public void HypergeometricQuadraticTransforms()
{
foreach (double a in abcs)
{
foreach (double b in abcs)
{
foreach (double x in xs)
{
if (!IsNonpositiveInteger(2.0 * b))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(a, b, 2.0 * b, x),
AdvancedMath.Hypergeometric2F1(0.5 * a, b - 0.5 * a, b + 0.5, x * x / (x - 1.0) / 4.0) / Math.Pow(1.0 - x, 0.5 * a),
TestUtilities.TargetPrecision * 1000.0
));
}
// Wrong sign, but Mathematica agrees with values!
/*
* if (!IsNonpositiveInteger(a - b + 1.0)) {
* Assert.IsTrue(TestUtilities.IsNearlyEqual(
* AdvancedMath.Hypergeometric2F1(a, b, a - b + 1.0, x),
* AdvancedMath.Hypergeometric2F1(0.5 * a, 0.5 * a - b + 0.5, a - b + 1.0, -4.0 * x / MoreMath.Sqr(1.0 - x)) / Math.Pow(1.0 - x, a)
* ));
* }
*/
}
}
}
}
public void HypergeometrticRecurrenceA()
{
// A&S 15.2.10
foreach (double a in abcs)
{
foreach (double b in abcs)
{
foreach (double c in abcs)
{
foreach (double x in xs)
{
double FM = AdvancedMath.Hypergeometric2F1(a - 1.0, b, c, x);
double F0 = AdvancedMath.Hypergeometric2F1(a, b, c, x);
double FP = AdvancedMath.Hypergeometric2F1(a + 1.0, b, c, x);
if ((c == a) && Double.IsNaN(FM))
{
continue;
}
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(
new double[] { (c - a) * FM, (2.0 * a - c - a * x + b * x) * F0 },
-a * (x - 1.0) * FP,
new EvaluationSettings()
{
RelativePrecision = 1.0E-12, AbsolutePrecision = 1.0E-15
}
));
}
}
}
}
}
public void HypergeometricAtMinusOne()
{
foreach (double a in abcs)
{
if ((a <= 0.0) && (Math.Round(a) == a))
{
continue;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(1.0, a, a + 1.0, -1.0),
a / 2.0 * (AdvancedMath.Psi((a + 1.0) / 2.0) - AdvancedMath.Psi(a / 2.0))
));
foreach (double b in abcs)
{
double c = a - b + 1.0;
if ((c <= 0.0) && (Math.Round(c) == c))
{
continue;
}
// If result vanishes, returned value may just be very small.
double R = AdvancedMath.Gamma(a - b + 1.0) * AdvancedMath.Gamma(a / 2.0 + 1.0) / AdvancedMath.Gamma(a + 1.0) / AdvancedMath.Gamma(a / 2.0 - b + 1.0);
if (R == 0.0)
{
Assert.IsTrue(Math.Abs(AdvancedMath.Hypergeometric2F1(a, b, a - b + 1.0, -1.0)) <= 1.0E-14);
}
else
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.Hypergeometric2F1(a, b, a - b + 1.0, -1.0), R));
}
}
}
}
public void HypergeometricAtOne()
{
// A&S 15.1.20
foreach (double a in abcs)
{
foreach (double b in abcs)
{
foreach (double c in abcs)
{
// Formula only hold for positive real c-a-b
if ((c - a - b) <= 0.0)
{
continue;
}
// Formula is still right for non-positive c, but to handle it we would need to deal with canceling infinite Gammas
if (IsNonpositiveInteger(c))
{
continue;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(a, b, c, 1.0),
AdvancedMath.Gamma(c) * AdvancedMath.Gamma(c - a - b) / AdvancedMath.Gamma(c - a) / AdvancedMath.Gamma(c - b)
));
}
}
}
}
public void HypergeometricIntegral()
{
// A&S 15.3.1
// F(a, b, c, x) = \frac{\Gamma(c)}{\Gamma(b) \Gamma(c - b)}
// \int_{0}^{1} \! dt \, t^{b-1} (1 - t)^{c - b - 1} (1 - x t)^{-a}
// Choose limits on a, b, c so that singularities of integrand are numerically integrable.
foreach (double a in TestUtilities.GenerateRealValues(1.0, 10.0, 2))
{
foreach (double b in TestUtilities.GenerateRealValues(0.6, 10.0, 2))
{
foreach (double c in TestUtilities.GenerateRealValues(b + 0.6, 10.0, 2))
{
foreach (double x in xs)
{
double I = FunctionMath.Integrate(
t => Math.Pow(t, b - 1.0) * Math.Pow(1.0 - t, c - b - 1.0) * Math.Pow(1.0 - x * t, -a),
Interval.FromEndpoints(0.0, 1.0)
);
double B = AdvancedMath.Beta(b, c - b);
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.Hypergeometric2F1(a, b, c, x), I / B));
}
}
}
}
}
public void HypergeometricIncompleteBeta()
{
foreach (double a in abcs)
{
if (a <= 0.0)
{
continue;
}
foreach (double b in abcs)
{
if (b <= 0.0)
{
continue;
}
foreach (double x in xs)
{
if ((x < 0.0) || (x > 1.0))
{
continue;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(
Math.Pow(x, a) * Math.Pow(1.0 - x, b) * AdvancedMath.Hypergeometric2F1(a + b, 1.0, a + 1.0, x) / a,
AdvancedMath.Beta(a, b, x)
));
}
}
}
}
public void HypergeometricAtSpecialPoints()
{
// Documented at http://mathworld.wolfram.com/HypergeometricFunction.html
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(1.0 / 3.0, 2.0 / 3.0, 5.0 / 6.0, 27.0 / 32.0),
8.0 / 5.0
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(0.25, 0.5, 0.75, 80.0 / 81.0),
9.0 / 5.0
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(1.0 / 8.0, 3.0 / 8.0, 1.0 / 2.0, 2400.0 / 2401.0),
2.0 / 3.0 * Math.Sqrt(7.0)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(1.0 / 6.0, 1.0 / 3.0, 1.0 / 2.0, 25.0 / 27.0),
3.0 / 4.0 * Math.Sqrt(3.0)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(1.0 / 6.0, 1.0 / 2.0, 2.0 / 3.0, 125.0 / 128.0),
4.0 / 3.0 * Math.Pow(2.0, 1.0 / 6.0)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(1.0 / 12.0, 5.0 / 12.0, 1.0 / 2.0, 1323.0 / 1331.0),
3.0 / 4.0 * Math.Pow(11.0, 1.0 / 4.0)
));
}
public void HypergeometricCubicTransforms()
{
foreach (double a in abcs)
{
foreach (double x in xs)
{
// DLMF 15.8.31
//if ((a == 7.0) && (x == 0.8)) continue;
if ((a == -3.0) && (x == 0.8))
{
continue;
}
if ((a == 3.1) && (x == -5.0))
{
continue;
}
if ((a == 4.5) && (x == -5.0))
{
continue;
}
if ((a == 7.0) && (x == -5.0))
{
continue;
}
if (x < 8.0 / 9.0)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(3.0 * a, 3.0 * a + 0.5, 4.0 * a + 2.0 / 3.0, x),
AdvancedMath.Hypergeometric2F1(a, a + 0.5, 2.0 * a + 5.0 / 6.0, 27.0 * x * x * (x - 1.0) / MoreMath.Sqr(9.0 * x - 8.0)) * Math.Pow(1.0 - 9.0 / 8.0 * x, -2.0 * a)
));
}
}
}
}
public void HypergeometricLinearTransforms()
{
foreach (double a in abcs)
{
foreach (double b in abcs)
{
foreach (double c in abcs)
{
if (IsNonpositiveInteger(c))
{
continue;
}
foreach (double x in xs)
{
double F = AdvancedMath.Hypergeometric2F1(a, b, c, x);
// A&S 15.3.3
Assert.IsTrue(TestUtilities.IsNearlyEqual(F, AdvancedMath.Hypergeometric2F1(c - a, c - b, c, x) * Math.Pow(1.0 - x, c - a - b)));
// A&S 15.3.4
Assert.IsTrue(TestUtilities.IsNearlyEqual(F, AdvancedMath.Hypergeometric2F1(a, c - b, c, x / (x - 1.0)) * Math.Pow(1.0 - x, -a)));
// A&S 15.3.5
Assert.IsTrue(TestUtilities.IsNearlyEqual(F, AdvancedMath.Hypergeometric2F1(b, c - a, c, x / (x - 1.0)) * Math.Pow(1.0 - x, -b)));
// A&S 15.3.6
if (!IsNonpositiveInteger(c - a - b) && !IsNonpositiveInteger(a + b - c) && (x > 0.0))
{
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(
new double[] {
AdvancedMath.Gamma(c) * AdvancedMath.Gamma(c - a - b) / AdvancedMath.Gamma(c - a) / AdvancedMath.Gamma(c - b) *
AdvancedMath.Hypergeometric2F1(a, b, a + b - c + 1.0, 1.0 - x),
AdvancedMath.Gamma(c) * AdvancedMath.Gamma(a + b - c) / AdvancedMath.Gamma(a) / AdvancedMath.Gamma(b) *
AdvancedMath.Hypergeometric2F1(c - a, c - b, c - a - b + 1.0, 1.0 - x) *
Math.Pow(1.0 - x, c - a - b)
}, F
));
}
// A&S 15.3.7
if (!IsNonpositiveInteger(a - b) && !IsNonpositiveInteger(b - a) && (x < 0.0))
{
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(
new double[] {
AdvancedMath.Gamma(c) * AdvancedMath.Gamma(b - a) / AdvancedMath.Gamma(b) / AdvancedMath.Gamma(c - a) *
AdvancedMath.Hypergeometric2F1(a, 1.0 - c + a, 1.0 - b + a, 1.0 / x) * Math.Pow(-x, -a),
AdvancedMath.Gamma(c) * AdvancedMath.Gamma(a - b) / AdvancedMath.Gamma(a) / AdvancedMath.Gamma(c - b) *
AdvancedMath.Hypergeometric2F1(b, 1.0 - c + b, 1.0 - a + b, 1.0 / x) * Math.Pow(-x, -b)
}, F
));
}
}
}
}
}
}
public void HypergeometricAtOneNinth()
{
foreach (double a in abcs)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(a, a + 0.5, 5.0 / 6.0 + 2.0 / 3.0 * a, 1.0 / 9.0),
Math.Sqrt(Math.PI) * Math.Pow(3.0 / 4.0, a) * AdvancedMath.Gamma(5.0 / 6.0 + 2.0 / 3.0 * a) / AdvancedMath.Gamma(1.0 / 2.0 + 1.0 / 3.0 * a) / AdvancedMath.Gamma(5.0 / 6.0 + 1.0 / 3.0 * a)
));
}
}
public void HypergeometricAtMinusOneThird()
{
foreach (double a in abcs)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(a, a + 0.5, 1.5 - 2.0 * a, -1.0 / 3.0),
Math.Pow(8.0 / 9.0, -2.0 * a) * AdvancedMath.Gamma(4.0 / 3.0) / AdvancedMath.Gamma(3.0 / 2.0) * AdvancedMath.Gamma(3.0 / 2.0 - 2.0 * a) / AdvancedMath.Gamma(4.0 / 3.0 - 2.0 * a)
));
}
}
public void HypergeometricPearsonTests()
{
// John Pearson wrote a thesis on the computation of Hypergeometric functions and assembled a list of test cases.
// (http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf)
foreach (var t in HypergeometricPearsonTestValues())
{
double F = AdvancedMath.Hypergeometric2F1(t.Item1, t.Item2, t.Item3, t.Item4);
Assert.IsTrue(TestUtilities.IsNearlyEqual(F, t.Item5));
}
}
public void CompleteEllipticHypergeometricAgreement()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-4, 1.0, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticK(k),
Math.PI / 2.0 * AdvancedMath.Hypergeometric2F1(0.5, 0.5, 1.0, k * k)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticE(k),
Math.PI / 2.0 * AdvancedMath.Hypergeometric2F1(0.5, -0.5, 1.0, k * k)
));
}
}
public void HypergeometricPolynomials()
{
foreach (int n in TestUtilities.GenerateUniformIntegerValues(0, 10, 4))
{
foreach (double x in TestUtilities.GenerateRealValues(1.0E-2, 1.0, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(-n, n, 0.5, x),
OrthogonalPolynomials.ChebyshevT(n, 1.0 - 2.0 * x)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(-n, n + 1, 1.0, x),
OrthogonalPolynomials.LegendreP(n, 1.0 - 2.0 * x)
));
}
}
}
public void HypergeometricBasicFunctions()
{
foreach (double x in xs)
{
// A&S 15.1.3
if (x < 1.0 && (x != 0.0))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(1.0, 1.0, 2.0, x), -Math.Log(1.0 - x) / x
));
}
// A&S 15.1.5
if (x != 0.0)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(0.5, 1.0, 1.5, -x * x), Math.Atan(x) / x
));
}
// A&S 15.1.6
if ((-1.0 <= x) && (x <= 1.0) && (x != 0.0))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(0.5, 0.5, 1.5, x * x), Math.Asin(x) / x
));
}
// Complete elliptic integrals
if ((0.0 <= x) && (x < 1.0))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
Math.PI / 2.0 * AdvancedMath.Hypergeometric2F1(0.5, 0.5, 1.0, x * x), AdvancedMath.EllipticK(x)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
Math.PI / 2.0 * AdvancedMath.Hypergeometric2F1(-0.5, 0.5, 1.0, x * x), AdvancedMath.EllipticE(x)
));
}
}
}
public void HypergeometricAtOneHalf()
{
foreach (double a in abcs)
{
foreach (double b in abcs)
{
double c = (a + b + 1.0) / 2.0;
if (!IsNonpositiveInteger(c))
{
if (IsNonpositiveInteger((a + 1.0) / 2.0) || IsNonpositiveInteger((b + 1.0) / 2.0))
{
Assert.IsTrue(Math.Abs(AdvancedMath.Hypergeometric2F1(a, b, c, 0.5)) < TestUtilities.TargetPrecision);
}
else
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(a, b, c, 0.5),
Math.Sqrt(Math.PI) * AdvancedMath.Gamma(c) / AdvancedMath.Gamma((a + 1.0) / 2.0) / AdvancedMath.Gamma((b + 1.0) / 2.0)
));
}
}
if (!IsNonpositiveInteger(b))
{
if (IsNonpositiveInteger((a + b) / 2.0) || IsNonpositiveInteger((b - a + 1.0) / 2.0))
{
Assert.IsTrue(Math.Abs(AdvancedMath.Hypergeometric2F1(a, 1.0 - a, b, 0.5)) < TestUtilities.TargetPrecision);
}
else
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(a, 1.0 - a, b, 0.5),
Math.Sqrt(Math.PI) * Math.Pow(2.0, 1.0 - b) * AdvancedMath.Gamma(b) / AdvancedMath.Gamma((a + b) / 2.0) / AdvancedMath.Gamma((b - a + 1.0) / 2.0)
));
}
}
}
}
}
public void HypergeometricRecurrenceC()
{
// A&S 15.2.12
foreach (double a in abcs)
{
foreach (double b in abcs)
{
foreach (double c in abcs)
{
foreach (double x in xs)
{
double FM = AdvancedMath.Hypergeometric2F1(a, b, c - 1.0, x);
double F0 = AdvancedMath.Hypergeometric2F1(a, b, c, x);
double FP = AdvancedMath.Hypergeometric2F1(a, b, c + 1.0, x);
if (Double.IsNaN(FM) && ((c == 0.0) || (c == 1.0)))
{
continue;
}
if (Double.IsNaN(FP) && ((c == a) || (c == b)))
{
continue;
}
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(
new double[] { c *(c - 1.0) * (x - 1.0) * FM, c * (c - 1.0 - (2.0 * c - a - b - 1.0) * x) * F0 },
-(c - a) * (c - b) * x * FP,
new EvaluationSettings()
{
RelativePrecision = 1.0E-12, AbsolutePrecision = 1.0E-16
}
));
}
}
}
}
}
