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.Pair;
26
27 /**
28 * Factory that creates Gauss-type quadrature rule using Legendre polynomials.
29 * In this implementation, the lower and upper bounds of the natural interval
30 * of integration are -1 and 1, respectively.
31 * The Legendre polynomials are evaluated using the recurrence relation
32 * presented in <a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun">
33 * Abramowitz and Stegun, 1964</a>.
34 *
35 */
36 public class LegendreRuleFactory extends AbstractRuleFactory {
37
38 /** Empty constructor.
39 * <p>
40 * This constructor is not strictly necessary, but it prevents spurious
41 * javadoc warnings with JDK 18 and later.
42 * </p>
43 * @since 3.0
44 */
45 public LegendreRuleFactory() { // NOPMD - unnecessary constructor added intentionally to make javadoc happy
46 // nothing to do
47 }
48
49 /** {@inheritDoc} */
50 @Override
51 protected Pair<double[], double[]> computeRule(int numberOfPoints)
52 throws MathIllegalArgumentException {
53
54 if (numberOfPoints == 1) {
55 // Break recursion.
56 return new Pair<>(new double[] { 0 } , new double[] { 2 });
57 }
58
59 // find nodes as roots of Legendre polynomial
60 final Legendre p = new Legendre(numberOfPoints);
61 final double[] points = findRoots(numberOfPoints, p::ratio);
62 enforceSymmetry(points);
63
64 // compute weights
65 final double[] weights = new double[numberOfPoints];
66 for (int i = 0; i <= numberOfPoints / 2; i++) {
67 final double c = points[i];
68 final double[] pKpKm1 = p.pNpNm1(c);
69 final double d = numberOfPoints * (pKpKm1[1] - c * pKpKm1[0]);
70 weights[i] = 2 * (1 - c * c) / (d * d);
71
72 // symmetrical point
73 final int idx = numberOfPoints - i - 1;
74 weights[idx] = weights[i];
75
76 }
77
78 return new Pair<>(points, weights);
79
80 }
81
82 /** Legendre polynomial. */
83 private static class Legendre {
84
85 /** Degree. */
86 private int degree;
87
88 /** Simple constructor.
89 * @param degree polynomial degree
90 */
91 Legendre(int degree) {
92 this.degree = degree;
93 }
94
95 /** Compute ratio P(x)/P'(x).
96 * @param x point at which ratio must be computed
97 * @return ratio P(x)/P'(x)
98 */
99 public double ratio(double x) {
100 double pm = 1;
101 double p = x;
102 double d = 1;
103 for (int n = 1; n < degree; n++) {
104 // apply recurrence relations (n+1) Pₙ₊₁(x) = (2n+1) x Pₙ(x) - n Pₙ₋₁(x)
105 // and P'ₙ₊₁(x) = (n+1) Pₙ(x) + x P'ₙ(x)
106 final double pp = (p * (x * (2 * n + 1)) - pm * n) / (n + 1);
107 d = p * (n + 1) + d * x;
108 pm = p;
109 p = pp;
110 }
111 return p / d;
112 }
113
114 /** Compute Pₙ(x) and Pₙ₋₁(x).
115 * @param x point at which polynomials are evaluated
116 * @return array containing Pₙ(x) at index 0 and Pₙ₋₁(x) at index 1
117 */
118 private double[] pNpNm1(final double x) {
119 double[] p = { x, 1 };
120 for (int n = 1; n < degree; n++) {
121 // apply recurrence relation (n+1) Pₙ₊₁(x) = (2n+1) x Pₙ(x) - n Pₙ₋₁(x)
122 final double pp = (p[0] * (x * (2 * n + 1)) - p[1] * n) / (n + 1);
123 p[1] = p[0];
124 p[0] = pp;
125 }
126 return p;
127 }
128
129 }
130
131 }