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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 a
27   * <a href="http://en.wikipedia.org/wiki/Gauss-Hermite_quadrature">
28   * Gauss-type quadrature rule using Hermite polynomials</a>
29   * of the first kind.
30   * Such a quadrature rule allows the calculation of improper integrals
31   * of a function
32   * <p>
33   *  \(f(x) e^{-x^2}\)
34   * </p><p>
35   * Recurrence relation and weights computation follow
36   * <a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun">
37   * Abramowitz and Stegun, 1964</a>.
38   * </p><p>
39   * The coefficients of the standard Hermite polynomials grow very rapidly.
40   * In order to avoid overflows, each Hermite polynomial is normalized with
41   * respect to the underlying scalar product.
42   * @param <T> Type of the number used to represent the points and weights of
43   * the quadrature rules.
44   * @since 2.0
45   */
46  public class FieldHermiteRuleFactory<T extends CalculusFieldElement<T>> extends FieldAbstractRuleFactory<T> {
47  
48      /** Simple constructor
49       * @param field field to which rule coefficients belong
50       */
51      public FieldHermiteRuleFactory(final Field<T> field) {
52          super(field);
53      }
54  
55      /** {@inheritDoc} */
56      @Override
57      protected Pair<T[], T[]> computeRule(int numberOfPoints)
58          throws MathIllegalArgumentException {
59  
60          final Field<T> field  = getField();
61          final T        sqrtPi = field.getZero().getPi().sqrt();
62  
63          if (numberOfPoints == 1) {
64              // Break recursion.
65              final T[] points  = MathArrays.buildArray(field, numberOfPoints);
66              final T[] weights = MathArrays.buildArray(field, numberOfPoints);
67              points[0]  = field.getZero();
68              weights[0] = sqrtPi;
69              return new Pair<>(points, weights);
70          }
71  
72          // find nodes as roots of Hermite polynomial
73          final T[] points = findRoots(numberOfPoints, new Hermite<>(field, numberOfPoints)::ratio);
74          enforceSymmetry(points);
75  
76          // compute weights
77          final T[] weights = MathArrays.buildArray(field, numberOfPoints);
78          final Hermite<T> hm1 = new Hermite<>(field, numberOfPoints - 1);
79          for (int i = 0; i < numberOfPoints; i++) {
80              final T y = hm1.hNhNm1(points[i])[0];
81              weights[i] = sqrtPi.divide(y.multiply(y).multiply(numberOfPoints));
82          }
83  
84          return new Pair<>(points, weights);
85  
86      }
87  
88      /** Hermite polynomial, normalized to avoid overflow.
89       * <p>
90       * The regular Hermite polynomials and associated weights are given by:
91       *   <pre>
92       *     H₀(x)   = 1
93       *     H₁(x)   = 2 x
94       *     Hₙ₊₁(x) = 2x Hₙ(x) - 2n Hₙ₋₁(x), and H'ₙ(x) = 2n Hₙ₋₁(x)
95       *     wₙ(xᵢ) = [2ⁿ⁻¹ n! √π]/[n Hₙ₋₁(xᵢ)]²
96       *   </pre>
97       * </p>
98       * <p>
99       * In order to avoid overflow with normalize the polynomials hₙ(x) = Hₙ(x) / √[2ⁿ n!]
100      * so the recurrence relations and weights become:
101      *   <pre>
102      *     h₀(x)   = 1
103      *     h₁(x)   = √2 x
104      *     hₙ₊₁(x) = [√2 x hₙ(x) - √n hₙ₋₁(x)]/√(n+1), and h'ₙ(x) = 2n hₙ₋₁(x)
105      *     uₙ(xᵢ) = √π/[n Nₙ₋₁(xᵢ)²]
106      *   </pre>
107      * </p>
108      * @param <T> Type of the field elements.
109      */
110     private static class Hermite<T extends CalculusFieldElement<T>> {
111 
112         /** √2. */
113         private final T sqrt2;
114 
115         /** Degree. */
116         private final int degree;
117 
118         /** Simple constructor.
119          * @param field field to which rule coefficients belong
120          * @param degree polynomial degree
121          */
122         Hermite(Field<T> field, int degree) {
123             this.sqrt2  = field.getZero().newInstance(2).sqrt();
124             this.degree = degree;
125         }
126 
127         /** Compute ratio H(x)/H'(x).
128          * @param x point at which ratio must be computed
129          * @return ratio H(x)/H'(x)
130          */
131         public T ratio(T x) {
132             T[] h = hNhNm1(x);
133             return h[0].divide(h[1].multiply(2 * degree));
134         }
135 
136         /** Compute Nₙ(x) and Nₙ₋₁(x).
137          * @param x point at which polynomials are evaluated
138          * @return array containing Nₙ(x) at index 0 and Nₙ₋₁(x) at index 1
139          */
140         private T[] hNhNm1(final T x) {
141             T[] h = MathArrays.buildArray(x.getField(), 2);
142             h[0] = sqrt2.multiply(x);
143             h[1] = x.getField().getOne();
144             T sqrtN = x.getField().getOne();
145             for (int n = 1; n < degree; n++) {
146                 // apply recurrence relation hₙ₊₁(x) = [√2 x hₙ(x) - √n hₙ₋₁(x)]/√(n+1)
147                 final T sqrtNp = x.getField().getZero().newInstance(n + 1).sqrt();
148                 final T hp = (h[0].multiply(x).multiply(sqrt2).subtract(h[1].multiply(sqrtN))).divide(sqrtNp);
149                 h[1]  = h[0];
150                 h[0]  = hp;
151                 sqrtN = sqrtNp;
152             }
153             return h;
154         }
155 
156     }
157 
158 }