @param degree polynomial degree
77           */
78          Laguerre(int degree) {
79              this.degree = degree;
80          }
81  
82          /** Evaluate polynomial.
83           * @param x point at which polynomial must be evaluated
84           * @return value of the polynomial
85           */
86          public T value(final T x) {
87              return lNlNm1(x)[0];
88          }
89  
90          /** Compute ratio L(x)/L'(x).
91           * @param x point at which ratio must be computed
92           * @return ratio L(x)/L'(x)
93           */
94          public T ratio(T x) {
95              T[] l = lNlNm1(x);
96              return x.multiply(l[0]).divide(l[0].subtract(l[1]).multiply(degree));
97          }
98  
99          /** Compute Lₙ(x) and Lₙ₋₁(x).
100          * @param x point at which polynomials are evaluated
101          * @return array containing Lₙ(x) at index 0 and Lₙ₋₁(x) at index 1
102          */
103         private T[] lNlNm1(final T x) {
104             T[] l = MathArrays.buildArray(x.getField(), 2);
105             l[0] = x.subtract(1).negate();
106             l[1] = x.getField().getOne();
107             for (int n = 1; n < degree; n++) {
108                 // apply recurrence relation (n+1) Lₙ₊₁(x) = (2n + 1 - x) Lₙ(x) - n Lₙ₋₁(x)
109                 final T lp = l[0].multiply(x.negate().add(2 * n + 1)).subtract(l[1].multiply(n)).divide(n + 1);
110                 l[1] = l[0];
111                 l[0] = lp;
112             }
113             return l;
114         }
115 
116     }
117 
118 }