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| author | dwinter |
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| date | Tue, 04 Jan 2011 10:02:07 +0100 |
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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.special;<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.MathException;<a name="line.19"></a> <FONT color="green">020</FONT> import org.apache.commons.math.MaxIterationsExceededException;<a name="line.20"></a> <FONT color="green">021</FONT> import org.apache.commons.math.util.ContinuedFraction;<a name="line.21"></a> <FONT color="green">022</FONT> <a name="line.22"></a> <FONT color="green">023</FONT> /**<a name="line.23"></a> <FONT color="green">024</FONT> * This is a utility class that provides computation methods related to the<a name="line.24"></a> <FONT color="green">025</FONT> * Gamma family of functions.<a name="line.25"></a> <FONT color="green">026</FONT> *<a name="line.26"></a> <FONT color="green">027</FONT> * @version $Revision: 920558 $ $Date: 2010-03-08 17:57:32 -0500 (Mon, 08 Mar 2010) $<a name="line.27"></a> <FONT color="green">028</FONT> */<a name="line.28"></a> <FONT color="green">029</FONT> public class Gamma {<a name="line.29"></a> <FONT color="green">030</FONT> <a name="line.30"></a> <FONT color="green">031</FONT> /**<a name="line.31"></a> <FONT color="green">032</FONT> * <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a><a name="line.32"></a> <FONT color="green">033</FONT> * @since 2.0<a name="line.33"></a> <FONT color="green">034</FONT> */<a name="line.34"></a> <FONT color="green">035</FONT> public static final double GAMMA = 0.577215664901532860606512090082;<a name="line.35"></a> <FONT color="green">036</FONT> <a name="line.36"></a> <FONT color="green">037</FONT> /** Maximum allowed numerical error. */<a name="line.37"></a> <FONT color="green">038</FONT> private static final double DEFAULT_EPSILON = 10e-15;<a name="line.38"></a> <FONT color="green">039</FONT> <a name="line.39"></a> <FONT color="green">040</FONT> /** Lanczos coefficients */<a name="line.40"></a> <FONT color="green">041</FONT> private static final double[] LANCZOS =<a name="line.41"></a> <FONT color="green">042</FONT> {<a name="line.42"></a> <FONT color="green">043</FONT> 0.99999999999999709182,<a name="line.43"></a> <FONT color="green">044</FONT> 57.156235665862923517,<a name="line.44"></a> <FONT color="green">045</FONT> -59.597960355475491248,<a name="line.45"></a> <FONT color="green">046</FONT> 14.136097974741747174,<a name="line.46"></a> <FONT color="green">047</FONT> -0.49191381609762019978,<a name="line.47"></a> <FONT color="green">048</FONT> .33994649984811888699e-4,<a name="line.48"></a> <FONT color="green">049</FONT> .46523628927048575665e-4,<a name="line.49"></a> <FONT color="green">050</FONT> -.98374475304879564677e-4,<a name="line.50"></a> <FONT color="green">051</FONT> .15808870322491248884e-3,<a name="line.51"></a> <FONT color="green">052</FONT> -.21026444172410488319e-3,<a name="line.52"></a> <FONT color="green">053</FONT> .21743961811521264320e-3,<a name="line.53"></a> <FONT color="green">054</FONT> -.16431810653676389022e-3,<a name="line.54"></a> <FONT color="green">055</FONT> .84418223983852743293e-4,<a name="line.55"></a> <FONT color="green">056</FONT> -.26190838401581408670e-4,<a name="line.56"></a> <FONT color="green">057</FONT> .36899182659531622704e-5,<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> /** Avoid repeated computation of log of 2 PI in logGamma */<a name="line.60"></a> <FONT color="green">061</FONT> private static final double HALF_LOG_2_PI = 0.5 * Math.log(2.0 * Math.PI);<a name="line.61"></a> <FONT color="green">062</FONT> <a name="line.62"></a> <FONT color="green">063</FONT> // limits for switching algorithm in digamma<a name="line.63"></a> <FONT color="green">064</FONT> /** C limit. */<a name="line.64"></a> <FONT color="green">065</FONT> private static final double C_LIMIT = 49;<a name="line.65"></a> <FONT color="green">066</FONT> <a name="line.66"></a> <FONT color="green">067</FONT> /** S limit. */<a name="line.67"></a> <FONT color="green">068</FONT> private static final double S_LIMIT = 1e-5;<a name="line.68"></a> <FONT color="green">069</FONT> <a name="line.69"></a> <FONT color="green">070</FONT> /**<a name="line.70"></a> <FONT color="green">071</FONT> * Default constructor. Prohibit instantiation.<a name="line.71"></a> <FONT color="green">072</FONT> */<a name="line.72"></a> <FONT color="green">073</FONT> private Gamma() {<a name="line.73"></a> <FONT color="green">074</FONT> super();<a name="line.74"></a> <FONT color="green">075</FONT> }<a name="line.75"></a> <FONT color="green">076</FONT> <a name="line.76"></a> <FONT color="green">077</FONT> /**<a name="line.77"></a> <FONT color="green">078</FONT> * Returns the natural logarithm of the gamma function &#915;(x).<a name="line.78"></a> <FONT color="green">079</FONT> *<a name="line.79"></a> <FONT color="green">080</FONT> * The implementation of this method is based on:<a name="line.80"></a> <FONT color="green">081</FONT> * <ul><a name="line.81"></a> <FONT color="green">082</FONT> * <li><a href="http://mathworld.wolfram.com/GammaFunction.html"><a name="line.82"></a> <FONT color="green">083</FONT> * Gamma Function</a>, equation (28).</li><a name="line.83"></a> <FONT color="green">084</FONT> * <li><a href="http://mathworld.wolfram.com/LanczosApproximation.html"><a name="line.84"></a> <FONT color="green">085</FONT> * Lanczos Approximation</a>, equations (1) through (5).</li><a name="line.85"></a> <FONT color="green">086</FONT> * <li><a href="http://my.fit.edu/~gabdo/gamma.txt">Paul Godfrey, A note on<a name="line.86"></a> <FONT color="green">087</FONT> * the computation of the convergent Lanczos complex Gamma approximation<a name="line.87"></a> <FONT color="green">088</FONT> * </a></li><a name="line.88"></a> <FONT color="green">089</FONT> * </ul><a name="line.89"></a> <FONT color="green">090</FONT> *<a name="line.90"></a> <FONT color="green">091</FONT> * @param x the value.<a name="line.91"></a> <FONT color="green">092</FONT> * @return log(&#915;(x))<a name="line.92"></a> <FONT color="green">093</FONT> */<a name="line.93"></a> <FONT color="green">094</FONT> public static double logGamma(double x) {<a name="line.94"></a> <FONT color="green">095</FONT> double ret;<a name="line.95"></a> <FONT color="green">096</FONT> <a name="line.96"></a> <FONT color="green">097</FONT> if (Double.isNaN(x) || (x <= 0.0)) {<a name="line.97"></a> <FONT color="green">098</FONT> ret = Double.NaN;<a name="line.98"></a> <FONT color="green">099</FONT> } else {<a name="line.99"></a> <FONT color="green">100</FONT> double g = 607.0 / 128.0;<a name="line.100"></a> <FONT color="green">101</FONT> <a name="line.101"></a> <FONT color="green">102</FONT> double sum = 0.0;<a name="line.102"></a> <FONT color="green">103</FONT> for (int i = LANCZOS.length - 1; i > 0; --i) {<a name="line.103"></a> <FONT color="green">104</FONT> sum = sum + (LANCZOS[i] / (x + i));<a name="line.104"></a> <FONT color="green">105</FONT> }<a name="line.105"></a> <FONT color="green">106</FONT> sum = sum + LANCZOS[0];<a name="line.106"></a> <FONT color="green">107</FONT> <a name="line.107"></a> <FONT color="green">108</FONT> double tmp = x + g + .5;<a name="line.108"></a> <FONT color="green">109</FONT> ret = ((x + .5) * Math.log(tmp)) - tmp +<a name="line.109"></a> <FONT color="green">110</FONT> HALF_LOG_2_PI + Math.log(sum / x);<a name="line.110"></a> <FONT color="green">111</FONT> }<a name="line.111"></a> <FONT color="green">112</FONT> <a name="line.112"></a> <FONT color="green">113</FONT> return ret;<a name="line.113"></a> <FONT color="green">114</FONT> }<a name="line.114"></a> <FONT color="green">115</FONT> <a name="line.115"></a> <FONT color="green">116</FONT> /**<a name="line.116"></a> <FONT color="green">117</FONT> * Returns the regularized gamma function P(a, x).<a name="line.117"></a> <FONT color="green">118</FONT> *<a name="line.118"></a> <FONT color="green">119</FONT> * @param a the a parameter.<a name="line.119"></a> <FONT color="green">120</FONT> * @param x the value.<a name="line.120"></a> <FONT color="green">121</FONT> * @return the regularized gamma function P(a, x)<a name="line.121"></a> <FONT color="green">122</FONT> * @throws MathException if the algorithm fails to converge.<a name="line.122"></a> <FONT color="green">123</FONT> */<a name="line.123"></a> <FONT color="green">124</FONT> public static double regularizedGammaP(double a, double x)<a name="line.124"></a> <FONT color="green">125</FONT> throws MathException<a name="line.125"></a> <FONT color="green">126</FONT> {<a name="line.126"></a> <FONT color="green">127</FONT> return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);<a name="line.127"></a> <FONT color="green">128</FONT> }<a name="line.128"></a> <FONT color="green">129</FONT> <a name="line.129"></a> <FONT color="green">130</FONT> <a name="line.130"></a> <FONT color="green">131</FONT> /**<a name="line.131"></a> <FONT color="green">132</FONT> * Returns the regularized gamma function P(a, x).<a name="line.132"></a> <FONT color="green">133</FONT> *<a name="line.133"></a> <FONT color="green">134</FONT> * The implementation of this method is based on:<a name="line.134"></a> <FONT color="green">135</FONT> * <ul><a name="line.135"></a> <FONT color="green">136</FONT> * <li><a name="line.136"></a> <FONT color="green">137</FONT> * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"><a name="line.137"></a> <FONT color="green">138</FONT> * Regularized Gamma Function</a>, equation (1).</li><a name="line.138"></a> <FONT color="green">139</FONT> * <li><a name="line.139"></a> <FONT color="green">140</FONT> * <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html"><a name="line.140"></a> <FONT color="green">141</FONT> * Incomplete Gamma Function</a>, equation (4).</li><a name="line.141"></a> <FONT color="green">142</FONT> * <li><a name="line.142"></a> <FONT color="green">143</FONT> * <a href="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html"><a name="line.143"></a> <FONT color="green">144</FONT> * Confluent Hypergeometric Function of the First Kind</a>, equation (1).<a name="line.144"></a> <FONT color="green">145</FONT> * </li><a name="line.145"></a> <FONT color="green">146</FONT> * </ul><a name="line.146"></a> <FONT color="green">147</FONT> *<a name="line.147"></a> <FONT color="green">148</FONT> * @param a the a parameter.<a name="line.148"></a> <FONT color="green">149</FONT> * @param x the value.<a name="line.149"></a> <FONT color="green">150</FONT> * @param epsilon When the absolute value of the nth item in the<a name="line.150"></a> <FONT color="green">151</FONT> * series is less than epsilon the approximation ceases<a name="line.151"></a> <FONT color="green">152</FONT> * to calculate further elements in the series.<a name="line.152"></a> <FONT color="green">153</FONT> * @param maxIterations Maximum number of "iterations" to complete.<a name="line.153"></a> <FONT color="green">154</FONT> * @return the regularized gamma function P(a, x)<a name="line.154"></a> <FONT color="green">155</FONT> * @throws MathException if the algorithm fails to converge.<a name="line.155"></a> <FONT color="green">156</FONT> */<a name="line.156"></a> <FONT color="green">157</FONT> public static double regularizedGammaP(double a,<a name="line.157"></a> <FONT color="green">158</FONT> double x,<a name="line.158"></a> <FONT color="green">159</FONT> double epsilon,<a name="line.159"></a> <FONT color="green">160</FONT> int maxIterations)<a name="line.160"></a> <FONT color="green">161</FONT> throws MathException<a name="line.161"></a> <FONT color="green">162</FONT> {<a name="line.162"></a> <FONT color="green">163</FONT> double ret;<a name="line.163"></a> <FONT color="green">164</FONT> <a name="line.164"></a> <FONT color="green">165</FONT> if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {<a name="line.165"></a> <FONT color="green">166</FONT> ret = Double.NaN;<a name="line.166"></a> <FONT color="green">167</FONT> } else if (x == 0.0) {<a name="line.167"></a> <FONT color="green">168</FONT> ret = 0.0;<a name="line.168"></a> <FONT color="green">169</FONT> } else if (x >= a + 1) {<a name="line.169"></a> <FONT color="green">170</FONT> // use regularizedGammaQ because it should converge faster in this<a name="line.170"></a> <FONT color="green">171</FONT> // case.<a name="line.171"></a> <FONT color="green">172</FONT> ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations);<a name="line.172"></a> <FONT color="green">173</FONT> } else {<a name="line.173"></a> <FONT color="green">174</FONT> // calculate series<a name="line.174"></a> <FONT color="green">175</FONT> double n = 0.0; // current element index<a name="line.175"></a> <FONT color="green">176</FONT> double an = 1.0 / a; // n-th element in the series<a name="line.176"></a> <FONT color="green">177</FONT> double sum = an; // partial sum<a name="line.177"></a> <FONT color="green">178</FONT> while (Math.abs(an/sum) > epsilon && n < maxIterations && sum < Double.POSITIVE_INFINITY) {<a name="line.178"></a> <FONT color="green">179</FONT> // compute next element in the series<a name="line.179"></a> <FONT color="green">180</FONT> n = n + 1.0;<a name="line.180"></a> <FONT color="green">181</FONT> an = an * (x / (a + n));<a name="line.181"></a> <FONT color="green">182</FONT> <a name="line.182"></a> <FONT color="green">183</FONT> // update partial sum<a name="line.183"></a> <FONT color="green">184</FONT> sum = sum + an;<a name="line.184"></a> <FONT color="green">185</FONT> }<a name="line.185"></a> <FONT color="green">186</FONT> if (n >= maxIterations) {<a name="line.186"></a> <FONT color="green">187</FONT> throw new MaxIterationsExceededException(maxIterations);<a name="line.187"></a> <FONT color="green">188</FONT> } else if (Double.isInfinite(sum)) {<a name="line.188"></a> <FONT color="green">189</FONT> ret = 1.0;<a name="line.189"></a> <FONT color="green">190</FONT> } else {<a name="line.190"></a> <FONT color="green">191</FONT> ret = Math.exp(-x + (a * Math.log(x)) - logGamma(a)) * sum;<a name="line.191"></a> <FONT color="green">192</FONT> }<a name="line.192"></a> <FONT color="green">193</FONT> }<a name="line.193"></a> <FONT color="green">194</FONT> <a name="line.194"></a> <FONT color="green">195</FONT> return ret;<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> * Returns the regularized gamma function Q(a, x) = 1 - P(a, x).<a name="line.199"></a> <FONT color="green">200</FONT> *<a name="line.200"></a> <FONT color="green">201</FONT> * @param a the a parameter.<a name="line.201"></a> <FONT color="green">202</FONT> * @param x the value.<a name="line.202"></a> <FONT color="green">203</FONT> * @return the regularized gamma function Q(a, x)<a name="line.203"></a> <FONT color="green">204</FONT> * @throws MathException if the algorithm fails to converge.<a name="line.204"></a> <FONT color="green">205</FONT> */<a name="line.205"></a> <FONT color="green">206</FONT> public static double regularizedGammaQ(double a, double x)<a name="line.206"></a> <FONT color="green">207</FONT> throws MathException<a name="line.207"></a> <FONT color="green">208</FONT> {<a name="line.208"></a> <FONT color="green">209</FONT> return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);<a name="line.209"></a> <FONT color="green">210</FONT> }<a name="line.210"></a> <FONT color="green">211</FONT> <a name="line.211"></a> <FONT color="green">212</FONT> /**<a name="line.212"></a> <FONT color="green">213</FONT> * Returns the regularized gamma function Q(a, x) = 1 - P(a, x).<a name="line.213"></a> <FONT color="green">214</FONT> *<a name="line.214"></a> <FONT color="green">215</FONT> * The implementation of this method is based on:<a name="line.215"></a> <FONT color="green">216</FONT> * <ul><a name="line.216"></a> <FONT color="green">217</FONT> * <li><a name="line.217"></a> <FONT color="green">218</FONT> * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"><a name="line.218"></a> <FONT color="green">219</FONT> * Regularized Gamma Function</a>, equation (1).</li><a name="line.219"></a> <FONT color="green">220</FONT> * <li><a name="line.220"></a> <FONT color="green">221</FONT> * <a href="http://functions.wolfram.com/GammaBetaErf/GammaRegularized/10/0003/"><a name="line.221"></a> <FONT color="green">222</FONT> * Regularized incomplete gamma function: Continued fraction representations (formula 06.08.10.0003)</a></li><a name="line.222"></a> <FONT color="green">223</FONT> * </ul><a name="line.223"></a> <FONT color="green">224</FONT> *<a name="line.224"></a> <FONT color="green">225</FONT> * @param a the a parameter.<a name="line.225"></a> <FONT color="green">226</FONT> * @param x the value.<a name="line.226"></a> <FONT color="green">227</FONT> * @param epsilon When the absolute value of the nth item in the<a name="line.227"></a> <FONT color="green">228</FONT> * series is less than epsilon the approximation ceases<a name="line.228"></a> <FONT color="green">229</FONT> * to calculate further elements in the series.<a name="line.229"></a> <FONT color="green">230</FONT> * @param maxIterations Maximum number of "iterations" to complete.<a name="line.230"></a> <FONT color="green">231</FONT> * @return the regularized gamma function P(a, x)<a name="line.231"></a> <FONT color="green">232</FONT> * @throws MathException if the algorithm fails to converge.<a name="line.232"></a> <FONT color="green">233</FONT> */<a name="line.233"></a> <FONT color="green">234</FONT> public static double regularizedGammaQ(final double a,<a name="line.234"></a> <FONT color="green">235</FONT> double x,<a name="line.235"></a> <FONT color="green">236</FONT> double epsilon,<a name="line.236"></a> <FONT color="green">237</FONT> int maxIterations)<a name="line.237"></a> <FONT color="green">238</FONT> throws MathException<a name="line.238"></a> <FONT color="green">239</FONT> {<a name="line.239"></a> <FONT color="green">240</FONT> double ret;<a name="line.240"></a> <FONT color="green">241</FONT> <a name="line.241"></a> <FONT color="green">242</FONT> if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {<a name="line.242"></a> <FONT color="green">243</FONT> ret = Double.NaN;<a name="line.243"></a> <FONT color="green">244</FONT> } else if (x == 0.0) {<a name="line.244"></a> <FONT color="green">245</FONT> ret = 1.0;<a name="line.245"></a> <FONT color="green">246</FONT> } else if (x < a + 1.0) {<a name="line.246"></a> <FONT color="green">247</FONT> // use regularizedGammaP because it should converge faster in this<a name="line.247"></a> <FONT color="green">248</FONT> // case.<a name="line.248"></a> <FONT color="green">249</FONT> ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations);<a name="line.249"></a> <FONT color="green">250</FONT> } else {<a name="line.250"></a> <FONT color="green">251</FONT> // create continued fraction<a name="line.251"></a> <FONT color="green">252</FONT> ContinuedFraction cf = new ContinuedFraction() {<a name="line.252"></a> <FONT color="green">253</FONT> <a name="line.253"></a> <FONT color="green">254</FONT> @Override<a name="line.254"></a> <FONT color="green">255</FONT> protected double getA(int n, double x) {<a name="line.255"></a> <FONT color="green">256</FONT> return ((2.0 * n) + 1.0) - a + x;<a name="line.256"></a> <FONT color="green">257</FONT> }<a name="line.257"></a> <FONT color="green">258</FONT> <a name="line.258"></a> <FONT color="green">259</FONT> @Override<a name="line.259"></a> <FONT color="green">260</FONT> protected double getB(int n, double x) {<a name="line.260"></a> <FONT color="green">261</FONT> return n * (a - n);<a name="line.261"></a> <FONT color="green">262</FONT> }<a name="line.262"></a> <FONT color="green">263</FONT> };<a name="line.263"></a> <FONT color="green">264</FONT> <a name="line.264"></a> <FONT color="green">265</FONT> ret = 1.0 / cf.evaluate(x, epsilon, maxIterations);<a name="line.265"></a> <FONT color="green">266</FONT> ret = Math.exp(-x + (a * Math.log(x)) - logGamma(a)) * ret;<a name="line.266"></a> <FONT color="green">267</FONT> }<a name="line.267"></a> <FONT color="green">268</FONT> <a name="line.268"></a> <FONT color="green">269</FONT> return ret;<a name="line.269"></a> <FONT color="green">270</FONT> }<a name="line.270"></a> <FONT color="green">271</FONT> <a name="line.271"></a> <FONT color="green">272</FONT> <a name="line.272"></a> <FONT color="green">273</FONT> /**<a name="line.273"></a> <FONT color="green">274</FONT> * <p>Computes the digamma function of x.</p><a name="line.274"></a> <FONT color="green">275</FONT> *<a name="line.275"></a> <FONT color="green">276</FONT> * <p>This is an independently written implementation of the algorithm described in<a name="line.276"></a> <FONT color="green">277</FONT> * Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.</p><a name="line.277"></a> <FONT color="green">278</FONT> *<a name="line.278"></a> <FONT color="green">279</FONT> * <p>Some of the constants have been changed to increase accuracy at the moderate expense<a name="line.279"></a> <FONT color="green">280</FONT> * of run-time. The result should be accurate to within 10^-8 absolute tolerance for<a name="line.280"></a> <FONT color="green">281</FONT> * x >= 10^-5 and within 10^-8 relative tolerance for x > 0.</p><a name="line.281"></a> <FONT color="green">282</FONT> *<a name="line.282"></a> <FONT color="green">283</FONT> * <p>Performance for large negative values of x will be quite expensive (proportional to<a name="line.283"></a> <FONT color="green">284</FONT> * |x|). Accuracy for negative values of x should be about 10^-8 absolute for results<a name="line.284"></a> <FONT color="green">285</FONT> * less than 10^5 and 10^-8 relative for results larger than that.</p><a name="line.285"></a> <FONT color="green">286</FONT> *<a name="line.286"></a> <FONT color="green">287</FONT> * @param x the argument<a name="line.287"></a> <FONT color="green">288</FONT> * @return digamma(x) to within 10-8 relative or absolute error whichever is smaller<a name="line.288"></a> <FONT color="green">289</FONT> * @see <a href="http://en.wikipedia.org/wiki/Digamma_function"> Digamma at wikipedia </a><a name="line.289"></a> <FONT color="green">290</FONT> * @see <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf"> Bernardo's original article </a><a name="line.290"></a> <FONT color="green">291</FONT> * @since 2.0<a name="line.291"></a> <FONT color="green">292</FONT> */<a name="line.292"></a> <FONT color="green">293</FONT> public static double digamma(double x) {<a name="line.293"></a> <FONT color="green">294</FONT> if (x > 0 && x <= S_LIMIT) {<a name="line.294"></a> <FONT color="green">295</FONT> // use method 5 from Bernardo AS103<a name="line.295"></a> <FONT color="green">296</FONT> // accurate to O(x)<a name="line.296"></a> <FONT color="green">297</FONT> return -GAMMA - 1 / x;<a name="line.297"></a> <FONT color="green">298</FONT> }<a name="line.298"></a> <FONT color="green">299</FONT> <a name="line.299"></a> <FONT color="green">300</FONT> if (x >= C_LIMIT) {<a name="line.300"></a> <FONT color="green">301</FONT> // use method 4 (accurate to O(1/x^8)<a name="line.301"></a> <FONT color="green">302</FONT> double inv = 1 / (x * x);<a name="line.302"></a> <FONT color="green">303</FONT> // 1 1 1 1<a name="line.303"></a> <FONT color="green">304</FONT> // log(x) - --- - ------ + ------- - -------<a name="line.304"></a> <FONT color="green">305</FONT> // 2 x 12 x^2 120 x^4 252 x^6<a name="line.305"></a> <FONT color="green">306</FONT> return Math.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252));<a name="line.306"></a> <FONT color="green">307</FONT> }<a name="line.307"></a> <FONT color="green">308</FONT> <a name="line.308"></a> <FONT color="green">309</FONT> return digamma(x + 1) - 1 / x;<a name="line.309"></a> <FONT color="green">310</FONT> }<a name="line.310"></a> <FONT color="green">311</FONT> <a name="line.311"></a> <FONT color="green">312</FONT> /**<a name="line.312"></a> <FONT color="green">313</FONT> * <p>Computes the trigamma function of x. This function is derived by taking the derivative of<a name="line.313"></a> <FONT color="green">314</FONT> * the implementation of digamma.</p><a name="line.314"></a> <FONT color="green">315</FONT> *<a name="line.315"></a> <FONT color="green">316</FONT> * @param x the argument<a name="line.316"></a> <FONT color="green">317</FONT> * @return trigamma(x) to within 10-8 relative or absolute error whichever is smaller<a name="line.317"></a> <FONT color="green">318</FONT> * @see <a href="http://en.wikipedia.org/wiki/Trigamma_function"> Trigamma at wikipedia </a><a name="line.318"></a> <FONT color="green">319</FONT> * @see Gamma#digamma(double)<a name="line.319"></a> <FONT color="green">320</FONT> * @since 2.0<a name="line.320"></a> <FONT color="green">321</FONT> */<a name="line.321"></a> <FONT color="green">322</FONT> public static double trigamma(double x) {<a name="line.322"></a> <FONT color="green">323</FONT> if (x > 0 && x <= S_LIMIT) {<a name="line.323"></a> <FONT color="green">324</FONT> return 1 / (x * x);<a name="line.324"></a> <FONT color="green">325</FONT> }<a name="line.325"></a> <FONT color="green">326</FONT> <a name="line.326"></a> <FONT color="green">327</FONT> if (x >= C_LIMIT) {<a name="line.327"></a> <FONT color="green">328</FONT> double inv = 1 / (x * x);<a name="line.328"></a> <FONT color="green">329</FONT> // 1 1 1 1 1<a name="line.329"></a> <FONT color="green">330</FONT> // - + ---- + ---- - ----- + -----<a name="line.330"></a> <FONT color="green">331</FONT> // x 2 3 5 7<a name="line.331"></a> <FONT color="green">332</FONT> // 2 x 6 x 30 x 42 x<a name="line.332"></a> <FONT color="green">333</FONT> return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42));<a name="line.333"></a> <FONT color="green">334</FONT> }<a name="line.334"></a> <FONT color="green">335</FONT> <a name="line.335"></a> <FONT color="green">336</FONT> return trigamma(x + 1) + 1 / (x * x);<a name="line.336"></a> <FONT color="green">337</FONT> }<a name="line.337"></a> <FONT color="green">338</FONT> }<a name="line.338"></a> </PRE> </BODY> </HTML>