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 }