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.square().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 }