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 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 DistributionRawMomentIntegral()
{
foreach (ContinuousDistribution distribution in distributions)
{
foreach (int r in TestUtilities.GenerateIntegerValues(2, 32, 8))
{
double M = distribution.RawMoment(r);
if (Double.IsInfinity(M) || Double.IsNaN(M))
{
continue; // don't try to do a non-convergent integral
}
Func <double, double> f = delegate(double x) {
return(distribution.ProbabilityDensity(x) * Math.Pow(x, r));
};
try {
IntegrationResult MI = FunctionMath.Integrate(f, distribution.Support);
Console.WriteLine("{0} {1} {2} {3}", distribution.GetType().Name, r, M, MI);
Assert.IsTrue(MI.Estimate.ConfidenceInterval(0.99).ClosedContains(M));
/*
* if (M == 0.0) {
* Assert.IsTrue(Math.Abs(MI) < TestUtilities.TargetPrecision);
* } else {
* Assert.IsTrue(TestUtilities.IsNearlyEqual(M, MI));
* }
*/
} catch (NonconvergenceException) {
Console.WriteLine("{0} {1} {2} NC", distribution.GetType().Name, r, M);
}
}
}
}
public void DistributionVarianceIntegral()
{
foreach (ContinuousDistribution distribution in distributions)
{
double v = distribution.Variance;
// Skip distributions with infinite variance
if (Double.IsNaN(v) || Double.IsInfinity(v))
{
continue;
}
// Determine target precision, which must be reduced for some cases where an integral singularity means that cannot achieve full precision
double e = TestUtilities.TargetPrecision;
GammaDistribution gammaDistribution = distribution as GammaDistribution;
if ((gammaDistribution != null) && (gammaDistribution.Shape < 1.0))
{
e = Math.Sqrt(e);
}
Func <double, double> f = delegate(double x) {
double z = x - distribution.Mean;
return(distribution.ProbabilityDensity(x) * z * z);
};
double C2 = FunctionMath.Integrate(f, distribution.Support, new IntegrationSettings()
{
RelativePrecision = e, AbsolutePrecision = 0.0
}).Value;
Assert.IsTrue(TestUtilities.IsNearlyEqual(C2, distribution.Variance, e));
}
}
public void FindExtremaNegativeGamma()
{
// https://en.wikipedia.org/wiki/Particular_values_of_the_Gamma_function#Other_constants
Tuple <double, double>[] extrema = new Tuple <double, double>[] {
Tuple.Create(-0.5040830082644554092582693045, -3.5446436111550050891219639933),
Tuple.Create(-1.5734984731623904587782860437, 2.3024072583396801358235820396),
Tuple.Create(-2.6107208684441446500015377157, -0.8881363584012419200955280294),
Tuple.Create(-3.6352933664369010978391815669, 0.2451275398343662504382300889)
};
foreach (Tuple <double, double> extremum in extrema)
{
// We use brackets since we know the extremum must lie between the singularities.
// We should be able to use the actual singularities at endpoints, but this doesn't work; look into it.
Interval bracket = Interval.FromEndpoints(Math.Floor(extremum.Item1) + 0.01, Math.Ceiling(extremum.Item1) - 0.01);
Extremum result;
if (extremum.Item2 < 0.0)
{
result = FunctionMath.FindMaximum(AdvancedMath.Gamma, bracket);
}
else
{
result = FunctionMath.FindMinimum(AdvancedMath.Gamma, bracket);
}
Assert.IsTrue(result.Bracket.ClosedContains(extremum.Item1));
Assert.IsTrue(result.Bracket.OpenContains(result.Location));
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Value, extremum.Item2));
}
}
public void AssociatedLegendreOrthonormalityL()
{
// don't let l and m get too big, or numerical integration will fail due to heavily oscilatory behavior
int[] ells = TestUtilities.GenerateIntegerValues(1, 10, 4);
for (int ki = 0; ki < ells.Length; ki++)
{
int k = ells[ki];
for (int li = 0; li <= ki; li++)
{
int l = ells[li];
foreach (int m in TestUtilities.GenerateUniformIntegerValues(0, Math.Min(k, l), 4))
{
Func <double, double> f = delegate(double x) {
return(OrthogonalPolynomials.LegendreP(k, m, x) * OrthogonalPolynomials.LegendreP(l, m, x));
};
double I = FunctionMath.Integrate(f, Interval.FromEndpoints(-1.0, 1.0));
Console.WriteLine("k={0} l={1} m={2} I={3}", k, l, m, I);
if (k == l)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
I, 2.0 / (2 * k + 1) * Math.Exp(AdvancedIntegerMath.LogFactorial(l + m) - AdvancedIntegerMath.LogFactorial(l - m))
));
}
else
{
Assert.IsTrue(Math.Abs(I) < TestUtilities.TargetPrecision);
}
}
}
}
}
public void RootOfEi()
{
double x = FunctionMath.FindZero(AdvancedMath.IntegralEi, Interval.FromEndpoints(0.1, 1.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(x, 0.37250741078136663446));
Assert.IsTrue(Math.Abs(AdvancedMath.IntegralEi(x)) < TestUtilities.TargetPrecision);
}
public void OdeAiry()
{
// This is the airy differential equation
Func <double, double, double> f = (double x, double y) => x * y;
// Solutions should be of the form f(x) = a Ai(x) + b Bi(x).
// Given initial value of f and f', this equation plus the Wronskian can be solved to give
// a = \pi ( f Bi' - f' Bi ) b = \pi ( f' Ai - f Ai' )
// Start with some initial conditions
double x0 = 0.0;
double y0 = 0.0;
double yp0 = 1.0;
// Find the a and b coefficients consistent with those values
SolutionPair s0 = AdvancedMath.Airy(x0);
double a = Math.PI * (y0 * s0.SecondSolutionDerivative - yp0 * s0.SecondSolutionValue);
double b = Math.PI * (yp0 * s0.FirstSolutionValue - y0 * s0.FirstSolutionDerivative);
Assert.IsTrue(TestUtilities.IsNearlyEqual(y0, a * s0.FirstSolutionValue + b * s0.SecondSolutionValue));
Assert.IsTrue(TestUtilities.IsNearlyEqual(yp0, a * s0.FirstSolutionDerivative + b * s0.SecondSolutionDerivative));
// Integrate to a new point (pick a negative one so we test left integration)
double x1 = -5.0;
OdeResult result = FunctionMath.IntegrateConservativeOde(f, x0, y0, yp0, x1);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.X, x1));
Console.WriteLine(result.EvaluationCount);
// The solution should still hold
SolutionPair s1 = AdvancedMath.Airy(x1);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Y, a * s1.FirstSolutionValue + b * s1.SecondSolutionValue, result.Settings));
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.YPrime, a * s1.FirstSolutionDerivative + b * s1.SecondSolutionDerivative, result.Settings));
}
public void DistributionVarianceIntegral()
{
foreach (Distribution distribution in distributions)
{
if (Double.IsNaN(distribution.Variance) || Double.IsInfinity(distribution.Variance))
{
continue;
}
double e = TestUtilities.TargetPrecision;
// since a Gamma distribution with \alpha < 1 has a power law singularity and numerical integration cannot achieve full precision with such a singularity,
// we reduce our precision requirement in this case
GammaDistribution gammaDistribution = distribution as GammaDistribution; if ((gammaDistribution != null) && (gammaDistribution.ShapeParameter < 1.0))
{
e = Math.Sqrt(e);
}
Console.WriteLine(distribution.GetType().Name);
Func <double, double> f = delegate(double x) {
double z = x - distribution.Mean;
return(distribution.ProbabilityDensity(x) * z * z);
};
double C2 = FunctionMath.Integrate(f, distribution.Support, new EvaluationSettings()
{
EvaluationBudget = 4096, RelativePrecision = e, AbsolutePrecision = 0.0
}).Value;
Console.WriteLine(" {0} {1}", distribution.StandardDeviation, Math.Sqrt(C2));
Assert.IsTrue(TestUtilities.IsNearlyEqual(C2, distribution.Variance, e));
}
}
public void DistributionRawMomentIntegral()
{
foreach (Distribution distribution in distributions)
{
// range of moments is about 3 to 30
foreach (int n in TestUtilities.GenerateIntegerValues(3, 30, 10))
{
double M = distribution.Moment(n);
if (Double.IsInfinity(M) || Double.IsNaN(M))
{
continue; // don't try to do a non-convergent integral
}
Func <double, double> f = delegate(double x) {
return(distribution.ProbabilityDensity(x) * Math.Pow(x, n));
};
try {
double MI = FunctionMath.Integrate(f, distribution.Support);
Console.WriteLine("{0} {1} {2} {3}", distribution.GetType().Name, n, M, MI);
if (M == 0.0)
{
Assert.IsTrue(Math.Abs(MI) < TestUtilities.TargetPrecision);
}
else
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(M, MI));
}
} catch (NonconvergenceException) {
Console.WriteLine("{0} {1} {2} NC", distribution.GetType().Name, n, M);
}
}
}
}
public void DifferentiationTest()
{
int i = 0;
foreach (TestDerivative test in derivatives)
{
i++;
foreach (double x in TestUtilities.GenerateRealValues(0.1, 100.0, 5))
{
UncertainValue nd = FunctionMath.Differentiate(test.Function, x);
double ed = test.Derivative(x);
Console.WriteLine("{0} f'({1}) = {2} = {3}", i, x, nd, ed);
Console.WriteLine(
//Assert.IsTrue(
nd.ConfidenceInterval(0.999).ClosedContains(ed)
);
/*
* TestUtilities.IsNearlyEqual(
* FunctionMath.Differentiate(test.Function, x),
* test.Derivative(x),
* Math.Pow(2, -42)
*/
}
}
}
public void RootOfPsi()
{
double x = FunctionMath.FindZero(AdvancedMath.Psi, 1.5);
Assert.IsTrue(TestUtilities.IsNearlyEqual(x, 1.46163214496836234126));
Assert.IsTrue(AdvancedMath.Psi(x) < TestUtilities.TargetPrecision);
}
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);
}
// COrrel Making for Hybrid Method!
public SymmetricMatrix MakeCorrel(int n, Grid grid)
{
//Correlation between Z , Z_1 and Z_2
//integrationCalcul
Func <double, double> func = t => (double)Math.Pow((1 * grid.get_Step() - t), (H - 0.5)) * Math.Pow((2 * grid.get_Step() - t), (H - 0.5));
EvaluationSettings settings = new EvaluationSettings();
settings.AbsolutePrecision = 1.0E-9;
settings.RelativePrecision = 0.0;
IntegrationResult integresult = FunctionMath.Integrate(func, Meta.Numerics.Interval.FromEndpoints(0.0, grid.get_Step()), settings);
#region Correlation Calcul (correl Matrix)
SymmetricMatrix correl = new SymmetricMatrix(3);
for (int i = 0; i < 3; i++)
{
correl[i, i] = 1.0;
}
double var_0 = (double)1 / n;
double var_1 = (double)1 / (Math.Pow(n, 2 * (H - 0.5) + 1) * (2 * (H - 0.5) + 1));
double var_2 = (double)Math.Pow(2, 2 * (H - 0.5) + 1) / (Math.Pow(n, 2 * (H - 0.5) + 1) * (2 * (H - 0.5) + 1));
correl[1, 0] = (double)1 / (Math.Pow(n, H + 0.5) * (H + 0.5)) / (Math.Pow(var_0 * var_1, 0.5));
correl[2, 0] = (double)(Math.Pow(2, H + 0.5) - 1) / (Math.Pow(n, H + 0.5) * (H + 0.5)) / (Math.Pow(var_0 * var_2, 0.5));
correl[2, 1] = integresult.Value / (Math.Pow(var_2 * var_1, 0.5));
#endregion
return(correl);
}
public void IntegralEiZero()
{
// Value of zero documented at https://dlmf.nist.gov/6.13
double x0 = FunctionMath.FindZero(AdvancedMath.IntegralEi, 1.0);
Assert.IsTrue(TestUtilities.IsNearlyEqual(x0, 0.37250741078136663446));
}
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 OdeSine()
{
// The sine and cosine functions satisfy
// y'' = - y
// This is perhaps the simplest conservative differential equation.
// (i.e. right hand side depends only on y, not y')
Func <double, double, double> f = (double x, double y) => - y;
int count = 0;
OdeSettings settings = new OdeSettings()
{
Listener = (OdeResult r) => {
Assert.IsTrue(TestUtilities.IsNearlyEqual(
MoreMath.Sqr(r.Y) + MoreMath.Sqr(r.YPrime), 1.0, r.Settings
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
r.Y, MoreMath.Sin(r.X), r.Settings
));
count++;
}
};
OdeResult result = FunctionMath.IntegrateConservativeOde(f, 0.0, 0.0, 1.0, 5.0, settings);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Y, MoreMath.Sin(5.0)));
Assert.IsTrue(count > 0);
}
public void IntegralE1Integral()
{
// A & S 5.1.33
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(t => MoreMath.Sqr(AdvancedMath.IntegralE(1, t)), 0.0, Double.PositiveInfinity),
2.0 * Math.Log(2.0)
));
}
public void OdeSine()
{
Func <double, double, double> f = (double x, double y) => - y;
double y1 = FunctionMath.SolveConservativeOde(f, 0.0, 0.0, 1.0, 5.0);
Console.WriteLine(y1);
Assert.IsTrue(TestUtilities.IsNearlyEqual(y1, MoreMath.Sin(5.0)));
}
/// <summary>
/// Computes the expectation value of the given function.
/// </summary>
/// <param name="f">The function.</param>
/// <returns>The expectation value of the function.</returns>
/// <exception cref="ArgumentNullException"><paramref name="f"/> is <see langword="null"/>.</exception>
public virtual double ExpectationValue(Func <double, double> f)
{
if (f == null)
{
throw new ArgumentNullException(nameof(f));
}
return(FunctionMath.Integrate(x => f(x) * ProbabilityDensity(x), Support).Estimate.Value);
}
public void Bug5505()
{
// finding the root of x^3 would fail with a NonconvergenceException because our termination criterion tested only
// for small relative changes in x; adding a test for small absolute changes too solved the problem
double x0 = FunctionMath.FindZero(delegate(double x) { return(x * x * x); }, 1.0);
Assert.IsTrue(Math.Abs(x0) < TestUtilities.TargetPrecision);
}
public void DistributionProbabilityIntegral()
{
Random rng = new Random(1);
// if integral is very small, we still want to get it very accurately
IntegrationSettings settings = new IntegrationSettings()
{
AbsolutePrecision = 0.0
};
foreach (ContinuousDistribution distribution in distributions)
{
if (distribution is BetaDistribution b && ((b.Alpha < 1.0) || (b.Beta < 1.0)))
{
continue;
}
for (int i = 0; i < 4; i++)
{
double x;
if (Double.IsNegativeInfinity(distribution.Support.LeftEndpoint) && Double.IsPositiveInfinity(distribution.Support.RightEndpoint))
{
// pick an exponentially distributed random point with a random sign
double y = rng.NextDouble();
x = -Math.Log(y);
if (rng.NextDouble() < 0.5)
{
x = -x;
}
}
else if (Double.IsPositiveInfinity(distribution.Support.RightEndpoint))
{
// pick an exponentially distributed random point
double y = rng.NextDouble();
x = distribution.Support.LeftEndpoint - Math.Log(y);
}
else
{
// pick a random point within the support
x = distribution.Support.LeftEndpoint + rng.NextDouble() * distribution.Support.Width;
}
double P = FunctionMath.Integrate(distribution.ProbabilityDensity, Interval.FromEndpoints(distribution.Support.LeftEndpoint, x), settings).Value;
double Q = FunctionMath.Integrate(distribution.ProbabilityDensity, Interval.FromEndpoints(x, distribution.Support.RightEndpoint), settings).Value;
if (!TestUtilities.IsNearlyEqual(P + Q, 1.0))
{
// the numerical integral for the triangular distribution can be inaccurate, because
// its locally low-polynomial behavior fools the integration routine into thinking it need
// not integrate as much near the inflection point as it must; this is a problem
// of the integration routine (or arguably the integral), not the triangular distribution,
// so skip it here
continue;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(P, distribution.LeftProbability(x)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(Q, distribution.RightProbability(x)));
}
}
}
public void BesselWeberIntegral()
{
Func <double, double> f = delegate(double t) {
return(Math.Exp(-t * t) * AdvancedMath.BesselJ(0, t) * t);
};
Interval r = Interval.FromEndpoints(0.0, Double.PositiveInfinity);
Assert.IsTrue(TestUtilities.IsNearlyEqual(FunctionMath.Integrate(f, r).Estimate.Value, Math.Exp(-1.0 / 4.0) / 2.0));
}
public void RootOfJ0()
{
Func <double, double> f = delegate(double x) {
return(AdvancedMath.BesselJ(0, x));
};
double y = FunctionMath.FindZero(f, Interval.FromEndpoints(2.0, 4.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(y, 2.40482555769577276862));
}
public void FindMaximaFromPoint()
{
foreach (TestExtremum testExtremum in testMaxima)
{
Extremum extremum = FunctionMath.FindMaximum(testExtremum.Function, testExtremum.Interval.Midpoint);
Assert.IsTrue(TestUtilities.IsNearlyEqual(extremum.Location, testExtremum.Location, Math.Sqrt(TestUtilities.TargetPrecision)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(extremum.Value, testExtremum.Value, TestUtilities.TargetPrecision));
}
}
public void IntegralEiE1Integrals()
{
// Here are a couple unusual integrals involving both Ei and E1
// https://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(x => Math.Exp(-x) * AdvancedMath.IntegralE(1, x) * AdvancedMath.IntegralEi(x), 0.0, Double.PositiveInfinity),
-Math.PI * Math.PI / 12.0
));
}
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 void IntegrateTestIntegrals()
{
for (int i = 0; i < integrals.Length; i++)
{
TestIntegral integral = integrals[i];
double result = FunctionMath.Integrate(integral.Integrand, integral.Range);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result, integral.Result));
}
}
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 FindMinimaFromPoint()
{
foreach (TestExtremum testExtremum in testMinima)
{
Extremum extremum = FunctionMath.FindMinimum(testExtremum.Function, testExtremum.Interval.Midpoint);
Assert.IsTrue(TestUtilities.IsNearlyEqual(extremum.Location, testExtremum.Location, Math.Sqrt(TestUtilities.TargetPrecision)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(extremum.Value, testExtremum.Value, TestUtilities.TargetPrecision));
//if (!Double.IsNaN(testMinimum.Curvature)) Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Curvature, testMinimum.Curvature, 0.05));
}
}
