001 /*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements. See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License. You may obtain a copy of the License at
008 *
009 * http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017 package org.apache.commons.math3.analysis.integration;
018
019 import org.apache.commons.math3.exception.MathIllegalArgumentException;
020 import org.apache.commons.math3.exception.MaxCountExceededException;
021 import org.apache.commons.math3.exception.NotStrictlyPositiveException;
022 import org.apache.commons.math3.exception.NumberIsTooSmallException;
023 import org.apache.commons.math3.exception.TooManyEvaluationsException;
024 import org.apache.commons.math3.exception.util.LocalizedFormats;
025 import org.apache.commons.math3.util.FastMath;
026
027 /**
028 * Implements the <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html">
029 * Legendre-Gauss</a> quadrature formula.
030 * <p>
031 * Legendre-Gauss integrators are efficient integrators that can
032 * accurately integrate functions with few function evaluations. A
033 * Legendre-Gauss integrator using an n-points quadrature formula can
034 * integrate 2n-1 degree polynomials exactly.
035 * </p>
036 * <p>
037 * These integrators evaluate the function on n carefully chosen
038 * abscissas in each step interval (mapped to the canonical [-1,1] interval).
039 * The evaluation abscissas are not evenly spaced and none of them are
040 * at the interval endpoints. This implies the function integrated can be
041 * undefined at integration interval endpoints.
042 * </p>
043 * <p>
044 * The evaluation abscissas x<sub>i</sub> are the roots of the degree n
045 * Legendre polynomial. The weights a<sub>i</sub> of the quadrature formula
046 * integrals from -1 to +1 ∫ Li<sup>2</sup> where Li (x) =
047 * ∏ (x-x<sub>k</sub>)/(x<sub>i</sub>-x<sub>k</sub>) for k != i.
048 * </p>
049 * <p>
050 * @version $Id: LegendreGaussIntegrator.java 1455194 2013-03-11 15:45:54Z luc $
051 * @since 1.2
052 * @deprecated As of 3.1 (to be removed in 4.0). Please use
053 * {@link IterativeLegendreGaussIntegrator} instead.
054 */
055 @Deprecated
056 public class LegendreGaussIntegrator extends BaseAbstractUnivariateIntegrator {
057
058 /** Abscissas for the 2 points method. */
059 private static final double[] ABSCISSAS_2 = {
060 -1.0 / FastMath.sqrt(3.0),
061 1.0 / FastMath.sqrt(3.0)
062 };
063
064 /** Weights for the 2 points method. */
065 private static final double[] WEIGHTS_2 = {
066 1.0,
067 1.0
068 };
069
070 /** Abscissas for the 3 points method. */
071 private static final double[] ABSCISSAS_3 = {
072 -FastMath.sqrt(0.6),
073 0.0,
074 FastMath.sqrt(0.6)
075 };
076
077 /** Weights for the 3 points method. */
078 private static final double[] WEIGHTS_3 = {
079 5.0 / 9.0,
080 8.0 / 9.0,
081 5.0 / 9.0
082 };
083
084 /** Abscissas for the 4 points method. */
085 private static final double[] ABSCISSAS_4 = {
086 -FastMath.sqrt((15.0 + 2.0 * FastMath.sqrt(30.0)) / 35.0),
087 -FastMath.sqrt((15.0 - 2.0 * FastMath.sqrt(30.0)) / 35.0),
088 FastMath.sqrt((15.0 - 2.0 * FastMath.sqrt(30.0)) / 35.0),
089 FastMath.sqrt((15.0 + 2.0 * FastMath.sqrt(30.0)) / 35.0)
090 };
091
092 /** Weights for the 4 points method. */
093 private static final double[] WEIGHTS_4 = {
094 (90.0 - 5.0 * FastMath.sqrt(30.0)) / 180.0,
095 (90.0 + 5.0 * FastMath.sqrt(30.0)) / 180.0,
096 (90.0 + 5.0 * FastMath.sqrt(30.0)) / 180.0,
097 (90.0 - 5.0 * FastMath.sqrt(30.0)) / 180.0
098 };
099
100 /** Abscissas for the 5 points method. */
101 private static final double[] ABSCISSAS_5 = {
102 -FastMath.sqrt((35.0 + 2.0 * FastMath.sqrt(70.0)) / 63.0),
103 -FastMath.sqrt((35.0 - 2.0 * FastMath.sqrt(70.0)) / 63.0),
104 0.0,
105 FastMath.sqrt((35.0 - 2.0 * FastMath.sqrt(70.0)) / 63.0),
106 FastMath.sqrt((35.0 + 2.0 * FastMath.sqrt(70.0)) / 63.0)
107 };
108
109 /** Weights for the 5 points method. */
110 private static final double[] WEIGHTS_5 = {
111 (322.0 - 13.0 * FastMath.sqrt(70.0)) / 900.0,
112 (322.0 + 13.0 * FastMath.sqrt(70.0)) / 900.0,
113 128.0 / 225.0,
114 (322.0 + 13.0 * FastMath.sqrt(70.0)) / 900.0,
115 (322.0 - 13.0 * FastMath.sqrt(70.0)) / 900.0
116 };
117
118 /** Abscissas for the current method. */
119 private final double[] abscissas;
120
121 /** Weights for the current method. */
122 private final double[] weights;
123
124 /**
125 * Build a Legendre-Gauss integrator with given accuracies and iterations counts.
126 * @param n number of points desired (must be between 2 and 5 inclusive)
127 * @param relativeAccuracy relative accuracy of the result
128 * @param absoluteAccuracy absolute accuracy of the result
129 * @param minimalIterationCount minimum number of iterations
130 * @param maximalIterationCount maximum number of iterations
131 * @exception MathIllegalArgumentException if number of points is out of [2; 5]
132 * @exception NotStrictlyPositiveException if minimal number of iterations
133 * is not strictly positive
134 * @exception NumberIsTooSmallException if maximal number of iterations
135 * is lesser than or equal to the minimal number of iterations
136 */
137 public LegendreGaussIntegrator(final int n,
138 final double relativeAccuracy,
139 final double absoluteAccuracy,
140 final int minimalIterationCount,
141 final int maximalIterationCount)
142 throws MathIllegalArgumentException, NotStrictlyPositiveException, NumberIsTooSmallException {
143 super(relativeAccuracy, absoluteAccuracy, minimalIterationCount, maximalIterationCount);
144 switch(n) {
145 case 2 :
146 abscissas = ABSCISSAS_2;
147 weights = WEIGHTS_2;
148 break;
149 case 3 :
150 abscissas = ABSCISSAS_3;
151 weights = WEIGHTS_3;
152 break;
153 case 4 :
154 abscissas = ABSCISSAS_4;
155 weights = WEIGHTS_4;
156 break;
157 case 5 :
158 abscissas = ABSCISSAS_5;
159 weights = WEIGHTS_5;
160 break;
161 default :
162 throw new MathIllegalArgumentException(
163 LocalizedFormats.N_POINTS_GAUSS_LEGENDRE_INTEGRATOR_NOT_SUPPORTED,
164 n, 2, 5);
165 }
166
167 }
168
169 /**
170 * Build a Legendre-Gauss integrator with given accuracies.
171 * @param n number of points desired (must be between 2 and 5 inclusive)
172 * @param relativeAccuracy relative accuracy of the result
173 * @param absoluteAccuracy absolute accuracy of the result
174 * @exception MathIllegalArgumentException if number of points is out of [2; 5]
175 */
176 public LegendreGaussIntegrator(final int n,
177 final double relativeAccuracy,
178 final double absoluteAccuracy)
179 throws MathIllegalArgumentException {
180 this(n, relativeAccuracy, absoluteAccuracy,
181 DEFAULT_MIN_ITERATIONS_COUNT, DEFAULT_MAX_ITERATIONS_COUNT);
182 }
183
184 /**
185 * Build a Legendre-Gauss integrator with given iteration counts.
186 * @param n number of points desired (must be between 2 and 5 inclusive)
187 * @param minimalIterationCount minimum number of iterations
188 * @param maximalIterationCount maximum number of iterations
189 * @exception MathIllegalArgumentException if number of points is out of [2; 5]
190 * @exception NotStrictlyPositiveException if minimal number of iterations
191 * is not strictly positive
192 * @exception NumberIsTooSmallException if maximal number of iterations
193 * is lesser than or equal to the minimal number of iterations
194 */
195 public LegendreGaussIntegrator(final int n,
196 final int minimalIterationCount,
197 final int maximalIterationCount)
198 throws MathIllegalArgumentException {
199 this(n, DEFAULT_RELATIVE_ACCURACY, DEFAULT_ABSOLUTE_ACCURACY,
200 minimalIterationCount, maximalIterationCount);
201 }
202
203 /** {@inheritDoc} */
204 @Override
205 protected double doIntegrate()
206 throws MathIllegalArgumentException, TooManyEvaluationsException, MaxCountExceededException {
207
208 // compute first estimate with a single step
209 double oldt = stage(1);
210
211 int n = 2;
212 while (true) {
213
214 // improve integral with a larger number of steps
215 final double t = stage(n);
216
217 // estimate error
218 final double delta = FastMath.abs(t - oldt);
219 final double limit =
220 FastMath.max(getAbsoluteAccuracy(),
221 getRelativeAccuracy() * (FastMath.abs(oldt) + FastMath.abs(t)) * 0.5);
222
223 // check convergence
224 if ((iterations.getCount() + 1 >= getMinimalIterationCount()) && (delta <= limit)) {
225 return t;
226 }
227
228 // prepare next iteration
229 double ratio = FastMath.min(4, FastMath.pow(delta / limit, 0.5 / abscissas.length));
230 n = FastMath.max((int) (ratio * n), n + 1);
231 oldt = t;
232 iterations.incrementCount();
233
234 }
235
236 }
237
238 /**
239 * Compute the n-th stage integral.
240 * @param n number of steps
241 * @return the value of n-th stage integral
242 * @throws TooManyEvaluationsException if the maximum number of evaluations
243 * is exceeded.
244 */
245 private double stage(final int n)
246 throws TooManyEvaluationsException {
247
248 // set up the step for the current stage
249 final double step = (getMax() - getMin()) / n;
250 final double halfStep = step / 2.0;
251
252 // integrate over all elementary steps
253 double midPoint = getMin() + halfStep;
254 double sum = 0.0;
255 for (int i = 0; i < n; ++i) {
256 for (int j = 0; j < abscissas.length; ++j) {
257 sum += weights[j] * computeObjectiveValue(midPoint + halfStep * abscissas[j]);
258 }
259 midPoint += step;
260 }
261
262 return halfStep * sum;
263
264 }
265
266 }