DividedDifferenceInterpolator.java

/*
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 * 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
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 *
 *      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,
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package org.apache.commons.math4.legacy.analysis.interpolation;

import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunctionLagrangeForm;
import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunctionNewtonForm;
import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException;
import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;

/**
 * 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 evaluation is in PolynomialFunctionLagrangeForm,
 * this class provides an easy-to-use interface to it.</p>
 *
 * @since 1.2
 */
public class DividedDifferenceInterpolator
    implements UnivariateInterpolator {
    /**
     * Compute an interpolating function for the dataset.
     *
     * @param x Interpolating points array.
     * @param y Interpolating values array.
     * @return a function which interpolates the dataset.
     * @throws DimensionMismatchException if the array lengths are different.
     * @throws NumberIsTooSmallException if the number of points is less than 2.
     * @throws NonMonotonicSequenceException if {@code x} is not sorted in
     * strictly increasing order.
     */
    @Override
    public PolynomialFunctionNewtonForm interpolate(double[] x, double[] y)
        throws DimensionMismatchException,
               NumberIsTooSmallException,
               NonMonotonicSequenceException {
        /*
         * 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])
         */
        PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);

        /*
         * 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>
         */
        final double[] c = new double[x.length-1];
        System.arraycopy(x, 0, c, 0, c.length);

        final double[] a = computeDividedDifference(x, y);
        return new PolynomialFunctionNewtonForm(a, c);
    }

    /**
     * Return 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>
     * The computational complexity is \(O(n^2)\) where \(n\) is the common
     * length of {@code x} and {@code y}.</p>
     *
     * @param x Interpolating points array.
     * @param y Interpolating values array.
     * @return a fresh copy of the divided difference array.
     * @throws DimensionMismatchException if the array lengths are different.
     * @throws NumberIsTooSmallException if the number of points is less than 2.
     * @throws NonMonotonicSequenceException
     * if {@code x} is not sorted in strictly increasing order.
     */
    protected static double[] computeDividedDifference(final double[] x, final double[] y)
        throws DimensionMismatchException,
               NumberIsTooSmallException,
               NonMonotonicSequenceException {
        PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);

        final double[] divdiff = y.clone(); // initialization

        final int n = x.length;
        final double[] a = new double [n];
        a[0] = divdiff[0];
        for (int i = 1; i < n; i++) {
            for (int j = 0; j < n-i; j++) {
                final double denominator = x[j+i] - x[j];
                divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator;
            }
            a[i] = divdiff[0];
        }

        return a;
    }
}