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 */
17package org.apache.commons.math.analysis.interpolation;
18
19import java.io.Serializable;
20
21import org.apache.commons.math.DuplicateSampleAbscissaException;
22import org.apache.commons.math.analysis.polynomials.PolynomialFunctionLagrangeForm;
23import org.apache.commons.math.analysis.polynomials.PolynomialFunctionNewtonForm;
24
25/**
26 * Implements the <a href="
27 * "http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html">
28 * Divided Difference Algorithm</a> for interpolation of real univariate
29 * functions. For reference, see <b>Introduction to Numerical Analysis</b>,
30 * ISBN 038795452X, chapter 2.
31 * <p>
32 * The actual code of Neville's evaluation is in PolynomialFunctionLagrangeForm,
33 * this class provides an easy-to-use interface to it.</p>
34 *
35 * @version $Revision: 825919 $ $Date: 2009-10-16 16:51:55 +0200 (ven. 16 oct. 2009) $
36 * @since 1.2
37 */
38public class DividedDifferenceInterpolator implements UnivariateRealInterpolator,
39    Serializable {
40
41    /** serializable version identifier */
42    private static final long serialVersionUID = 107049519551235069L;
43
44    /**
45     * Computes an interpolating function for the data set.
46     *
47     * @param x the interpolating points array
48     * @param y the interpolating values array
49     * @return a function which interpolates the data set
50     * @throws DuplicateSampleAbscissaException if arguments are invalid
51     */
52    public PolynomialFunctionNewtonForm interpolate(double x[], double y[]) throws
53        DuplicateSampleAbscissaException {
54
55        /**
56         * a[] and c[] are defined in the general formula of Newton form:
57         * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
58         *        a[n](x-c[0])(x-c[1])...(x-c[n-1])
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        final double[] c = new double[x.length-1];
71        System.arraycopy(x, 0, c, 0, c.length);
72
73        final double[] a = computeDividedDifference(x, y);
74        return new PolynomialFunctionNewtonForm(a, c);
75
76    }
77
78    /**
79     * Returns a copy of the divided difference array.
80     * <p>
81     * The divided difference array is defined recursively by <pre>
82     * f[x0] = f(x0)
83     * f[x0,x1,...,xk] = (f(x1,...,xk) - f(x0,...,x[k-1])) / (xk - x0)
84     * </pre></p>
85     * <p>
86     * The computational complexity is O(N^2).</p>
87     *
88     * @param x the interpolating points array
89     * @param y the interpolating values array
90     * @return a fresh copy of the divided difference array
91     * @throws DuplicateSampleAbscissaException if any abscissas coincide
92     */
93    protected static double[] computeDividedDifference(final double x[], final double y[])
94        throws DuplicateSampleAbscissaException {
95
96        PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y);
97
98        final double[] divdiff = y.clone(); // initialization
99
100        final int n = x.length;
101        final double[] a = new double [n];
102        a[0] = divdiff[0];
103        for (int i = 1; i < n; i++) {
104            for (int j = 0; j < n-i; j++) {
105                final double denominator = x[j+i] - x[j];
106                if (denominator == 0.0) {
107                    // This happens only when two abscissas are identical.
108                    throw new DuplicateSampleAbscissaException(x[j], j, j+i);
109                }
110                divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator;
111            }
112            a[i] = divdiff[0];
113        }
114
115        return a;
116    }
117}
118