public void EllipticNomeAgreement()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-2, 1.0, 4))
{
double K = AdvancedMath.EllipticK(k);
double k1 = Math.Sqrt((1.0 - k) * (1.0 + k));
double K1 = AdvancedMath.EllipticK(k1);
double q = AdvancedMath.EllipticNome(k);
// For k << 1, this test fails if formulated as q = e^{\pi K' / K}.
// Problem is that computation of k' looses some digits to cancellation,
// then K' is near singularity so error is amplified, then
// exp function amplifies error some more. Investigation indicates
// that q agrees with Mathematica to nearly all digits, so problem
// isn't in nome function. Run test in log space and don't let k get
// extremely small.
Assert.IsTrue(TestUtilities.IsNearlyEqual(
Math.Log(q),
-Math.PI * K1 / K
));
}
}
public void JacobiIntegrals()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-1, 1.0, 4))
{
// DLMF 22.14.18 says
// \int_{0}^{K(k)} \ln(\dn(t, k)) \, dt = \frac{1}{2} K(k) \ln k'
double K = AdvancedMath.EllipticK(k);
double k1 = Math.Sqrt(1.0 - k * k);
double I1 = FunctionMath.Integrate(
x => Math.Log(AdvancedMath.JacobiDn(x, k)),
Interval.FromEndpoints(0.0, K)
);
Assert.IsTrue(TestUtilities.IsNearlyEqual(I1, K / 2.0 * Math.Log(k1)));
// If k is small, log(k1) is subject to cancellation error, so don't pick k too small.
// It also gives values for the analogous integrals with \sn and \cn,
// but their values involve K'(k), for which we don't have a method.
// Mathworld (http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html) says
// \int_{0}^{K(k)} \dn^2(t, k) = E(k)
double I2 = FunctionMath.Integrate(
u => MoreMath.Sqr(AdvancedMath.JacobiDn(u, k)),
Interval.FromEndpoints(0.0, K)
);
Assert.IsTrue(TestUtilities.IsNearlyEqual(I2, AdvancedMath.EllipticE(k)));
}
}
public void EllipticKCatalanIntegral()
{
System.Diagnostics.Stopwatch timer = System.Diagnostics.Stopwatch.StartNew();
Interval i = Interval.FromEndpoints(0.0, 1.0);
// http://mathworld.wolfram.com/CatalansConstant.html equation 37
Func <double, double> f1 = delegate(double k) {
return(AdvancedMath.EllipticK(k));
};
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(f1, i), 2.0 * AdvancedMath.Catalan
));
Func <double, double> f2 = delegate(double k) {
return(AdvancedMath.EllipticK(k) * k);
};
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(f2, i), 1.0
));
timer.Stop();
Console.WriteLine(timer.ElapsedTicks);
}
public void EllipticLandenTransform()
{
foreach (double k in TestUtilities.GenerateUniformRealValues(0.0, 1.0, 8))
{
double kp = Math.Sqrt(1.0 - k * k);
double k1 = (1.0 - kp) / (1.0 + kp);
// For complete 1st kind
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticK(k),
(1.0 + k1) * AdvancedMath.EllipticK(k1)
));
// For complete 2nd kind
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticE(k) + kp * AdvancedMath.EllipticK(k),
(1.0 + kp) * AdvancedMath.EllipticE(k1)
));
/*
* foreach (double t in TestUtilities.GenerateUniformRealValues(0.0, Math.PI / 2.0, 4)) {
*
* double s = Math.Sin(t);
* double t1 = Math.Asin((1.0 + kp) * s * Math.Sqrt(1.0 - s * s) / Math.Sqrt(1.0 - k * k * s * s));
*
* // Since t1 can be > pi/2, we need to handle a larger range of angles before we can test the incomplete cases
*
*
* }
*
*/
}
}
public void EllipticKIntegrals()
{
Interval i = Interval.FromEndpoints(0.0, 1.0);
// http://mathworld.wolfram.com/CatalansConstant.html equation 37
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(
k => AdvancedMath.EllipticK(k),
Interval.FromEndpoints(0.0, 1.0)),
2.0 * AdvancedMath.Catalan
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(
k => AdvancedMath.EllipticK(k) * k,
Interval.FromEndpoints(0.0, 1.0)),
1.0
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(
k => AdvancedMath.EllipticK(k) / (1.0 + k),
Interval.FromEndpoints(0.0, 1.0)),
Math.PI * Math.PI / 8.0
));
}
public void JacobiPeriodic()
{
double u = 3.1;
double k = 0.2;
double K = AdvancedMath.EllipticK(k);
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.JacobiSn(u, k),
-AdvancedMath.JacobiSn(u + 2.0 * K, k)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.JacobiSn(u, k),
AdvancedMath.JacobiSn(u + 4.0 * K, k)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.JacobiCn(u, k),
-AdvancedMath.JacobiCn(u + 2.0 * K, k)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.JacobiCn(u, k),
AdvancedMath.JacobiCn(u + 4.0 * K, k)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.JacobiDn(u, k),
AdvancedMath.JacobiDn(u + 2.0 * K, k)
));
}
public void EllipticKInequality()
{
// DLMF 19.9.1
foreach (double k in TestUtilities.GenerateUniformRealValues(0.0, 1.0, 4))
{
double S = AdvancedMath.EllipticK(k) + Math.Log(Math.Sqrt(1.0 - k * k));
Assert.IsTrue((Math.Log(4.0) <= S) && (S <= Math.PI / 2.0));
}
}
public void EllipticFCompleteAgreement()
{
foreach (double k in TestUtilities.GenerateRealValues(0.01, 1.0, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticF(Math.PI / 2.0, k),
AdvancedMath.EllipticK(k)
));
}
}
public void EllipticLegendreRelation()
{
foreach (double k in TestUtilities.GenerateRealValues(0.01, 1.0, 8))
{
double kp = Math.Sqrt(1.0 - k * k);
double E = AdvancedMath.EllipticE(k);
double EP = AdvancedMath.EllipticE(kp);
double K = AdvancedMath.EllipticK(k);
double KP = AdvancedMath.EllipticK(kp);
Assert.IsTrue(TestUtilities.IsNearlyEqual(K * EP + KP * E, Math.PI / 2.0 + K * KP));
}
}
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 EllipticFDisplacement()
{
foreach (int m in TestUtilities.GenerateUniformIntegerValues(-100, 100, 4))
{
foreach (double phi in TestUtilities.GenerateUniformRealValues(-Math.PI / 2.0, +Math.PI / 2.0, 4))
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-4, 1.0, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticF(m * Math.PI + phi, k),
2 * m * AdvancedMath.EllipticK(k) + AdvancedMath.EllipticF(phi, k)
));
}
}
}
}
public void EllipticKIntegration()
{
// The defining integral for K
Interval i = Interval.FromEndpoints(0.0, Math.PI / 2.0);
foreach (double k in TestUtilities.GenerateRealValues(0.01, 1.0, 8))
{
Func <double, double> f = delegate(double t) {
double z = k * Math.Sin(t);
return(1.0 / Math.Sqrt(1.0 - z * z));
};
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(f, i), AdvancedMath.EllipticK(k)
));
}
}
public void OdePendulum()
{
// Without the small-angle approximation, the period of a pendulum is not
// independent of angle. Instead, it is given by
// P = 4 K(\sin(\phi_0) / 2)
// where \phi_0 is the release angle and K is the complete elliptic function
// of the first kind.
// See https://en.wikipedia.org/wiki/Pendulum_(mathematics)
// We compute for an initial angle of 45 degrees, far beyond a small angle.
Func <double, double, double> rhs = (double t, double u) => - MoreMath.Sin(u);
double u0 = Math.PI / 4.0;
double p = 4.0 * AdvancedMath.EllipticK(MoreMath.Sin(u0 / 2.0));
OdeResult r = FunctionMath.IntegrateConservativeOde(rhs, 0.0, u0, 0.0, p);
Assert.IsTrue(TestUtilities.IsNearlyEqual(r.Y, u0));
Console.WriteLine(r.EvaluationCount);
}
public void JacobiSpecial()
{
// Verify values at u = 0, K/2, K given in DMLF 22.5i and A&S
double k = 1.0 / 3.1;
double k1 = Math.Sqrt(1.0 - k * k);
double K = AdvancedMath.EllipticK(k);
// u = 0
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiSn(0.0, k), 0.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiCn(0.0, k), 1.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiDn(0.0, k), 1.0));
// u = K / 2
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiSn(K / 2.0, k), 1.0 / Math.Sqrt(1.0 + k1)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiCn(K / 2.0, k), Math.Sqrt(k1 / (1.0 + k1))));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiDn(K / 2.0, k), Math.Sqrt(k1)));
// u = K
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiSn(K, k), 1.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiCn(K, k), 0.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.JacobiDn(K, k), k1));
}
public void EllipticKExtremeValues()
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticK(Double.Epsilon), Math.PI / 2.0));
Assert.IsTrue(AdvancedMath.EllipticK(1.0 - 2.0E-16) < Double.PositiveInfinity);
Assert.IsTrue(Double.IsNaN(AdvancedMath.EllipticK(Double.NaN)));
}
public void OdeEuler()
{
// Euler's equations for rigid body motion (without external forces) are
// I_1 \dot{\omega}_1 = (I_2 - I_3) \omega_2 \omega_3
// I_2 \dot{\omega}_2 = (I_3 - I_1) \omega_3 \omega_1
// I_3 \dot{\omega}_3 = (I_1 - I_2) \omega_1 \omega_2
// where \omega's are rotations about the principal axes and I's are the moments
// of inertia about those axes. The rotational kinetic energy
// I_1 \omega_1^2 + I_2 \omega_2^2 + I_3 \omega_3^2 = 2 E
// and angular momentum
// I_1^2 \omega_1^2 + I_2^2 \omega_2^2 + I_3^2 \omega_3^2 = L^2
// are conserved quantities.
// Landau & Lifshitz, Mechanics (3rd edition) solves these equations.
// Note
// (\cn u)' = - (\sn u)(\dn u)
// (\sn u)' = (\dn u)(\cn u)
// (\dn u)' = -k^2 (\cn u)(\sn u)
// So a solution with \omega_1 ~ \cn, \omega_2 ~ \sn, \omega_3 ~ \dn
// would seem to work, if we can re-scale variables to eliminate factors.
// In fact, this turns out to be the case.
// Label axis so that I_3 > I_2 > I_1. If L^2 > 2 E I_2, define
// v^2 = \frac{(I_3 - I_2)(L^2 - 2E I_1)}{I_1 I_2 I_3}
// k^2 = \frac{(I_2 - I_1)(2E I_3 - L^2)}{(I_3 - I_2)(L^2 - 2E I_1)}
// A_1^2 = \frac{2E I_3 - L^2}{I_1 (I_3 - I_1)}
// A_2^2 = \frac{2E I_3 - L^2}{I_2 (I_3 - I_2)}
// A_3^2 = \frac{L^2 - 2E I_1}{I_3 (I_3 - I_1)}
// (If L^2 < 2 E I_2, just switch subscripts 1 <-> 3). Then
// \omega_1 = A_1 \cn (c t, k)
// \omega_2 = A_2 \sn (c t, k)
// \omega_3 = A_3 \dn (c t, k)
// Period is complete when v T = 4 K.
ColumnVector I = new ColumnVector(3.0, 4.0, 5.0);
Func <double, IReadOnlyList <double>, IReadOnlyList <double> > rhs = (double t, IReadOnlyList <double> w) => {
return(new ColumnVector((I[1] - I[2]) * w[1] * w[2] / I[0], (I[2] - I[0]) * w[2] * w[0] / I[1], (I[0] - I[1]) * w[0] * w[1] / I[2]));
};
ColumnVector w0 = new ColumnVector(6.0, 0.0, 1.0);
// Determine L^2 and 2 E
double L2 = EulerAngularMomentum(I, w0);
double E2 = EulerKineticEnergy(I, w0);
// As Landau points out, these inequalities should be ensured by the definitions of E and L.
Debug.Assert(E2 * I[0] < L2);
Debug.Assert(L2 < E2 * I[2]);
double v, k;
Func <double, ColumnVector> sln;
if (L2 > E2 * I[1])
{
v = Math.Sqrt((I[2] - I[1]) * (L2 - E2 * I[0]) / I[0] / I[1] / I[2]);
k = Math.Sqrt((I[1] - I[0]) * (E2 * I[2] - L2) / (I[2] - I[1]) / (L2 - E2 * I[0]));
ColumnVector A = new ColumnVector(
Math.Sqrt((E2 * I[2] - L2) / I[0] / (I[2] - I[0])),
Math.Sqrt((E2 * I[2] - L2) / I[1] / (I[2] - I[1])),
Math.Sqrt((L2 - E2 * I[0]) / I[2] / (I[2] - I[0]))
);
sln = t => new ColumnVector(
A[0] * AdvancedMath.JacobiCn(v * t, k),
A[1] * AdvancedMath.JacobiSn(v * t, k),
A[2] * AdvancedMath.JacobiDn(v * t, k)
);
}
else
{
v = Math.Sqrt((I[0] - I[1]) * (L2 - E2 * I[2]) / I[0] / I[1] / I[2]);
k = Math.Sqrt((I[1] - I[2]) * (E2 * I[0] - L2) / (I[0] - I[1]) / (L2 - E2 * I[2]));
ColumnVector A = new ColumnVector(
Math.Sqrt((L2 - E2 * I[2]) / I[0] / (I[0] - I[2])),
Math.Sqrt((E2 * I[0] - L2) / I[1] / (I[0] - I[1])),
Math.Sqrt((E2 * I[0] - L2) / I[2] / (I[0] - I[2]))
);
sln = t => new ColumnVector(
A[0] * AdvancedMath.JacobiDn(v * t, k),
A[1] * AdvancedMath.JacobiSn(v * t, k),
A[2] * AdvancedMath.JacobiCn(v * t, k)
);
}
Debug.Assert(k < 1.0);
double T = 4.0 * AdvancedMath.EllipticK(k) / v;
int listenerCount = 0;
MultiOdeSettings settings = new MultiOdeSettings()
{
Listener = (MultiOdeResult q) => {
listenerCount++;
// Verify that energy and angular momentum conservation is respected
Assert.IsTrue(TestUtilities.IsNearlyEqual(EulerKineticEnergy(I, q.Y), E2, q.Settings));
Assert.IsTrue(TestUtilities.IsNearlyEqual(EulerAngularMomentum(I, q.Y), L2, q.Settings));
// Verify that the result agrees with the analytic solution
//ColumnVector YP = new ColumnVector(
// A[0] * AdvancedMath.JacobiCn(v * q.X, k),
// A[1] * AdvancedMath.JacobiSn(v * q.X, k),
// A[2] * AdvancedMath.JacobiDn(v * q.X, k)
//);
Assert.IsTrue(TestUtilities.IsNearlyEqual(q.Y, sln(q.X), q.Settings));
}
};
MultiOdeResult result = MultiFunctionMath.IntegrateOde(rhs, 0.0, w0, T, settings);
Console.WriteLine(result.EvaluationCount);
//MultiOdeResult r2 = NewMethods.IntegrateOde(rhs, 0.0, w0, T, settings);
// Verify that one period of evolution has brought us back to our initial state
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.X, T));
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Y, w0, settings));
Assert.IsTrue(listenerCount > 0);
}
public void EllipticKSpecialCases()
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticK(0.0), Math.PI / 2.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticK(1.0 / Math.Sqrt(2.0)), Math.Pow(AdvancedMath.Gamma(1.0 / 4.0), 2.0) / Math.Sqrt(Math.PI) / 4.0));
Assert.IsTrue(Double.IsPositiveInfinity(AdvancedMath.EllipticK(1.0)));
}
public void RambleIntegrals()
{
// W_{n}(s) = \int_{0}^{1} dx_1 \cdots dx_n \vert \sum_k e^{2 \pi i x_k} \vert^{s}
// appears in problems of random walk in n dimensions. This is an oscilatory integral.
// Analytic values are known for W_3(1) and W_3(-1) (https://www.carma.newcastle.edu.au/jon/walks.pdf).
// Also W_2(1) = 4 / \pi and even arguments (http://www.carma.newcastle.edu.au/jon/walkstalk.pdf).
// W_4(-1) can be related to a one-dimensional integral involving elliptic functions
// (https://www.carma.newcastle.edu.au/jon/walks2.pdf and http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/combat.pdf).
// This is an oscilatory integral, so drop the required precision.
IntegrationSettings settings2 = new IntegrationSettings()
{
RelativePrecision = 1.0E-4
};
IntegrationSettings settings3 = new IntegrationSettings()
{
RelativePrecision = 1.0E-3
};
IntegrationSettings settings4 = new IntegrationSettings()
{
RelativePrecision = 1.0E-2
};
Assert.IsTrue(RambleIntegral(2, 1, settings3).Estimate.ConfidenceInterval(0.95).ClosedContains(4.0 / Math.PI));
Assert.IsTrue(RambleIntegral(3, -1, settings3).Estimate.ConfidenceInterval(0.95).ClosedContains(
3.0 / 16.0 * Math.Pow(2.0, 1.0 / 3.0) / MoreMath.Pow(Math.PI, 4) * MoreMath.Pow(AdvancedMath.Gamma(1.0 / 3.0), 6)
));
Assert.IsTrue(RambleIntegral(3, 1, settings3).Estimate.ConfidenceInterval(0.95).ClosedContains(
3.0 / 16.0 * Math.Pow(2.0, 1.0 / 3.0) / MoreMath.Pow(Math.PI, 4) * MoreMath.Pow(AdvancedMath.Gamma(1.0 / 3.0), 6) +
27.0 / 4.0 * Math.Pow(2.0, 2.0 / 3.0) / MoreMath.Pow(Math.PI, 4) * MoreMath.Pow(AdvancedMath.Gamma(2.0 / 3.0), 6)
));
Assert.IsTrue(RambleIntegral(4, -1, settings4).Estimate.ConfidenceInterval(0.95).ClosedContains(
8.0 / MoreMath.Pow(Math.PI, 3) * FunctionMath.Integrate(k => MoreMath.Sqr(AdvancedMath.EllipticK(k)), Interval.FromEndpoints(0.0, 1.0))
));
}
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 EllipticPiSpecialCharacteristic()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-4, 1.0, 4))
{
// DLMF 19.6.1
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticPi(k * k, k), AdvancedMath.EllipticE(k) / (1.0 - k * k)));
// DLMF 19.6.2
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticPi(-k, k), Math.PI / 4.0 / (1.0 + k) + AdvancedMath.EllipticK(k) / 2.0));
// DLMF 19.6.3
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticPi(0.0, k), AdvancedMath.EllipticK(k)));
Assert.IsTrue(Double.IsPositiveInfinity(AdvancedMath.EllipticPi(1.0, k)));
}
}
