public void OdeErf()
{
Func <double, double, double> rhs = (double t, double u) =>
2.0 / Math.Sqrt(Math.PI) * Math.Exp(-t * t);
OdeResult r = FunctionMath.IntegrateOde(rhs, 0.0, 0.0, 5.0);
Console.WriteLine(r.Y);
}
public static void IntegrateOde()
{
Func <double, double, double> rhs = (x, y) => - x * y;
OdeResult sln = FunctionMath.IntegrateOde(rhs, 0.0, 1.0, 2.0);
Console.WriteLine($"Numeric solution y({sln.X}) = {sln.Y}.");
Console.WriteLine($"Required {sln.EvaluationCount} evaluations.");
Console.WriteLine($"Analytic solution y({sln.X}) = {Math.Exp(-MoreMath.Sqr(sln.X) / 2.0)}");
// Lotka-Volterra equations
double A = 0.1;
double B = 0.02;
double C = 0.4;
double D = 0.02;
Func <double, IReadOnlyList <double>, IReadOnlyList <double> > lkRhs = (t, y) => {
return(new double[] {
A *y[0] - B * y[0] * y[1], D *y[0] * y[1] - C * y[1]
});
};
MultiOdeSettings lkSettings = new MultiOdeSettings()
{
Listener = r => { Console.WriteLine($"t={r.X} rabbits={r.Y[0]}, foxes={r.Y[1]}"); }
};
MultiFunctionMath.IntegrateOde(lkRhs, 0.0, new double[] { 20.0, 10.0 }, 50.0, lkSettings);
Func <double, IReadOnlyList <double>, IReadOnlyList <double> > rhs1 = (x, u) => {
return(new double[] { u[1], -u[0] });
};
MultiOdeSettings settings1 = new MultiOdeSettings()
{
EvaluationBudget = 100000
};
MultiOdeResult result1 = MultiFunctionMath.IntegrateOde(
rhs1, 0.0, new double[] { 0.0, 1.0 }, 500.0, settings1
);
double s1 = MoreMath.Sqr(result1.Y[0]) + MoreMath.Sqr(result1.Y[1]);
Console.WriteLine($"y({result1.X}) = {result1.Y[0]}, (y)^2 + (y')^2 = {s1}");
Console.WriteLine($"Required {result1.EvaluationCount} evaluations.");
Func <double, double, double> rhs2 = (x, y) => - y;
OdeSettings settings2 = new OdeSettings()
{
EvaluationBudget = 100000
};
OdeResult result2 = FunctionMath.IntegrateConservativeOde(
rhs2, 0.0, 0.0, 1.0, 500.0, settings2
);
double s2 = MoreMath.Sqr(result2.Y) + MoreMath.Sqr(result2.YPrime);
Console.WriteLine($"y({result2.X}) = {result2.Y}, (y)^2 + (y')^2 = {s2}");
Console.WriteLine($"Required {result2.EvaluationCount} evaluations");
Console.WriteLine(MoreMath.Sin(500.0));
}
public void OdeExponential()
{
// The exponential function y = e^x satisfies
// y' = y
// This is perhaps the simplest differential equation.
Func <double, double, double> f = (double x, double y) => y;
OdeResult r = FunctionMath.IntegrateOde(f, 0.0, 1.0, 2.0);
Assert.IsTrue(TestUtilities.IsNearlyEqual(r.Y, MoreMath.Sqr(Math.E)));
Console.WriteLine(r.EvaluationCount);
}
public void OdeExample()
{
Func <double, double, double> rhs = (double t, double u) => (1.0 - 2.0 * t) * u;
Func <double, double, double> solution = (double u0, double t) => u0 *Math.Exp(t - t *t);
int count = 0;
foreach (double t in new double[] { 0.5, 0.75, 1.50, 2.25, 3.25 })
{
OdeResult r = FunctionMath.IntegrateOde(rhs, 0.0, 1.0, t);
double y1 = r.Y;
Assert.IsTrue(TestUtilities.IsNearlyEqual(y1, solution(1.0, t)));
count += r.EvaluationCount;
}
Console.WriteLine(count);
}
public void OdeLogistic()
{
// y = \frac{y_0}{y_0 + (1 - y_0) e^{-(x - x_0)}
Func <double, double, double> rhs = (double x, double y) => y * (1.0 - y);
Func <double, double, double> solution = (double y0, double x) => y0 / (y0 + (1.0 - y0) * Math.Exp(-x));
int count = 0;
foreach (double y0 in new double[] { -0.1, 0.0, 0.4, 1.0, 1.6 })
{
Console.WriteLine(y0);
Interval r = Interval.FromEndpoints(0.0, 2.0);
OdeResult s = FunctionMath.IntegrateOde(rhs, 0.0, y0, 2.0);
double y1 = s.Y;
Console.WriteLine(y1);
Assert.IsTrue(TestUtilities.IsNearlyEqual(y1, solution(y0, 2.0)));
count += s.EvaluationCount;
}
Console.WriteLine(count);
}
public void OdeNonlinear()
{
// y = \frac{y_0}{1 - y_0 (x - x_0)}
Func <double, double, double> f = (double x, double y) => MoreMath.Sqr(y);
int count = 0;
OdeSettings settings = new OdeSettings()
{
RelativePrecision = 1.0E-8,
EvaluationBudget = 1024,
Listener = (OdeResult) => count++
};
OdeResult result = FunctionMath.IntegrateOde(f, 0.0, 1.0, 0.99, settings);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Y, 1.0 / (1.0 - 1.0 * (0.99 - 0.0)), result.Settings));
Assert.IsTrue(count > 0);
Console.WriteLine(result.EvaluationCount);
}
public void OdeDawson()
{
// The Dawson function fulfills a simple ODE.
// \frac{dF}{dx} + 2 x F = 1 \qquad F(0) = 0
// See e.g. https://en.wikipedia.org/wiki/Dawson_function
// Verify that we get correct values via ODE integration.
Func <double, double, double> rhs = (double x, double F) => 1.0 - 2.0 * x * F;
foreach (double x1 in TestUtilities.GenerateRealValues(0.1, 10.0, 8))
{
EvaluationSettings s = new EvaluationSettings()
{
RelativePrecision = 1.0E-13,
AbsolutePrecision = 0.0
};
OdeResult r = FunctionMath.IntegrateOde(rhs, 0.0, 0.0, x1);
Debug.WriteLine("{0}: {1} {2}: {3}", x1, r.Y, AdvancedMath.Dawson(x1), r.EvaluationCount);
Assert.IsTrue(TestUtilities.IsNearlyEqual(r.Y, AdvancedMath.Dawson(x1), s));
}
}