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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 org.apache.commons.math.FunctionEvaluationException;<a name="line.19"></a> <FONT color="green">020</FONT> import org.apache.commons.math.MathRuntimeException;<a name="line.20"></a> <FONT color="green">021</FONT> import org.apache.commons.math.analysis.UnivariateRealFunction;<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> * Implements the representation of a real polynomial function in<a name="line.24"></a> <FONT color="green">025</FONT> * Newton Form. For reference, see <b>Elementary Numerical Analysis</b>,<a name="line.25"></a> <FONT color="green">026</FONT> * ISBN 0070124477, chapter 2.<a name="line.26"></a> <FONT color="green">027</FONT> * <p><a name="line.27"></a> <FONT color="green">028</FONT> * The formula of polynomial in Newton form is<a name="line.28"></a> <FONT color="green">029</FONT> * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +<a name="line.29"></a> <FONT color="green">030</FONT> * a[n](x-c[0])(x-c[1])...(x-c[n-1])<a name="line.30"></a> <FONT color="green">031</FONT> * Note that the length of a[] is one more than the length of c[]</p><a name="line.31"></a> <FONT color="green">032</FONT> *<a name="line.32"></a> <FONT color="green">033</FONT> * @version $Revision: 922708 $ $Date: 2010-03-13 20:15:47 -0500 (Sat, 13 Mar 2010) $<a name="line.33"></a> <FONT color="green">034</FONT> * @since 1.2<a name="line.34"></a> <FONT color="green">035</FONT> */<a name="line.35"></a> <FONT color="green">036</FONT> public class PolynomialFunctionNewtonForm implements UnivariateRealFunction {<a name="line.36"></a> <FONT color="green">037</FONT> <a name="line.37"></a> <FONT color="green">038</FONT> /**<a name="line.38"></a> <FONT color="green">039</FONT> * The coefficients of the polynomial, ordered by degree -- i.e.<a name="line.39"></a> <FONT color="green">040</FONT> * coefficients[0] is the constant term and coefficients[n] is the<a name="line.40"></a> <FONT color="green">041</FONT> * coefficient of x^n where n is the degree of the polynomial.<a name="line.41"></a> <FONT color="green">042</FONT> */<a name="line.42"></a> <FONT color="green">043</FONT> private double coefficients[];<a name="line.43"></a> <FONT color="green">044</FONT> <a name="line.44"></a> <FONT color="green">045</FONT> /**<a name="line.45"></a> <FONT color="green">046</FONT> * Centers of the Newton polynomial.<a name="line.46"></a> <FONT color="green">047</FONT> */<a name="line.47"></a> <FONT color="green">048</FONT> private final double c[];<a name="line.48"></a> <FONT color="green">049</FONT> <a name="line.49"></a> <FONT color="green">050</FONT> /**<a name="line.50"></a> <FONT color="green">051</FONT> * When all c[i] = 0, a[] becomes normal polynomial coefficients,<a name="line.51"></a> <FONT color="green">052</FONT> * i.e. a[i] = coefficients[i].<a name="line.52"></a> <FONT color="green">053</FONT> */<a name="line.53"></a> <FONT color="green">054</FONT> private final double a[];<a name="line.54"></a> <FONT color="green">055</FONT> <a name="line.55"></a> <FONT color="green">056</FONT> /**<a name="line.56"></a> <FONT color="green">057</FONT> * Whether the polynomial coefficients are available.<a name="line.57"></a> <FONT color="green">058</FONT> */<a name="line.58"></a> <FONT color="green">059</FONT> private boolean coefficientsComputed;<a name="line.59"></a> <FONT color="green">060</FONT> <a name="line.60"></a> <FONT color="green">061</FONT> /**<a name="line.61"></a> <FONT color="green">062</FONT> * Construct a Newton polynomial with the given a[] and c[]. The order of<a name="line.62"></a> <FONT color="green">063</FONT> * centers are important in that if c[] shuffle, then values of a[] would<a name="line.63"></a> <FONT color="green">064</FONT> * completely change, not just a permutation of old a[].<a name="line.64"></a> <FONT color="green">065</FONT> * <p><a name="line.65"></a> <FONT color="green">066</FONT> * The constructor makes copy of the input arrays and assigns them.</p><a name="line.66"></a> <FONT color="green">067</FONT> *<a name="line.67"></a> <FONT color="green">068</FONT> * @param a the coefficients in Newton form formula<a name="line.68"></a> <FONT color="green">069</FONT> * @param c the centers<a name="line.69"></a> <FONT color="green">070</FONT> * @throws IllegalArgumentException if input arrays are not valid<a name="line.70"></a> <FONT color="green">071</FONT> */<a name="line.71"></a> <FONT color="green">072</FONT> public PolynomialFunctionNewtonForm(double a[], double c[])<a name="line.72"></a> <FONT color="green">073</FONT> throws IllegalArgumentException {<a name="line.73"></a> <FONT color="green">074</FONT> <a name="line.74"></a> <FONT color="green">075</FONT> verifyInputArray(a, c);<a name="line.75"></a> <FONT color="green">076</FONT> this.a = new double[a.length];<a name="line.76"></a> <FONT color="green">077</FONT> this.c = new double[c.length];<a name="line.77"></a> <FONT color="green">078</FONT> System.arraycopy(a, 0, this.a, 0, a.length);<a name="line.78"></a> <FONT color="green">079</FONT> System.arraycopy(c, 0, this.c, 0, c.length);<a name="line.79"></a> <FONT color="green">080</FONT> coefficientsComputed = false;<a name="line.80"></a> <FONT color="green">081</FONT> }<a name="line.81"></a> <FONT color="green">082</FONT> <a name="line.82"></a> <FONT color="green">083</FONT> /**<a name="line.83"></a> <FONT color="green">084</FONT> * Calculate the function value at the given point.<a name="line.84"></a> <FONT color="green">085</FONT> *<a name="line.85"></a> <FONT color="green">086</FONT> * @param z the point at which the function value is to be computed<a name="line.86"></a> <FONT color="green">087</FONT> * @return the function value<a name="line.87"></a> <FONT color="green">088</FONT> * @throws FunctionEvaluationException if a runtime error occurs<a name="line.88"></a> <FONT color="green">089</FONT> * @see UnivariateRealFunction#value(double)<a name="line.89"></a> <FONT color="green">090</FONT> */<a name="line.90"></a> <FONT color="green">091</FONT> public double value(double z) throws FunctionEvaluationException {<a name="line.91"></a> <FONT color="green">092</FONT> return evaluate(a, c, z);<a name="line.92"></a> <FONT color="green">093</FONT> }<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> * Returns the degree of the polynomial.<a name="line.96"></a> <FONT color="green">097</FONT> *<a name="line.97"></a> <FONT color="green">098</FONT> * @return the degree of the polynomial<a name="line.98"></a> <FONT color="green">099</FONT> */<a name="line.99"></a> <FONT color="green">100</FONT> public int degree() {<a name="line.100"></a> <FONT color="green">101</FONT> return c.length;<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> * Returns a copy of coefficients in Newton form formula.<a name="line.105"></a> <FONT color="green">106</FONT> * <p><a name="line.106"></a> <FONT color="green">107</FONT> * Changes made to the returned copy will not affect the polynomial.</p><a name="line.107"></a> <FONT color="green">108</FONT> *<a name="line.108"></a> <FONT color="green">109</FONT> * @return a fresh copy of coefficients in Newton form formula<a name="line.109"></a> <FONT color="green">110</FONT> */<a name="line.110"></a> <FONT color="green">111</FONT> public double[] getNewtonCoefficients() {<a name="line.111"></a> <FONT color="green">112</FONT> double[] out = new double[a.length];<a name="line.112"></a> <FONT color="green">113</FONT> System.arraycopy(a, 0, out, 0, a.length);<a name="line.113"></a> <FONT color="green">114</FONT> return out;<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> /**<a name="line.117"></a> <FONT color="green">118</FONT> * Returns a copy of the centers array.<a name="line.118"></a> <FONT color="green">119</FONT> * <p><a name="line.119"></a> <FONT color="green">120</FONT> * Changes made to the returned copy will not affect the polynomial.</p><a name="line.120"></a> <FONT color="green">121</FONT> *<a name="line.121"></a> <FONT color="green">122</FONT> * @return a fresh copy of the centers array<a name="line.122"></a> <FONT color="green">123</FONT> */<a name="line.123"></a> <FONT color="green">124</FONT> public double[] getCenters() {<a name="line.124"></a> <FONT color="green">125</FONT> double[] out = new double[c.length];<a name="line.125"></a> <FONT color="green">126</FONT> System.arraycopy(c, 0, out, 0, c.length);<a name="line.126"></a> <FONT color="green">127</FONT> return out;<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> * Returns a copy of the coefficients array.<a name="line.131"></a> <FONT color="green">132</FONT> * <p><a name="line.132"></a> <FONT color="green">133</FONT> * Changes made to the returned copy will not affect the polynomial.</p><a name="line.133"></a> <FONT color="green">134</FONT> *<a name="line.134"></a> <FONT color="green">135</FONT> * @return a fresh copy of the coefficients array<a name="line.135"></a> <FONT color="green">136</FONT> */<a name="line.136"></a> <FONT color="green">137</FONT> public double[] getCoefficients() {<a name="line.137"></a> <FONT color="green">138</FONT> if (!coefficientsComputed) {<a name="line.138"></a> <FONT color="green">139</FONT> computeCoefficients();<a name="line.139"></a> <FONT color="green">140</FONT> }<a name="line.140"></a> <FONT color="green">141</FONT> double[] out = new double[coefficients.length];<a name="line.141"></a> <FONT color="green">142</FONT> System.arraycopy(coefficients, 0, out, 0, coefficients.length);<a name="line.142"></a> <FONT color="green">143</FONT> return out;<a name="line.143"></a> <FONT color="green">144</FONT> }<a name="line.144"></a> <FONT color="green">145</FONT> <a name="line.145"></a> <FONT color="green">146</FONT> /**<a name="line.146"></a> <FONT color="green">147</FONT> * Evaluate the Newton polynomial using nested multiplication. It is<a name="line.147"></a> <FONT color="green">148</FONT> * also called <a href="http://mathworld.wolfram.com/HornersRule.html"><a name="line.148"></a> <FONT color="green">149</FONT> * Horner's Rule</a> and takes O(N) time.<a name="line.149"></a> <FONT color="green">150</FONT> *<a name="line.150"></a> <FONT color="green">151</FONT> * @param a the coefficients in Newton form formula<a name="line.151"></a> <FONT color="green">152</FONT> * @param c the centers<a name="line.152"></a> <FONT color="green">153</FONT> * @param z the point at which the function value is to be computed<a name="line.153"></a> <FONT color="green">154</FONT> * @return the function value<a name="line.154"></a> <FONT color="green">155</FONT> * @throws FunctionEvaluationException if a runtime error occurs<a name="line.155"></a> <FONT color="green">156</FONT> * @throws IllegalArgumentException if inputs are not valid<a name="line.156"></a> <FONT color="green">157</FONT> */<a name="line.157"></a> <FONT color="green">158</FONT> public static double evaluate(double a[], double c[], double z) throws<a name="line.158"></a> <FONT color="green">159</FONT> FunctionEvaluationException, IllegalArgumentException {<a name="line.159"></a> <FONT color="green">160</FONT> <a name="line.160"></a> <FONT color="green">161</FONT> verifyInputArray(a, c);<a name="line.161"></a> <FONT color="green">162</FONT> <a name="line.162"></a> <FONT color="green">163</FONT> int n = c.length;<a name="line.163"></a> <FONT color="green">164</FONT> double value = a[n];<a name="line.164"></a> <FONT color="green">165</FONT> for (int i = n-1; i >= 0; i--) {<a name="line.165"></a> <FONT color="green">166</FONT> value = a[i] + (z - c[i]) * value;<a name="line.166"></a> <FONT color="green">167</FONT> }<a name="line.167"></a> <FONT color="green">168</FONT> <a name="line.168"></a> <FONT color="green">169</FONT> return value;<a name="line.169"></a> <FONT color="green">170</FONT> }<a name="line.170"></a> <FONT color="green">171</FONT> <a name="line.171"></a> <FONT color="green">172</FONT> /**<a name="line.172"></a> <FONT color="green">173</FONT> * Calculate the normal polynomial coefficients given the Newton form.<a name="line.173"></a> <FONT color="green">174</FONT> * It also uses nested multiplication but takes O(N^2) time.<a name="line.174"></a> <FONT color="green">175</FONT> */<a name="line.175"></a> <FONT color="green">176</FONT> protected void computeCoefficients() {<a name="line.176"></a> <FONT color="green">177</FONT> final int n = degree();<a name="line.177"></a> <FONT color="green">178</FONT> <a name="line.178"></a> <FONT color="green">179</FONT> coefficients = new double[n+1];<a name="line.179"></a> <FONT color="green">180</FONT> for (int i = 0; i <= n; i++) {<a name="line.180"></a> <FONT color="green">181</FONT> coefficients[i] = 0.0;<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> coefficients[0] = a[n];<a name="line.184"></a> <FONT color="green">185</FONT> for (int i = n-1; i >= 0; i--) {<a name="line.185"></a> <FONT color="green">186</FONT> for (int j = n-i; j > 0; j--) {<a name="line.186"></a> <FONT color="green">187</FONT> coefficients[j] = coefficients[j-1] - c[i] * coefficients[j];<a name="line.187"></a> <FONT color="green">188</FONT> }<a name="line.188"></a> <FONT color="green">189</FONT> coefficients[0] = a[i] - c[i] * coefficients[0];<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> coefficientsComputed = true;<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> /**<a name="line.195"></a> <FONT color="green">196</FONT> * Verifies that the input arrays are valid.<a name="line.196"></a> <FONT color="green">197</FONT> * <p><a name="line.197"></a> <FONT color="green">198</FONT> * The centers must be distinct for interpolation purposes, but not<a name="line.198"></a> <FONT color="green">199</FONT> * for general use. Thus it is not verified here.</p><a name="line.199"></a> <FONT color="green">200</FONT> *<a name="line.200"></a> <FONT color="green">201</FONT> * @param a the coefficients in Newton form formula<a name="line.201"></a> <FONT color="green">202</FONT> * @param c the centers<a name="line.202"></a> <FONT color="green">203</FONT> * @throws IllegalArgumentException if not valid<a name="line.203"></a> <FONT color="green">204</FONT> * @see org.apache.commons.math.analysis.interpolation.DividedDifferenceInterpolator#computeDividedDifference(double[],<a name="line.204"></a> <FONT color="green">205</FONT> * double[])<a name="line.205"></a> <FONT color="green">206</FONT> */<a name="line.206"></a> <FONT color="green">207</FONT> protected static void verifyInputArray(double a[], double c[]) throws<a name="line.207"></a> <FONT color="green">208</FONT> IllegalArgumentException {<a name="line.208"></a> <FONT color="green">209</FONT> <a name="line.209"></a> <FONT color="green">210</FONT> if (a.length < 1 || c.length < 1) {<a name="line.210"></a> <FONT color="green">211</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.211"></a> <FONT color="green">212</FONT> "empty polynomials coefficients array");<a name="line.212"></a> <FONT color="green">213</FONT> }<a name="line.213"></a> <FONT color="green">214</FONT> if (a.length != c.length + 1) {<a name="line.214"></a> <FONT color="green">215</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.215"></a> <FONT color="green">216</FONT> "array sizes should have difference 1 ({0} != {1} + 1)",<a name="line.216"></a> <FONT color="green">217</FONT> a.length, c.length);<a name="line.217"></a> <FONT color="green">218</FONT> }<a name="line.218"></a> <FONT color="green">219</FONT> }<a name="line.219"></a> <FONT color="green">220</FONT> }<a name="line.220"></a> </PRE> </BODY> </HTML>