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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      http://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  package org.apache.commons.math.analysis;
18  
19  import java.io.Serializable;
20  
21  import org.apache.commons.math.DuplicateSampleAbscissaException;
22  
23  /**
24   * Implements the <a href="
25   * "http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html">
26   * Divided Difference Algorithm</a> for interpolation of real univariate
27   * functions. For reference, see <b>Introduction to Numerical Analysis</b>,
28   * ISBN 038795452X, chapter 2.
29   * <p>
30   * The actual code of Neville's evalution is in PolynomialFunctionLagrangeForm,
31   * this class provides an easy-to-use interface to it.</p>
32   *
33   * @version $Revision: 620312 $ $Date: 2008-02-10 12:28:59 -0700 (Sun, 10 Feb 2008) $
34   * @since 1.2
35   */
36  public class DividedDifferenceInterpolator implements UnivariateRealInterpolator,
37      Serializable {
38  
39      /** serializable version identifier */
40      private static final long serialVersionUID = 107049519551235069L;
41  
42      /**
43       * Computes an interpolating function for the data set.
44       *
45       * @param x the interpolating points array
46       * @param y the interpolating values array
47       * @return a function which interpolates the data set
48       * @throws DuplicateSampleAbscissaException if arguments are invalid
49       */
50      public UnivariateRealFunction interpolate(double x[], double y[]) throws
51          DuplicateSampleAbscissaException {
52  
53          /**
54           * a[] and c[] are defined in the general formula of Newton form:
55           * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
56           *        a[n](x-c[0])(x-c[1])...(x-c[n-1])
57           */
58          double a[], c[];
59  
60          PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y);
61  
62          /**
63           * When used for interpolation, the Newton form formula becomes
64           * p(x) = f[x0] + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ... +
65           *        f[x0,x1,...,x[n-1]](x-x0)(x-x1)...(x-x[n-2])
66           * Therefore, a[k] = f[x0,x1,...,xk], c[k] = x[k].
67           * <p>
68           * Note x[], y[], a[] have the same length but c[]'s size is one less.</p>
69           */
70          c = new double[x.length-1];
71          for (int i = 0; i < c.length; i++) {
72              c[i] = x[i];
73          }
74          a = computeDividedDifference(x, y);
75  
76          PolynomialFunctionNewtonForm p;
77          p = new PolynomialFunctionNewtonForm(a, c);
78          return p;
79      }
80  
81      /**
82       * Returns a copy of the divided difference array.
83       * <p> 
84       * The divided difference array is defined recursively by <pre>
85       * f[x0] = f(x0)
86       * f[x0,x1,...,xk] = (f(x1,...,xk) - f(x0,...,x[k-1])) / (xk - x0)
87       * </pre></p>
88       * <p>
89       * The computational complexity is O(N^2).</p>
90       *
91       * @param x the interpolating points array
92       * @param y the interpolating values array
93       * @return a fresh copy of the divided difference array
94       * @throws DuplicateSampleAbscissaException if any abscissas coincide
95       */
96      protected static double[] computeDividedDifference(double x[], double y[])
97          throws DuplicateSampleAbscissaException {
98  
99          int i, j, n;
100         double divdiff[], a[], denominator;
101 
102         PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y);
103 
104         n = x.length;
105         divdiff = new double[n];
106         for (i = 0; i < n; i++) {
107             divdiff[i] = y[i];      // initialization
108         }
109 
110         a = new double [n];
111         a[0] = divdiff[0];
112         for (i = 1; i < n; i++) {
113             for (j = 0; j < n-i; j++) {
114                 denominator = x[j+i] - x[j];
115                 if (denominator == 0.0) {
116                     // This happens only when two abscissas are identical.
117                     throw new DuplicateSampleAbscissaException(x[j], j, j+i);
118                 }
119                 divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator;
120             }
121             a[i] = divdiff[0];
122         }
123 
124         return a;
125     }
126 }