    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    }