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)); } }