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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.polynomials;<a name="line.17"></a> <FONT color="green">018</FONT> <a name="line.18"></a> <FONT color="green">019</FONT> import java.util.ArrayList;<a name="line.19"></a> <FONT color="green">020</FONT> <a name="line.20"></a> <FONT color="green">021</FONT> import org.apache.commons.math.fraction.BigFraction;<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> * A collection of static methods that operate on or return polynomials.<a name="line.24"></a> <FONT color="green">025</FONT> *<a name="line.25"></a> <FONT color="green">026</FONT> * @version $Revision: 811685 $ $Date: 2009-09-05 13:36:48 -0400 (Sat, 05 Sep 2009) $<a name="line.26"></a> <FONT color="green">027</FONT> * @since 2.0<a name="line.27"></a> <FONT color="green">028</FONT> */<a name="line.28"></a> <FONT color="green">029</FONT> public class PolynomialsUtils {<a name="line.29"></a> <FONT color="green">030</FONT> <a name="line.30"></a> <FONT color="green">031</FONT> /** Coefficients for Chebyshev polynomials. */<a name="line.31"></a> <FONT color="green">032</FONT> private static final ArrayList<BigFraction> CHEBYSHEV_COEFFICIENTS;<a name="line.32"></a> <FONT color="green">033</FONT> <a name="line.33"></a> <FONT color="green">034</FONT> /** Coefficients for Hermite polynomials. */<a name="line.34"></a> <FONT color="green">035</FONT> private static final ArrayList<BigFraction> HERMITE_COEFFICIENTS;<a name="line.35"></a> <FONT color="green">036</FONT> <a name="line.36"></a> <FONT color="green">037</FONT> /** Coefficients for Laguerre polynomials. */<a name="line.37"></a> <FONT color="green">038</FONT> private static final ArrayList<BigFraction> LAGUERRE_COEFFICIENTS;<a name="line.38"></a> <FONT color="green">039</FONT> <a name="line.39"></a> <FONT color="green">040</FONT> /** Coefficients for Legendre polynomials. */<a name="line.40"></a> <FONT color="green">041</FONT> private static final ArrayList<BigFraction> LEGENDRE_COEFFICIENTS;<a name="line.41"></a> <FONT color="green">042</FONT> <a name="line.42"></a> <FONT color="green">043</FONT> static {<a name="line.43"></a> <FONT color="green">044</FONT> <a name="line.44"></a> <FONT color="green">045</FONT> // initialize recurrence for Chebyshev polynomials<a name="line.45"></a> <FONT color="green">046</FONT> // T0(X) = 1, T1(X) = 0 + 1 * X<a name="line.46"></a> <FONT color="green">047</FONT> CHEBYSHEV_COEFFICIENTS = new ArrayList<BigFraction>();<a name="line.47"></a> <FONT color="green">048</FONT> CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE);<a name="line.48"></a> <FONT color="green">049</FONT> CHEBYSHEV_COEFFICIENTS.add(BigFraction.ZERO);<a name="line.49"></a> <FONT color="green">050</FONT> CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE);<a name="line.50"></a> <FONT color="green">051</FONT> <a name="line.51"></a> <FONT color="green">052</FONT> // initialize recurrence for Hermite polynomials<a name="line.52"></a> <FONT color="green">053</FONT> // H0(X) = 1, H1(X) = 0 + 2 * X<a name="line.53"></a> <FONT color="green">054</FONT> HERMITE_COEFFICIENTS = new ArrayList<BigFraction>();<a name="line.54"></a> <FONT color="green">055</FONT> HERMITE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.55"></a> <FONT color="green">056</FONT> HERMITE_COEFFICIENTS.add(BigFraction.ZERO);<a name="line.56"></a> <FONT color="green">057</FONT> HERMITE_COEFFICIENTS.add(BigFraction.TWO);<a name="line.57"></a> <FONT color="green">058</FONT> <a name="line.58"></a> <FONT color="green">059</FONT> // initialize recurrence for Laguerre polynomials<a name="line.59"></a> <FONT color="green">060</FONT> // L0(X) = 1, L1(X) = 1 - 1 * X<a name="line.60"></a> <FONT color="green">061</FONT> LAGUERRE_COEFFICIENTS = new ArrayList<BigFraction>();<a name="line.61"></a> <FONT color="green">062</FONT> LAGUERRE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.62"></a> <FONT color="green">063</FONT> LAGUERRE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.63"></a> <FONT color="green">064</FONT> LAGUERRE_COEFFICIENTS.add(BigFraction.MINUS_ONE);<a name="line.64"></a> <FONT color="green">065</FONT> <a name="line.65"></a> <FONT color="green">066</FONT> // initialize recurrence for Legendre polynomials<a name="line.66"></a> <FONT color="green">067</FONT> // P0(X) = 1, P1(X) = 0 + 1 * X<a name="line.67"></a> <FONT color="green">068</FONT> LEGENDRE_COEFFICIENTS = new ArrayList<BigFraction>();<a name="line.68"></a> <FONT color="green">069</FONT> LEGENDRE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.69"></a> <FONT color="green">070</FONT> LEGENDRE_COEFFICIENTS.add(BigFraction.ZERO);<a name="line.70"></a> <FONT color="green">071</FONT> LEGENDRE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.71"></a> <FONT color="green">072</FONT> <a name="line.72"></a> <FONT color="green">073</FONT> }<a name="line.73"></a> <FONT color="green">074</FONT> <a name="line.74"></a> <FONT color="green">075</FONT> /**<a name="line.75"></a> <FONT color="green">076</FONT> * Private constructor, to prevent instantiation.<a name="line.76"></a> <FONT color="green">077</FONT> */<a name="line.77"></a> <FONT color="green">078</FONT> private PolynomialsUtils() {<a name="line.78"></a> <FONT color="green">079</FONT> }<a name="line.79"></a> <FONT color="green">080</FONT> <a name="line.80"></a> <FONT color="green">081</FONT> /**<a name="line.81"></a> <FONT color="green">082</FONT> * Create a Chebyshev polynomial of the first kind.<a name="line.82"></a> <FONT color="green">083</FONT> * <p><a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev<a name="line.83"></a> <FONT color="green">084</FONT> * polynomials of the first kind</a> are orthogonal polynomials.<a name="line.84"></a> <FONT color="green">085</FONT> * They can be defined by the following recurrence relations:<a name="line.85"></a> <FONT color="green">086</FONT> * <pre><a name="line.86"></a> <FONT color="green">087</FONT> * T<sub>0</sub>(X) = 1<a name="line.87"></a> <FONT color="green">088</FONT> * T<sub>1</sub>(X) = X<a name="line.88"></a> <FONT color="green">089</FONT> * T<sub>k+1</sub>(X) = 2X T<sub>k</sub>(X) - T<sub>k-1</sub>(X)<a name="line.89"></a> <FONT color="green">090</FONT> * </pre></p><a name="line.90"></a> <FONT color="green">091</FONT> * @param degree degree of the polynomial<a name="line.91"></a> <FONT color="green">092</FONT> * @return Chebyshev polynomial of specified degree<a name="line.92"></a> <FONT color="green">093</FONT> */<a name="line.93"></a> <FONT color="green">094</FONT> public static PolynomialFunction createChebyshevPolynomial(final int degree) {<a name="line.94"></a> <FONT color="green">095</FONT> return buildPolynomial(degree, CHEBYSHEV_COEFFICIENTS,<a name="line.95"></a> <FONT color="green">096</FONT> new RecurrenceCoefficientsGenerator() {<a name="line.96"></a> <FONT color="green">097</FONT> private final BigFraction[] coeffs = { BigFraction.ZERO, BigFraction.TWO, BigFraction.ONE };<a name="line.97"></a> <FONT color="green">098</FONT> /** {@inheritDoc} */<a name="line.98"></a> <FONT color="green">099</FONT> public BigFraction[] generate(int k) {<a name="line.99"></a> <FONT color="green">100</FONT> return coeffs;<a name="line.100"></a> <FONT color="green">101</FONT> }<a name="line.101"></a> <FONT color="green">102</FONT> });<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> /**<a name="line.105"></a> <FONT color="green">106</FONT> * Create a Hermite polynomial.<a name="line.106"></a> <FONT color="green">107</FONT> * <p><a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite<a name="line.107"></a> <FONT color="green">108</FONT> * polynomials</a> are orthogonal polynomials.<a name="line.108"></a> <FONT color="green">109</FONT> * They can be defined by the following recurrence relations:<a name="line.109"></a> <FONT color="green">110</FONT> * <pre><a name="line.110"></a> <FONT color="green">111</FONT> * H<sub>0</sub>(X) = 1<a name="line.111"></a> <FONT color="green">112</FONT> * H<sub>1</sub>(X) = 2X<a name="line.112"></a> <FONT color="green">113</FONT> * H<sub>k+1</sub>(X) = 2X H<sub>k</sub>(X) - 2k H<sub>k-1</sub>(X)<a name="line.113"></a> <FONT color="green">114</FONT> * </pre></p><a name="line.114"></a> <FONT color="green">115</FONT> <a name="line.115"></a> <FONT color="green">116</FONT> * @param degree degree of the polynomial<a name="line.116"></a> <FONT color="green">117</FONT> * @return Hermite polynomial of specified degree<a name="line.117"></a> <FONT color="green">118</FONT> */<a name="line.118"></a> <FONT color="green">119</FONT> public static PolynomialFunction createHermitePolynomial(final int degree) {<a name="line.119"></a> <FONT color="green">120</FONT> return buildPolynomial(degree, HERMITE_COEFFICIENTS,<a name="line.120"></a> <FONT color="green">121</FONT> new RecurrenceCoefficientsGenerator() {<a name="line.121"></a> <FONT color="green">122</FONT> /** {@inheritDoc} */<a name="line.122"></a> <FONT color="green">123</FONT> public BigFraction[] generate(int k) {<a name="line.123"></a> <FONT color="green">124</FONT> return new BigFraction[] {<a name="line.124"></a> <FONT color="green">125</FONT> BigFraction.ZERO,<a name="line.125"></a> <FONT color="green">126</FONT> BigFraction.TWO,<a name="line.126"></a> <FONT color="green">127</FONT> new BigFraction(2 * k)};<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> /**<a name="line.132"></a> <FONT color="green">133</FONT> * Create a Laguerre polynomial.<a name="line.133"></a> <FONT color="green">134</FONT> * <p><a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre<a name="line.134"></a> <FONT color="green">135</FONT> * polynomials</a> are orthogonal polynomials.<a name="line.135"></a> <FONT color="green">136</FONT> * They can be defined by the following recurrence relations:<a name="line.136"></a> <FONT color="green">137</FONT> * <pre><a name="line.137"></a> <FONT color="green">138</FONT> * L<sub>0</sub>(X) = 1<a name="line.138"></a> <FONT color="green">139</FONT> * L<sub>1</sub>(X) = 1 - X<a name="line.139"></a> <FONT color="green">140</FONT> * (k+1) L<sub>k+1</sub>(X) = (2k + 1 - X) L<sub>k</sub>(X) - k L<sub>k-1</sub>(X)<a name="line.140"></a> <FONT color="green">141</FONT> * </pre></p><a name="line.141"></a> <FONT color="green">142</FONT> * @param degree degree of the polynomial<a name="line.142"></a> <FONT color="green">143</FONT> * @return Laguerre polynomial of specified degree<a name="line.143"></a> <FONT color="green">144</FONT> */<a name="line.144"></a> <FONT color="green">145</FONT> public static PolynomialFunction createLaguerrePolynomial(final int degree) {<a name="line.145"></a> <FONT color="green">146</FONT> return buildPolynomial(degree, LAGUERRE_COEFFICIENTS,<a name="line.146"></a> <FONT color="green">147</FONT> new RecurrenceCoefficientsGenerator() {<a name="line.147"></a> <FONT color="green">148</FONT> /** {@inheritDoc} */<a name="line.148"></a> <FONT color="green">149</FONT> public BigFraction[] generate(int k) {<a name="line.149"></a> <FONT color="green">150</FONT> final int kP1 = k + 1;<a name="line.150"></a> <FONT color="green">151</FONT> return new BigFraction[] {<a name="line.151"></a> <FONT color="green">152</FONT> new BigFraction(2 * k + 1, kP1),<a name="line.152"></a> <FONT color="green">153</FONT> new BigFraction(-1, kP1),<a name="line.153"></a> <FONT color="green">154</FONT> new BigFraction(k, kP1)};<a name="line.154"></a> <FONT color="green">155</FONT> }<a name="line.155"></a> <FONT color="green">156</FONT> });<a name="line.156"></a> <FONT color="green">157</FONT> }<a name="line.157"></a> <FONT color="green">158</FONT> <a name="line.158"></a> <FONT color="green">159</FONT> /**<a name="line.159"></a> <FONT color="green">160</FONT> * Create a Legendre polynomial.<a name="line.160"></a> <FONT color="green">161</FONT> * <p><a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre<a name="line.161"></a> <FONT color="green">162</FONT> * polynomials</a> are orthogonal polynomials.<a name="line.162"></a> <FONT color="green">163</FONT> * They can be defined by the following recurrence relations:<a name="line.163"></a> <FONT color="green">164</FONT> * <pre><a name="line.164"></a> <FONT color="green">165</FONT> * P<sub>0</sub>(X) = 1<a name="line.165"></a> <FONT color="green">166</FONT> * P<sub>1</sub>(X) = X<a name="line.166"></a> <FONT color="green">167</FONT> * (k+1) P<sub>k+1</sub>(X) = (2k+1) X P<sub>k</sub>(X) - k P<sub>k-1</sub>(X)<a name="line.167"></a> <FONT color="green">168</FONT> * </pre></p><a name="line.168"></a> <FONT color="green">169</FONT> * @param degree degree of the polynomial<a name="line.169"></a> <FONT color="green">170</FONT> * @return Legendre polynomial of specified degree<a name="line.170"></a> <FONT color="green">171</FONT> */<a name="line.171"></a> <FONT color="green">172</FONT> public static PolynomialFunction createLegendrePolynomial(final int degree) {<a name="line.172"></a> <FONT color="green">173</FONT> return buildPolynomial(degree, LEGENDRE_COEFFICIENTS,<a name="line.173"></a> <FONT color="green">174</FONT> new RecurrenceCoefficientsGenerator() {<a name="line.174"></a> <FONT color="green">175</FONT> /** {@inheritDoc} */<a name="line.175"></a> <FONT color="green">176</FONT> public BigFraction[] generate(int k) {<a name="line.176"></a> <FONT color="green">177</FONT> final int kP1 = k + 1;<a name="line.177"></a> <FONT color="green">178</FONT> return new BigFraction[] {<a name="line.178"></a> <FONT color="green">179</FONT> BigFraction.ZERO,<a name="line.179"></a> <FONT color="green">180</FONT> new BigFraction(k + kP1, kP1),<a name="line.180"></a> <FONT color="green">181</FONT> new BigFraction(k, kP1)};<a name="line.181"></a> <FONT color="green">182</FONT> }<a name="line.182"></a> <FONT color="green">183</FONT> });<a name="line.183"></a> <FONT color="green">184</FONT> }<a name="line.184"></a> <FONT color="green">185</FONT> <a name="line.185"></a> <FONT color="green">186</FONT> /** Get the coefficients array for a given degree.<a name="line.186"></a> <FONT color="green">187</FONT> * @param degree degree of the polynomial<a name="line.187"></a> <FONT color="green">188</FONT> * @param coefficients list where the computed coefficients are stored<a name="line.188"></a> <FONT color="green">189</FONT> * @param generator recurrence coefficients generator<a name="line.189"></a> <FONT color="green">190</FONT> * @return coefficients array<a name="line.190"></a> <FONT color="green">191</FONT> */<a name="line.191"></a> <FONT color="green">192</FONT> private static PolynomialFunction buildPolynomial(final int degree,<a name="line.192"></a> <FONT color="green">193</FONT> final ArrayList<BigFraction> coefficients,<a name="line.193"></a> <FONT color="green">194</FONT> final RecurrenceCoefficientsGenerator generator) {<a name="line.194"></a> <FONT color="green">195</FONT> <a name="line.195"></a> <FONT color="green">196</FONT> final int maxDegree = (int) Math.floor(Math.sqrt(2 * coefficients.size())) - 1;<a name="line.196"></a> <FONT color="green">197</FONT> synchronized (PolynomialsUtils.class) {<a name="line.197"></a> <FONT color="green">198</FONT> if (degree > maxDegree) {<a name="line.198"></a> <FONT color="green">199</FONT> computeUpToDegree(degree, maxDegree, generator, coefficients);<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> // coefficient for polynomial 0 is l [0]<a name="line.203"></a> <FONT color="green">204</FONT> // coefficients for polynomial 1 are l [1] ... l [2] (degrees 0 ... 1)<a name="line.204"></a> <FONT color="green">205</FONT> // coefficients for polynomial 2 are l [3] ... l [5] (degrees 0 ... 2)<a name="line.205"></a> <FONT color="green">206</FONT> // coefficients for polynomial 3 are l [6] ... l [9] (degrees 0 ... 3)<a name="line.206"></a> <FONT color="green">207</FONT> // coefficients for polynomial 4 are l[10] ... l[14] (degrees 0 ... 4)<a name="line.207"></a> <FONT color="green">208</FONT> // coefficients for polynomial 5 are l[15] ... l[20] (degrees 0 ... 5)<a name="line.208"></a> <FONT color="green">209</FONT> // coefficients for polynomial 6 are l[21] ... l[27] (degrees 0 ... 6)<a name="line.209"></a> <FONT color="green">210</FONT> // ...<a name="line.210"></a> <FONT color="green">211</FONT> final int start = degree * (degree + 1) / 2;<a name="line.211"></a> <FONT color="green">212</FONT> <a name="line.212"></a> <FONT color="green">213</FONT> final double[] a = new double[degree + 1];<a name="line.213"></a> <FONT color="green">214</FONT> for (int i = 0; i <= degree; ++i) {<a name="line.214"></a> <FONT color="green">215</FONT> a[i] = coefficients.get(start + i).doubleValue();<a name="line.215"></a> <FONT color="green">216</FONT> }<a name="line.216"></a> <FONT color="green">217</FONT> <a name="line.217"></a> <FONT color="green">218</FONT> // build the polynomial<a name="line.218"></a> <FONT color="green">219</FONT> return new PolynomialFunction(a);<a name="line.219"></a> <FONT color="green">220</FONT> <a name="line.220"></a> <FONT color="green">221</FONT> }<a name="line.221"></a> <FONT color="green">222</FONT> <a name="line.222"></a> <FONT color="green">223</FONT> /** Compute polynomial coefficients up to a given degree.<a name="line.223"></a> <FONT color="green">224</FONT> * @param degree maximal degree<a name="line.224"></a> <FONT color="green">225</FONT> * @param maxDegree current maximal degree<a name="line.225"></a> <FONT color="green">226</FONT> * @param generator recurrence coefficients generator<a name="line.226"></a> <FONT color="green">227</FONT> * @param coefficients list where the computed coefficients should be appended<a name="line.227"></a> <FONT color="green">228</FONT> */<a name="line.228"></a> <FONT color="green">229</FONT> private static void computeUpToDegree(final int degree, final int maxDegree,<a name="line.229"></a> <FONT color="green">230</FONT> final RecurrenceCoefficientsGenerator generator,<a name="line.230"></a> <FONT color="green">231</FONT> final ArrayList<BigFraction> coefficients) {<a name="line.231"></a> <FONT color="green">232</FONT> <a name="line.232"></a> <FONT color="green">233</FONT> int startK = (maxDegree - 1) * maxDegree / 2;<a name="line.233"></a> <FONT color="green">234</FONT> for (int k = maxDegree; k < degree; ++k) {<a name="line.234"></a> <FONT color="green">235</FONT> <a name="line.235"></a> <FONT color="green">236</FONT> // start indices of two previous polynomials Pk(X) and Pk-1(X)<a name="line.236"></a> <FONT color="green">237</FONT> int startKm1 = startK;<a name="line.237"></a> <FONT color="green">238</FONT> startK += k;<a name="line.238"></a> <FONT color="green">239</FONT> <a name="line.239"></a> <FONT color="green">240</FONT> // Pk+1(X) = (a[0] + a[1] X) Pk(X) - a[2] Pk-1(X)<a name="line.240"></a> <FONT color="green">241</FONT> BigFraction[] ai = generator.generate(k);<a name="line.241"></a> <FONT color="green">242</FONT> <a name="line.242"></a> <FONT color="green">243</FONT> BigFraction ck = coefficients.get(startK);<a name="line.243"></a> <FONT color="green">244</FONT> BigFraction ckm1 = coefficients.get(startKm1);<a name="line.244"></a> <FONT color="green">245</FONT> <a name="line.245"></a> <FONT color="green">246</FONT> // degree 0 coefficient<a name="line.246"></a> <FONT color="green">247</FONT> coefficients.add(ck.multiply(ai[0]).subtract(ckm1.multiply(ai[2])));<a name="line.247"></a> <FONT color="green">248</FONT> <a name="line.248"></a> <FONT color="green">249</FONT> // degree 1 to degree k-1 coefficients<a name="line.249"></a> <FONT color="green">250</FONT> for (int i = 1; i < k; ++i) {<a name="line.250"></a> <FONT color="green">251</FONT> final BigFraction ckPrev = ck;<a name="line.251"></a> <FONT color="green">252</FONT> ck = coefficients.get(startK + i);<a name="line.252"></a> <FONT color="green">253</FONT> ckm1 = coefficients.get(startKm1 + i);<a name="line.253"></a> <FONT color="green">254</FONT> coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1])).subtract(ckm1.multiply(ai[2])));<a name="line.254"></a> <FONT color="green">255</FONT> }<a name="line.255"></a> <FONT color="green">256</FONT> <a name="line.256"></a> <FONT color="green">257</FONT> // degree k coefficient<a name="line.257"></a> <FONT color="green">258</FONT> final BigFraction ckPrev = ck;<a name="line.258"></a> <FONT color="green">259</FONT> ck = coefficients.get(startK + k);<a name="line.259"></a> <FONT color="green">260</FONT> coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1])));<a name="line.260"></a> <FONT color="green">261</FONT> <a name="line.261"></a> <FONT color="green">262</FONT> // degree k+1 coefficient<a name="line.262"></a> <FONT color="green">263</FONT> coefficients.add(ck.multiply(ai[1]));<a name="line.263"></a> <FONT color="green">264</FONT> <a name="line.264"></a> <FONT color="green">265</FONT> }<a name="line.265"></a> <FONT color="green">266</FONT> <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> /** Interface for recurrence coefficients generation. */<a name="line.269"></a> <FONT color="green">270</FONT> private static interface RecurrenceCoefficientsGenerator {<a name="line.270"></a> <FONT color="green">271</FONT> /**<a name="line.271"></a> <FONT color="green">272</FONT> * Generate recurrence coefficients.<a name="line.272"></a> <FONT color="green">273</FONT> * @param k highest degree of the polynomials used in the recurrence<a name="line.273"></a> <FONT color="green">274</FONT> * @return an array of three coefficients such that<a name="line.274"></a> <FONT color="green">275</FONT> * P<sub>k+1</sub>(X) = (a[0] + a[1] X) P<sub>k</sub>(X) - a[2] P<sub>k-1</sub>(X)<a name="line.275"></a> <FONT color="green">276</FONT> */<a name="line.276"></a> <FONT color="green">277</FONT> BigFraction[] generate(int k);<a name="line.277"></a> <FONT color="green">278</FONT> }<a name="line.278"></a> <FONT color="green">279</FONT> <a name="line.279"></a> <FONT color="green">280</FONT> }<a name="line.280"></a> </PRE> </BODY> </HTML>