@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 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));
}
}
}
public ComplexNumber integrate(final ComplexNumber fromPoint, final ComplexNumber toPoint) {
final PolynomialFunction<ComplexNumber> tmpPrim = this.buildPrimitive();
final ComplexNumber tmpFromVal = tmpPrim.invoke(fromPoint);
final ComplexNumber tmpToVal = tmpPrim.invoke(toPoint);
return tmpToVal.subtract(tmpFromVal);
}
@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 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 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);
}
}
@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)));
}
}
private void checkNullPolynomial(PolynomialFunction p) {
for (double coefficient : p.getCoefficients()) {
Assert.assertEquals(0, coefficient, 1e-13);
}
}
private void checkPolynomial(PolynomialFunction p, String reference) {
Assert.assertEquals(reference, p.toString());
}
private void checkPolynomial(PolynomialFunction p, long denominator, String reference) {
PolynomialFunction q = new PolynomialFunction(new double[] {denominator});
Assert.assertEquals(reference, p.multiply(q).toString());
}
@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 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 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)));
}
}
