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<HTML> <BODY BGCOLOR="white"> <PRE> <FONT color="green">001</FONT> /*<a name="line.1"></a> <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a> <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a> <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a> <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a> <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a> <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a> <FONT color="green">008</FONT> *<a name="line.8"></a> <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a> <FONT color="green">010</FONT> *<a name="line.10"></a> <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a> <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a> <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a> <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a> <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a> <FONT color="green">016</FONT> */<a name="line.16"></a> <FONT color="green">017</FONT> package org.apache.commons.math.analysis.integration;<a name="line.17"></a> <FONT color="green">018</FONT> <a name="line.18"></a> <FONT color="green">019</FONT> import org.apache.commons.math.ConvergenceException;<a name="line.19"></a> <FONT color="green">020</FONT> import org.apache.commons.math.FunctionEvaluationException;<a name="line.20"></a> <FONT color="green">021</FONT> import org.apache.commons.math.MathRuntimeException;<a name="line.21"></a> <FONT color="green">022</FONT> import org.apache.commons.math.MaxIterationsExceededException;<a name="line.22"></a> <FONT color="green">023</FONT> import org.apache.commons.math.analysis.UnivariateRealFunction;<a name="line.23"></a> <FONT color="green">024</FONT> <a name="line.24"></a> <FONT color="green">025</FONT> /**<a name="line.25"></a> <FONT color="green">026</FONT> * Implements the <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html"><a name="line.26"></a> <FONT color="green">027</FONT> * Legendre-Gauss</a> quadrature formula.<a name="line.27"></a> <FONT color="green">028</FONT> * <p><a name="line.28"></a> <FONT color="green">029</FONT> * Legendre-Gauss integrators are efficient integrators that can<a name="line.29"></a> <FONT color="green">030</FONT> * accurately integrate functions with few functions evaluations. A<a name="line.30"></a> <FONT color="green">031</FONT> * Legendre-Gauss integrator using an n-points quadrature formula can<a name="line.31"></a> <FONT color="green">032</FONT> * integrate exactly 2n-1 degree polynomialss.<a name="line.32"></a> <FONT color="green">033</FONT> * </p><a name="line.33"></a> <FONT color="green">034</FONT> * <p><a name="line.34"></a> <FONT color="green">035</FONT> * These integrators evaluate the function on n carefully chosen<a name="line.35"></a> <FONT color="green">036</FONT> * abscissas in each step interval (mapped to the canonical [-1 1] interval).<a name="line.36"></a> <FONT color="green">037</FONT> * The evaluation abscissas are not evenly spaced and none of them are<a name="line.37"></a> <FONT color="green">038</FONT> * at the interval endpoints. This implies the function integrated can be<a name="line.38"></a> <FONT color="green">039</FONT> * undefined at integration interval endpoints.<a name="line.39"></a> <FONT color="green">040</FONT> * </p><a name="line.40"></a> <FONT color="green">041</FONT> * <p><a name="line.41"></a> <FONT color="green">042</FONT> * The evaluation abscissas x<sub>i</sub> are the roots of the degree n<a name="line.42"></a> <FONT color="green">043</FONT> * Legendre polynomial. The weights a<sub>i</sub> of the quadrature formula<a name="line.43"></a> <FONT color="green">044</FONT> * integrals from -1 to +1 &int; Li<sup>2</sup> where Li (x) =<a name="line.44"></a> <FONT color="green">045</FONT> * &prod; (x-x<sub>k</sub>)/(x<sub>i</sub>-x<sub>k</sub>) for k != i.<a name="line.45"></a> <FONT color="green">046</FONT> * </p><a name="line.46"></a> <FONT color="green">047</FONT> * <p><a name="line.47"></a> <FONT color="green">048</FONT> * @version $Revision: 811685 $ $Date: 2009-09-05 13:36:48 -0400 (Sat, 05 Sep 2009) $<a name="line.48"></a> <FONT color="green">049</FONT> * @since 1.2<a name="line.49"></a> <FONT color="green">050</FONT> */<a name="line.50"></a> <FONT color="green">051</FONT> <a name="line.51"></a> <FONT color="green">052</FONT> public class LegendreGaussIntegrator extends UnivariateRealIntegratorImpl {<a name="line.52"></a> <FONT color="green">053</FONT> <a name="line.53"></a> <FONT color="green">054</FONT> /** Abscissas for the 2 points method. */<a name="line.54"></a> <FONT color="green">055</FONT> private static final double[] ABSCISSAS_2 = {<a name="line.55"></a> <FONT color="green">056</FONT> -1.0 / Math.sqrt(3.0),<a name="line.56"></a> <FONT color="green">057</FONT> 1.0 / Math.sqrt(3.0)<a name="line.57"></a> <FONT color="green">058</FONT> };<a name="line.58"></a> <FONT color="green">059</FONT> <a name="line.59"></a> <FONT color="green">060</FONT> /** Weights for the 2 points method. */<a name="line.60"></a> <FONT color="green">061</FONT> private static final double[] WEIGHTS_2 = {<a name="line.61"></a> <FONT color="green">062</FONT> 1.0,<a name="line.62"></a> <FONT color="green">063</FONT> 1.0<a name="line.63"></a> <FONT color="green">064</FONT> };<a name="line.64"></a> <FONT color="green">065</FONT> <a name="line.65"></a> <FONT color="green">066</FONT> /** Abscissas for the 3 points method. */<a name="line.66"></a> <FONT color="green">067</FONT> private static final double[] ABSCISSAS_3 = {<a name="line.67"></a> <FONT color="green">068</FONT> -Math.sqrt(0.6),<a name="line.68"></a> <FONT color="green">069</FONT> 0.0,<a name="line.69"></a> <FONT color="green">070</FONT> Math.sqrt(0.6)<a name="line.70"></a> <FONT color="green">071</FONT> };<a name="line.71"></a> <FONT color="green">072</FONT> <a name="line.72"></a> <FONT color="green">073</FONT> /** Weights for the 3 points method. */<a name="line.73"></a> <FONT color="green">074</FONT> private static final double[] WEIGHTS_3 = {<a name="line.74"></a> <FONT color="green">075</FONT> 5.0 / 9.0,<a name="line.75"></a> <FONT color="green">076</FONT> 8.0 / 9.0,<a name="line.76"></a> <FONT color="green">077</FONT> 5.0 / 9.0<a name="line.77"></a> <FONT color="green">078</FONT> };<a name="line.78"></a> <FONT color="green">079</FONT> <a name="line.79"></a> <FONT color="green">080</FONT> /** Abscissas for the 4 points method. */<a name="line.80"></a> <FONT color="green">081</FONT> private static final double[] ABSCISSAS_4 = {<a name="line.81"></a> <FONT color="green">082</FONT> -Math.sqrt((15.0 + 2.0 * Math.sqrt(30.0)) / 35.0),<a name="line.82"></a> <FONT color="green">083</FONT> -Math.sqrt((15.0 - 2.0 * Math.sqrt(30.0)) / 35.0),<a name="line.83"></a> <FONT color="green">084</FONT> Math.sqrt((15.0 - 2.0 * Math.sqrt(30.0)) / 35.0),<a name="line.84"></a> <FONT color="green">085</FONT> Math.sqrt((15.0 + 2.0 * Math.sqrt(30.0)) / 35.0)<a name="line.85"></a> <FONT color="green">086</FONT> };<a name="line.86"></a> <FONT color="green">087</FONT> <a name="line.87"></a> <FONT color="green">088</FONT> /** Weights for the 4 points method. */<a name="line.88"></a> <FONT color="green">089</FONT> private static final double[] WEIGHTS_4 = {<a name="line.89"></a> <FONT color="green">090</FONT> (90.0 - 5.0 * Math.sqrt(30.0)) / 180.0,<a name="line.90"></a> <FONT color="green">091</FONT> (90.0 + 5.0 * Math.sqrt(30.0)) / 180.0,<a name="line.91"></a> <FONT color="green">092</FONT> (90.0 + 5.0 * Math.sqrt(30.0)) / 180.0,<a name="line.92"></a> <FONT color="green">093</FONT> (90.0 - 5.0 * Math.sqrt(30.0)) / 180.0<a name="line.93"></a> <FONT color="green">094</FONT> };<a name="line.94"></a> <FONT color="green">095</FONT> <a name="line.95"></a> <FONT color="green">096</FONT> /** Abscissas for the 5 points method. */<a name="line.96"></a> <FONT color="green">097</FONT> private static final double[] ABSCISSAS_5 = {<a name="line.97"></a> <FONT color="green">098</FONT> -Math.sqrt((35.0 + 2.0 * Math.sqrt(70.0)) / 63.0),<a name="line.98"></a> <FONT color="green">099</FONT> -Math.sqrt((35.0 - 2.0 * Math.sqrt(70.0)) / 63.0),<a name="line.99"></a> <FONT color="green">100</FONT> 0.0,<a name="line.100"></a> <FONT color="green">101</FONT> Math.sqrt((35.0 - 2.0 * Math.sqrt(70.0)) / 63.0),<a name="line.101"></a> <FONT color="green">102</FONT> Math.sqrt((35.0 + 2.0 * Math.sqrt(70.0)) / 63.0)<a name="line.102"></a> <FONT color="green">103</FONT> };<a name="line.103"></a> <FONT color="green">104</FONT> <a name="line.104"></a> <FONT color="green">105</FONT> /** Weights for the 5 points method. */<a name="line.105"></a> <FONT color="green">106</FONT> private static final double[] WEIGHTS_5 = {<a name="line.106"></a> <FONT color="green">107</FONT> (322.0 - 13.0 * Math.sqrt(70.0)) / 900.0,<a name="line.107"></a> <FONT color="green">108</FONT> (322.0 + 13.0 * Math.sqrt(70.0)) / 900.0,<a name="line.108"></a> <FONT color="green">109</FONT> 128.0 / 225.0,<a name="line.109"></a> <FONT color="green">110</FONT> (322.0 + 13.0 * Math.sqrt(70.0)) / 900.0,<a name="line.110"></a> <FONT color="green">111</FONT> (322.0 - 13.0 * Math.sqrt(70.0)) / 900.0<a name="line.111"></a> <FONT color="green">112</FONT> };<a name="line.112"></a> <FONT color="green">113</FONT> <a name="line.113"></a> <FONT color="green">114</FONT> /** Abscissas for the current method. */<a name="line.114"></a> <FONT color="green">115</FONT> private final double[] abscissas;<a name="line.115"></a> <FONT color="green">116</FONT> <a name="line.116"></a> <FONT color="green">117</FONT> /** Weights for the current method. */<a name="line.117"></a> <FONT color="green">118</FONT> private final double[] weights;<a name="line.118"></a> <FONT color="green">119</FONT> <a name="line.119"></a> <FONT color="green">120</FONT> /** Build a Legendre-Gauss integrator.<a name="line.120"></a> <FONT color="green">121</FONT> * @param n number of points desired (must be between 2 and 5 inclusive)<a name="line.121"></a> <FONT color="green">122</FONT> * @param defaultMaximalIterationCount maximum number of iterations<a name="line.122"></a> <FONT color="green">123</FONT> * @exception IllegalArgumentException if the number of points is not<a name="line.123"></a> <FONT color="green">124</FONT> * in the supported range<a name="line.124"></a> <FONT color="green">125</FONT> */<a name="line.125"></a> <FONT color="green">126</FONT> public LegendreGaussIntegrator(final int n, final int defaultMaximalIterationCount)<a name="line.126"></a> <FONT color="green">127</FONT> throws IllegalArgumentException {<a name="line.127"></a> <FONT color="green">128</FONT> super(defaultMaximalIterationCount);<a name="line.128"></a> <FONT color="green">129</FONT> switch(n) {<a name="line.129"></a> <FONT color="green">130</FONT> case 2 :<a name="line.130"></a> <FONT color="green">131</FONT> abscissas = ABSCISSAS_2;<a name="line.131"></a> <FONT color="green">132</FONT> weights = WEIGHTS_2;<a name="line.132"></a> <FONT color="green">133</FONT> break;<a name="line.133"></a> <FONT color="green">134</FONT> case 3 :<a name="line.134"></a> <FONT color="green">135</FONT> abscissas = ABSCISSAS_3;<a name="line.135"></a> <FONT color="green">136</FONT> weights = WEIGHTS_3;<a name="line.136"></a> <FONT color="green">137</FONT> break;<a name="line.137"></a> <FONT color="green">138</FONT> case 4 :<a name="line.138"></a> <FONT color="green">139</FONT> abscissas = ABSCISSAS_4;<a name="line.139"></a> <FONT color="green">140</FONT> weights = WEIGHTS_4;<a name="line.140"></a> <FONT color="green">141</FONT> break;<a name="line.141"></a> <FONT color="green">142</FONT> case 5 :<a name="line.142"></a> <FONT color="green">143</FONT> abscissas = ABSCISSAS_5;<a name="line.143"></a> <FONT color="green">144</FONT> weights = WEIGHTS_5;<a name="line.144"></a> <FONT color="green">145</FONT> break;<a name="line.145"></a> <FONT color="green">146</FONT> default :<a name="line.146"></a> <FONT color="green">147</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.147"></a> <FONT color="green">148</FONT> "{0} points Legendre-Gauss integrator not supported, " +<a name="line.148"></a> <FONT color="green">149</FONT> "number of points must be in the {1}-{2} range",<a name="line.149"></a> <FONT color="green">150</FONT> n, 2, 5);<a name="line.150"></a> <FONT color="green">151</FONT> }<a name="line.151"></a> <FONT color="green">152</FONT> <a name="line.152"></a> <FONT color="green">153</FONT> }<a name="line.153"></a> <FONT color="green">154</FONT> <a name="line.154"></a> <FONT color="green">155</FONT> /** {@inheritDoc} */<a name="line.155"></a> <FONT color="green">156</FONT> @Deprecated<a name="line.156"></a> <FONT color="green">157</FONT> public double integrate(final double min, final double max)<a name="line.157"></a> <FONT color="green">158</FONT> throws ConvergenceException, FunctionEvaluationException, IllegalArgumentException {<a name="line.158"></a> <FONT color="green">159</FONT> return integrate(f, min, max);<a name="line.159"></a> <FONT color="green">160</FONT> }<a name="line.160"></a> <FONT color="green">161</FONT> <a name="line.161"></a> <FONT color="green">162</FONT> /** {@inheritDoc} */<a name="line.162"></a> <FONT color="green">163</FONT> public double integrate(final UnivariateRealFunction f,<a name="line.163"></a> <FONT color="green">164</FONT> final double min, final double max)<a name="line.164"></a> <FONT color="green">165</FONT> throws ConvergenceException, FunctionEvaluationException, IllegalArgumentException {<a name="line.165"></a> <FONT color="green">166</FONT> <a name="line.166"></a> <FONT color="green">167</FONT> clearResult();<a name="line.167"></a> <FONT color="green">168</FONT> verifyInterval(min, max);<a name="line.168"></a> <FONT color="green">169</FONT> verifyIterationCount();<a name="line.169"></a> <FONT color="green">170</FONT> <a name="line.170"></a> <FONT color="green">171</FONT> // compute first estimate with a single step<a name="line.171"></a> <FONT color="green">172</FONT> double oldt = stage(f, min, max, 1);<a name="line.172"></a> <FONT color="green">173</FONT> <a name="line.173"></a> <FONT color="green">174</FONT> int n = 2;<a name="line.174"></a> <FONT color="green">175</FONT> for (int i = 0; i < maximalIterationCount; ++i) {<a name="line.175"></a> <FONT color="green">176</FONT> <a name="line.176"></a> <FONT color="green">177</FONT> // improve integral with a larger number of steps<a name="line.177"></a> <FONT color="green">178</FONT> final double t = stage(f, min, max, n);<a name="line.178"></a> <FONT color="green">179</FONT> <a name="line.179"></a> <FONT color="green">180</FONT> // estimate error<a name="line.180"></a> <FONT color="green">181</FONT> final double delta = Math.abs(t - oldt);<a name="line.181"></a> <FONT color="green">182</FONT> final double limit =<a name="line.182"></a> <FONT color="green">183</FONT> Math.max(absoluteAccuracy,<a name="line.183"></a> <FONT color="green">184</FONT> relativeAccuracy * (Math.abs(oldt) + Math.abs(t)) * 0.5);<a name="line.184"></a> <FONT color="green">185</FONT> <a name="line.185"></a> <FONT color="green">186</FONT> // check convergence<a name="line.186"></a> <FONT color="green">187</FONT> if ((i + 1 >= minimalIterationCount) && (delta <= limit)) {<a name="line.187"></a> <FONT color="green">188</FONT> setResult(t, i);<a name="line.188"></a> <FONT color="green">189</FONT> return result;<a name="line.189"></a> <FONT color="green">190</FONT> }<a name="line.190"></a> <FONT color="green">191</FONT> <a name="line.191"></a> <FONT color="green">192</FONT> // prepare next iteration<a name="line.192"></a> <FONT color="green">193</FONT> double ratio = Math.min(4, Math.pow(delta / limit, 0.5 / abscissas.length));<a name="line.193"></a> <FONT color="green">194</FONT> n = Math.max((int) (ratio * n), n + 1);<a name="line.194"></a> <FONT color="green">195</FONT> oldt = t;<a name="line.195"></a> <FONT color="green">196</FONT> <a name="line.196"></a> <FONT color="green">197</FONT> }<a name="line.197"></a> <FONT color="green">198</FONT> <a name="line.198"></a> <FONT color="green">199</FONT> throw new MaxIterationsExceededException(maximalIterationCount);<a name="line.199"></a> <FONT color="green">200</FONT> <a name="line.200"></a> <FONT color="green">201</FONT> }<a name="line.201"></a> <FONT color="green">202</FONT> <a name="line.202"></a> <FONT color="green">203</FONT> /**<a name="line.203"></a> <FONT color="green">204</FONT> * Compute the n-th stage integral.<a name="line.204"></a> <FONT color="green">205</FONT> * @param f the integrand function<a name="line.205"></a> <FONT color="green">206</FONT> * @param min the lower bound for the interval<a name="line.206"></a> <FONT color="green">207</FONT> * @param max the upper bound for the interval<a name="line.207"></a> <FONT color="green">208</FONT> * @param n number of steps<a name="line.208"></a> <FONT color="green">209</FONT> * @return the value of n-th stage integral<a name="line.209"></a> <FONT color="green">210</FONT> * @throws FunctionEvaluationException if an error occurs evaluating the<a name="line.210"></a> <FONT color="green">211</FONT> * function<a name="line.211"></a> <FONT color="green">212</FONT> */<a name="line.212"></a> <FONT color="green">213</FONT> private double stage(final UnivariateRealFunction f,<a name="line.213"></a> <FONT color="green">214</FONT> final double min, final double max, final int n)<a name="line.214"></a> <FONT color="green">215</FONT> throws FunctionEvaluationException {<a name="line.215"></a> <FONT color="green">216</FONT> <a name="line.216"></a> <FONT color="green">217</FONT> // set up the step for the current stage<a name="line.217"></a> <FONT color="green">218</FONT> final double step = (max - min) / n;<a name="line.218"></a> <FONT color="green">219</FONT> final double halfStep = step / 2.0;<a name="line.219"></a> <FONT color="green">220</FONT> <a name="line.220"></a> <FONT color="green">221</FONT> // integrate over all elementary steps<a name="line.221"></a> <FONT color="green">222</FONT> double midPoint = min + halfStep;<a name="line.222"></a> <FONT color="green">223</FONT> double sum = 0.0;<a name="line.223"></a> <FONT color="green">224</FONT> for (int i = 0; i < n; ++i) {<a name="line.224"></a> <FONT color="green">225</FONT> for (int j = 0; j < abscissas.length; ++j) {<a name="line.225"></a> <FONT color="green">226</FONT> sum += weights[j] * f.value(midPoint + halfStep * abscissas[j]);<a name="line.226"></a> <FONT color="green">227</FONT> }<a name="line.227"></a> <FONT color="green">228</FONT> midPoint += step;<a name="line.228"></a> <FONT color="green">229</FONT> }<a name="line.229"></a> <FONT color="green">230</FONT> <a name="line.230"></a> <FONT color="green">231</FONT> return halfStep * sum;<a name="line.231"></a> <FONT color="green">232</FONT> <a name="line.232"></a> <FONT color="green">233</FONT> }<a name="line.233"></a> <FONT color="green">234</FONT> <a name="line.234"></a> <FONT color="green">235</FONT> }<a name="line.235"></a> </PRE> </BODY> </HTML>