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/*
 * 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.
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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 Elementary Numerical Analysis,
 * ISBN 0070124477, chapter 2.
 * 

* 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[]

* * @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[]. *

* The constructor makes copy of the input arrays and assigns them.

* * @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. *

* Changes made to the returned copy will not affect the polynomial.

* * @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. *

* Changes made to the returned copy will not affect the polynomial.

* * @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. *

* Changes made to the returned copy will not affect the polynomial.

* * @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 * Horner's Rule 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. *

* The centers must be distinct for interpolation purposes, but not * for general use. Thus it is not verified here.

* * @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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