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Java example source code file (PolynomialsUtilsTest.java)

This example Java source code file (PolynomialsUtilsTest.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

hk1g1, iterativelegendregaussintegrator, lk1g1, lk2g2, pk0, pk0g0, pk1, polynomialfunction, string, test, tk0, tk1g1, tk2, univariatefunction

The PolynomialsUtilsTest.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.UnivariateFunction;
import org.apache.commons.math3.analysis.integration.IterativeLegendreGaussIntegrator;
import org.apache.commons.math3.util.CombinatoricsUtils;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.Precision;
import org.junit.Assert;
import org.junit.Test;

/**
 * Tests the PolynomialsUtils class.
 *
 */
public class PolynomialsUtilsTest {

    @Test
    public void testFirstChebyshevPolynomials() {
        checkPolynomial(PolynomialsUtils.createChebyshevPolynomial(3), "-3 x + 4 x^3");
        checkPolynomial(PolynomialsUtils.createChebyshevPolynomial(2), "-1 + 2 x^2");
        checkPolynomial(PolynomialsUtils.createChebyshevPolynomial(1), "x");
        checkPolynomial(PolynomialsUtils.createChebyshevPolynomial(0), "1");

        checkPolynomial(PolynomialsUtils.createChebyshevPolynomial(7), "-7 x + 56 x^3 - 112 x^5 + 64 x^7");
        checkPolynomial(PolynomialsUtils.createChebyshevPolynomial(6), "-1 + 18 x^2 - 48 x^4 + 32 x^6");
        checkPolynomial(PolynomialsUtils.createChebyshevPolynomial(5), "5 x - 20 x^3 + 16 x^5");
        checkPolynomial(PolynomialsUtils.createChebyshevPolynomial(4), "1 - 8 x^2 + 8 x^4");

    }

    @Test
    public void testChebyshevBounds() {
        for (int k = 0; k < 12; ++k) {
            PolynomialFunction Tk = PolynomialsUtils.createChebyshevPolynomial(k);
            for (double x = -1; x <= 1; x += 0.02) {
                Assert.assertTrue(k + " " + Tk.value(x), FastMath.abs(Tk.value(x)) < (1 + 1e-12));
            }
        }
    }

    @Test
    public void testChebyshevDifferentials() {
        for (int k = 0; k < 12; ++k) {

            PolynomialFunction Tk0 = PolynomialsUtils.createChebyshevPolynomial(k);
            PolynomialFunction Tk1 = Tk0.polynomialDerivative();
            PolynomialFunction Tk2 = Tk1.polynomialDerivative();

            PolynomialFunction g0 = new PolynomialFunction(new double[] { k * k });
            PolynomialFunction g1 = new PolynomialFunction(new double[] { 0, -1});
            PolynomialFunction g2 = new PolynomialFunction(new double[] { 1, 0, -1 });

            PolynomialFunction Tk0g0 = Tk0.multiply(g0);
            PolynomialFunction Tk1g1 = Tk1.multiply(g1);
            PolynomialFunction Tk2g2 = Tk2.multiply(g2);

            checkNullPolynomial(Tk0g0.add(Tk1g1.add(Tk2g2)));

        }
    }

    @Test
    public void testChebyshevOrthogonality() {
        UnivariateFunction weight = new UnivariateFunction() {
            public double value(double x) {
                return 1 / FastMath.sqrt(1 - x * x);
            }
        };
        for (int i = 0; i < 10; ++i) {
            PolynomialFunction pi = PolynomialsUtils.createChebyshevPolynomial(i);
            for (int j = 0; j <= i; ++j) {
                PolynomialFunction pj = PolynomialsUtils.createChebyshevPolynomial(j);
                checkOrthogonality(pi, pj, weight, -0.9999, 0.9999, 1.5, 0.03);
            }
        }
    }

    @Test
    public void testFirstHermitePolynomials() {
        checkPolynomial(PolynomialsUtils.createHermitePolynomial(3), "-12 x + 8 x^3");
        checkPolynomial(PolynomialsUtils.createHermitePolynomial(2), "-2 + 4 x^2");
        checkPolynomial(PolynomialsUtils.createHermitePolynomial(1), "2 x");
        checkPolynomial(PolynomialsUtils.createHermitePolynomial(0), "1");

        checkPolynomial(PolynomialsUtils.createHermitePolynomial(7), "-1680 x + 3360 x^3 - 1344 x^5 + 128 x^7");
        checkPolynomial(PolynomialsUtils.createHermitePolynomial(6), "-120 + 720 x^2 - 480 x^4 + 64 x^6");
        checkPolynomial(PolynomialsUtils.createHermitePolynomial(5), "120 x - 160 x^3 + 32 x^5");
        checkPolynomial(PolynomialsUtils.createHermitePolynomial(4), "12 - 48 x^2 + 16 x^4");

    }

    @Test
    public void testHermiteDifferentials() {
        for (int k = 0; k < 12; ++k) {

            PolynomialFunction Hk0 = PolynomialsUtils.createHermitePolynomial(k);
            PolynomialFunction Hk1 = Hk0.polynomialDerivative();
            PolynomialFunction Hk2 = Hk1.polynomialDerivative();

            PolynomialFunction g0 = new PolynomialFunction(new double[] { 2 * k });
            PolynomialFunction g1 = new PolynomialFunction(new double[] { 0, -2 });
            PolynomialFunction g2 = new PolynomialFunction(new double[] { 1 });

            PolynomialFunction Hk0g0 = Hk0.multiply(g0);
            PolynomialFunction Hk1g1 = Hk1.multiply(g1);
            PolynomialFunction Hk2g2 = Hk2.multiply(g2);

            checkNullPolynomial(Hk0g0.add(Hk1g1.add(Hk2g2)));

        }
    }

    @Test
    public void testHermiteOrthogonality() {
        UnivariateFunction weight = new UnivariateFunction() {
            public double value(double x) {
                return FastMath.exp(-x * x);
            }
        };
        for (int i = 0; i < 10; ++i) {
            PolynomialFunction pi = PolynomialsUtils.createHermitePolynomial(i);
            for (int j = 0; j <= i; ++j) {
                PolynomialFunction pj = PolynomialsUtils.createHermitePolynomial(j);
                checkOrthogonality(pi, pj, weight, -50, 50, 1.5, 1.0e-8);
            }
        }
    }

    @Test
    public void testFirstLaguerrePolynomials() {
        checkPolynomial(PolynomialsUtils.createLaguerrePolynomial(3), 6l, "6 - 18 x + 9 x^2 - x^3");
        checkPolynomial(PolynomialsUtils.createLaguerrePolynomial(2), 2l, "2 - 4 x + x^2");
        checkPolynomial(PolynomialsUtils.createLaguerrePolynomial(1), 1l, "1 - x");
        checkPolynomial(PolynomialsUtils.createLaguerrePolynomial(0), 1l, "1");

        checkPolynomial(PolynomialsUtils.createLaguerrePolynomial(7), 5040l,
                "5040 - 35280 x + 52920 x^2 - 29400 x^3"
                + " + 7350 x^4 - 882 x^5 + 49 x^6 - x^7");
        checkPolynomial(PolynomialsUtils.createLaguerrePolynomial(6),  720l,
                "720 - 4320 x + 5400 x^2 - 2400 x^3 + 450 x^4"
                + " - 36 x^5 + x^6");
        checkPolynomial(PolynomialsUtils.createLaguerrePolynomial(5),  120l,
        "120 - 600 x + 600 x^2 - 200 x^3 + 25 x^4 - x^5");
        checkPolynomial(PolynomialsUtils.createLaguerrePolynomial(4),   24l,
        "24 - 96 x + 72 x^2 - 16 x^3 + x^4");

    }

    @Test
    public void testLaguerreDifferentials() {
        for (int k = 0; k < 12; ++k) {

            PolynomialFunction Lk0 = PolynomialsUtils.createLaguerrePolynomial(k);
            PolynomialFunction Lk1 = Lk0.polynomialDerivative();
            PolynomialFunction Lk2 = Lk1.polynomialDerivative();

            PolynomialFunction g0 = new PolynomialFunction(new double[] { k });
            PolynomialFunction g1 = new PolynomialFunction(new double[] { 1, -1 });
            PolynomialFunction g2 = new PolynomialFunction(new double[] { 0, 1 });

            PolynomialFunction Lk0g0 = Lk0.multiply(g0);
            PolynomialFunction Lk1g1 = Lk1.multiply(g1);
            PolynomialFunction Lk2g2 = Lk2.multiply(g2);

            checkNullPolynomial(Lk0g0.add(Lk1g1.add(Lk2g2)));

        }
    }

    @Test
    public void testLaguerreOrthogonality() {
        UnivariateFunction weight = new UnivariateFunction() {
            public double value(double x) {
                return FastMath.exp(-x);
            }
        };
        for (int i = 0; i < 10; ++i) {
            PolynomialFunction pi = PolynomialsUtils.createLaguerrePolynomial(i);
            for (int j = 0; j <= i; ++j) {
                PolynomialFunction pj = PolynomialsUtils.createLaguerrePolynomial(j);
                checkOrthogonality(pi, pj, weight, 0.0, 100.0, 0.99999, 1.0e-13);
            }
        }
    }

    @Test
    public void testFirstLegendrePolynomials() {
        checkPolynomial(PolynomialsUtils.createLegendrePolynomial(3),  2l, "-3 x + 5 x^3");
        checkPolynomial(PolynomialsUtils.createLegendrePolynomial(2),  2l, "-1 + 3 x^2");
        checkPolynomial(PolynomialsUtils.createLegendrePolynomial(1),  1l, "x");
        checkPolynomial(PolynomialsUtils.createLegendrePolynomial(0),  1l, "1");

        checkPolynomial(PolynomialsUtils.createLegendrePolynomial(7), 16l, "-35 x + 315 x^3 - 693 x^5 + 429 x^7");
        checkPolynomial(PolynomialsUtils.createLegendrePolynomial(6), 16l, "-5 + 105 x^2 - 315 x^4 + 231 x^6");
        checkPolynomial(PolynomialsUtils.createLegendrePolynomial(5),  8l, "15 x - 70 x^3 + 63 x^5");
        checkPolynomial(PolynomialsUtils.createLegendrePolynomial(4),  8l, "3 - 30 x^2 + 35 x^4");

    }

    @Test
    public void testLegendreDifferentials() {
        for (int k = 0; k < 12; ++k) {

            PolynomialFunction Pk0 = PolynomialsUtils.createLegendrePolynomial(k);
            PolynomialFunction Pk1 = Pk0.polynomialDerivative();
            PolynomialFunction Pk2 = Pk1.polynomialDerivative();

            PolynomialFunction g0 = new PolynomialFunction(new double[] { k * (k + 1) });
            PolynomialFunction g1 = new PolynomialFunction(new double[] { 0, -2 });
            PolynomialFunction g2 = new PolynomialFunction(new double[] { 1, 0, -1 });

            PolynomialFunction Pk0g0 = Pk0.multiply(g0);
            PolynomialFunction Pk1g1 = Pk1.multiply(g1);
            PolynomialFunction Pk2g2 = Pk2.multiply(g2);

            checkNullPolynomial(Pk0g0.add(Pk1g1.add(Pk2g2)));

        }
    }

    @Test
    public void testLegendreOrthogonality() {
        UnivariateFunction weight = new UnivariateFunction() {
            public double value(double x) {
                return 1;
            }
        };
        for (int i = 0; i < 10; ++i) {
            PolynomialFunction pi = PolynomialsUtils.createLegendrePolynomial(i);
            for (int j = 0; j <= i; ++j) {
                PolynomialFunction pj = PolynomialsUtils.createLegendrePolynomial(j);
                checkOrthogonality(pi, pj, weight, -1, 1, 0.1, 1.0e-13);
            }
        }
    }

    @Test
    public void testHighDegreeLegendre() {
        PolynomialsUtils.createLegendrePolynomial(40);
        double[] l40 = PolynomialsUtils.createLegendrePolynomial(40).getCoefficients();
        double denominator = 274877906944d;
        double[] numerators = new double[] {
                          +34461632205d,            -28258538408100d,          +3847870979902950d,        -207785032914759300d,
                  +5929294332103310025d,     -103301483474866556880d,    +1197358103913226000200d,    -9763073770369381232400d,
              +58171647881784229843050d,  -260061484647976556945400d,  +888315281771246239250340d, -2345767627188139419665400d,
            +4819022625419112503443050d, -7710436200670580005508880d, +9566652323054238154983240d, -9104813935044723209570256d,
            +6516550296251767619752905d, -3391858621221953912598660d, +1211378079007840683070950d,  -265365894974690562152100d,
              +26876802183334044115405d
        };
        for (int i = 0; i < l40.length; ++i) {
            if (i % 2 == 0) {
                double ci = numerators[i / 2] / denominator;
                Assert.assertEquals(ci, l40[i], FastMath.abs(ci) * 1e-15);
            } else {
                Assert.assertEquals(0, l40[i], 0);
            }
        }
    }

    @Test
    public void testJacobiLegendre() {
        for (int i = 0; i < 10; ++i) {
            PolynomialFunction legendre = PolynomialsUtils.createLegendrePolynomial(i);
            PolynomialFunction jacobi   = PolynomialsUtils.createJacobiPolynomial(i, 0, 0);
            checkNullPolynomial(legendre.subtract(jacobi));
        }
    }

    @Test
    public void testJacobiEvaluationAt1() {
        for (int v = 0; v < 10; ++v) {
            for (int w = 0; w < 10; ++w) {
                for (int i = 0; i < 10; ++i) {
                    PolynomialFunction jacobi = PolynomialsUtils.createJacobiPolynomial(i, v, w);
                    double binomial = CombinatoricsUtils.binomialCoefficient(v + i, i);
                    Assert.assertTrue(Precision.equals(binomial, jacobi.value(1.0), 1));
                }
            }
        }
    }

    @Test
    public void testJacobiOrthogonality() {
        for (int v = 0; v < 5; ++v) {
            for (int w = v; w < 5; ++w) {
                final int vv = v;
                final int ww = w;
                UnivariateFunction weight = new UnivariateFunction() {
                    public double value(double x) {
                        return FastMath.pow(1 - x, vv) * FastMath.pow(1 + x, ww);
                    }
                };
                for (int i = 0; i < 10; ++i) {
                    PolynomialFunction pi = PolynomialsUtils.createJacobiPolynomial(i, v, w);
                    for (int j = 0; j <= i; ++j) {
                        PolynomialFunction pj = PolynomialsUtils.createJacobiPolynomial(j, v, w);
                        checkOrthogonality(pi, pj, weight, -1, 1, 0.1, 1.0e-12);
                    }
                }
            }
        }
    }

    @Test
    public void testShift() {
        // f1(x) = 1 + x + 2 x^2
        PolynomialFunction f1x = new PolynomialFunction(new double[] { 1, 1, 2 });

        PolynomialFunction f1x1
            = new PolynomialFunction(PolynomialsUtils.shift(f1x.getCoefficients(), 1));
        checkPolynomial(f1x1, "4 + 5 x + 2 x^2");

        PolynomialFunction f1xM1
            = new PolynomialFunction(PolynomialsUtils.shift(f1x.getCoefficients(), -1));
        checkPolynomial(f1xM1, "2 - 3 x + 2 x^2");

        PolynomialFunction f1x3
            = new PolynomialFunction(PolynomialsUtils.shift(f1x.getCoefficients(), 3));
        checkPolynomial(f1x3, "22 + 13 x + 2 x^2");

        // f2(x) = 2 + 3 x^2 + 8 x^3 + 121 x^5
        PolynomialFunction f2x = new PolynomialFunction(new double[]{2, 0, 3, 8, 0, 121});

        PolynomialFunction f2x1
            = new PolynomialFunction(PolynomialsUtils.shift(f2x.getCoefficients(), 1));
        checkPolynomial(f2x1, "134 + 635 x + 1237 x^2 + 1218 x^3 + 605 x^4 + 121 x^5");

        PolynomialFunction f2x3
            = new PolynomialFunction(PolynomialsUtils.shift(f2x.getCoefficients(), 3));
        checkPolynomial(f2x3, "29648 + 49239 x + 32745 x^2 + 10898 x^3 + 1815 x^4 + 121 x^5");
    }


    private void checkPolynomial(PolynomialFunction p, long denominator, String reference) {
        PolynomialFunction q = new PolynomialFunction(new double[] { denominator});
        Assert.assertEquals(reference, p.multiply(q).toString());
    }

    private void checkPolynomial(PolynomialFunction p, String reference) {
        Assert.assertEquals(reference, p.toString());
    }

    private void checkNullPolynomial(PolynomialFunction p) {
        for (double coefficient : p.getCoefficients()) {
            Assert.assertEquals(0, coefficient, 1e-13);
        }
    }

    private void checkOrthogonality(final PolynomialFunction p1,
                                    final PolynomialFunction p2,
                                    final UnivariateFunction weight,
                                    final double a, final double b,
                                    final double nonZeroThreshold,
                                    final double zeroThreshold) {
        UnivariateFunction f = new UnivariateFunction() {
            public double value(double x) {
                return weight.value(x) * p1.value(x) * p2.value(x);
            }
        };
        double dotProduct =
                new IterativeLegendreGaussIntegrator(5, 1.0e-9, 1.0e-8, 2, 15).integrate(1000000, f, a, b);
        if (p1.degree() == p2.degree()) {
            // integral should be non-zero
            Assert.assertTrue("I(" + p1.degree() + ", " + p2.degree() + ") = "+ dotProduct,
                              FastMath.abs(dotProduct) > nonZeroThreshold);
        } else {
            // integral should be zero
            Assert.assertEquals("I(" + p1.degree() + ", " + p2.degree() + ") = "+ dotProduct,
                                0.0, FastMath.abs(dotProduct), zeroThreshold);
        }
    }
}

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