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Java example source code file (PolynomialsUtils.java)

This example Java source code file (PolynomialsUtils.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

arraylist, bigfraction, chebyshev_coefficients, hermite_coefficients, jacobi_coefficients, jacobikey, laguerre_coefficients, legendre_coefficients, list, object, override, polynomialfunction, polynomialsutils, recurrencecoefficientsgenerator, util

The PolynomialsUtils.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 java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;

import org.apache.commons.math3.fraction.BigFraction;
import org.apache.commons.math3.util.CombinatoricsUtils;
import org.apache.commons.math3.util.FastMath;

/**
 * A collection of static methods that operate on or return polynomials.
 *
 * @since 2.0
 */
public class PolynomialsUtils {

    /** Coefficients for Chebyshev polynomials. */
    private static final List<BigFraction> CHEBYSHEV_COEFFICIENTS;

    /** Coefficients for Hermite polynomials. */
    private static final List<BigFraction> HERMITE_COEFFICIENTS;

    /** Coefficients for Laguerre polynomials. */
    private static final List<BigFraction> LAGUERRE_COEFFICIENTS;

    /** Coefficients for Legendre polynomials. */
    private static final List<BigFraction> LEGENDRE_COEFFICIENTS;

    /** Coefficients for Jacobi polynomials. */
    private static final Map<JacobiKey, List JACOBI_COEFFICIENTS;

    static {

        // initialize recurrence for Chebyshev polynomials
        // T0(X) = 1, T1(X) = 0 + 1 * X
        CHEBYSHEV_COEFFICIENTS = new ArrayList<BigFraction>();
        CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE);
        CHEBYSHEV_COEFFICIENTS.add(BigFraction.ZERO);
        CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE);

        // initialize recurrence for Hermite polynomials
        // H0(X) = 1, H1(X) = 0 + 2 * X
        HERMITE_COEFFICIENTS = new ArrayList<BigFraction>();
        HERMITE_COEFFICIENTS.add(BigFraction.ONE);
        HERMITE_COEFFICIENTS.add(BigFraction.ZERO);
        HERMITE_COEFFICIENTS.add(BigFraction.TWO);

        // initialize recurrence for Laguerre polynomials
        // L0(X) = 1, L1(X) = 1 - 1 * X
        LAGUERRE_COEFFICIENTS = new ArrayList<BigFraction>();
        LAGUERRE_COEFFICIENTS.add(BigFraction.ONE);
        LAGUERRE_COEFFICIENTS.add(BigFraction.ONE);
        LAGUERRE_COEFFICIENTS.add(BigFraction.MINUS_ONE);

        // initialize recurrence for Legendre polynomials
        // P0(X) = 1, P1(X) = 0 + 1 * X
        LEGENDRE_COEFFICIENTS = new ArrayList<BigFraction>();
        LEGENDRE_COEFFICIENTS.add(BigFraction.ONE);
        LEGENDRE_COEFFICIENTS.add(BigFraction.ZERO);
        LEGENDRE_COEFFICIENTS.add(BigFraction.ONE);

        // initialize map for Jacobi polynomials
        JACOBI_COEFFICIENTS = new HashMap<JacobiKey, List();

    }

    /**
     * Private constructor, to prevent instantiation.
     */
    private PolynomialsUtils() {
    }

    /**
     * Create a Chebyshev polynomial of the first kind.
     * <p>Chebyshev
     * polynomials of the first kind</a> are orthogonal polynomials.
     * They can be defined by the following recurrence relations:</p>

* \( * T_0(x) = 1 \\ * T_1(x) = x \\ * T_{k+1}(x) = 2x T_k(x) - T_{k-1}(x) * \) * </p> * @param degree degree of the polynomial * @return Chebyshev polynomial of specified degree */ public static PolynomialFunction createChebyshevPolynomial(final int degree) { return buildPolynomial(degree, CHEBYSHEV_COEFFICIENTS, new RecurrenceCoefficientsGenerator() { /** Fixed recurrence coefficients. */ private final BigFraction[] coeffs = { BigFraction.ZERO, BigFraction.TWO, BigFraction.ONE }; /** {@inheritDoc} */ public BigFraction[] generate(int k) { return coeffs; } }); } /** * Create a Hermite polynomial. * <p>Hermite * polynomials</a> are orthogonal polynomials. * They can be defined by the following recurrence relations:</p>

* \( * H_0(x) = 1 \\ * H_1(x) = 2x \\ * H_{k+1}(x) = 2x H_k(X) - 2k H_{k-1}(x) * \) * </p> * @param degree degree of the polynomial * @return Hermite polynomial of specified degree */ public static PolynomialFunction createHermitePolynomial(final int degree) { return buildPolynomial(degree, HERMITE_COEFFICIENTS, new RecurrenceCoefficientsGenerator() { /** {@inheritDoc} */ public BigFraction[] generate(int k) { return new BigFraction[] { BigFraction.ZERO, BigFraction.TWO, new BigFraction(2 * k)}; } }); } /** * Create a Laguerre polynomial. * <p>Laguerre * polynomials</a> are orthogonal polynomials. * They can be defined by the following recurrence relations:</p>

* \( * L_0(x) = 1 \\ * L_1(x) = 1 - x \\ * (k+1) L_{k+1}(x) = (2k + 1 - x) L_k(x) - k L_{k-1}(x) * \) * </p> * @param degree degree of the polynomial * @return Laguerre polynomial of specified degree */ public static PolynomialFunction createLaguerrePolynomial(final int degree) { return buildPolynomial(degree, LAGUERRE_COEFFICIENTS, new RecurrenceCoefficientsGenerator() { /** {@inheritDoc} */ public BigFraction[] generate(int k) { final int kP1 = k + 1; return new BigFraction[] { new BigFraction(2 * k + 1, kP1), new BigFraction(-1, kP1), new BigFraction(k, kP1)}; } }); } /** * Create a Legendre polynomial. * <p>Legendre * polynomials</a> are orthogonal polynomials. * They can be defined by the following recurrence relations:</p>

* \( * P_0(x) = 1 \\ * P_1(x) = x \\ * (k+1) P_{k+1}(x) = (2k+1) x P_k(x) - k P_{k-1}(x) * \) * </p> * @param degree degree of the polynomial * @return Legendre polynomial of specified degree */ public static PolynomialFunction createLegendrePolynomial(final int degree) { return buildPolynomial(degree, LEGENDRE_COEFFICIENTS, new RecurrenceCoefficientsGenerator() { /** {@inheritDoc} */ public BigFraction[] generate(int k) { final int kP1 = k + 1; return new BigFraction[] { BigFraction.ZERO, new BigFraction(k + kP1, kP1), new BigFraction(k, kP1)}; } }); } /** * Create a Jacobi polynomial. * <p>Jacobi * polynomials</a> are orthogonal polynomials. * They can be defined by the following recurrence relations:</p>

* \( * P_0^{vw}(x) = 1 \\ * P_{-1}^{vw}(x) = 0 \\ * 2k(k + v + w)(2k + v + w - 2) P_k^{vw}(x) = \\ * (2k + v + w - 1)[(2k + v + w)(2k + v + w - 2) x + v^2 - w^2] P_{k-1}^{vw}(x) \\ * - 2(k + v - 1)(k + w - 1)(2k + v + w) P_{k-2}^{vw}(x) * \) * </p> * @param degree degree of the polynomial * @param v first exponent * @param w second exponent * @return Jacobi polynomial of specified degree */ public static PolynomialFunction createJacobiPolynomial(final int degree, final int v, final int w) { // select the appropriate list final JacobiKey key = new JacobiKey(v, w); if (!JACOBI_COEFFICIENTS.containsKey(key)) { // allocate a new list for v, w final List<BigFraction> list = new ArrayList(); JACOBI_COEFFICIENTS.put(key, list); // Pv,w,0(x) = 1; list.add(BigFraction.ONE); // P1(x) = (v - w) / 2 + (2 + v + w) * X / 2 list.add(new BigFraction(v - w, 2)); list.add(new BigFraction(2 + v + w, 2)); } return buildPolynomial(degree, JACOBI_COEFFICIENTS.get(key), new RecurrenceCoefficientsGenerator() { /** {@inheritDoc} */ public BigFraction[] generate(int k) { k++; final int kvw = k + v + w; final int twoKvw = kvw + k; final int twoKvwM1 = twoKvw - 1; final int twoKvwM2 = twoKvw - 2; final int den = 2 * k * kvw * twoKvwM2; return new BigFraction[] { new BigFraction(twoKvwM1 * (v * v - w * w), den), new BigFraction(twoKvwM1 * twoKvw * twoKvwM2, den), new BigFraction(2 * (k + v - 1) * (k + w - 1) * twoKvw, den) }; } }); } /** Inner class for Jacobi polynomials keys. */ private static class JacobiKey { /** First exponent. */ private final int v; /** Second exponent. */ private final int w; /** Simple constructor. * @param v first exponent * @param w second exponent */ JacobiKey(final int v, final int w) { this.v = v; this.w = w; } /** Get hash code. * @return hash code */ @Override public int hashCode() { return (v << 16) ^ w; } /** Check if the instance represent the same key as another instance. * @param key other key * @return true if the instance and the other key refer to the same polynomial */ @Override public boolean equals(final Object key) { if ((key == null) || !(key instanceof JacobiKey)) { return false; } final JacobiKey otherK = (JacobiKey) key; return (v == otherK.v) && (w == otherK.w); } } /** * Compute the coefficients of the polynomial \(P_s(x)\) * whose values at point {@code x} will be the same as the those from the * original polynomial \(P(x)\) when computed at {@code x + shift}. * <p> * More precisely, let \(\Delta = \) {@code shift} and let * \(P_s(x) = P(x + \Delta)\). The returned array * consists of the coefficients of \(P_s\). So if \(a_0, ..., a_{n-1}\) * are the coefficients of \(P\), then the returned array * \(b_0, ..., b_{n-1}\) satisfies the identity * \(\sum_{i=0}^{n-1} b_i x^i = \sum_{i=0}^{n-1} a_i (x + \Delta)^i\) for all \(x\). * * @param coefficients Coefficients of the original polynomial. * @param shift Shift value. * @return the coefficients \(b_i\) of the shifted * polynomial. */ public static double[] shift(final double[] coefficients, final double shift) { final int dp1 = coefficients.length; final double[] newCoefficients = new double[dp1]; // Pascal triangle. final int[][] coeff = new int[dp1][dp1]; for (int i = 0; i < dp1; i++){ for(int j = 0; j <= i; j++){ coeff[i][j] = (int) CombinatoricsUtils.binomialCoefficient(i, j); } } // First polynomial coefficient. for (int i = 0; i < dp1; i++){ newCoefficients[0] += coefficients[i] * FastMath.pow(shift, i); } // Superior order. final int d = dp1 - 1; for (int i = 0; i < d; i++) { for (int j = i; j < d; j++){ newCoefficients[i + 1] += coeff[j + 1][j - i] * coefficients[j + 1] * FastMath.pow(shift, j - i); } } return newCoefficients; } /** Get the coefficients array for a given degree. * @param degree degree of the polynomial * @param coefficients list where the computed coefficients are stored * @param generator recurrence coefficients generator * @return coefficients array */ private static PolynomialFunction buildPolynomial(final int degree, final List<BigFraction> coefficients, final RecurrenceCoefficientsGenerator generator) { synchronized (coefficients) { final int maxDegree = (int) FastMath.floor(FastMath.sqrt(2 * coefficients.size())) - 1; if (degree > maxDegree) { computeUpToDegree(degree, maxDegree, generator, coefficients); } } // coefficient for polynomial 0 is l [0] // coefficients for polynomial 1 are l [1] ... l [2] (degrees 0 ... 1) // coefficients for polynomial 2 are l [3] ... l [5] (degrees 0 ... 2) // coefficients for polynomial 3 are l [6] ... l [9] (degrees 0 ... 3) // coefficients for polynomial 4 are l[10] ... l[14] (degrees 0 ... 4) // coefficients for polynomial 5 are l[15] ... l[20] (degrees 0 ... 5) // coefficients for polynomial 6 are l[21] ... l[27] (degrees 0 ... 6) // ... final int start = degree * (degree + 1) / 2; final double[] a = new double[degree + 1]; for (int i = 0; i <= degree; ++i) { a[i] = coefficients.get(start + i).doubleValue(); } // build the polynomial return new PolynomialFunction(a); } /** Compute polynomial coefficients up to a given degree. * @param degree maximal degree * @param maxDegree current maximal degree * @param generator recurrence coefficients generator * @param coefficients list where the computed coefficients should be appended */ private static void computeUpToDegree(final int degree, final int maxDegree, final RecurrenceCoefficientsGenerator generator, final List<BigFraction> coefficients) { int startK = (maxDegree - 1) * maxDegree / 2; for (int k = maxDegree; k < degree; ++k) { // start indices of two previous polynomials Pk(X) and Pk-1(X) int startKm1 = startK; startK += k; // Pk+1(X) = (a[0] + a[1] X) Pk(X) - a[2] Pk-1(X) BigFraction[] ai = generator.generate(k); BigFraction ck = coefficients.get(startK); BigFraction ckm1 = coefficients.get(startKm1); // degree 0 coefficient coefficients.add(ck.multiply(ai[0]).subtract(ckm1.multiply(ai[2]))); // degree 1 to degree k-1 coefficients for (int i = 1; i < k; ++i) { final BigFraction ckPrev = ck; ck = coefficients.get(startK + i); ckm1 = coefficients.get(startKm1 + i); coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1])).subtract(ckm1.multiply(ai[2]))); } // degree k coefficient final BigFraction ckPrev = ck; ck = coefficients.get(startK + k); coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1]))); // degree k+1 coefficient coefficients.add(ck.multiply(ai[1])); } } /** Interface for recurrence coefficients generation. */ private interface RecurrenceCoefficientsGenerator { /** * Generate recurrence coefficients. * @param k highest degree of the polynomials used in the recurrence * @return an array of three coefficients such that * \( P_{k+1}(x) = (a[0] + a[1] x) P_k(x) - a[2] P_{k-1}(x) \) */ BigFraction[] generate(int k); } }



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