/// <summary>
/// Isolates a root within a given interval.
/// </summary>
/// <param name="f">The function whoose zero is sought.</param>
/// <param name="bracket">An interval bracketing the root.</param>
/// <returns>An ordinate within the bracket at which the function has a zero.</returns>
/// <exception cref="InvalidOperationException">The function does not change sign across the given interval.</exception>
public static double FindZero(Func<double, double> f, Interval bracket)
{
if (f == null) throw new ArgumentNullException("f");
double x1 = bracket.LeftEndpoint;
double x2 = bracket.RightEndpoint;
double f1 = f(x1);
double f2 = f(x2);
// make sure the bracket points really do bracket a root
if (Math.Sign(f1) == Math.Sign(f2)) throw new InvalidOperationException();
return (FindZero(f, x1, f1, x2, f2));
}
public void Ackley()
{
// Ackley's function has many local minima, and a global minimum at (0, 0) -> 0.
Func<IList<double>, double> function = (IList<double> x) => {
double s = 0.0;
double c = 0.0;
for (int i = 0; i < x.Count; i++) {
s += x[i] * x[i];
c += Math.Cos(2.0 * Math.PI * x[i]);
}
return (-20.0 * Math.Exp(-0.2 * Math.Sqrt(s / x.Count)) - Math.Exp(c / x.Count) + 20.0 + Math.E);
};
EvaluationSettings settings = new EvaluationSettings() { AbsolutePrecision = 1.0E-8, EvaluationBudget = 10000000 };
for (int n = 2; n < 16; n = (int) Math.Round(AdvancedMath.GoldenRatio * n)) {
Console.WriteLine("n={0}", n);
Interval[] box = new Interval[n];
for (int i = 0; i < box.Length; i++) box[i] = Interval.FromEndpoints(-32.0, 32.0);
MultiExtremum minimum = MultiFunctionMath.FindGlobalMinimum(function, box, settings);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine(minimum.Value);
foreach (double coordinate in minimum.Location) Console.WriteLine(coordinate);
Assert.IsTrue(minimum.Dimension == n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, 0.0, new EvaluationSettings() { AbsolutePrecision = 2.0 * minimum.Precision }));
ColumnVector solution = new ColumnVector(n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Location, solution, new EvaluationSettings() { AbsolutePrecision = 2.0 * Math.Sqrt(minimum.Precision) }));
}
}
public void PackCirclesInCircle()
{
// Put n points into the unit circle. Place them so as to maximize the minimum distance between them.
// http://en.wikipedia.org/wiki/Circle_packing_in_a_circle, http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html
// This can also be ecoded via a box constraint if we use polar coordinates.
// In that case, 0 < r < 1 and -\pi < \theta < +\pi are the coordinate bounds.
double[] solutions = {
Double.NaN,
Double.NaN,
2.0, /* uniformly along edge, i.e. opposite sides */
Math.Sqrt(3.0), /* uniformly along edge, i.e. triangle */
Math.Sqrt(2.0), /* uniformly along edge, i.e. square */
Math.Sqrt(2.0 / (1.0 + 1.0 / Math.Sqrt(5.0))), /* uniformly along edge, i.e. pentagon */
1.0, /* uniformly along edge, i.e. hexagon */
1.0, /* six uniformly along edge plus one in center */
2.0 * Math.Sin(Math.PI / 7.0),
Math.Sqrt(2.0 / (2.0 + Math.Sqrt(2.0))),
0.710978235561246550830725976904,
2.0 * Math.Sin(Math.PI / 9.0)
};
for (int n = 2; n < 8; n++)
{
// We have a problem with n = 5. Chooses to put one in middle and others at 90 degree angles around it instead of distributing uniformly along edge.
if (n == 5) continue;
Console.WriteLine(n);
// We assume that the point r=1, t=0 exists and encode only the remaining n-1 points in a 2(n-1)-length vector.
Func<IList<double>, double> f = (IList<double> u) => {
double sMin = Double.MaxValue;
for (int i = 0; i < (n - 1); i++) {
double ri = u[2 * i];
double ti = u[2 * i + 1];
double xi = ri * Math.Cos(ti);
double yi = ri * Math.Sin(ti);
for (int j = 0; j < i; j++) {
double rj = u[2 * j];
double tj = u[2 * j + 1];
double xj = rj * Math.Cos(tj);
double yj = rj * Math.Sin(tj);
double s = MoreMath.Hypot(xi - xj, yi - yj);
if (s < sMin) sMin = s;
}
// also compare distance to fixed point (1, 0)
double s1 = MoreMath.Hypot(xi - 1.0, yi);
if (s1 < sMin) sMin = s1;
}
return (sMin);
};
Interval[] box = new Interval[2 * (n - 1)];
for (int i = 0; i < (n - 1); i++) {
box[2 * i] = Interval.FromEndpoints(0.0, 1.0);
box[2 * i + 1] = Interval.FromEndpoints(-Math.PI, Math.PI);
}
MultiExtremum maxmimum = MultiFunctionMath.FindGlobalMaximum(f, box);
Console.WriteLine(maxmimum.EvaluationCount);
Console.WriteLine("{0} {1}", maxmimum.Value, maxmimum.Precision);
for (int i = 0; i < (n - 1); i++) Console.WriteLine("{0} {1}", maxmimum.Location[2 * i], maxmimum.Location[2 * i + 1]);
Assert.IsTrue(maxmimum.Dimension == 2 * (n - 1));
Assert.IsTrue(TestUtilities.IsNearlyEqual(maxmimum.Value, solutions[n], new EvaluationSettings() { AbsolutePrecision = 2.0 * maxmimum.Precision }));
}
}
public void Griewank()
{
// See http://mathworld.wolfram.com/GriewankFunction.html
for (int n = 2; n < 8; n++) {
Console.WriteLine(n);
Func<IList<double>, double> function = (IList<double> x) => {
double s = 0.0;
double p = 1.0;
for (int i = 0; i < x.Count; i++) {
s += MoreMath.Sqr(x[i]);
p *= Math.Cos(x[i] / Math.Sqrt(i + 1.0));
}
return (1.0 + s / 4000.0 - p);
};
Interval[] box = new Interval[n];
for (int i = 0; i < n; i++) box[i] = Interval.FromEndpoints(-100.0, 100.0);
EvaluationSettings settings = new EvaluationSettings() { AbsolutePrecision = 1.0E-6, EvaluationBudget = 1000000 };
MultiExtremum minimum = MultiFunctionMath.FindGlobalMinimum(function, box, settings);
Console.WriteLine(minimum.Dimension);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} ({1}) ?= 0.0", minimum.Value, minimum.Precision);
Console.WriteLine("{0} {1} ...", minimum.Location[0], minimum.Location[1]);
// We usually find the minimum at 0, but sometimes land a valley or two over.
}
}
public void EasomGlobal()
{
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)));
IList<Interval> box = new Interval[] { Interval.FromEndpoints(-10.0, 10.0), Interval.FromEndpoints(-10.0, 10.0) };
MultiExtremum maximum = MultiFunctionMath.FindGlobalMaximum(function, box);
Console.WriteLine(maximum.EvaluationCount);
Console.WriteLine(maximum.Value);
Assert.IsTrue(TestUtilities.IsNearlyEqual(maximum.Value, 1.0, new EvaluationSettings() { AbsolutePrecision = 2.0 * maximum.Precision }));
Assert.IsTrue(TestUtilities.IsNearlyEqual(maximum.Location, new ColumnVector(Math.PI, Math.PI), new EvaluationSettings { AbsolutePrecision = 2.0 * Math.Sqrt(maximum.Precision) }));
}
public void Bukin()
{
// Burkin has a narrow valley, not aligned with any axis, punctuated with many tiny "wells" along its bottom.
// The deepest well is at (-10,1)-> 0.
Func<IList<double>, double> function = (IList<double> x) => 100.0 * Math.Sqrt(Math.Abs(x[1] - 0.01 * x[0] * x[0])) + 0.01 * Math.Abs(x[0] + 10.0);
IList<Interval> box = new Interval[] { Interval.FromEndpoints(-15.0, 0.0), Interval.FromEndpoints(-3.0, 3.0) };
EvaluationSettings settings = new EvaluationSettings() { RelativePrecision = 0.0, AbsolutePrecision = 1.0E-4, EvaluationBudget = 1000000 };
/*
settings.Update += (object result) => {
MultiExtremum e = (MultiExtremum) result;
Console.WriteLine("After {0} evaluations, best value {1}", e.EvaluationCount, e.Value);
};
*/
MultiExtremum minimum = MultiFunctionMath.FindGlobalMinimum(function, box, settings);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} ({1})", minimum.Value, minimum.Precision);
Console.WriteLine("{0} {1}", minimum.Location[0], minimum.Location[1]);
// We do not end up finding the global minimum.
}
public void TammesGlobal()
{
double[] tammesSolutions = new double[] { Double.NaN, Double.NaN, 2.0, Math.Sqrt(3.0), Math.Sqrt(8.0 / 3.0), Math.Sqrt(2.0), Math.Sqrt(2.0) };
for (int n = 2; n < 8; n++) {
Console.WriteLine("n = {0}", n);
// Define a function that returns the minimum distance between points
Func<IList<double>, double> f = (IList<double> u) => {
// add a point at 0,0
double[] v = new double[2 * n];
u.CopyTo(v, 0);
double dmin = Double.PositiveInfinity;
// 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);
if (d < dmin) dmin = d;
}
}
return (dmin);
};
/*
// Start with a random arrangement
Random r = new Random(1001110000);
double[] start = new double[2 * n];
for (int i = 0; i < n; i++) {
start[2 * i] = -Math.PI + 2.0 * r.NextDouble() * Math.PI;
start[2 * i + 1] = Math.Asin(2.0 * r.NextDouble() - 1.0);
}
// Maximize the minimum distance
MultiExtremum maximum = MultiFunctionMath.FindLocalMinimum(f, start);
*/
Interval[] limits = new Interval[2 * (n - 1)];
for (int i = 0; i < (n - 1); i++) {
limits[2 * i] = Interval.FromEndpoints(-Math.PI, +Math.PI);
limits[2 * i + 1] = Interval.FromEndpoints(-Math.PI / 2.0, +Math.PI / 2.0);
}
MultiExtremum maximum = MultiFunctionMath.FindGlobalMaximum(f, limits);
Console.WriteLine(maximum.EvaluationCount);
Console.WriteLine("{0} ({1})", maximum.Value, maximum.Precision);
Assert.IsTrue(maximum.Dimension == 2 * (n - 1));
if (n < tammesSolutions.Length) Assert.IsTrue(TestUtilities.IsNearlyEqual(maximum.Value, tammesSolutions[n], new EvaluationSettings() { AbsolutePrecision = 2.0 * maximum.Precision }));
}
}
public void WatsonIntegrals()
{
// Watson defined and analytically integrated three complicated triple integrals related to random walks in three dimension
// See http://mathworld.wolfram.com/WatsonsTripleIntegrals.html
// These integrals are difficult, so up the budget to about 1,000,000 and reduce the target accuracy to about 10^{-4}
EvaluationSettings settings = new EvaluationSettings() { RelativePrecision = 1.0E-4, EvaluationBudget = 1000000 };
Interval watsonWidth = Interval.FromEndpoints(0.0, Math.PI);
Interval[] watsonBox = new Interval[] { watsonWidth, watsonWidth, watsonWidth };
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IList<double> x) => 1.0 / (1.0 - Math.Cos(x[0]) * Math.Cos(x[1]) * Math.Cos(x[2])), watsonBox, settings
).Estimate.ConfidenceInterval(0.99).ClosedContains(
MoreMath.Pow(AdvancedMath.Gamma(1.0 / 4.0), 4) / 4.0
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IList<double> x) => 1.0 / (3.0 - Math.Cos(x[0]) * Math.Cos(x[1]) - Math.Cos(x[1]) * Math.Cos(x[2]) - Math.Cos(x[0]) * Math.Cos(x[2])), watsonBox, settings
).Estimate.ConfidenceInterval(0.99).ClosedContains(
3.0 * MoreMath.Pow(AdvancedMath.Gamma(1.0 / 3.0), 6) / Math.Pow(2.0, 14.0 / 3.0) / Math.PI
)
);
Assert.IsTrue(
MultiFunctionMath.Integrate(
(IList<double> x) => 1.0 / (3.0 - Math.Cos(x[0]) - Math.Cos(x[1]) - Math.Cos(x[2])), watsonBox, settings
).Estimate.ConfidenceInterval(0.99).ClosedContains(
Math.Sqrt(6.0) / 96.0 * AdvancedMath.Gamma(1.0 / 24.0) * AdvancedMath.Gamma(5.0 / 24.0) * AdvancedMath.Gamma(7.0 / 24.0) * AdvancedMath.Gamma(11.0 / 24.0)
)
);
}
public void IsingIntegrals()
{
// See http://www.davidhbailey.com/dhbpapers/ising.pdf.
EvaluationSettings settings = new EvaluationSettings() { RelativePrecision = 1.0E-3, EvaluationBudget = 1000000 };
int d = 4;
Interval[] volume = new Interval[d];
for (int i = 0; i < volume.Length; i++) {
volume[i] = Interval.FromEndpoints(0.0, Double.PositiveInfinity);
}
IntegrationResult r = MultiFunctionMath.Integrate((IList<double> x) => {
double p = 1.0;
double q = 0.0;
for (int i = 0; i < x.Count; i++) {
double u = x[i];
double v = 1.0 / u;
q += (u + v);
p *= v;
}
return (p / MoreMath.Sqr(q));
}, volume, settings);
double c = 4.0 / AdvancedIntegerMath.Factorial(d);
Console.WriteLine("{0} {1}", c * r.Estimate, r.EvaluationCount);
Console.WriteLine(7.0 * AdvancedMath.RiemannZeta(3.0) / 12.0);
Assert.IsTrue((c * r.Estimate).ConfidenceInterval(0.99).ClosedContains(7.0 * AdvancedMath.RiemannZeta(3.0) / 12.0));
}
// Returns a d-dimensional unit cube [0,1]^d
public Interval[] UnitCube(int d)
{
Interval[] box = new Interval[d];
for (int j = 0; j < d; j++) {
box[j] = Interval.FromEndpoints(0.0, 1.0);
}
return (box);
}
public void GaussianIntegrals()
{
Random rng = new Random(1);
for (int d = 2; d < 4; d++) {
if (d == 4 || d == 5 || d == 6) continue;
Console.WriteLine(d);
// Create a symmetric matrix
SymmetricMatrix A = new SymmetricMatrix(d);
for (int r = 0; r < d; r++) {
for (int c = 0; c < r; c++) {
A[r, c] = rng.NextDouble();
}
// Ensure it is positive definite by diagonal dominance
A[r, r] = r + 1.0;
}
// Compute its determinant, which appears in the analytic value of the integral
CholeskyDecomposition CD = A.CholeskyDecomposition();
double detA = CD.Determinant();
// Compute the integral
Func<IList<double>, double> f = (IList<double> x) => {
ColumnVector v = new ColumnVector(x);
double s = v.Transpose() * (A * v);
return (Math.Exp(-s));
};
Interval[] volume = new Interval[d];
for (int i = 0; i < d; i++) volume[i] = Interval.FromEndpoints(Double.NegativeInfinity, Double.PositiveInfinity);
IntegrationResult I = MultiFunctionMath.Integrate(f, volume);
// Compare to the analytic result
Console.WriteLine("{0} ({1}) {2}", I.Value, I.Precision, Math.Sqrt(MoreMath.Pow(Math.PI, d) / detA));
Assert.IsTrue(TestUtilities.IsNearlyEqual(I.Value, Math.Sqrt(MoreMath.Pow(Math.PI, d) / detA), new EvaluationSettings() { AbsolutePrecision = 2.0 * I.Precision }));
}
}
/// <summary>
/// Maximizes a function on the given interval.
/// </summary>
/// <param name="f">The function.</param>
/// <param name="r">The interval.</param>
/// <returns>The maximum.</returns>
/// <exception cref="ArgumentNullException"><paramref name="f"/> is null.</exception>
/// <exception cref="NonconvergenceException">More than the maximum allowed number of function evaluations occured without a minimum being determined.</exception>
public static Extremum FindMaximum(Func<double, double> f, Interval r)
{
return (FindMaximum(f, r, DefaultExtremaSettings));
}
/// <summary>
/// Minimizes a function on the given interval, subject to the given evaluation settings.
/// </summary>
/// <param name="f">The function.</param>
/// <param name="r">The interval.</param>
/// <param name="settings">The settings used when searching for the minimum.</param>
/// <returns>The minimum.</returns>
/// <exception cref="ArgumentNullException"><paramref name="f"/> is null or <paramref name="settings"/> is null.</exception>
/// <exception cref="NonconvergenceException">More than the maximum allowed number of function evaluations occured without a minimum being determined to the prescribed precision.</exception>
/// <remarks>
/// <para>When you supply <paramref name="settings"/>, note that the supplied <see cref="EvaluationSettings.RelativePrecision"/> and <see cref="EvaluationSettings.AbsolutePrecision"/>
/// values refer to argument (i.e. x) values, not function (i.e. f) values. Note also that, for typical functions, the best attainable relative precision is of the order of the
/// square root of machine precision (about 10<sup>-7</sup>), i.e. half the number of digits in a <see cref="Double"/>. This is because to identify an extremum we need to resolve changes
/// in the function value, and near an extremum δf ∼ (δx)<sup>2</sup>, so changes in the function value δf ∼ ε correspond to changes in the
/// argument value δx ∼ √ε. If you supply zero values for both precision settings, the method will adaptively approximate the best attainable precision for
/// the supplied function and locate the extremum to that resolution. This is our suggested practice unless you know that you require a less precise determination.</para>
/// </remarks>
public static Extremum FindMinimum(Func<double, double> f, Interval r, EvaluationSettings settings)
{
if (f == null) throw new ArgumentNullException("f");
if (settings == null) throw new ArgumentNullException("settings");
return (FindMinimum(new Functor(f), r.LeftEndpoint, r.RightEndpoint, settings));
}
public void PackCirclesInSquare()
{
// Put n points into the unit square. Place them so as to maximize the minimum distance between them.
// See http://en.wikipedia.org/wiki/Circle_packing_in_a_square, http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html
// This is a great test problem because it is simple to understand, hard to solve,
// solutions are known/proven for many n, and it covers many dimensions.
double[] solutions = new double[] {
Double.NaN,
Double.NaN,
Math.Sqrt(2.0), /* at opposite corners */
Math.Sqrt(6.0) - Math.Sqrt(2.0),
1.0, /* at each corner */
Math.Sqrt(2.0) / 2.0 /* at each corner and in center */,
Math.Sqrt(13.0) / 6.0,
4.0 - 2.0 * Math.Sqrt(3.0),
(Math.Sqrt(6.0) - Math.Sqrt(2.0)) / 2.0,
1.0 / 2.0, /* 3 X 3 grid */
0.421279543983903432768821760651,
0.398207310236844165221512929748,
Math.Sqrt(34.0) / 15.0,
0.366096007696425085295389370603,
2.0 / (4.0 + Math.Sqrt(3.0)),
2.0 / (Math.Sqrt(6.0) + 2.0 + Math.Sqrt(2.0)),
1.0 / 3.0
};
//int n = 11;
for (int n = 2; n < 8; n++)
{
Console.WriteLine("n={0}", n);
Func<IList<double>, double> function = (IList<double> x) => {
// Intrepret coordinates as (x_1, y_1, x_2, y_2, \cdots, x_n, y_n)
// Iterate over all pairs of points, finding smallest distance betwen any pair of points.
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);
};
IList<Interval> box = new Interval[2 * n];
for (int i = 0; i < box.Count; i++) box[i] = Interval.FromEndpoints(0.0, 1.0);
EvaluationSettings settings = new EvaluationSettings() { RelativePrecision = 1.0E-4, AbsolutePrecision = 1.0E-6, EvaluationBudget = 10000000 };
MultiExtremum maximum = MultiFunctionMath.FindGlobalMaximum(function, box, settings);
Console.WriteLine(maximum.EvaluationCount);
Console.WriteLine("{0} {1}", solutions[n], maximum.Value);
Assert.IsTrue(maximum.Dimension == 2 * n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(maximum.Value, solutions[n], new EvaluationSettings() { AbsolutePrecision = 2.0 * maximum.Precision }));
}
}
public TestIntegral (Func<double, double> f, Interval r, double I) {
this.Integrand = f;
this.Range = r;
this.Result = I;
}
public void PackSpheresInCube()
{
// See http://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r33/pdf
double[] solutions = new double[] {
Double.NaN,
Double.NaN,
Math.Sqrt(3.0), /* opposite corners */
Math.Sqrt(2.0), /* three non-adjacent corners*/
Math.Sqrt(2.0), /* on opposite faces in opposite corners */
Math.Sqrt(5.0) / 2.0,
3.0 * Math.Sqrt(2.0) / 4.0,
Math.Sqrt(2.0 / 3.0 * (8.0 + Math.Sqrt(3.0) - 2.0 * Math.Sqrt(10.0 + 4.0 * Math.Sqrt(3.0)))),
1.0, /* in each corner */
Math.Sqrt(3.0) / 2.0, /* in each corner and at center */
3.0 / 4.0,
0.710116382462,
0.707106806467,
1.0 / Math.Sqrt(2.0),
1.0 / Math.Sqrt(2.0),
5.0 / 8.0
};
//int n = 11;
for (int n = 2; n < 8; n++)
{
Func<IList<double>, double> function = (IList<double> x) => {
// Intrepret coordinates as (x_1, y_1, z_1, x_2, y_2, z_2, \cdots, x_n, y_n, z_n)
// Iterate over all pairs of points, finding smallest distance betwen any pair of points.
double sMin = Double.MaxValue;
for (int i = 0; i < n; i++) {
for (int j = 0; j < i; j++) {
double s = Math.Sqrt(
MoreMath.Sqr(x[3 * i] - x[3 * j]) +
MoreMath.Sqr(x[3 * i + 1] - x[3 * j + 1]) +
MoreMath.Sqr(x[3 * i + 2] - x[3 * j + 2])
);
if (s < sMin) sMin = s;
}
}
return (sMin);
};
IList<Interval> box = new Interval[3 * n];
for (int i = 0; i < box.Count; i++) box[i] = Interval.FromEndpoints(0.0, 1.0);
MultiExtremum maximum = MultiFunctionMath.FindGlobalMaximum(function, box);
Console.WriteLine(maximum.EvaluationCount);
Console.WriteLine("{0} {1} ({2})", solutions[n], maximum.Value, maximum.Precision);
Assert.IsTrue(maximum.Dimension == 3 * n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(maximum.Value, solutions[n], new EvaluationSettings() { AbsolutePrecision = 2.0 * maximum.Precision }));
}
}
public TestExtremum(Func<double, double> function, Interval interval, double location, double value, double curvature)
{
this.Function = function;
this.Interval = interval;
this.Location = location;
this.Value = value;
this.Curvature = curvature;
}
public void Schwefel()
{
// Test over [-500,500], minimum at (420.969,...) -> -418.983*d, many local minima
Func<IList<double>, double> function = (IList<double> x) => {
double s = 0.0;
for (int i = 0; i < x.Count; i++) {
s += x[i] * Math.Sin(Math.Sqrt(Math.Abs(x[i])));
}
return (-s);
};
for (int n = 2; n < 32; n = (int) Math.Round(AdvancedMath.GoldenRatio * n)) {
Console.WriteLine("n={0}", n);
Interval[] box = new Interval[n];
for (int i = 0; i < box.Length; i++) box[i] = Interval.FromEndpoints(-500.0, 500.0);
MultiExtremum minimum = MultiFunctionMath.FindGlobalMinimum(function, box);
Assert.IsTrue(minimum.Dimension == n);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} ({1}) ?= {2}", minimum.Value, minimum.Precision, -418.983 * n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, -418.983 * n, new EvaluationSettings() { AbsolutePrecision = minimum.Precision }));
foreach (double coordinate in minimum.Location) Console.WriteLine(coordinate);
ColumnVector solution = new ColumnVector(n);
solution.Fill((int i, int j) => 420.969);
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Location, solution, new EvaluationSettings() { RelativePrecision = 2.0 * Math.Sqrt(Math.Abs(minimum.Precision / minimum.Value)) }));
}
}
public void STA()
{
// Has local minimum when any coordinate is at one of two values, global when all coordinates at one of them.
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);
};
// Asymptotic "minima" at infinity, and (3,1/2)->0
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]);
Func<IList<double>, double> fEggholder = (IList<double> x) => {
double y47 = x[1] + 47.0;
return(-y47 * Math.Sin(Math.Sqrt(Math.Abs(y47 + x[0] / 2.0))) - x[0] * Math.Sin(Math.Sqrt(Math.Abs(x[0] - y47))));
};
// Many local minima, global minimum at (0,0)->0
Func<IList<double>, double> fAckley = (IList<double> x) => {
double s = 0.0;
double c = 0.0;
for (int i = 0; i < x.Count; i++) {
s += x[i] * x[i];
c += Math.Cos(2.0 * Math.PI * x[i]);
}
return (-20.0 * Math.Exp(-0.2 * Math.Sqrt(s / x.Count)) - Math.Exp(c / x.Count) + 20.0 + Math.E);
};
// Burkin has a narrow valley, not aligned with any axis, punctuated with many tiny "wells" along its bottom.
// The deepest well is at (-10,1)-> 0.
Func<IList<double>, double> fBurkin = (IList<double> x) => 100.0 * Math.Sqrt(Math.Abs(x[1] - 0.01 * x[0] * x[0])) + 0.01 * Math.Abs(x[0] + 10.0);
Func<IList<double>, double> threeHumpCamel = (IList<double> x) => {
double x2 = x[0] * x[0];
return (2.0 * x2 - 1.05 * x2 * x2 + x2 * x2 * x2 / 6.0 + (x[0] + x[1]) * x[1]);
};
// Easom has many lobal extra ripples and then a deep but narrow global minimum at (\pi, \pi) -> -1
Func<IList<double>, double> fEasom = (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)));
// Test over [-500,500], minimum at (420.969,...) -> -418.983*d, many local minima
Func<IList<double>, double> fSchwefel = (IList<double> x) => {
double s = 0.0;
for (int i = 0; i < x.Count; i++) {
s += x[i] * Math.Sin(Math.Sqrt(Math.Abs(x[i])));
}
return (-s);
};
Func<IList<double>, double> function = fSchwefel;
int n = 4;
ColumnVector start = new ColumnVector(n);
//for (int i = 0; i < start.Dimension; i++) start[i] = 1.0;
Interval[] box = new Interval[n];
for (int i = 0; i < box.Length; i++) {
box[i] = Interval.FromEndpoints(-500.0, 500.0);
}
//box[0] = Interval.FromEndpoints(-15.0, -5.0);
//box[1] = Interval.FromEndpoints(-3.0, 3.0);
MultiFunctionMath.FindGlobalMinimum(function, box);
//MultiFunctionMath.FindExtremum_Amobea(fStyblinskiTang, start, new EvaluationSettings() { RelativePrecision = 1.0E-10, AbsolutePrecision = 1.0E-12, EvaluationBudget = 1000 });
}
public void ThompsonGlobal()
{
for (int n = 2; n < 8; n++)
{
Console.WriteLine("n={0}", n);
Func<IList<double>, double> f = GetThompsonFunction(n);
Interval[] box = new Interval[2 * (n - 1)];
for (int i = 0; i < (n - 1); i++) {
box[2 * i] = Interval.FromEndpoints(-Math.PI, Math.PI);
box[2 * i + 1] = Interval.FromEndpoints(-Math.PI / 2.0, Math.PI / 2.0);
}
MultiExtremum minimum = MultiFunctionMath.FindGlobalMinimum(f, box);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} ({1})", minimum.Value, minimum.Precision);
for (int i = 0; i < (n - 1); i++) {
Console.WriteLine("{0} {1}", minimum.Location[2 * i], minimum.Location[2 * i + 1]);
}
Console.WriteLine("0.0 0.0");
Assert.IsTrue(minimum.Dimension == 2 * (n - 1));
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, thompsonSolutions[n], new EvaluationSettings() { AbsolutePrecision = 2.0 * minimum.Precision }));
}
}
private void button1_Click(object sender, EventArgs e)
{
var watch = System.Diagnostics.Stopwatch.StartNew();
ConsolaTB.Text = "\nIniciando cálculos para el modelo de Mandl & Volek\n";
progressBar1.Value = 0;
double Qiny, hw, hr, CalVapor, Lv, por, densRoca, SGOil, calEspRoca, SatOil, Sw, densAgua, calEspAgua, espesor, difTermForm, condTermForm, tempVapor, tempYac, sor, Tiny;
Qiny = propiedades.Qiny;
hw = propiedades.hw;
hr = propiedades.hr;
CalVapor = propiedades.CalVapor;
Lv = propiedades.Lv;
por = propiedades.por;
densRoca = propiedades.densRoca;
SGOil = propiedades.SGOil;
calEspRoca = propiedades.calEspRoca;
SatOil = propiedades.SatOil;
densAgua = propiedades.densAgua;
calEspAgua = propiedades.calEspAgua;
espesor = propiedades.espesor;
difTermForm = propiedades.difTermForm;
condTermForm = propiedades.condTermForm;
tempVapor = propiedades.tempVapor;
tempYac = propiedades.tempYac;
sor = propiedades.sor;
Tiny = propiedades.Tiny;
Sw = 1 - SatOil;
bool Calculo = propiedades.Calculo;
double capCalForm;
if (!Calculo)
{
capCalForm = propiedades.capCalRoca;
ConsolaTB.AppendText("\nCapacidad Calorífica de la Formación = " + capCalForm.ToString("F5") + " BTU/ft3-°F");
}
else
{
double densidadCrudo = SGOil * 62.4;
double calEspCrudo = 0.388 + (0.00045 * tempYac) / Math.Pow(propiedades.SGOil, 0.5);
ConsolaTB.AppendText("\nLa capacidad calorífica será calculada\n");
ConsolaTB.AppendText("\nCalor específico del crudo = " + calEspCrudo.ToString("F5"));
capCalForm = (1 - por) * densRoca * calEspRoca + por * (densidadCrudo * calEspCrudo * SatOil + densAgua * calEspAgua * (Sw));
ConsolaTB.AppendText("\nCap Calorífica Formación = " + capCalForm.ToString("F5") + " BTU/ft3-°F");
}
beta = Math.Pow(1 + (Lv * CalVapor / (calEspAgua * (tempVapor - tempYac))), -1);
ConsolaTB.AppendText("\nParámetro Mandi & Volek= " + beta.ToString("F3"));
MN.Interval inter = MN.Interval.FromEndpoints(0, 1);
double xi = MN.Analysis.FunctionMath.FindZero(errorComp, inter);
string erp = (100 * (MN.Functions.AdvancedMath.Erfc(xi) - beta) / MN.Functions.AdvancedMath.Erfc(xi)).ToString("F5");
ConsolaTB.AppendText("\n% Error Xiterado vs Parámetro Mandi & Volek = " + erp);
double tc = Math.Pow((xi * capCalForm * espesor * Math.Pow(difTermForm, 0.5)) / (2 * condTermForm), 2);
ConsolaTB.AppendText("\nTiempo critico = " + tc.ToString("F3") + " horas");
double Qi = (350.0 / 24.0) * Qiny * ((hw - hr) + CalVapor * Lv); //Tasa de inyección de calor
ConsolaTB.AppendText("\nTasa de Inyección de Calor = " + Qi.ToString("F3") + " BTU/día\n");
progressBar1.Value = 20;
/* Comienzo de la corrida de calculos */
DataTable caudales = OfficeHandler.createTable("Tiempo", "Caudal");
double tiempo = 0;
/*Calculo con Marx y Langhenheim */
ConsolaTB.AppendText("\nCalculando tasas de flujo en región menor a Tc con Marx & Langhenheim:\n\n");
while (tiempo < tc && tiempo < (Tiny * 24))
{
tiempo = tiempo + 24;
double X = ((2 * propiedades.condTermForm) / (capCalForm * propiedades.espesor * Math.Pow(propiedades.difTermForm, 0.5))) * Math.Pow(tiempo, 0.5);
double error = MN.Functions.AdvancedMath.Erfc(X);
double q0 = 4.274 * (Qi * propiedades.por * (propiedades.SatOil - propiedades.sor) * error) / (capCalForm * (propiedades.tempVapor - propiedades.tempYac));
ConsolaTB.AppendText("\nT=" + tiempo + " horas \t Q=" + q0.ToString("F3") + " BBL/dia");
caudales.Rows.Add(tiempo, q0);
}
progressBar1.Value = 70;
if (tiempo <= Tiny * 24)
{
ConsolaTB.AppendText("\n\nCalculando tasas de flujo en región mayor a Tc con Mandl & Volek: ");
}
while (tiempo <= Tiny * 24)
{
tiempo = tiempo + 24;
double X = ((2 * propiedades.condTermForm) / (capCalForm * propiedades.espesor * Math.Pow(propiedades.difTermForm, 0.5))) * Math.Pow(tiempo, 0.5);
double error = MN.Functions.AdvancedMath.Erfc(X);
double t1 = 4.274 * (Qi * propiedades.por * (propiedades.SatOil - propiedades.sor)) / (capCalForm * (propiedades.tempVapor - propiedades.tempYac));
double xdiff = Math.Pow(X, 2) - Math.Pow(MN.Functions.AdvancedMath.Erfc(xi), 2);
double t2 = (1 - ((xdiff - 2) * Math.Pow(xdiff, 0.5) / (3 * Math.Pow(Math.PI, 0.5))) - ((xdiff - 3) / (6 * Math.Pow(Math.PI * xdiff, 0.5)))) * error;
double t3 = ((1 / (3 * Math.PI)) * (xdiff - 2 + (Math.Pow(MN.Functions.AdvancedMath.Erfc(xi), 2) / 2 * Math.Pow(X, 2))) * Math.Pow(xdiff / X, 0.5)) - (beta / (2 * Math.Pow(Math.PI * xdiff, 0.5)));
double q0 = t1 * (t2 + t3);
ConsolaTB.AppendText("\nT=" + tiempo + " horas \t Q=" + q0.ToString("F3") + " BBL/dia");
caudales.Rows.Add(tiempo, q0);
}
progressBar1.Value = 95;
double Acumulado = 0;
foreach (DataRow row in caudales.Rows)
{
Acumulado = Acumulado + Double.Parse(row["Caudal"].ToString());
}
watch.Stop();
double elapsedMs = watch.ElapsedMilliseconds;
ConsolaTB.AppendText("\n\nCálculos finalizados. \nTiempo total = " + propiedades.Tiny.ToString("F2") + " días\nAcumulado= " + Acumulado.ToString("F3") + " Barriles \nTiempo de calculo=" + elapsedMs / 1000 + " segundos");
progressBar1.Value = 100;
if (MessageBox.Show("Cálculos finalizados, ¿ Desea exportar a excel los resultados ?", "Cálculos finalizados", MessageBoxButtons.YesNo, MessageBoxIcon.Information) == DialogResult.Yes)
{
OfficeHandler.exportExcel(caudales);
}
}