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.math3.analysis.interpolation;
018
019 import java.io.Serializable;
020 import org.apache.commons.math3.analysis.polynomials.PolynomialFunctionLagrangeForm;
021 import org.apache.commons.math3.analysis.polynomials.PolynomialFunctionNewtonForm;
022 import org.apache.commons.math3.exception.DimensionMismatchException;
023 import org.apache.commons.math3.exception.NumberIsTooSmallException;
024 import org.apache.commons.math3.exception.NonMonotonicSequenceException;
025
026 /**
027 * Implements the <a href="
028 * http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html">
029 * Divided Difference Algorithm</a> for interpolation of real univariate
030 * functions. For reference, see <b>Introduction to Numerical Analysis</b>,
031 * ISBN 038795452X, chapter 2.
032 * <p>
033 * The actual code of Neville's evaluation is in PolynomialFunctionLagrangeForm,
034 * this class provides an easy-to-use interface to it.</p>
035 *
036 * @version $Id: DividedDifferenceInterpolator.java 1385313 2012-09-16 16:35:23Z tn $
037 * @since 1.2
038 */
039 public class DividedDifferenceInterpolator
040 implements UnivariateInterpolator, Serializable {
041 /** serializable version identifier */
042 private static final long serialVersionUID = 107049519551235069L;
043
044 /**
045 * Compute an interpolating function for the dataset.
046 *
047 * @param x Interpolating points array.
048 * @param y Interpolating values array.
049 * @return a function which interpolates the dataset.
050 * @throws DimensionMismatchException if the array lengths are different.
051 * @throws NumberIsTooSmallException if the number of points is less than 2.
052 * @throws NonMonotonicSequenceException if {@code x} is not sorted in
053 * strictly increasing order.
054 */
055 public PolynomialFunctionNewtonForm interpolate(double x[], double y[])
056 throws DimensionMismatchException,
057 NumberIsTooSmallException,
058 NonMonotonicSequenceException {
059 /**
060 * a[] and c[] are defined in the general formula of Newton form:
061 * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
062 * a[n](x-c[0])(x-c[1])...(x-c[n-1])
063 */
064 PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);
065
066 /**
067 * When used for interpolation, the Newton form formula becomes
068 * p(x) = f[x0] + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ... +
069 * f[x0,x1,...,x[n-1]](x-x0)(x-x1)...(x-x[n-2])
070 * Therefore, a[k] = f[x0,x1,...,xk], c[k] = x[k].
071 * <p>
072 * Note x[], y[], a[] have the same length but c[]'s size is one less.</p>
073 */
074 final double[] c = new double[x.length-1];
075 System.arraycopy(x, 0, c, 0, c.length);
076
077 final double[] a = computeDividedDifference(x, y);
078 return new PolynomialFunctionNewtonForm(a, c);
079 }
080
081 /**
082 * Return a copy of the divided difference array.
083 * <p>
084 * The divided difference array is defined recursively by <pre>
085 * f[x0] = f(x0)
086 * f[x0,x1,...,xk] = (f[x1,...,xk] - f[x0,...,x[k-1]]) / (xk - x0)
087 * </pre></p>
088 * <p>
089 * The computational complexity is O(N^2).</p>
090 *
091 * @param x Interpolating points array.
092 * @param y Interpolating values array.
093 * @return a fresh copy of the divided difference array.
094 * @throws DimensionMismatchException if the array lengths are different.
095 * @throws NumberIsTooSmallException if the number of points is less than 2.
096 * @throws NonMonotonicSequenceException
097 * if {@code x} is not sorted in strictly increasing order.
098 */
099 protected static double[] computeDividedDifference(final double x[], final double y[])
100 throws DimensionMismatchException,
101 NumberIsTooSmallException,
102 NonMonotonicSequenceException {
103 PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);
104
105 final double[] divdiff = y.clone(); // initialization
106
107 final int n = x.length;
108 final double[] a = new double [n];
109 a[0] = divdiff[0];
110 for (int i = 1; i < n; i++) {
111 for (int j = 0; j < n-i; j++) {
112 final double denominator = x[j+i] - x[j];
113 divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator;
114 }
115 a[i] = divdiff[0];
116 }
117
118 return a;
119 }
120 }