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1   /*
2    * Licensed to the Apache Software Foundation (ASF) 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 ASF 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  
18  /*
19   * This is not the original file distributed by the Apache Software Foundation
20   * It has been modified by the Hipparchus project
21   */
22  package org.hipparchus.analysis.integration.gauss;
23  
24  import org.hipparchus.exception.MathIllegalArgumentException;
25  import org.hipparchus.util.FastMath;
26  import org.hipparchus.util.Pair;
27  
28  /**
29   * Factory that creates a
30   * <a href="http://en.wikipedia.org/wiki/Gauss-Hermite_quadrature">
31   * Gauss-type quadrature rule using Hermite polynomials</a>
32   * of the first kind.
33   * Such a quadrature rule allows the calculation of improper integrals
34   * of a function
35   * <p>
36   *  \(f(x) e^{-x^2}\)
37   * </p>
38   * <p>
39   * Recurrence relation and weights computation follow
40   * <a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun">
41   * Abramowitz and Stegun, 1964</a>.
42   * </p>
43   *
44   */
45  public class HermiteRuleFactory extends AbstractRuleFactory {
46  
47      /** √π. */
48      private static final double SQRT_PI = 1.77245385090551602729;
49  
50      /** Empty constructor.
51       * <p>
52       * This constructor is not strictly necessary, but it prevents spurious
53       * javadoc warnings with JDK 18 and later.
54       * </p>
55       * @since 3.0
56       */
57      public HermiteRuleFactory() { // NOPMD - unnecessary constructor added intentionally to make javadoc happy
58          // nothing to do
59      }
60  
61      /** {@inheritDoc} */
62      @Override
63      protected Pair<double[], double[]> computeRule(int numberOfPoints)
64          throws MathIllegalArgumentException {
65  
66          if (numberOfPoints == 1) {
67              // Break recursion.
68              return new Pair<>(new double[] { 0 } , new double[] { SQRT_PI });
69          }
70  
71          // find nodes as roots of Hermite polynomial
72          final double[] points = findRoots(numberOfPoints, new Hermite(numberOfPoints)::ratio);
73          enforceSymmetry(points);
74  
75          // compute weights
76          final double[] weights = new double[numberOfPoints];
77          final Hermite hm1 = new Hermite(numberOfPoints - 1);
78          for (int i = 0; i < numberOfPoints; i++) {
79              final double y = hm1.hNhNm1(points[i])[0];
80              weights[i] = SQRT_PI / (numberOfPoints * y * y);
81          }
82  
83          return new Pair<>(points, weights);
84  
85      }
86  
87      /** Hermite polynomial, normalized to avoid overflow.
88       * <p>
89       * The regular Hermite polynomials and associated weights are given by:
90       *   <pre>
91       *     H₀(x)   = 1
92       *     H₁(x)   = 2 x
93       *     Hₙ₊₁(x) = 2x Hₙ(x) - 2n Hₙ₋₁(x), and H'ₙ(x) = 2n Hₙ₋₁(x)
94       *     wₙ(xᵢ) = [2ⁿ⁻¹ n! √π]/[n Hₙ₋₁(xᵢ)]²
95       *   </pre>
96       * </p>
97       * <p>
98       * In order to avoid overflow with normalize the polynomials hₙ(x) = Hₙ(x) / √[2ⁿ n!]
99       * so the recurrence relations and weights become:
100      *   <pre>
101      *     h₀(x)   = 1
102      *     h₁(x)   = √2 x
103      *     hₙ₊₁(x) = [√2 x hₙ(x) - √n hₙ₋₁(x)]/√(n+1), and h'ₙ(x) = 2n hₙ₋₁(x)
104      *     uₙ(xᵢ) = √π/[n Nₙ₋₁(xᵢ)²]
105      *   </pre>
106      * </p>
107      */
108     private static class Hermite {
109 
110         /** √2. */
111         private static final double SQRT2 = FastMath.sqrt(2);
112 
113         /** Degree. */
114         private final int degree;
115 
116         /** Simple constructor.
117          * @param degree polynomial degree
118          */
119         Hermite(int degree) {
120             this.degree = degree;
121         }
122 
123         /** Compute ratio H(x)/H'(x).
124          * @param x point at which ratio must be computed
125          * @return ratio H(x)/H'(x)
126          */
127         public double ratio(double x) {
128             double[] h = hNhNm1(x);
129             return h[0] / (h[1] * 2 * degree);
130         }
131 
132         /** Compute Nₙ(x) and Nₙ₋₁(x).
133          * @param x point at which polynomials are evaluated
134          * @return array containing Nₙ(x) at index 0 and Nₙ₋₁(x) at index 1
135          */
136         private double[] hNhNm1(final double x) {
137             double[] h = { SQRT2 * x, 1 };
138             double sqrtN = 1;
139             for (int n = 1; n < degree; n++) {
140                 // apply recurrence relation hₙ₊₁(x) = [√2 x hₙ(x) - √n hₙ₋₁(x)]/√(n+1)
141                 final double sqrtNp = FastMath.sqrt(n + 1);
142                 final double hp = (h[0] * x * SQRT2 - h[1] * sqrtN) / sqrtNp;
143                 h[1]  = h[0];
144                 h[0]  = hp;
145                 sqrtN = sqrtNp;
146             }
147             return h;
148         }
149 
150     }
151 
152 }