public void GoldsteinPrice()
{
// Goldstein-Price has a valley with a complicated shape and a global minimum value of 3 at (0,-1).
// It also has local minima, so we have to start close to this minimum if we expect to end at it.
Func <IList <double>, double> fGoldsteinPrice = (IList <double> v) => {
double x = v[0];
double y = v[1];
return(
(1 + MoreMath.Pow(x + y + 1, 2) * (19 - 14 * x + 3 * x * x - 14 * y + 6 * x * y + 6 * y * y)) *
(30 + MoreMath.Pow(2 * x - 3 * y, 2) * (18 - 32 * x + 12 * x * x + 48 * y - 36 * x * y + 27 * y * y))
);
};
ColumnVector start = new ColumnVector(0.5, -0.5);
MultiExtremum min = MultiFunctionMath.FindLocalMinimum(fGoldsteinPrice, start);
Console.WriteLine(min.EvaluationCount);
Console.WriteLine(min.Value);
Assert.IsTrue(min.Dimension == 2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(min.Value, 3.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(min.Location, new ColumnVector(0.0, -1.0), Math.Sqrt(TestUtilities.TargetPrecision)));
MultiExtremum min2 = MultiFunctionMath.FindGlobalMinimum(fGoldsteinPrice, new Interval[] { Interval.FromEndpoints(-2.0, 2.0), Interval.FromEndpoints(-2.0, 2.0) });
Assert.IsTrue(min2.Dimension == 2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(min2.Value, 3.0, new EvaluationSettings()
{
AbsolutePrecision = min2.Precision
}));
//Assert.IsTrue(TestUtilities.IsNearlyEqual(min2.Location, new ColumnVector(0.0, -1.0), Math.Sqrt(TestUtilities.TargetPrecision)));
}
public void Perm()
{
Func <IList <double>, double> fPerm = (IList <double> x) => {
double s = 0.0;
for (int i = 1; i <= x.Count; i++)
{
double t = 0.0;
for (int j = 0; j < x.Count; j++)
{
t += (j + 1) * (MoreMath.Pow(x[j], i) - 1.0 / MoreMath.Pow(j + 1, i));
}
s += MoreMath.Sqr(t);
}
return(s);
};
int n = 4;
ColumnVector start = new ColumnVector(n);
MultiExtremum minimum = MultiFunctionMath.FindLocalMinimum(fPerm, start);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine(minimum.Value);
for (int i = 0; i < minimum.Dimension; i++)
{
Console.WriteLine(minimum.Location[i]);
}
}
public void Beale()
{
// Beale is a very interesting function.
// The only local minimum is at (3,1/2) where the function value is 0.
// But along the lines (0,+\infty) and (-\infty,1) the function value decreases toward 0 at infinity.
// From the point of view of a local minimizer those are perfectly valid downhill directions and
// if the minimzer gets caught in them it will move toward infinity until its evaluation budget is
// exhausted. Starting from y > 0 will probably keep us safe, or we can do bounded optimization.
Func <IList <double>, double> fBeale = (IList <double> x) =>
MoreMath.Sqr(1.5 - x[0] + x[0] * x[1]) +
MoreMath.Sqr(2.25 - x[0] + x[0] * x[1] * x[1]) +
MoreMath.Sqr(2.625 - x[0] + x[0] * x[1] * x[1] * x[1]);
ColumnVector start = new ColumnVector(2.0, 1.0);
MultiExtremum min = MultiFunctionMath.FindLocalMinimum(fBeale, start);
Console.WriteLine(min.EvaluationCount);
Console.WriteLine(min.Value);
ColumnVector solution = new ColumnVector(3.0, 0.5);
Assert.IsTrue(min.Dimension == 2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(min.Value, 0.0, new EvaluationSettings()
{
AbsolutePrecision = min.Precision
}));
Assert.IsTrue(TestUtilities.IsNearlyEqual(min.Location, solution, Math.Sqrt(TestUtilities.TargetPrecision)));
}
public void MinimizePerturbedQuadratic2D()
{
int count = 0;
MultiExtremumSettings s = new MultiExtremumSettings()
{
EvaluationBudget = 100,
RelativePrecision = 1.0E-10,
Listener = r => { count++; }
};
Func <IReadOnlyList <double>, double> f = (IReadOnlyList <double> x) =>
1.0 + 2.0 * MoreMath.Sqr(x[0] - 3.0) + 4.0 * (x[0] - 3.0) * (x[1] - 5.0) + 6.0 * MoreMath.Sqr(x[1] - 5.0) +
7.0 * MoreMath.Pow(x[0] - 3.0, 4) + 8.0 * MoreMath.Pow(x[1] - 5.0, 4);
MultiExtremum m = MultiFunctionMath.FindLocalMinimum(f, new double[] { 1.0, 1.0 }, s);
Assert.IsTrue(m.EvaluationCount <= s.EvaluationBudget);
Assert.IsTrue(m.Dimension == 2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(m.Value, 1.0, s));
Assert.IsTrue(TestUtilities.IsNearlyEqual(m.Location, new ColumnVector(3.0, 5.0), new EvaluationSettings()
{
RelativePrecision = Math.Sqrt(s.RelativePrecision)
}));
Assert.IsTrue(count > 0);
}
public void MinimizeQuadratic1D()
{
Func <IList <double>, double> f = (IList <double> x) => 1.0 + MoreMath.Sqr(x[0] - 3.0);
MultiExtremum m = MultiFunctionMath.FindLocalMinimum(f, new double[] { 1.0 });
Assert.IsTrue(TestUtilities.IsNearlyEqual(1.0, m.Value));
}
internal NonlinearRegressionResult(
IReadOnlyList <double> x, IReadOnlyList <double> y,
Func <IReadOnlyList <double>, double, double> function,
IReadOnlyList <double> start, IReadOnlyList <string> names)
{
Debug.Assert(x != null);
Debug.Assert(y != null);
Debug.Assert(function != null);
Debug.Assert(start != null);
Debug.Assert(names != null);
Debug.Assert(x.Count == y.Count);
Debug.Assert(start.Count > 0);
Debug.Assert(names.Count == start.Count);
int n = x.Count;
int d = start.Count;
if (n <= d)
{
throw new InsufficientDataException();
}
MultiExtremum min = MultiFunctionMath.FindLocalMinimum((IReadOnlyList <double> a) => {
double ss = 0.0;
for (int i = 0; i < n; i++)
{
double r = y[i] - function(a, x[i]);
ss += r * r;
}
return(ss);
}, start);
CholeskyDecomposition cholesky = min.HessianMatrix.CholeskyDecomposition();
if (cholesky == null)
{
throw new DivideByZeroException();
}
b = min.Location;
C = cholesky.Inverse();
C = (2.0 * min.Value / (n - d)) * C;
sumOfSquaredResiduals = 0.0;
residuals = new List <double>(n);
for (int i = 0; i < n; i++)
{
double z = y[i] - function(b, x[i]);
sumOfSquaredResiduals += z * z;
residuals.Add(z);
}
this.names = names;
this.function = function;
}
public void SmoothedEasom()
{
// This function is mostly flat except very near (\pi, \pi).
// For (1,1) or (0,0) "converges" to minimum of 0 at (1.30, 1.30). This is probably a local minimum of the cosine product.
//Func<IList<double>, double> function = (IList<double> x) => -Math.Exp(-(MoreMath.Sqr(x[0] - Math.PI) + MoreMath.Sqr(x[1] - Math.PI)));
Func <IList <double>, double> function = (IList <double> x) => - Math.Cos(x[0]) * Math.Cos(x[1]) * Math.Exp(-(MoreMath.Sqr(x[0] - Math.PI) + MoreMath.Sqr(x[1] - Math.PI)));
ColumnVector start = new ColumnVector(2.0, 2.0);
MultiExtremum minimum = MultiFunctionMath.FindLocalMinimum(function, start);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine(minimum.Value);
Console.WriteLine("{0} {1}", minimum.Location[0], minimum.Location[1]);
}
public void ThomsonLocal()
{
for (int n = 2; n < 8; n++)
{
Console.WriteLine(n);
// define the thompson metric
Func <IList <double>, double> f = GetThompsonFunction(n);
// random distribution to start
// using antipodal pairs gives us a better starting configuration
Random r = new Random(1001110000);
double[] start = new double[2 * (n - 1)];
for (int i = 0; i < (n - 1) / 2; i++)
{
int j = 4 * i;
start[j] = -Math.PI + 2.0 * r.NextDouble() * Math.PI;
start[j + 1] = Math.Asin(2.0 * r.NextDouble() - 1.0);
start[j + 2] = -(Math.PI - start[j]);
start[j + 3] = -start[j + 1];
}
// add one more point if necessary
if (n % 2 == 0)
{
start[2 * n - 4] = -Math.PI + 2.0 * r.NextDouble() * Math.PI;
start[2 * n - 3] = Math.Asin(2.0 * r.NextDouble() - 1.0);
}
EvaluationSettings set = new EvaluationSettings()
{
RelativePrecision = 1.0E-9
};
MultiExtremum min = MultiFunctionMath.FindLocalMinimum(f, start, set);
Console.WriteLine(min.Dimension);
Console.WriteLine(min.EvaluationCount);
Console.WriteLine("{0} ({1}) ?= {2}", min.Value, min.Precision, thompsonSolutions[n]);
Assert.IsTrue(min.Dimension == 2 * (n - 1));
Assert.IsTrue(TestUtilities.IsNearlyEqual(min.Value, thompsonSolutions[n], new EvaluationSettings()
{
AbsolutePrecision = 4.0 * min.Precision
}));
}
}
/// <summary>
/// Finds the parameterized function that best fits the data.
/// </summary>
/// <param name="f">The parameterized function.</param>
/// <param name="start">An initial guess for the parameters.</param>
/// <returns>The fit result.</returns>
/// <remarks>
/// <para>
/// In the returned <see cref="FitResult"/>, the parameters appear in the same order as in
/// the supplied fit function and initial guess vector. No goodness-of-fit test is returned.
/// </para>
/// </remarks>
/// <exception cref="ArgumentNullException"><paramref name="f"/> or <paramref name="start"/> is null.</exception>
/// <exception cref="InsufficientDataException">There are not more data points than fit parameters.</exception>
/// <exception cref="DivideByZeroException">The curvature matrix is singular, indicating that the data is independent of
/// one or more parameters, or that two or more parameters are linearly dependent.</exception>
public FitResult NonlinearRegression(Func <IList <double>, double, double> f, IList <double> start)
{
if (f == null)
{
throw new ArgumentNullException(nameof(f));
}
if (start == null)
{
throw new ArgumentNullException(nameof(start));
}
int n = this.Count;
int d = start.Count;
if (n <= d)
{
throw new InsufficientDataException();
}
MultiExtremum min = MultiFunctionMath.FindLocalMinimum((IList <double> a) => {
double ss = 0.0;
for (int i = 0; i < n; i++)
{
double r = yData[i] - f(a, xData[i]);
ss += r * r;
}
return(ss);
}, start);
CholeskyDecomposition cholesky = min.HessianMatrix.CholeskyDecomposition();
if (cholesky == null)
{
throw new DivideByZeroException();
}
SymmetricMatrix curvature = cholesky.Inverse();
curvature = (2.0 * min.Value / (n - d)) * curvature;
FitResult result = new FitResult(min.Location, curvature, null);
return(result);
}
public void Rosenbrock()
{
// This is a multi-dimensional generalization of the famous Rosenbrock, aka banana function,
// which has a narrow parabolic valley whose floor slopes only gently to the minimum.
Func <IList <double>, double> fRosenbrock = delegate(IList <double> x) {
double s = 0.0;
for (int i = 0; i < (x.Count - 1); i++)
{
s += 100.0 * MoreMath.Pow(x[i + 1] - x[i] * x[i], 2) + MoreMath.Pow(1.0 - x[i], 2);
}
return(s);
};
for (int n = 2; n < 8; n++)
{
Console.WriteLine("n={0}", n);
double[] start = new double[n];
MultiExtremum minimum = MultiFunctionMath.FindLocalMinimum(fRosenbrock, start);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} {1} ?= 0.0", minimum.Value, minimum.Precision);
ColumnVector solution = new ColumnVector(n);
for (int i = 0; i < solution.Dimension; i++)
{
solution[i] = 1.0;
}
Assert.IsTrue(minimum.Dimension == n);
Assert.IsTrue(minimum.Precision > 0.0);
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, 0.0, new EvaluationSettings()
{
AbsolutePrecision = 2.0 * minimum.Precision
}));
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Location, solution, new EvaluationSettings()
{
RelativePrecision = 2.0 * Math.Sqrt(minimum.Precision)
}));
}
}
public void SumOfPowers()
{
// This test is difficult because the minimum is emphatically not quadratic.
// We do get close to the minimum but we massivly underestimate our error.
Func <IList <double>, double> function = (IList <double> x) => {
double s = 0.0;
for (int i = 0; i < x.Count; i++)
{
s += MoreMath.Pow(Math.Abs(x[i]), i + 2);
}
return(s);
};
for (int n = 2; n < 8; n++)
{
Console.WriteLine(n);
ColumnVector start = new ColumnVector(n);
for (int i = 0; i < n; i++)
{
start[i] = 1.0;
}
EvaluationSettings settings = new EvaluationSettings()
{
AbsolutePrecision = 1.0E-8, EvaluationBudget = 32 * n * n * n
};
MultiExtremum minimum = MultiFunctionMath.FindLocalMinimum(function, start, settings);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} {1}", minimum.Value, minimum.Precision);
Console.WriteLine("|| {0} {1} ... || = {2}", minimum.Location[0], minimum.Location[1], minimum.Location.FrobeniusNorm());
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, 0.0, new EvaluationSettings()
{
AbsolutePrecision = 1.0E-4
}));
//Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Location, new ColumnVector(n), new EvaluationSettings() { AbsolutePrecision = 1.0E-2 }));
}
}
public void ThreeHumpCamel()
{
// This function has three local minima, so not at all starting points should be expected to bring us to the global minimum at the origin.
Func <IList <double>, double> function = (IList <double> x) => 2.0 * MoreMath.Pow(x[0], 2) - 1.05 * MoreMath.Pow(x[0], 4) + MoreMath.Pow(x[0], 6) / 6.0 + x[0] * x[1] + MoreMath.Pow(x[1], 2);
ColumnVector start = new ColumnVector(1.0, 1.0);
MultiExtremum minimum = MultiFunctionMath.FindLocalMinimum(function, start);
Console.WriteLine("{0} ({1}) ?= 0.0", minimum.Value, minimum.Precision);
Console.WriteLine("{0} {1}", minimum.Location[0], minimum.Location[1]);
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, 0.0, new EvaluationSettings()
{
AbsolutePrecision = 2.0 * minimum.Precision
}));
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Location, new ColumnVector(2), new EvaluationSettings {
AbsolutePrecision = 2.0 * Math.Sqrt(minimum.Precision)
}));
}
public void StylblinskiTang()
{
Func <IList <double>, double> fStyblinskiTang = (IList <double> x) => {
double fst = 0.0;
for (int i = 0; i < x.Count; i++)
{
double x1 = x[i];
double x2 = MoreMath.Sqr(x1);
fst += x2 * (x2 - 16.0) + 5.0 * x1;
}
return(fst / 2.0);
};
// solution coordinate is root of 5 - 32 x + 4 x^3 = 0 with negative second derivative.
// There are two such roots.
double root1 = -2.9035340277711770951;
double root2 = 2.7468027709908369925;
// tested up to n=16, works with slightly decreasing accuracy of Location
for (int n = 2; n < 8; n++)
{
Console.WriteLine(n);
ColumnVector start = new ColumnVector(n);
//ColumnVector start = new ColumnVector(-1.0, -2.0, -3.0, -4.0, -5.0, -6.0);
MultiExtremum minimum = MultiFunctionMath.FindLocalMinimum(fStyblinskiTang, start);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine(minimum.Value);
for (int i = 0; i < minimum.Dimension; i++)
{
Console.WriteLine(minimum.Location[i]);
Assert.IsTrue(
TestUtilities.IsNearlyEqual(minimum.Location[i], root1, Math.Sqrt(Math.Sqrt(TestUtilities.TargetPrecision))) ||
TestUtilities.IsNearlyEqual(minimum.Location[i], root2, Math.Sqrt(Math.Sqrt(TestUtilities.TargetPrecision)))
);
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, fStyblinskiTang(minimum.Location)));
}
}
public void Vardim()
{
// This function is described by Powell in his NEWOUA paper as one that caused problems for the simplest formulation of that minimizer.
Func <IList <double>, double> f = (IList <double> x) => {
double s1 = 0.0;
double s2 = 0.0;
for (int i = 0; i < x.Count; i++)
{
s1 += i * (x[i] - 1.0);
s2 += MoreMath.Sqr(x[i] - 1.0);
}
return(s2 + s1 * s1 + s1 * s1 * s1 * s1);
};
for (int n = 2; n < 8; n++)
{
Console.WriteLine("n = {0}", n);
ColumnVector start = new ColumnVector(n);
ColumnVector solution = new ColumnVector(n);
solution.Fill((i, j) => 1.0);
MultiExtremum min = MultiFunctionMath.FindLocalMinimum(f, start);
Console.WriteLine(min.EvaluationCount);
Console.WriteLine("{0} ({1}) ?= 0.0", min.Value, min.Precision);
Assert.IsTrue(min.Dimension == n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(min.Value, 0.0, new EvaluationSettings()
{
AbsolutePrecision = 4.0 * min.Precision
}));
Assert.IsTrue(TestUtilities.IsNearlyEqual(min.Location, solution, new EvaluationSettings()
{
RelativePrecision = Math.Sqrt(4.0 * min.Precision)
}));
}
}
public void MinimizeQuadratic2D()
{
Func <IList <double>, double> f = (IList <double> x) => 1.0 + 2.0 * MoreMath.Sqr(x[0] - 3.0) + 4.0 * (x[0] - 3.0) * (x[1] - 5.0) + 6.0 * MoreMath.Sqr(x[1] - 5.0);
MultiExtremum m = MultiFunctionMath.FindLocalMinimum(f, new double[] { 1.0, 1.0 });
Console.WriteLine(m.Value);
SymmetricMatrix H = new SymmetricMatrix(2);
H[0, 0] = 4.0;
H[0, 1] = 4.0;
H[1, 1] = 12.0;
Assert.IsTrue(m.Dimension == 2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(m.Value, 1.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(m.Location, new ColumnVector(3.0, 5.0), new EvaluationSettings()
{
RelativePrecision = Math.Sqrt(TestUtilities.TargetPrecision)
}));
Assert.IsTrue(TestUtilities.IsNearlyEqual(m.HessianMatrix, H, new EvaluationSettings()
{
RelativePrecision = 8.0 * Math.Sqrt(Math.Sqrt(TestUtilities.TargetPrecision))
}));
}
/// <summary>
/// Computes the best-fit linear logistic regression from the data.
/// </summary>
/// <returns>The fit result.</returns>
/// <remarks>
/// <para>Linear logistic regression is a way to fit binary outcome data to a linear model.</para>
/// <para>The method assumes that binary outcomes are encoded as 0 and 1. If any y-values other than
/// 0 and 1 are encountered, it throws an <see cref="InvalidOperationException"/>.</para>
/// <para>The fit result is two-dimensional. The first parameter is a, the second b.</para>
/// </remarks>
/// <exception cref="InsufficientDataException">There are fewer than three data points.</exception>
/// <exception cref="InvalidOperationException">There is a y-value other than 0 or 1.</exception>
public FitResult LinearLogisticRegression()
{
// check size of data set
if (Count < 3)
{
throw new InsufficientDataException();
}
// check limits on Y
double p = Y.Mean; double q = 1.0 - p;
if ((p <= 0.0) || (q <= 0.0))
{
throw new InvalidOperationException();
}
// Define the log-likelyhood as a function of the parameters
Func <IList <double>, double> f = delegate(IList <double> a) {
double L = 0.0;
for (int i = 0; i < Count; i++)
{
double x = xData[i];
double z = a[0] + a[1] * x;
double ez = Math.Exp(z);
double y = yData[i];
if (y == 0.0)
{
L += Math.Log(1.0 + ez);
}
else if (y == 1.0)
{
L += Math.Log(1.0 + 1.0 / ez);
}
else
{
throw new InvalidOperationException();
}
}
return(L);
};
// We need an initial guess at the parameters.
// This particular initial guess is driven by the fact that, in a logistic model
// \frac{\partial p}{\partial x} = \beta p ( 1 - p)
// Evaluating at means, and noting that p (1 - p) = var(y) and that, in a development around the means,
// cov(p, x) = \frac{\partial p}{\partial x} var(x)
// we get
// \beta = \frac{cov(y, x)}{var(x) var(y)}
// This approximation gets the sign right, but it looks like it usually gets the magnitude quite wrong.
double b0 = Covariance / X.Variance / Y.Variance;
double a0 = Math.Log(p / q) - b0 * X.Mean;
MultiExtremum m = MultiFunctionMath.FindLocalMinimum(f, new double[2] {
a0, b0
});
return(new FitResult(m.Location, m.HessianMatrix.Inverse(), null));
//SpaceExtremum m = FunctionMath.FindMinimum(f, new double[2] { a0, b0 });
//return (new FitResult(m.Location(), m.Curvature().Inverse(), null));
}
public static void Optimization()
{
Interval range = Interval.FromEndpoints(0.0, 6.28);
Extremum min = FunctionMath.FindMinimum(x => Math.Sin(x), range);
Console.WriteLine($"Found minimum of {min.Value} at {min.Location}.");
Console.WriteLine($"Required {min.EvaluationCount} evaluations.");
MultiExtremum rosenbrock = MultiFunctionMath.FindLocalMinimum(
x => MoreMath.Sqr(2.0 - x[0]) + 100.0 * MoreMath.Sqr(x[1] - x[0] * x[0]),
new ColumnVector(0.0, 0.0)
);
ColumnVector xm = rosenbrock.Location;
Console.WriteLine($"Found minimum of {rosenbrock.Value} at ({xm[0]},{xm[1]}).");
Console.WriteLine($"Required {rosenbrock.EvaluationCount} evaluations.");
Func <IReadOnlyList <double>, double> leviFunction = z => {
double x = z[0];
double y = z[1];
return(
MoreMath.Sqr(MoreMath.Sin(3.0 * Math.PI * x)) +
MoreMath.Sqr(x - 1.0) * (1.0 + MoreMath.Sqr(MoreMath.Sin(3.0 * Math.PI * y))) +
MoreMath.Sqr(y - 1.0) * (1.0 + MoreMath.Sqr(MoreMath.Sin(2.0 * Math.PI * y)))
);
};
Interval[] leviRegion = new Interval[] {
Interval.FromEndpoints(-10.0, +10.0),
Interval.FromEndpoints(-10.0, +10.0)
};
MultiExtremum levi = MultiFunctionMath.FindGlobalMinimum(leviFunction, leviRegion);
ColumnVector zm = levi.Location;
Console.WriteLine($"Found minimum of {levi.Value} at ({zm[0]},{zm[1]}).");
Console.WriteLine($"Required {levi.EvaluationCount} evaluations.");
// Select a dimension
int n = 6;
// Define a function that takes 2n polar coordinates in the form
// phi_1, theta_1, phi_2, theta_2, ..., phi_n, theta_n, computes
// the sum of the potential energy 1/d for all pairs.
Func <IReadOnlyList <double>, double> thompson = (IReadOnlyList <double> v) => {
double e = 0.0;
// iterate over all distinct pairs of points
for (int i = 0; i < n; i++)
{
for (int j = 0; j < i; j++)
{
// compute the chord length between points i and j
double dx = Math.Cos(v[2 * j]) * Math.Cos(v[2 * j + 1]) - Math.Cos(v[2 * i]) * Math.Cos(v[2 * i + 1]);
double dy = Math.Cos(v[2 * j]) * Math.Sin(v[2 * j + 1]) - Math.Cos(v[2 * i]) * Math.Sin(v[2 * i + 1]);
double dz = Math.Sin(v[2 * j]) - Math.Sin(v[2 * i]);
double d = Math.Sqrt(dx * dx + dy * dy + dz * dz);
e += 1.0 / d;
}
}
return(e);
};
// Limit all polar coordinates to their standard ranges.
Interval[] box = new Interval[2 * n];
for (int i = 0; i < n; i++)
{
box[2 * i] = Interval.FromEndpoints(-Math.PI, Math.PI);
box[2 * i + 1] = Interval.FromEndpoints(-Math.PI / 2.0, Math.PI / 2.0);
}
// Use settings to monitor proress toward a rough solution.
MultiExtremumSettings roughSettings = new MultiExtremumSettings()
{
RelativePrecision = 0.05, AbsolutePrecision = 0.0,
Listener = r => {
Console.WriteLine($"Minimum {r.Value} after {r.EvaluationCount} evaluations.");
}
};
MultiExtremum roughThompson = MultiFunctionMath.FindGlobalMinimum(thompson, box, roughSettings);
// Use settings to monitor proress toward a refined solution.
MultiExtremumSettings refinedSettings = new MultiExtremumSettings()
{
RelativePrecision = 1.0E-5, AbsolutePrecision = 0.0,
Listener = r => {
Console.WriteLine($"Minimum {r.Value} after {r.EvaluationCount} evaluations.");
}
};
MultiExtremum refinedThompson = MultiFunctionMath.FindLocalMinimum(thompson, roughThompson.Location, refinedSettings);
Console.WriteLine($"Minimum potential energy {refinedThompson.Value}.");
Console.WriteLine($"Required {roughThompson.EvaluationCount} + {refinedThompson.EvaluationCount} evaluations.");
/*
* // Define a function that takes 2n coordinates x1, y1, x2, y2, ... xn, yn
* // and finds the smallest distance between two coordinate pairs.
* Func<IReadOnlyList<double>, double> function = (IReadOnlyList<double> x) => {
* double sMin = Double.MaxValue;
* for (int i = 0; i < n; i++) {
* for (int j = 0; j < i; j++) {
* double s = MoreMath.Hypot(x[2 * i] - x[2 * j], x[2 * i + 1] - x[2 * j + 1]);
* if (s < sMin) sMin = s;
* }
* }
* return (sMin);
* };
*
* // Limit all coordinates to the unit box.
* Interval[] box = new Interval[2 * n];
* for (int i = 0; i < box.Length; i++) box[i] = Interval.FromEndpoints(0.0, 1.0);
*
* // Use settings to monitor proress toward a rough solution.
* MultiExtremumSettings roughSettings = new MultiExtremumSettings() {
* RelativePrecision = 1.0E-2, AbsolutePrecision = 0.0,
* Listener = r => {
* Console.WriteLine($"Minimum {r.Value} after {r.EvaluationCount} evaluations.");
* }
* };
* MultiExtremum roughMaximum = MultiFunctionMath.FindGlobalMaximum(function, box, roughSettings);
*
* // Use settings to monitor proress toward a rough solution.
* MultiExtremumSettings refinedSettings = new MultiExtremumSettings() {
* RelativePrecision = 1.0E-8, AbsolutePrecision = 0.0,
* Listener = r => {
* Console.WriteLine($"Minimum {r.Value} after {r.EvaluationCount} evaluations.");
* }
* };
* MultiExtremum refinedMaximum = MultiFunctionMath.FindLocalMaximum(function, roughMaximum.Location, refinedSettings);
*/
}
