Java tutorial
/* * 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.polynomials; import org.apache.commons.math.DuplicateSampleAbscissaException; import org.apache.commons.math.MathRuntimeException; import org.apache.commons.math.analysis.UnivariateRealFunction; import org.apache.commons.math.FunctionEvaluationException; import org.apache.commons.math.exception.util.LocalizedFormats; import org.apache.commons.math.util.FastMath; /** * Implements the representation of a real polynomial function in * <a href="http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html"> * Lagrange Form</a>. For reference, see <b>Introduction to Numerical * Analysis</b>, ISBN 038795452X, chapter 2. * <p> * The approximated function should be smooth enough for Lagrange polynomial * to work well. Otherwise, consider using splines instead.</p> * * @version $Revision: 1073498 $ $Date: 2011-02-22 21:57:26 +0100 (mar. 22 fvr. 2011) $ * @since 1.2 */ public class PolynomialFunctionLagrangeForm implements UnivariateRealFunction { /** * 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[]; /** * Interpolating points (abscissas). */ private final double x[]; /** * Function values at interpolating points. */ private final double y[]; /** * Whether the polynomial coefficients are available. */ private boolean coefficientsComputed; /** * Construct a Lagrange polynomial with the given abscissas and function * values. The order of interpolating points are not important. * <p> * The constructor makes copy of the input arrays and assigns them.</p> * * @param x interpolating points * @param y function values at interpolating points * @throws IllegalArgumentException if input arrays are not valid */ public PolynomialFunctionLagrangeForm(double x[], double y[]) throws IllegalArgumentException { verifyInterpolationArray(x, y); this.x = new double[x.length]; this.y = new double[y.length]; System.arraycopy(x, 0, this.x, 0, x.length); System.arraycopy(y, 0, this.y, 0, y.length); coefficientsComputed = false; } /** {@inheritDoc} */ public double value(double z) throws FunctionEvaluationException { try { return evaluate(x, y, z); } catch (DuplicateSampleAbscissaException e) { throw new FunctionEvaluationException(z, e.getSpecificPattern(), e.getGeneralPattern(), e.getArguments()); } } /** * Returns the degree of the polynomial. * * @return the degree of the polynomial */ public int degree() { return x.length - 1; } /** * Returns a copy of the interpolating points array. * <p> * Changes made to the returned copy will not affect the polynomial.</p> * * @return a fresh copy of the interpolating points array */ public double[] getInterpolatingPoints() { double[] out = new double[x.length]; System.arraycopy(x, 0, out, 0, x.length); return out; } /** * Returns a copy of the interpolating values array. * <p> * Changes made to the returned copy will not affect the polynomial.</p> * * @return a fresh copy of the interpolating values array */ public double[] getInterpolatingValues() { double[] out = new double[y.length]; System.arraycopy(y, 0, out, 0, y.length); return out; } /** * Returns a copy of the coefficients array. * <p> * Changes made to the returned copy will not affect the polynomial.</p> * <p> * Note that coefficients computation can be ill-conditioned. Use with caution * and only when it is necessary.</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 Lagrange polynomial using * <a href="http://mathworld.wolfram.com/NevillesAlgorithm.html"> * Neville's Algorithm</a>. It takes O(N^2) time. * <p> * This function is made public static so that users can call it directly * without instantiating PolynomialFunctionLagrangeForm object.</p> * * @param x the interpolating points array * @param y the interpolating values array * @param z the point at which the function value is to be computed * @return the function value * @throws DuplicateSampleAbscissaException if the sample has duplicate abscissas * @throws IllegalArgumentException if inputs are not valid */ public static double evaluate(double x[], double y[], double z) throws DuplicateSampleAbscissaException, IllegalArgumentException { verifyInterpolationArray(x, y); int nearest = 0; final int n = x.length; final double[] c = new double[n]; final double[] d = new double[n]; double min_dist = Double.POSITIVE_INFINITY; for (int i = 0; i < n; i++) { // initialize the difference arrays c[i] = y[i]; d[i] = y[i]; // find out the abscissa closest to z final double dist = FastMath.abs(z - x[i]); if (dist < min_dist) { nearest = i; min_dist = dist; } } // initial approximation to the function value at z double value = y[nearest]; for (int i = 1; i < n; i++) { for (int j = 0; j < n - i; j++) { final double tc = x[j] - z; final double td = x[i + j] - z; final double divider = x[j] - x[i + j]; if (divider == 0.0) { // This happens only when two abscissas are identical. throw new DuplicateSampleAbscissaException(x[i], i, i + j); } // update the difference arrays final double w = (c[j + 1] - d[j]) / divider; c[j] = tc * w; d[j] = td * w; } // sum up the difference terms to get the final value if (nearest < 0.5 * (n - i + 1)) { value += c[nearest]; // fork down } else { nearest--; value += d[nearest]; // fork up } } return value; } /** * Calculate the coefficients of Lagrange polynomial from the * interpolation data. It takes O(N^2) time. * <p> * Note this computation can be ill-conditioned. Use with caution * and only when it is necessary.</p> * * @throws ArithmeticException if any abscissas coincide */ protected void computeCoefficients() throws ArithmeticException { final int n = degree() + 1; coefficients = new double[n]; for (int i = 0; i < n; i++) { coefficients[i] = 0.0; } // c[] are the coefficients of P(x) = (x-x[0])(x-x[1])...(x-x[n-1]) final double[] c = new double[n + 1]; c[0] = 1.0; for (int i = 0; i < n; i++) { for (int j = i; j > 0; j--) { c[j] = c[j - 1] - c[j] * x[i]; } c[0] *= -x[i]; c[i + 1] = 1; } final double[] tc = new double[n]; for (int i = 0; i < n; i++) { // d = (x[i]-x[0])...(x[i]-x[i-1])(x[i]-x[i+1])...(x[i]-x[n-1]) double d = 1; for (int j = 0; j < n; j++) { if (i != j) { d *= x[i] - x[j]; } } if (d == 0.0) { // This happens only when two abscissas are identical. for (int k = 0; k < n; ++k) { if ((i != k) && (x[i] == x[k])) { throw MathRuntimeException.createArithmeticException( LocalizedFormats.IDENTICAL_ABSCISSAS_DIVISION_BY_ZERO, i, k, x[i]); } } } final double t = y[i] / d; // Lagrange polynomial is the sum of n terms, each of which is a // polynomial of degree n-1. tc[] are the coefficients of the i-th // numerator Pi(x) = (x-x[0])...(x-x[i-1])(x-x[i+1])...(x-x[n-1]). tc[n - 1] = c[n]; // actually c[n] = 1 coefficients[n - 1] += t * tc[n - 1]; for (int j = n - 2; j >= 0; j--) { tc[j] = c[j + 1] + tc[j + 1] * x[i]; coefficients[j] += t * tc[j]; } } coefficientsComputed = true; } /** * Verifies that the interpolation arrays are valid. * <p> * The arrays features checked by this method are that both arrays have the * same length and this length is at least 2. * </p> * <p> * The interpolating points must be distinct. However it is not * verified here, it is checked in evaluate() and computeCoefficients(). * </p> * * @param x the interpolating points array * @param y the interpolating values array * @throws IllegalArgumentException if not valid * @see #evaluate(double[], double[], double) * @see #computeCoefficients() */ public static void verifyInterpolationArray(double x[], double y[]) throws IllegalArgumentException { if (x.length != y.length) { throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.DIMENSIONS_MISMATCH_SIMPLE, x.length, y.length); } if (x.length < 2) { throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.WRONG_NUMBER_OF_POINTS, 2, x.length); } } }