1dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond/* 2dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Licensed to the Apache Software Foundation (ASF) under one or more 3dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * contributor license agreements. See the NOTICE file distributed with 4dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * this work for additional information regarding copyright ownership. 5dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * The ASF licenses this file to You under the Apache License, Version 2.0 6dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * (the "License"); you may not use this file except in compliance with 7dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * the License. You may obtain a copy of the License at 8dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * 9dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * http://www.apache.org/licenses/LICENSE-2.0 10dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * 11dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Unless required by applicable law or agreed to in writing, software 12dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * distributed under the License is distributed on an "AS IS" BASIS, 13dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 14dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * See the License for the specific language governing permissions and 15dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * limitations under the License. 16dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 17dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondpackage org.apache.commons.math.analysis.integration; 18dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 19dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.ConvergenceException; 20dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.FunctionEvaluationException; 21dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.MathRuntimeException; 22dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.MaxIterationsExceededException; 23dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.analysis.UnivariateRealFunction; 24dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.exception.util.LocalizedFormats; 25dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.util.FastMath; 26dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 27dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond/** 28dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Implements the <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html"> 29dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Legendre-Gauss</a> quadrature formula. 30dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <p> 31dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Legendre-Gauss integrators are efficient integrators that can 32dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * accurately integrate functions with few functions evaluations. A 33dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Legendre-Gauss integrator using an n-points quadrature formula can 34dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * integrate exactly 2n-1 degree polynomials. 35dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * </p> 36dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <p> 37dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * These integrators evaluate the function on n carefully chosen 38dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * abscissas in each step interval (mapped to the canonical [-1 1] interval). 39dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * The evaluation abscissas are not evenly spaced and none of them are 40dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * at the interval endpoints. This implies the function integrated can be 41dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * undefined at integration interval endpoints. 42dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * </p> 43dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <p> 44dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * The evaluation abscissas x<sub>i</sub> are the roots of the degree n 45dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Legendre polynomial. The weights a<sub>i</sub> of the quadrature formula 46dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * integrals from -1 to +1 ∫ Li<sup>2</sup> where Li (x) = 47dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * ∏ (x-x<sub>k</sub>)/(x<sub>i</sub>-x<sub>k</sub>) for k != i. 48dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * </p> 49dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <p> 50dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @version $Revision: 1070725 $ $Date: 2011-02-15 02:31:12 +0100 (mar. 15 févr. 2011) $ 51dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @since 1.2 52dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 53dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 54dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondpublic class LegendreGaussIntegrator extends UnivariateRealIntegratorImpl { 55dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 56dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Abscissas for the 2 points method. */ 57dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final double[] ABSCISSAS_2 = { 58dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond -1.0 / FastMath.sqrt(3.0), 59dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 1.0 / FastMath.sqrt(3.0) 60dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }; 61dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 62dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Weights for the 2 points method. */ 63dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final double[] WEIGHTS_2 = { 64dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 1.0, 65dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 1.0 66dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }; 67dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 68dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Abscissas for the 3 points method. */ 69dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final double[] ABSCISSAS_3 = { 70dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond -FastMath.sqrt(0.6), 71dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 0.0, 72dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond FastMath.sqrt(0.6) 73dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }; 74dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 75dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Weights for the 3 points method. */ 76dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final double[] WEIGHTS_3 = { 77dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 5.0 / 9.0, 78dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 8.0 / 9.0, 79dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 5.0 / 9.0 80dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }; 81dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 82dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Abscissas for the 4 points method. */ 83dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final double[] ABSCISSAS_4 = { 84dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond -FastMath.sqrt((15.0 + 2.0 * FastMath.sqrt(30.0)) / 35.0), 85dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond -FastMath.sqrt((15.0 - 2.0 * FastMath.sqrt(30.0)) / 35.0), 86dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond FastMath.sqrt((15.0 - 2.0 * FastMath.sqrt(30.0)) / 35.0), 87dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond FastMath.sqrt((15.0 + 2.0 * FastMath.sqrt(30.0)) / 35.0) 88dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }; 89dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 90dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Weights for the 4 points method. */ 91dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final double[] WEIGHTS_4 = { 92dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond (90.0 - 5.0 * FastMath.sqrt(30.0)) / 180.0, 93dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond (90.0 + 5.0 * FastMath.sqrt(30.0)) / 180.0, 94dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond (90.0 + 5.0 * FastMath.sqrt(30.0)) / 180.0, 95dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond (90.0 - 5.0 * FastMath.sqrt(30.0)) / 180.0 96dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }; 97dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 98dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Abscissas for the 5 points method. */ 99dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final double[] ABSCISSAS_5 = { 100dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond -FastMath.sqrt((35.0 + 2.0 * FastMath.sqrt(70.0)) / 63.0), 101dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond -FastMath.sqrt((35.0 - 2.0 * FastMath.sqrt(70.0)) / 63.0), 102dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 0.0, 103dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond FastMath.sqrt((35.0 - 2.0 * FastMath.sqrt(70.0)) / 63.0), 104dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond FastMath.sqrt((35.0 + 2.0 * FastMath.sqrt(70.0)) / 63.0) 105dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }; 106dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 107dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Weights for the 5 points method. */ 108dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final double[] WEIGHTS_5 = { 109dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond (322.0 - 13.0 * FastMath.sqrt(70.0)) / 900.0, 110dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond (322.0 + 13.0 * FastMath.sqrt(70.0)) / 900.0, 111dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 128.0 / 225.0, 112dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond (322.0 + 13.0 * FastMath.sqrt(70.0)) / 900.0, 113dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond (322.0 - 13.0 * FastMath.sqrt(70.0)) / 900.0 114dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }; 115dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 116dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Abscissas for the current method. */ 117dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private final double[] abscissas; 118dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 119dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Weights for the current method. */ 120dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private final double[] weights; 121dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 122dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** 123dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Build a Legendre-Gauss integrator. 124dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param n number of points desired (must be between 2 and 5 inclusive) 125dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param defaultMaximalIterationCount maximum number of iterations 126dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @exception IllegalArgumentException if the number of points is not 127dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * in the supported range 128dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 129dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public LegendreGaussIntegrator(final int n, final int defaultMaximalIterationCount) 130dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond throws IllegalArgumentException { 131dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond super(defaultMaximalIterationCount); 132dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond switch(n) { 133dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond case 2 : 134dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond abscissas = ABSCISSAS_2; 135dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond weights = WEIGHTS_2; 136dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond break; 137dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond case 3 : 138dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond abscissas = ABSCISSAS_3; 139dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond weights = WEIGHTS_3; 140dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond break; 141dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond case 4 : 142dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond abscissas = ABSCISSAS_4; 143dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond weights = WEIGHTS_4; 144dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond break; 145dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond case 5 : 146dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond abscissas = ABSCISSAS_5; 147dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond weights = WEIGHTS_5; 148dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond break; 149dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond default : 150dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond throw MathRuntimeException.createIllegalArgumentException( 151dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LocalizedFormats.N_POINTS_GAUSS_LEGENDRE_INTEGRATOR_NOT_SUPPORTED, 152dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond n, 2, 5); 153dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 154dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 155dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 156dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 157dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** {@inheritDoc} */ 158dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond @Deprecated 159dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public double integrate(final double min, final double max) 160dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond throws ConvergenceException, FunctionEvaluationException, IllegalArgumentException { 161dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return integrate(f, min, max); 162dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 163dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 164dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** {@inheritDoc} */ 165dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public double integrate(final UnivariateRealFunction f, final double min, final double max) 166dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond throws ConvergenceException, FunctionEvaluationException, IllegalArgumentException { 167dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 168dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond clearResult(); 169dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond verifyInterval(min, max); 170dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond verifyIterationCount(); 171dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 172dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // compute first estimate with a single step 173dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond double oldt = stage(f, min, max, 1); 174dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 175dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond int n = 2; 176dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond for (int i = 0; i < maximalIterationCount; ++i) { 177dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 178dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // improve integral with a larger number of steps 179dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final double t = stage(f, min, max, n); 180dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 181dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // estimate error 182dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final double delta = FastMath.abs(t - oldt); 183dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final double limit = 184dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond FastMath.max(absoluteAccuracy, 185dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond relativeAccuracy * (FastMath.abs(oldt) + FastMath.abs(t)) * 0.5); 186dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 187dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // check convergence 188dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond if ((i + 1 >= minimalIterationCount) && (delta <= limit)) { 189dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond setResult(t, i); 190dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return result; 191dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 192dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 193dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // prepare next iteration 194dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond double ratio = FastMath.min(4, FastMath.pow(delta / limit, 0.5 / abscissas.length)); 195dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond n = FastMath.max((int) (ratio * n), n + 1); 196dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond oldt = t; 197dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 198dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 199dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 200dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond throw new MaxIterationsExceededException(maximalIterationCount); 201dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 202dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 203dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 204dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** 205dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Compute the n-th stage integral. 206dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param f the integrand function 207dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param min the lower bound for the interval 208dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param max the upper bound for the interval 209dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param n number of steps 210dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @return the value of n-th stage integral 211dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @throws FunctionEvaluationException if an error occurs evaluating the 212dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * function 213dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 214dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private double stage(final UnivariateRealFunction f, 215dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final double min, final double max, final int n) 216dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond throws FunctionEvaluationException { 217dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 218dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // set up the step for the current stage 219dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final double step = (max - min) / n; 220dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final double halfStep = step / 2.0; 221dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 222dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // integrate over all elementary steps 223dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond double midPoint = min + halfStep; 224dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond double sum = 0.0; 225dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond for (int i = 0; i < n; ++i) { 226dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond for (int j = 0; j < abscissas.length; ++j) { 227dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond sum += weights[j] * f.value(midPoint + halfStep * abscissas[j]); 228dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 229dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond midPoint += step; 230dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 231dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 232dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return halfStep * sum; 233dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 234dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 235dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 236dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond} 237