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Commons Math example source code file (PolynomialFunctionLagrangeForm.java)

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

Java - Commons Math tags/keywords

arithmeticexception, duplicatesampleabscissaexception, duplicatesampleabscissaexception, functionevaluationexception, functionevaluationexception, illegalargumentexception, illegalargumentexception, polynomialfunctionlagrangeform, polynomialfunctionlagrangeform, univariaterealfunction

The Commons Math PolynomialFunctionLagrangeForm.java 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.math.analysis.polynomials;

import org.apache.commons.math.DuplicateSampleAbscissaException;
import org.apache.commons.math.FunctionEvaluationException;
import org.apache.commons.math.MathRuntimeException;
import org.apache.commons.math.analysis.UnivariateRealFunction;

/**
 * Implements the representation of a real polynomial function in
 * <a href="http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html">
 * Lagrange Form</a>. For reference, see 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: 922708 $ $Date: 2010-03-13 20:15:47 -0500 (Sat, 13 Mar 2010) $
 * @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;
    }

    /**
     * Calculate the function value at the given point.
     *
     * @param z the point at which the function value is to be computed
     * @return the function value
     * @throws FunctionEvaluationException if a runtime error occurs
     * @see UnivariateRealFunction#value(double)
     */
    public double value(double z) throws FunctionEvaluationException {
        try {
            return evaluate(x, y, z);
        } catch (DuplicateSampleAbscissaException e) {
            throw new FunctionEvaluationException(e, z, e.getPattern(), 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 = Math.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("identical abscissas x[{0}] == x[{1}] == {2} cause 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(
                  "dimension mismatch {0} != {1}", x.length, y.length);
        }

        if (x.length < 2) {
            throw MathRuntimeException.createIllegalArgumentException(
                  "{0} points are required, got only {1}", 2, x.length);
        }

    }
}

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