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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.
*/
package org.apache.commons.math4.analysis.polynomials;
import org.apache.commons.math4.analysis.UnivariateFunction;
import org.apache.commons.math4.analysis.integration.IterativeLegendreGaussIntegrator;
import org.apache.commons.math4.util.CombinatoricsUtils;
import org.apache.commons.math4.util.FastMath;
import org.apache.commons.numbers.core.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() {
@Override
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() {
@Override
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() {
@Override
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() {
@Override
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() {
@Override
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() {
@Override
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);
}
}
}