public void HermiteSpecialCases()
{
foreach (double x in aArguments)
{
Assert.IsTrue(OrthogonalPolynomials.HermiteH(0, x) == 1.0);
Assert.IsTrue(OrthogonalPolynomials.HermiteH(1, x) == 2.0 * x);
}
foreach (int n in orders)
{
if (n > 100)
{
continue;
}
if (n % 2 == 0)
{
int m = n / 2;
int s = 1;
if (m % 2 != 0)
{
s = -s;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(OrthogonalPolynomials.HermiteH(n, 0.0), s * AdvancedIntegerMath.Factorial(n) / AdvancedIntegerMath.Factorial(m)));
}
else
{
Assert.IsTrue(OrthogonalPolynomials.HermiteH(n, 0.0) == 0.0);
}
}
}
public void HermiteAddition()
{
foreach (int n in orders)
{
if (n > 100)
{
continue;
}
foreach (double x in aArguments)
{
foreach (double y in aArguments)
{
double value = OrthogonalPolynomials.HermiteH(n, x + y);
double[] terms = new double[n + 1]; double sum = 0.0;
for (int k = 0; k <= n; k++)
{
terms[k] = AdvancedIntegerMath.BinomialCoefficient(n, k) * OrthogonalPolynomials.HermiteH(k, x) * Math.Pow(2 * y, n - k);
sum += terms[k];
}
Console.WriteLine("n={0}, x={1}, y={2}, H(x+y)={3}, sum={4}", n, x, y, value, sum);
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(terms, value));
}
}
}
}
public void HermiteRecurrence()
{
foreach (int n in TestUtilities.GenerateIntegerValues(1, 100, 10))
{
foreach (double x in TestUtilities.GenerateRealValues(1.0, 1000.0, 10))
{
Console.WriteLine("n={0} x={1}", n, x);
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(
OrthogonalPolynomials.HermiteH(n + 1, x), 2.0 * n * OrthogonalPolynomials.HermiteH(n - 1, x),
2.0 * x * OrthogonalPolynomials.HermiteH(n, x)
));
}
}
}
public void HermiteReflection()
{
foreach (int n in TestUtilities.GenerateIntegerValues(1, 100, 10))
{
foreach (double x in TestUtilities.GenerateRealValues(1.0, 1000.0, 10))
{
Console.WriteLine("n={0} x={1}", n, x);
if (n % 2 == 0)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(OrthogonalPolynomials.HermiteH(n, -x), OrthogonalPolynomials.HermiteH(n, x)));
}
else
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(OrthogonalPolynomials.HermiteH(n, -x), -OrthogonalPolynomials.HermiteH(n, x)));
}
}
}
}
public void HermiteSum()
{
// accuracy of equality falls at high n because of cancelations among large terms
for (int n = 0; n < 12; n++)
{
double sum = 0.0;
for (int k = 0; k <= n; k++)
{
double term = AdvancedIntegerMath.BinomialCoefficient(n, k) * OrthogonalPolynomials.HermiteH(n, k);
Console.WriteLine(" k={0}, term={1}", k, term);
if ((n - k) % 2 == 0)
{
sum += term;
}
else
{
sum -= term;
}
}
Console.WriteLine("n={0}, sum={1}", n, sum);
Assert.IsTrue(TestUtilities.IsNearlyEqual(sum, Math.Pow(2, n) * AdvancedIntegerMath.Factorial(n)));
}
}
public void HermiteOrthonormality()
{
foreach (int n in TestUtilities.GenerateIntegerValues(1, 30, 3))
{
foreach (int m in TestUtilities.GenerateIntegerValues(1, 30, 3))
{
Func <double, double> f = delegate(double x) {
return(Math.Exp(-x * x) * OrthogonalPolynomials.HermiteH(n, x) * OrthogonalPolynomials.HermiteH(m, x));
};
Interval r = Interval.FromEndpoints(Double.NegativeInfinity, Double.PositiveInfinity);
double I = FunctionMath.Integrate(f, r);
double N = Math.Sqrt(Math.PI) * Math.Pow(2.0, n) * AdvancedIntegerMath.Factorial(n);
Console.WriteLine("{0} {1} {2} {3}", n, m, I, N);
if (n == m)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(I, N));
}
else
{
Assert.IsTrue(Math.Abs(I) <= TestUtilities.TargetPrecision);
}
}
}
}
public void HermiteInvalidOrder()
{
OrthogonalPolynomials.HermiteH(-1, 1.0);
}