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Java example source code file (PolynomialFunctionNewtonForm.java)

This example Java source code file (PolynomialFunctionNewtonForm.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

Learn more about this Java project at its project page.

Java - Java tags/keywords

derivativestructure, dimensionmismatchexception, nodataexception, nullargumentexception, polynomialfunctionnewtonform, univariatedifferentiablefunction

The PolynomialFunctionNewtonForm.java Java example source code

/*
 * 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.math3.analysis.polynomials;

import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.MathUtils;

/**
 * Implements the representation of a real polynomial function in
 * Newton Form. For reference, see <b>Elementary Numerical Analysis,
 * ISBN 0070124477, chapter 2.
 * <p>
 * The formula of polynomial in Newton form is
 *     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])
 * Note that the length of a[] is one more than the length of c[]</p>
 *
 * @since 1.2
 */
public class PolynomialFunctionNewtonForm implements UnivariateDifferentiableFunction {

    /**
     * The coefficients of the polynomial, ordered by degree -- i.e.
     * coefficients[0] is the constant term and coefficients[n] is the
     * coefficient of x^n where n is the degree of the polynomial.
     */
    private double coefficients[];

    /**
     * Centers of the Newton polynomial.
     */
    private final double c[];

    /**
     * When all c[i] = 0, a[] becomes normal polynomial coefficients,
     * i.e. a[i] = coefficients[i].
     */
    private final double a[];

    /**
     * Whether the polynomial coefficients are available.
     */
    private boolean coefficientsComputed;

    /**
     * Construct a Newton polynomial with the given a[] and c[]. The order of
     * centers are important in that if c[] shuffle, then values of a[] would
     * completely change, not just a permutation of old a[].
     * <p>
     * The constructor makes copy of the input arrays and assigns them.</p>
     *
     * @param a Coefficients in Newton form formula.
     * @param c Centers.
     * @throws NullArgumentException if any argument is {@code null}.
     * @throws NoDataException if any array has zero length.
     * @throws DimensionMismatchException if the size difference between
     * {@code a} and {@code c} is not equal to 1.
     */
    public PolynomialFunctionNewtonForm(double a[], double c[])
        throws NullArgumentException, NoDataException, DimensionMismatchException {

        verifyInputArray(a, c);
        this.a = new double[a.length];
        this.c = new double[c.length];
        System.arraycopy(a, 0, this.a, 0, a.length);
        System.arraycopy(c, 0, this.c, 0, c.length);
        coefficientsComputed = false;
    }

    /**
     * Calculate the function value at the given point.
     *
     * @param z Point at which the function value is to be computed.
     * @return the function value.
     */
    public double value(double z) {
       return evaluate(a, c, z);
    }

    /**
     * {@inheritDoc}
     * @since 3.1
     */
    public DerivativeStructure value(final DerivativeStructure t) {
        verifyInputArray(a, c);

        final int n = c.length;
        DerivativeStructure value = new DerivativeStructure(t.getFreeParameters(), t.getOrder(), a[n]);
        for (int i = n - 1; i >= 0; i--) {
            value = t.subtract(c[i]).multiply(value).add(a[i]);
        }

        return value;

    }

    /**
     * Returns the degree of the polynomial.
     *
     * @return the degree of the polynomial
     */
    public int degree() {
        return c.length;
    }

    /**
     * Returns a copy of coefficients in Newton form formula.
     * <p>
     * Changes made to the returned copy will not affect the polynomial.</p>
     *
     * @return a fresh copy of coefficients in Newton form formula
     */
    public double[] getNewtonCoefficients() {
        double[] out = new double[a.length];
        System.arraycopy(a, 0, out, 0, a.length);
        return out;
    }

    /**
     * Returns a copy of the centers array.
     * <p>
     * Changes made to the returned copy will not affect the polynomial.</p>
     *
     * @return a fresh copy of the centers array.
     */
    public double[] getCenters() {
        double[] out = new double[c.length];
        System.arraycopy(c, 0, out, 0, c.length);
        return out;
    }

    /**
     * Returns a copy of the coefficients array.
     * <p>
     * Changes made to the returned copy will not affect the polynomial.</p>
     *
     * @return a fresh copy of the coefficients array.
     */
    public double[] getCoefficients() {
        if (!coefficientsComputed) {
            computeCoefficients();
        }
        double[] out = new double[coefficients.length];
        System.arraycopy(coefficients, 0, out, 0, coefficients.length);
        return out;
    }

    /**
     * Evaluate the Newton polynomial using nested multiplication. It is
     * also called <a href="http://mathworld.wolfram.com/HornersRule.html">
     * Horner's Rule</a> and takes O(N) time.
     *
     * @param a Coefficients in Newton form formula.
     * @param c Centers.
     * @param z Point at which the function value is to be computed.
     * @return the function value.
     * @throws NullArgumentException if any argument is {@code null}.
     * @throws NoDataException if any array has zero length.
     * @throws DimensionMismatchException if the size difference between
     * {@code a} and {@code c} is not equal to 1.
     */
    public static double evaluate(double a[], double c[], double z)
        throws NullArgumentException, DimensionMismatchException, NoDataException {
        verifyInputArray(a, c);

        final int n = c.length;
        double value = a[n];
        for (int i = n - 1; i >= 0; i--) {
            value = a[i] + (z - c[i]) * value;
        }

        return value;
    }

    /**
     * Calculate the normal polynomial coefficients given the Newton form.
     * It also uses nested multiplication but takes O(N^2) time.
     */
    protected void computeCoefficients() {
        final int n = degree();

        coefficients = new double[n+1];
        for (int i = 0; i <= n; i++) {
            coefficients[i] = 0.0;
        }

        coefficients[0] = a[n];
        for (int i = n-1; i >= 0; i--) {
            for (int j = n-i; j > 0; j--) {
                coefficients[j] = coefficients[j-1] - c[i] * coefficients[j];
            }
            coefficients[0] = a[i] - c[i] * coefficients[0];
        }

        coefficientsComputed = true;
    }

    /**
     * Verifies that the input arrays are valid.
     * <p>
     * The centers must be distinct for interpolation purposes, but not
     * for general use. Thus it is not verified here.</p>
     *
     * @param a the coefficients in Newton form formula
     * @param c the centers
     * @throws NullArgumentException if any argument is {@code null}.
     * @throws NoDataException if any array has zero length.
     * @throws DimensionMismatchException if the size difference between
     * {@code a} and {@code c} is not equal to 1.
     * @see org.apache.commons.math3.analysis.interpolation.DividedDifferenceInterpolator#computeDividedDifference(double[],
     * double[])
     */
    protected static void verifyInputArray(double a[], double c[])
        throws NullArgumentException, NoDataException, DimensionMismatchException {
        MathUtils.checkNotNull(a);
        MathUtils.checkNotNull(c);
        if (a.length == 0 || c.length == 0) {
            throw new NoDataException(LocalizedFormats.EMPTY_POLYNOMIALS_COEFFICIENTS_ARRAY);
        }
        if (a.length != c.length + 1) {
            throw new DimensionMismatchException(LocalizedFormats.ARRAY_SIZES_SHOULD_HAVE_DIFFERENCE_1,
                                                 a.length, c.length);
        }
    }

}

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