001 /*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements. See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License. You may obtain a copy of the License at
008 *
009 * http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017 package org.apache.commons.math.analysis.interpolation;
018
019 import java.io.Serializable;
020
021 import org.apache.commons.math.DuplicateSampleAbscissaException;
022 import org.apache.commons.math.analysis.polynomials.PolynomialFunctionLagrangeForm;
023 import org.apache.commons.math.analysis.polynomials.PolynomialFunctionNewtonForm;
024
025 /**
026 * Implements the <a href="
027 * "http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html">
028 * Divided Difference Algorithm</a> for interpolation of real univariate
029 * functions. For reference, see <b>Introduction to Numerical Analysis</b>,
030 * ISBN 038795452X, chapter 2.
031 * <p>
032 * The actual code of Neville's evaluation is in PolynomialFunctionLagrangeForm,
033 * this class provides an easy-to-use interface to it.</p>
034 *
035 * @version $Revision: 825919 $ $Date: 2009-10-16 16:51:55 +0200 (ven. 16 oct. 2009) $
036 * @since 1.2
037 */
038 public class DividedDifferenceInterpolator implements UnivariateRealInterpolator,
039 Serializable {
040
041 /** serializable version identifier */
042 private static final long serialVersionUID = 107049519551235069L;
043
044 /**
045 * Computes an interpolating function for the data set.
046 *
047 * @param x the interpolating points array
048 * @param y the interpolating values array
049 * @return a function which interpolates the data set
050 * @throws DuplicateSampleAbscissaException if arguments are invalid
051 */
052 public PolynomialFunctionNewtonForm interpolate(double x[], double y[]) throws
053 DuplicateSampleAbscissaException {
054
055 /**
056 * a[] and c[] are defined in the general formula of Newton form:
057 * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
058 * a[n](x-c[0])(x-c[1])...(x-c[n-1])
059 */
060 PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y);
061
062 /**
063 * When used for interpolation, the Newton form formula becomes
064 * p(x) = f[x0] + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ... +
065 * f[x0,x1,...,x[n-1]](x-x0)(x-x1)...(x-x[n-2])
066 * Therefore, a[k] = f[x0,x1,...,xk], c[k] = x[k].
067 * <p>
068 * Note x[], y[], a[] have the same length but c[]'s size is one less.</p>
069 */
070 final double[] c = new double[x.length-1];
071 System.arraycopy(x, 0, c, 0, c.length);
072
073 final double[] a = computeDividedDifference(x, y);
074 return new PolynomialFunctionNewtonForm(a, c);
075
076 }
077
078 /**
079 * Returns a copy of the divided difference array.
080 * <p>
081 * The divided difference array is defined recursively by <pre>
082 * f[x0] = f(x0)
083 * f[x0,x1,...,xk] = (f(x1,...,xk) - f(x0,...,x[k-1])) / (xk - x0)
084 * </pre></p>
085 * <p>
086 * The computational complexity is O(N^2).</p>
087 *
088 * @param x the interpolating points array
089 * @param y the interpolating values array
090 * @return a fresh copy of the divided difference array
091 * @throws DuplicateSampleAbscissaException if any abscissas coincide
092 */
093 protected static double[] computeDividedDifference(final double x[], final double y[])
094 throws DuplicateSampleAbscissaException {
095
096 PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y);
097
098 final double[] divdiff = y.clone(); // initialization
099
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 }