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1 <HTML>
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2 <BODY BGCOLOR="white">
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3 <PRE>
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4 <FONT color="green">001</FONT> /*<a name="line.1"></a>
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5 <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
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6 <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a>
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7 <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a>
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8 <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
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9 <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
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10 <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a>
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11 <FONT color="green">008</FONT> *<a name="line.8"></a>
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12 <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
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13 <FONT color="green">010</FONT> *<a name="line.10"></a>
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14 <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
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15 <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
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16 <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
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17 <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a>
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18 <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a>
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19 <FONT color="green">016</FONT> */<a name="line.16"></a>
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20 <FONT color="green">017</FONT> package org.apache.commons.math.analysis.polynomials;<a name="line.17"></a>
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21 <FONT color="green">018</FONT> <a name="line.18"></a>
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22 <FONT color="green">019</FONT> import org.apache.commons.math.DuplicateSampleAbscissaException;<a name="line.19"></a>
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23 <FONT color="green">020</FONT> import org.apache.commons.math.FunctionEvaluationException;<a name="line.20"></a>
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24 <FONT color="green">021</FONT> import org.apache.commons.math.MathRuntimeException;<a name="line.21"></a>
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25 <FONT color="green">022</FONT> import org.apache.commons.math.analysis.UnivariateRealFunction;<a name="line.22"></a>
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26 <FONT color="green">023</FONT> <a name="line.23"></a>
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27 <FONT color="green">024</FONT> /**<a name="line.24"></a>
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28 <FONT color="green">025</FONT> * Implements the representation of a real polynomial function in<a name="line.25"></a>
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29 <FONT color="green">026</FONT> * <a href="http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html"><a name="line.26"></a>
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30 <FONT color="green">027</FONT> * Lagrange Form</a>. For reference, see <b>Introduction to Numerical<a name="line.27"></a>
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31 <FONT color="green">028</FONT> * Analysis</b>, ISBN 038795452X, chapter 2.<a name="line.28"></a>
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32 <FONT color="green">029</FONT> * <p><a name="line.29"></a>
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33 <FONT color="green">030</FONT> * The approximated function should be smooth enough for Lagrange polynomial<a name="line.30"></a>
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34 <FONT color="green">031</FONT> * to work well. Otherwise, consider using splines instead.</p><a name="line.31"></a>
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35 <FONT color="green">032</FONT> *<a name="line.32"></a>
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36 <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>
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37 <FONT color="green">034</FONT> * @since 1.2<a name="line.34"></a>
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38 <FONT color="green">035</FONT> */<a name="line.35"></a>
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39 <FONT color="green">036</FONT> public class PolynomialFunctionLagrangeForm implements UnivariateRealFunction {<a name="line.36"></a>
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40 <FONT color="green">037</FONT> <a name="line.37"></a>
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41 <FONT color="green">038</FONT> /**<a name="line.38"></a>
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42 <FONT color="green">039</FONT> * The coefficients of the polynomial, ordered by degree -- i.e.<a name="line.39"></a>
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43 <FONT color="green">040</FONT> * coefficients[0] is the constant term and coefficients[n] is the<a name="line.40"></a>
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44 <FONT color="green">041</FONT> * coefficient of x^n where n is the degree of the polynomial.<a name="line.41"></a>
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45 <FONT color="green">042</FONT> */<a name="line.42"></a>
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46 <FONT color="green">043</FONT> private double coefficients[];<a name="line.43"></a>
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47 <FONT color="green">044</FONT> <a name="line.44"></a>
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48 <FONT color="green">045</FONT> /**<a name="line.45"></a>
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49 <FONT color="green">046</FONT> * Interpolating points (abscissas).<a name="line.46"></a>
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50 <FONT color="green">047</FONT> */<a name="line.47"></a>
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51 <FONT color="green">048</FONT> private final double x[];<a name="line.48"></a>
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52 <FONT color="green">049</FONT> <a name="line.49"></a>
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53 <FONT color="green">050</FONT> /**<a name="line.50"></a>
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54 <FONT color="green">051</FONT> * Function values at interpolating points.<a name="line.51"></a>
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55 <FONT color="green">052</FONT> */<a name="line.52"></a>
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56 <FONT color="green">053</FONT> private final double y[];<a name="line.53"></a>
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57 <FONT color="green">054</FONT> <a name="line.54"></a>
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58 <FONT color="green">055</FONT> /**<a name="line.55"></a>
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59 <FONT color="green">056</FONT> * Whether the polynomial coefficients are available.<a name="line.56"></a>
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60 <FONT color="green">057</FONT> */<a name="line.57"></a>
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61 <FONT color="green">058</FONT> private boolean coefficientsComputed;<a name="line.58"></a>
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62 <FONT color="green">059</FONT> <a name="line.59"></a>
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63 <FONT color="green">060</FONT> /**<a name="line.60"></a>
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64 <FONT color="green">061</FONT> * Construct a Lagrange polynomial with the given abscissas and function<a name="line.61"></a>
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65 <FONT color="green">062</FONT> * values. The order of interpolating points are not important.<a name="line.62"></a>
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66 <FONT color="green">063</FONT> * <p><a name="line.63"></a>
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67 <FONT color="green">064</FONT> * The constructor makes copy of the input arrays and assigns them.</p><a name="line.64"></a>
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68 <FONT color="green">065</FONT> *<a name="line.65"></a>
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69 <FONT color="green">066</FONT> * @param x interpolating points<a name="line.66"></a>
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70 <FONT color="green">067</FONT> * @param y function values at interpolating points<a name="line.67"></a>
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71 <FONT color="green">068</FONT> * @throws IllegalArgumentException if input arrays are not valid<a name="line.68"></a>
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72 <FONT color="green">069</FONT> */<a name="line.69"></a>
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73 <FONT color="green">070</FONT> public PolynomialFunctionLagrangeForm(double x[], double y[])<a name="line.70"></a>
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74 <FONT color="green">071</FONT> throws IllegalArgumentException {<a name="line.71"></a>
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75 <FONT color="green">072</FONT> <a name="line.72"></a>
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76 <FONT color="green">073</FONT> verifyInterpolationArray(x, y);<a name="line.73"></a>
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77 <FONT color="green">074</FONT> this.x = new double[x.length];<a name="line.74"></a>
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78 <FONT color="green">075</FONT> this.y = new double[y.length];<a name="line.75"></a>
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79 <FONT color="green">076</FONT> System.arraycopy(x, 0, this.x, 0, x.length);<a name="line.76"></a>
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80 <FONT color="green">077</FONT> System.arraycopy(y, 0, this.y, 0, y.length);<a name="line.77"></a>
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81 <FONT color="green">078</FONT> coefficientsComputed = false;<a name="line.78"></a>
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82 <FONT color="green">079</FONT> }<a name="line.79"></a>
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83 <FONT color="green">080</FONT> <a name="line.80"></a>
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84 <FONT color="green">081</FONT> /**<a name="line.81"></a>
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85 <FONT color="green">082</FONT> * Calculate the function value at the given point.<a name="line.82"></a>
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86 <FONT color="green">083</FONT> *<a name="line.83"></a>
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87 <FONT color="green">084</FONT> * @param z the point at which the function value is to be computed<a name="line.84"></a>
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88 <FONT color="green">085</FONT> * @return the function value<a name="line.85"></a>
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89 <FONT color="green">086</FONT> * @throws FunctionEvaluationException if a runtime error occurs<a name="line.86"></a>
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90 <FONT color="green">087</FONT> * @see UnivariateRealFunction#value(double)<a name="line.87"></a>
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91 <FONT color="green">088</FONT> */<a name="line.88"></a>
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92 <FONT color="green">089</FONT> public double value(double z) throws FunctionEvaluationException {<a name="line.89"></a>
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93 <FONT color="green">090</FONT> try {<a name="line.90"></a>
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94 <FONT color="green">091</FONT> return evaluate(x, y, z);<a name="line.91"></a>
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95 <FONT color="green">092</FONT> } catch (DuplicateSampleAbscissaException e) {<a name="line.92"></a>
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96 <FONT color="green">093</FONT> throw new FunctionEvaluationException(e, z, e.getPattern(), e.getArguments());<a name="line.93"></a>
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97 <FONT color="green">094</FONT> }<a name="line.94"></a>
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98 <FONT color="green">095</FONT> }<a name="line.95"></a>
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99 <FONT color="green">096</FONT> <a name="line.96"></a>
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100 <FONT color="green">097</FONT> /**<a name="line.97"></a>
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101 <FONT color="green">098</FONT> * Returns the degree of the polynomial.<a name="line.98"></a>
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102 <FONT color="green">099</FONT> *<a name="line.99"></a>
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103 <FONT color="green">100</FONT> * @return the degree of the polynomial<a name="line.100"></a>
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104 <FONT color="green">101</FONT> */<a name="line.101"></a>
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105 <FONT color="green">102</FONT> public int degree() {<a name="line.102"></a>
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106 <FONT color="green">103</FONT> return x.length - 1;<a name="line.103"></a>
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107 <FONT color="green">104</FONT> }<a name="line.104"></a>
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108 <FONT color="green">105</FONT> <a name="line.105"></a>
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109 <FONT color="green">106</FONT> /**<a name="line.106"></a>
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110 <FONT color="green">107</FONT> * Returns a copy of the interpolating points array.<a name="line.107"></a>
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111 <FONT color="green">108</FONT> * <p><a name="line.108"></a>
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112 <FONT color="green">109</FONT> * Changes made to the returned copy will not affect the polynomial.</p><a name="line.109"></a>
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113 <FONT color="green">110</FONT> *<a name="line.110"></a>
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114 <FONT color="green">111</FONT> * @return a fresh copy of the interpolating points array<a name="line.111"></a>
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115 <FONT color="green">112</FONT> */<a name="line.112"></a>
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116 <FONT color="green">113</FONT> public double[] getInterpolatingPoints() {<a name="line.113"></a>
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117 <FONT color="green">114</FONT> double[] out = new double[x.length];<a name="line.114"></a>
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118 <FONT color="green">115</FONT> System.arraycopy(x, 0, out, 0, x.length);<a name="line.115"></a>
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119 <FONT color="green">116</FONT> return out;<a name="line.116"></a>
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120 <FONT color="green">117</FONT> }<a name="line.117"></a>
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121 <FONT color="green">118</FONT> <a name="line.118"></a>
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122 <FONT color="green">119</FONT> /**<a name="line.119"></a>
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123 <FONT color="green">120</FONT> * Returns a copy of the interpolating values array.<a name="line.120"></a>
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124 <FONT color="green">121</FONT> * <p><a name="line.121"></a>
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125 <FONT color="green">122</FONT> * Changes made to the returned copy will not affect the polynomial.</p><a name="line.122"></a>
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126 <FONT color="green">123</FONT> *<a name="line.123"></a>
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127 <FONT color="green">124</FONT> * @return a fresh copy of the interpolating values array<a name="line.124"></a>
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128 <FONT color="green">125</FONT> */<a name="line.125"></a>
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129 <FONT color="green">126</FONT> public double[] getInterpolatingValues() {<a name="line.126"></a>
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130 <FONT color="green">127</FONT> double[] out = new double[y.length];<a name="line.127"></a>
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131 <FONT color="green">128</FONT> System.arraycopy(y, 0, out, 0, y.length);<a name="line.128"></a>
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132 <FONT color="green">129</FONT> return out;<a name="line.129"></a>
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133 <FONT color="green">130</FONT> }<a name="line.130"></a>
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134 <FONT color="green">131</FONT> <a name="line.131"></a>
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135 <FONT color="green">132</FONT> /**<a name="line.132"></a>
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136 <FONT color="green">133</FONT> * Returns a copy of the coefficients array.<a name="line.133"></a>
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137 <FONT color="green">134</FONT> * <p><a name="line.134"></a>
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138 <FONT color="green">135</FONT> * Changes made to the returned copy will not affect the polynomial.</p><a name="line.135"></a>
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139 <FONT color="green">136</FONT> * <p><a name="line.136"></a>
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140 <FONT color="green">137</FONT> * Note that coefficients computation can be ill-conditioned. Use with caution<a name="line.137"></a>
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141 <FONT color="green">138</FONT> * and only when it is necessary.</p><a name="line.138"></a>
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142 <FONT color="green">139</FONT> *<a name="line.139"></a>
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143 <FONT color="green">140</FONT> * @return a fresh copy of the coefficients array<a name="line.140"></a>
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144 <FONT color="green">141</FONT> */<a name="line.141"></a>
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145 <FONT color="green">142</FONT> public double[] getCoefficients() {<a name="line.142"></a>
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146 <FONT color="green">143</FONT> if (!coefficientsComputed) {<a name="line.143"></a>
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147 <FONT color="green">144</FONT> computeCoefficients();<a name="line.144"></a>
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148 <FONT color="green">145</FONT> }<a name="line.145"></a>
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149 <FONT color="green">146</FONT> double[] out = new double[coefficients.length];<a name="line.146"></a>
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150 <FONT color="green">147</FONT> System.arraycopy(coefficients, 0, out, 0, coefficients.length);<a name="line.147"></a>
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151 <FONT color="green">148</FONT> return out;<a name="line.148"></a>
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152 <FONT color="green">149</FONT> }<a name="line.149"></a>
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153 <FONT color="green">150</FONT> <a name="line.150"></a>
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154 <FONT color="green">151</FONT> /**<a name="line.151"></a>
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155 <FONT color="green">152</FONT> * Evaluate the Lagrange polynomial using<a name="line.152"></a>
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156 <FONT color="green">153</FONT> * <a href="http://mathworld.wolfram.com/NevillesAlgorithm.html"><a name="line.153"></a>
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157 <FONT color="green">154</FONT> * Neville's Algorithm</a>. It takes O(N^2) time.<a name="line.154"></a>
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158 <FONT color="green">155</FONT> * <p><a name="line.155"></a>
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159 <FONT color="green">156</FONT> * This function is made public static so that users can call it directly<a name="line.156"></a>
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160 <FONT color="green">157</FONT> * without instantiating PolynomialFunctionLagrangeForm object.</p><a name="line.157"></a>
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161 <FONT color="green">158</FONT> *<a name="line.158"></a>
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162 <FONT color="green">159</FONT> * @param x the interpolating points array<a name="line.159"></a>
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163 <FONT color="green">160</FONT> * @param y the interpolating values array<a name="line.160"></a>
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164 <FONT color="green">161</FONT> * @param z the point at which the function value is to be computed<a name="line.161"></a>
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165 <FONT color="green">162</FONT> * @return the function value<a name="line.162"></a>
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166 <FONT color="green">163</FONT> * @throws DuplicateSampleAbscissaException if the sample has duplicate abscissas<a name="line.163"></a>
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167 <FONT color="green">164</FONT> * @throws IllegalArgumentException if inputs are not valid<a name="line.164"></a>
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168 <FONT color="green">165</FONT> */<a name="line.165"></a>
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169 <FONT color="green">166</FONT> public static double evaluate(double x[], double y[], double z) throws<a name="line.166"></a>
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170 <FONT color="green">167</FONT> DuplicateSampleAbscissaException, IllegalArgumentException {<a name="line.167"></a>
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171 <FONT color="green">168</FONT> <a name="line.168"></a>
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172 <FONT color="green">169</FONT> verifyInterpolationArray(x, y);<a name="line.169"></a>
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173 <FONT color="green">170</FONT> <a name="line.170"></a>
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174 <FONT color="green">171</FONT> int nearest = 0;<a name="line.171"></a>
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175 <FONT color="green">172</FONT> final int n = x.length;<a name="line.172"></a>
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176 <FONT color="green">173</FONT> final double[] c = new double[n];<a name="line.173"></a>
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177 <FONT color="green">174</FONT> final double[] d = new double[n];<a name="line.174"></a>
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178 <FONT color="green">175</FONT> double min_dist = Double.POSITIVE_INFINITY;<a name="line.175"></a>
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179 <FONT color="green">176</FONT> for (int i = 0; i < n; i++) {<a name="line.176"></a>
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180 <FONT color="green">177</FONT> // initialize the difference arrays<a name="line.177"></a>
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181 <FONT color="green">178</FONT> c[i] = y[i];<a name="line.178"></a>
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182 <FONT color="green">179</FONT> d[i] = y[i];<a name="line.179"></a>
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183 <FONT color="green">180</FONT> // find out the abscissa closest to z<a name="line.180"></a>
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184 <FONT color="green">181</FONT> final double dist = Math.abs(z - x[i]);<a name="line.181"></a>
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185 <FONT color="green">182</FONT> if (dist < min_dist) {<a name="line.182"></a>
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186 <FONT color="green">183</FONT> nearest = i;<a name="line.183"></a>
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187 <FONT color="green">184</FONT> min_dist = dist;<a name="line.184"></a>
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188 <FONT color="green">185</FONT> }<a name="line.185"></a>
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189 <FONT color="green">186</FONT> }<a name="line.186"></a>
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190 <FONT color="green">187</FONT> <a name="line.187"></a>
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191 <FONT color="green">188</FONT> // initial approximation to the function value at z<a name="line.188"></a>
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192 <FONT color="green">189</FONT> double value = y[nearest];<a name="line.189"></a>
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193 <FONT color="green">190</FONT> <a name="line.190"></a>
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194 <FONT color="green">191</FONT> for (int i = 1; i < n; i++) {<a name="line.191"></a>
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195 <FONT color="green">192</FONT> for (int j = 0; j < n-i; j++) {<a name="line.192"></a>
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196 <FONT color="green">193</FONT> final double tc = x[j] - z;<a name="line.193"></a>
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197 <FONT color="green">194</FONT> final double td = x[i+j] - z;<a name="line.194"></a>
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198 <FONT color="green">195</FONT> final double divider = x[j] - x[i+j];<a name="line.195"></a>
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199 <FONT color="green">196</FONT> if (divider == 0.0) {<a name="line.196"></a>
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200 <FONT color="green">197</FONT> // This happens only when two abscissas are identical.<a name="line.197"></a>
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201 <FONT color="green">198</FONT> throw new DuplicateSampleAbscissaException(x[i], i, i+j);<a name="line.198"></a>
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202 <FONT color="green">199</FONT> }<a name="line.199"></a>
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203 <FONT color="green">200</FONT> // update the difference arrays<a name="line.200"></a>
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204 <FONT color="green">201</FONT> final double w = (c[j+1] - d[j]) / divider;<a name="line.201"></a>
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205 <FONT color="green">202</FONT> c[j] = tc * w;<a name="line.202"></a>
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206 <FONT color="green">203</FONT> d[j] = td * w;<a name="line.203"></a>
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207 <FONT color="green">204</FONT> }<a name="line.204"></a>
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208 <FONT color="green">205</FONT> // sum up the difference terms to get the final value<a name="line.205"></a>
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209 <FONT color="green">206</FONT> if (nearest < 0.5*(n-i+1)) {<a name="line.206"></a>
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210 <FONT color="green">207</FONT> value += c[nearest]; // fork down<a name="line.207"></a>
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211 <FONT color="green">208</FONT> } else {<a name="line.208"></a>
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212 <FONT color="green">209</FONT> nearest--;<a name="line.209"></a>
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213 <FONT color="green">210</FONT> value += d[nearest]; // fork up<a name="line.210"></a>
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214 <FONT color="green">211</FONT> }<a name="line.211"></a>
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215 <FONT color="green">212</FONT> }<a name="line.212"></a>
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216 <FONT color="green">213</FONT> <a name="line.213"></a>
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217 <FONT color="green">214</FONT> return value;<a name="line.214"></a>
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218 <FONT color="green">215</FONT> }<a name="line.215"></a>
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219 <FONT color="green">216</FONT> <a name="line.216"></a>
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220 <FONT color="green">217</FONT> /**<a name="line.217"></a>
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221 <FONT color="green">218</FONT> * Calculate the coefficients of Lagrange polynomial from the<a name="line.218"></a>
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222 <FONT color="green">219</FONT> * interpolation data. It takes O(N^2) time.<a name="line.219"></a>
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223 <FONT color="green">220</FONT> * <p><a name="line.220"></a>
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224 <FONT color="green">221</FONT> * Note this computation can be ill-conditioned. Use with caution<a name="line.221"></a>
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225 <FONT color="green">222</FONT> * and only when it is necessary.</p><a name="line.222"></a>
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226 <FONT color="green">223</FONT> *<a name="line.223"></a>
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227 <FONT color="green">224</FONT> * @throws ArithmeticException if any abscissas coincide<a name="line.224"></a>
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228 <FONT color="green">225</FONT> */<a name="line.225"></a>
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229 <FONT color="green">226</FONT> protected void computeCoefficients() throws ArithmeticException {<a name="line.226"></a>
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230 <FONT color="green">227</FONT> <a name="line.227"></a>
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231 <FONT color="green">228</FONT> final int n = degree() + 1;<a name="line.228"></a>
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232 <FONT color="green">229</FONT> coefficients = new double[n];<a name="line.229"></a>
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233 <FONT color="green">230</FONT> for (int i = 0; i < n; i++) {<a name="line.230"></a>
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234 <FONT color="green">231</FONT> coefficients[i] = 0.0;<a name="line.231"></a>
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235 <FONT color="green">232</FONT> }<a name="line.232"></a>
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236 <FONT color="green">233</FONT> <a name="line.233"></a>
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237 <FONT color="green">234</FONT> // c[] are the coefficients of P(x) = (x-x[0])(x-x[1])...(x-x[n-1])<a name="line.234"></a>
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238 <FONT color="green">235</FONT> final double[] c = new double[n+1];<a name="line.235"></a>
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239 <FONT color="green">236</FONT> c[0] = 1.0;<a name="line.236"></a>
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240 <FONT color="green">237</FONT> for (int i = 0; i < n; i++) {<a name="line.237"></a>
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241 <FONT color="green">238</FONT> for (int j = i; j > 0; j--) {<a name="line.238"></a>
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242 <FONT color="green">239</FONT> c[j] = c[j-1] - c[j] * x[i];<a name="line.239"></a>
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243 <FONT color="green">240</FONT> }<a name="line.240"></a>
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244 <FONT color="green">241</FONT> c[0] *= -x[i];<a name="line.241"></a>
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245 <FONT color="green">242</FONT> c[i+1] = 1;<a name="line.242"></a>
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246 <FONT color="green">243</FONT> }<a name="line.243"></a>
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247 <FONT color="green">244</FONT> <a name="line.244"></a>
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248 <FONT color="green">245</FONT> final double[] tc = new double[n];<a name="line.245"></a>
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249 <FONT color="green">246</FONT> for (int i = 0; i < n; i++) {<a name="line.246"></a>
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250 <FONT color="green">247</FONT> // d = (x[i]-x[0])...(x[i]-x[i-1])(x[i]-x[i+1])...(x[i]-x[n-1])<a name="line.247"></a>
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251 <FONT color="green">248</FONT> double d = 1;<a name="line.248"></a>
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252 <FONT color="green">249</FONT> for (int j = 0; j < n; j++) {<a name="line.249"></a>
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253 <FONT color="green">250</FONT> if (i != j) {<a name="line.250"></a>
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254 <FONT color="green">251</FONT> d *= x[i] - x[j];<a name="line.251"></a>
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255 <FONT color="green">252</FONT> }<a name="line.252"></a>
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256 <FONT color="green">253</FONT> }<a name="line.253"></a>
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257 <FONT color="green">254</FONT> if (d == 0.0) {<a name="line.254"></a>
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258 <FONT color="green">255</FONT> // This happens only when two abscissas are identical.<a name="line.255"></a>
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259 <FONT color="green">256</FONT> for (int k = 0; k < n; ++k) {<a name="line.256"></a>
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260 <FONT color="green">257</FONT> if ((i != k) && (x[i] == x[k])) {<a name="line.257"></a>
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261 <FONT color="green">258</FONT> throw MathRuntimeException.createArithmeticException("identical abscissas x[{0}] == x[{1}] == {2} cause division by zero",<a name="line.258"></a>
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262 <FONT color="green">259</FONT> i, k, x[i]);<a name="line.259"></a>
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263 <FONT color="green">260</FONT> }<a name="line.260"></a>
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264 <FONT color="green">261</FONT> }<a name="line.261"></a>
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265 <FONT color="green">262</FONT> }<a name="line.262"></a>
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266 <FONT color="green">263</FONT> final double t = y[i] / d;<a name="line.263"></a>
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267 <FONT color="green">264</FONT> // Lagrange polynomial is the sum of n terms, each of which is a<a name="line.264"></a>
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268 <FONT color="green">265</FONT> // polynomial of degree n-1. tc[] are the coefficients of the i-th<a name="line.265"></a>
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269 <FONT color="green">266</FONT> // numerator Pi(x) = (x-x[0])...(x-x[i-1])(x-x[i+1])...(x-x[n-1]).<a name="line.266"></a>
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270 <FONT color="green">267</FONT> tc[n-1] = c[n]; // actually c[n] = 1<a name="line.267"></a>
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271 <FONT color="green">268</FONT> coefficients[n-1] += t * tc[n-1];<a name="line.268"></a>
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272 <FONT color="green">269</FONT> for (int j = n-2; j >= 0; j--) {<a name="line.269"></a>
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273 <FONT color="green">270</FONT> tc[j] = c[j+1] + tc[j+1] * x[i];<a name="line.270"></a>
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274 <FONT color="green">271</FONT> coefficients[j] += t * tc[j];<a name="line.271"></a>
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275 <FONT color="green">272</FONT> }<a name="line.272"></a>
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276 <FONT color="green">273</FONT> }<a name="line.273"></a>
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277 <FONT color="green">274</FONT> <a name="line.274"></a>
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278 <FONT color="green">275</FONT> coefficientsComputed = true;<a name="line.275"></a>
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279 <FONT color="green">276</FONT> }<a name="line.276"></a>
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280 <FONT color="green">277</FONT> <a name="line.277"></a>
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281 <FONT color="green">278</FONT> /**<a name="line.278"></a>
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282 <FONT color="green">279</FONT> * Verifies that the interpolation arrays are valid.<a name="line.279"></a>
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283 <FONT color="green">280</FONT> * <p><a name="line.280"></a>
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284 <FONT color="green">281</FONT> * The arrays features checked by this method are that both arrays have the<a name="line.281"></a>
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285 <FONT color="green">282</FONT> * same length and this length is at least 2.<a name="line.282"></a>
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286 <FONT color="green">283</FONT> * </p><a name="line.283"></a>
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287 <FONT color="green">284</FONT> * <p><a name="line.284"></a>
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288 <FONT color="green">285</FONT> * The interpolating points must be distinct. However it is not<a name="line.285"></a>
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289 <FONT color="green">286</FONT> * verified here, it is checked in evaluate() and computeCoefficients().<a name="line.286"></a>
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290 <FONT color="green">287</FONT> * </p><a name="line.287"></a>
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291 <FONT color="green">288</FONT> *<a name="line.288"></a>
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292 <FONT color="green">289</FONT> * @param x the interpolating points array<a name="line.289"></a>
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293 <FONT color="green">290</FONT> * @param y the interpolating values array<a name="line.290"></a>
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294 <FONT color="green">291</FONT> * @throws IllegalArgumentException if not valid<a name="line.291"></a>
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295 <FONT color="green">292</FONT> * @see #evaluate(double[], double[], double)<a name="line.292"></a>
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296 <FONT color="green">293</FONT> * @see #computeCoefficients()<a name="line.293"></a>
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297 <FONT color="green">294</FONT> */<a name="line.294"></a>
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298 <FONT color="green">295</FONT> public static void verifyInterpolationArray(double x[], double y[])<a name="line.295"></a>
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299 <FONT color="green">296</FONT> throws IllegalArgumentException {<a name="line.296"></a>
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300 <FONT color="green">297</FONT> <a name="line.297"></a>
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301 <FONT color="green">298</FONT> if (x.length != y.length) {<a name="line.298"></a>
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302 <FONT color="green">299</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.299"></a>
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303 <FONT color="green">300</FONT> "dimension mismatch {0} != {1}", x.length, y.length);<a name="line.300"></a>
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304 <FONT color="green">301</FONT> }<a name="line.301"></a>
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305 <FONT color="green">302</FONT> <a name="line.302"></a>
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306 <FONT color="green">303</FONT> if (x.length < 2) {<a name="line.303"></a>
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307 <FONT color="green">304</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.304"></a>
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308 <FONT color="green">305</FONT> "{0} points are required, got only {1}", 2, x.length);<a name="line.305"></a>
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309 <FONT color="green">306</FONT> }<a name="line.306"></a>
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310 <FONT color="green">307</FONT> <a name="line.307"></a>
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311 <FONT color="green">308</FONT> }<a name="line.308"></a>
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312 <FONT color="green">309</FONT> }<a name="line.309"></a>
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373 </PRE>
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374 </BODY>
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375 </HTML>
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