divideddifferenceinterpolator.java

Apache的common math数学软件包

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
package org.apache.commons.math.analysis;

import java.io.Serializable;

import org.apache.commons.math.DuplicateSampleAbscissaException;

/**
 * Implements the <a href="
 * "http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html">
 * Divided Difference Algorithm</a> for interpolation of real univariate
 * functions. For reference, see <b>Introduction to Numerical Analysis</b>,
 * ISBN 038795452X, chapter 2.
 * <p>
 * The actual code of Neville's evalution is in PolynomialFunctionLagrangeForm,
 * this class provides an easy-to-use interface to it.</p>
 *
 * @version $Revision: 620312 $ $Date: 2008-02-10 12:28:59 -0700 (Sun, 10 Feb 2008) $
 * @since 1.2
 */
public class DividedDifferenceInterpolator implements UnivariateRealInterpolator,
    Serializable {

    /** serializable version identifier */
    private static final long serialVersionUID = 107049519551235069L;

    /**
     * Computes an interpolating function for the data set.
     *
     * @param x the interpolating points array
     * @param y the interpolating values array
     * @return a function which interpolates the data set
     * @throws DuplicateSampleAbscissaException if arguments are invalid
     */
    public UnivariateRealFunction interpolate(double x[], double y[]) throws
        DuplicateSampleAbscissaException {

        /**
         * a[] and c[] are defined in the general formula of Newton form:
         * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
         *        a[n](x-c[0])(x-c[1])...(x-c[n-1])
         */
        double a[], c[];

        PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y);

        /**
         * When used for interpolation, the Newton form formula becomes
         * p(x) = f[x0] + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ... +
         *        f[x0,x1,...,x[n-1]](x-x0)(x-x1)...(x-x[n-2])
         * Therefore, a[k] = f[x0,x1,...,xk], c[k] = x[k].
         * <p>
         * Note x[], y[], a[] have the same length but c[]'s size is one less.</p>
         */
        c = new double[x.length-1];
        for (int i = 0; i < c.length; i++) {
            c[i] = x[i];
        }
        a = computeDividedDifference(x, y);

        PolynomialFunctionNewtonForm p;
        p = new PolynomialFunctionNewtonForm(a, c);
        return p;
    }

    /**
     * Returns a copy of the divided difference array.
     * <p> 
     * The divided difference array is defined recursively by <pre>
     * f[x0] = f(x0)
     * f[x0,x1,...,xk] = (f(x1,...,xk) - f(x0,...,x[k-1])) / (xk - x0)
     * </pre></p>
     * <p>
     * The computational complexity is O(N^2).</p>
     *
     * @param x the interpolating points array
     * @param y the interpolating values array
     * @return a fresh copy of the divided difference array
     * @throws DuplicateSampleAbscissaException if any abscissas coincide
     */
    protected static double[] computeDividedDifference(double x[], double y[])
        throws DuplicateSampleAbscissaException {

        int i, j, n;
        double divdiff[], a[], denominator;

        PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y);

        n = x.length;
        divdiff = new double[n];
        for (i = 0; i < n; i++) {
            divdiff[i] = y[i];      // initialization
        }

        a = new double [n];
        a[0] = divdiff[0];
        for (i = 1; i < n; i++) {
            for (j = 0; j < n-i; j++) {
                denominator = x[j+i] - x[j];
                if (denominator == 0.0) {
                    // This happens only when two abscissas are identical.
                    throw new DuplicateSampleAbscissaException(x[j], j, j+i);
                }
                divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator;
            }
            a[i] = divdiff[0];
        }

        return a;
    }
}