private static UncertainValue Integrate_Adaptive(MultiFunctor f, CoordinateTransform[] map, IntegrationRegion r, EvaluationSettings settings)
{
// Create an evaluation rule
GenzMalik75Rule rule = new GenzMalik75Rule(r.Dimension);
// Use it on the whole region and put the answer in a linked list
rule.Evaluate(f, map, r);
LinkedList<IntegrationRegion> regionList = new LinkedList<IntegrationRegion>();
regionList.AddFirst(r);
// Iterate until convergence
while (f.EvaluationCount < settings.EvaluationBudget) {
// Add up value and errors in all regions.
// While we are at it, take note of the region with the largest error.
double value = 0.0;
double error = 0.0;
LinkedListNode<IntegrationRegion> regionNode = regionList.First;
double maxError = 0.0;
LinkedListNode<IntegrationRegion> maxErrorNode = null;
while (regionNode != null) {
IntegrationRegion region = regionNode.Value;
value += region.Value;
error += region.Error;
//error += MoreMath.Sqr(region.Error);
if (region.Error > maxError) {
maxError = region.Error;
maxErrorNode = regionNode;
}
regionNode = regionNode.Next;
}
// Check for convergence.
if ((error <= settings.AbsolutePrecision) || (error <= settings.RelativePrecision * Math.Abs(value))) {
return (new UncertainValue(value, error));
}
// Split the region with the largest error, and evaluate each subregion.
IntegrationRegion maxErrorRegion = maxErrorNode.Value;
regionList.Remove(maxErrorNode);
IList<IntegrationRegion> subRegions = maxErrorRegion.Split(maxErrorRegion.SplitIndex);
/*
Countdown cnt = new Countdown(2);
ThreadPool.QueueUserWorkItem((object state) => { rule.Evaluate(f, subRegions[0]); cnt.Signal(); });
ThreadPool.QueueUserWorkItem((object state) => { rule.Evaluate(f, subRegions[1]); cnt.Signal(); });
cnt.Wait();
foreach (IntegrationRegion subRegion in subRegions) {
regionList.AddLast(subRegion);
}
*/
foreach (IntegrationRegion subRegion in subRegions) {
rule.Evaluate(f, map, subRegion);
regionList.AddLast(subRegion);
}
}
throw new NonconvergenceException();
}
internal EvaluationResult(int count, EvaluationSettings settings)
{
Debug.Assert(count > 0);
Debug.Assert(settings != null);
this.count = count;
this.settings = settings;
}
/// <summary>
/// Finds a local maximum of a multi-dimensional function in the vincinity of the given starting location, subject to the given evaluation constraints.
/// </summary>
/// <param name="function">The multi-dimensional function to maximize.</param>
/// <param name="start">The starting location for the search.</param>
/// <param name="settings">The evaluation settings that govern the search for the maximum.</param>
/// <returns>The local maximum.</returns>
public static MultiExtremum FindLocalMaximum(Func<IList<double>, double> function, IList<double> start, EvaluationSettings settings)
{
if (function == null) throw new ArgumentNullException("function");
if (start == null) throw new ArgumentNullException("start");
if (settings == null) throw new ArgumentNullException("settings");
return(FindLocalExtremum(function, start, settings, true));
}
internal Extremum(double x, double f, double f2, int count, EvaluationSettings settings)
: base(count, settings)
{
this.x = x;
this.f = f;
this.f2 = f2;
}
internal MultiExtremum(int count, EvaluationSettings settings, double[] point, double value, double precision, double[][] hessian)
: base(count, settings)
{
Debug.Assert(point != null);
this.point = point;
this.value = value;
this.precision = precision;
this.hessian = hessian;
}
private static MultiExtremum FindGlobalExtremum(Func<IList<double>, double> function, IList<Interval> volume, EvaluationSettings settings, bool negate)
{
if (function == null) throw new ArgumentNullException("function");
if (volume == null) throw new ArgumentNullException("volume");
MultiFunctor f = new MultiFunctor(function, negate);
DifferentialEvolutionSettings deSettings = GetDefaultSettings(settings, volume.Count);
MultiExtremum extremum = FindGlobalExtremum(f, volume, deSettings);
return (extremum);
}
public void BoxIntegrals()
{
// The box integrals
// B_n(r) = \int{0}^{1} dx_1 \cdots dx_n ( x_1^2 + \cdots x_n^2 )^{r/2}
// D_n(r) = \int{0}^{1} dx_1 \cdots dx_n dy_1 \cdots dy_n \left[ (x_1 - y_1)^2 + \cdots (x_n - y_n)^2 \right]^{r/2}
// Give the mean distance of a point in a box from its center or from other points. Various of these are known analytically.
// see Bailey, Borwein, Crandall, "Box Integrals", Journal of Computational and Applied Mathematics 206 (2007) 196
// http://www.davidhbailey.com/dhbpapers/boxintegrals.pdf and http://www.davidhbailey.com/dhbpapers/bbbz-conmath.pdf
EvaluationSettings settings = new EvaluationSettings() { EvaluationBudget = 1000000, RelativePrecision = 1.0E-3 };
// 2D integrals
Assert.IsTrue(TestUtilities.IsNearlyEqual(
BoxIntegralB(2, -1, settings), 2.0 * Math.Log(1.0 + Math.Sqrt(2.0)), settings.RelativePrecision * 2
));
// Note 2 \ln(1 + \sqrt{2}) = \ln(3 + 2 \sqrt{2}) because (1 + \sqrt{2})^2 = 3 + 2 \sqrt{2}
Assert.IsTrue(TestUtilities.IsNearlyEqual(
BoxIntegralB(2, 1, settings), (Math.Sqrt(2.0) + Math.Log(Math.Sqrt(2.0) + 1.0)) / 3.0, settings.RelativePrecision * 2
));
/*
Assert.IsTrue(TestUtilities.IsNearlyEqual(
BoxIntegralD(1, -1, new EvaluationSettings() { EvaluationBudget = 100000, RelativePrecision = 1.0E-2 }),
(2.0 - 4.0 * Math.Sqrt(2.0)) / 3.0 + 4.0 * Math.Log(1.0 + Math.Sqrt(2.0)), settings.RelativePrecision * 2
));
*/
Assert.IsTrue(TestUtilities.IsNearlyEqual(
BoxIntegralD(1, 1, settings), 1.0 / 3.0, settings.RelativePrecision * 2
));
// 3D integrals
Assert.IsTrue(TestUtilities.IsNearlyEqual(
BoxIntegralB(3, -1, settings), Math.Log(5.0 + 3.0 * Math.Sqrt(3.0)) - Math.Log(2.0) / 2.0 - Math.PI / 4.0, settings.RelativePrecision * 4
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
BoxIntegralB(3, 1, settings), Math.Log(2.0 + Math.Sqrt(3.0)) / 2.0 + Math.Sqrt(3.0) / 4.0 - Math.PI / 24.0, settings.RelativePrecision * 2
));
// 4D integrals
Assert.IsTrue(TestUtilities.IsNearlyEqual(
BoxIntegralD(2, 1, settings), (2.0 + Math.Sqrt(2.0) + 5.0 * Math.Log(1.0 + Math.Sqrt(2.0))) / 15.0, settings.RelativePrecision * 4
));
// 6D integrals
Assert.IsTrue(TestUtilities.IsNearlyEqual(
BoxIntegralD(3, 1, settings),
4.0 / 105.0 + 17.0 * Math.Sqrt(2.0) / 105.0 - 2.0 * Math.Sqrt(3.0) / 35.0 + Math.Log(1.0 + Math.Sqrt(2.0)) / 5.0 + 2.0 * Math.Log(2.0 + Math.Sqrt(3.0)) / 5.0 - Math.PI / 15.0,
settings.RelativePrecision * 2
));
}
public double BoxIntegralB(int d, int r, EvaluationSettings settings)
{
return (MultiFunctionMath.Integrate((IList<double> x) => {
double s = 0.0;
for (int k = 0; k < d; k++) {
s += x[k] * x[k];
}
return (Math.Pow(s, r / 2.0));
}, UnitCube(d), settings).Value);
}
public BaseOdeStepper(Func <double, T, T> rhs, double x0, T y0, EvaluationSettings settings)
{
if (rhs == null)
{
throw new ArgumentNullException("rhs");
}
if (settings == null)
{
throw new ArgumentNullException("settings");
}
this.rhs = rhs;
this.X = x0;
this.Y = y0;
this.Settings = settings;
}
private static void SetDefaultOptimizationSettings(EvaluationSettings settings, int d)
{
if (settings.RelativePrecision < 0.0)
{
settings.RelativePrecision = Math.Pow(10.0, -(10.0 + 4.0 / d));
}
if (settings.AbsolutePrecision < 0.0)
{
settings.AbsolutePrecision = Math.Pow(10.0, -(10.0 + 4.0 / d));
}
if (settings.EvaluationBudget < 0)
{
settings.EvaluationBudget = 16 * (d + 1) * (d + 2) * (d + 3);
}
}
internal static void SetOdeDefaults(EvaluationSettings settings)
{
if (settings.RelativePrecision < 0)
{
settings.RelativePrecision = 1.0E-12;
}
if (settings.AbsolutePrecision < 0)
{
settings.AbsolutePrecision = 1.0E-24;
}
if (settings.EvaluationBudget < 0)
{
settings.EvaluationBudget = 8192;
}
}
private static MultiExtremum FindLocalExtremum(Func<IList<double>, double> function, IList<double> start, EvaluationSettings settings, bool negate)
{
MultiFunctor f = new MultiFunctor(function, negate);
// Pick an initial radius; we need to do this better.
/*
double s = Double.MaxValue;
foreach (double x in start) s = Math.Min((Math.Abs(x) + 1.0 / 8.0) / 8.0, s);
*/
double s = 0.0;
foreach (double x in start) s += (Math.Abs(x) + 1.0 / 4.0) / 4.0;
s = s / start.Count;
//double s = 0.2;
Debug.WriteLine("s={0}", s);
return (FindMinimum_ModelTrust(f, start, s, settings));
}
private static DifferentialEvolutionSettings GetDefaultSettings(EvaluationSettings settings, int d)
{
DifferentialEvolutionSettings deSettings = new DifferentialEvolutionSettings();
deSettings.Population = 8 * d + 4;
deSettings.CrossoverProbability = 1.0 - 1.0 / 8.0 - 1.0 / d;
if (settings == null)
{
deSettings.RelativePrecision = Math.Pow(10.0, -(2.0 + 4.0 / d));
deSettings.AbsolutePrecision = MoreMath.Sqr(deSettings.RelativePrecision);
deSettings.EvaluationBudget = 128 * d * d * d * d;
}
else
{
deSettings.RelativePrecision = settings.RelativePrecision;
deSettings.AbsolutePrecision = settings.AbsolutePrecision;
deSettings.EvaluationBudget = settings.EvaluationBudget;
}
return(deSettings);
}
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 static double SolveConservativeOde(Func <double, double, double> rhs, double x0, double y0, double yp0, double x1, EvaluationSettings settings)
{
if (rhs == null)
{
throw new ArgumentNullException("rhs");
}
BaseOdeStepper <double> stepper = new StoermerStepper(rhs, x0, y0, yp0, settings);
stepper.Integrate(x1);
return(stepper.Y);
}
/// <summary>
/// Maximizes a function on a multi-dimensional space in the vicinity of a given point, subject to the given settings.
/// </summary>
/// <param name="f">The function.</param>
/// <param name="x">The starting point for the search.</param>
/// <param name="settings">The evaluation settings.</param>
/// <returns>The maximum.</returns>
/// <exception cref="ArgumentNullException"><paramref name="f"/>, <paramref name="x"/>, or <paramref name="settings"/> is null.</exception>
/// <exception cref="NonconvergenceException">The maximum was not found to the required precision within the budgeted number of function evaluations.</exception>
internal static SpaceExtremum FindMaximum(Func <double[], double> f, double[] x, EvaluationSettings settings)
{
return(FindMinimum((double[] p) => - f(p), x, settings));
}
public void DistributionCentralMomentIntegral()
{
foreach (Distribution distribution in distributions) {
foreach (int n in TestUtilities.GenerateIntegerValues(2, 24, 8)) {
// get the predicted central moment
double C = distribution.MomentAboutMean(n);
// don't try to integrate infinite moments
if (Double.IsInfinity(C) || Double.IsNaN(C)) continue;
if (C == 0.0) continue;
EvaluationSettings settings = new EvaluationSettings();
if (C == 0.0) {
// if moment is zero, use absolute precision
settings.AbsolutePrecision = TestUtilities.TargetPrecision;
settings.RelativePrecision = 0.0;
} else {
// if moment in non-zero, use relative precision
settings.AbsolutePrecision = 0.0;
settings.RelativePrecision = TestUtilities.TargetPrecision;
}
// do the integral
double m = distribution.Mean;
Func<double, double> f = delegate(double x) {
return (distribution.ProbabilityDensity(x) * MoreMath.Pow(x - m, n));
};
try {
double CI = FunctionMath.Integrate(f, distribution.Support, settings).Value;
Console.WriteLine("{0} {1} {2} {3}", distribution.GetType().Name, n, C, CI);
if (C == 0.0) {
Assert.IsTrue(Math.Abs(CI) < TestUtilities.TargetPrecision);
} else {
double e = TestUtilities.TargetPrecision;
// reduce required precision, because some distributions (e.g. Kolmogorov, Weibull)
// have no analytic expressions for central moments, which must therefore be
// determined via raw moments and are thus subject to cancelation error
// can we revisit this later?
if (distribution is WeibullDistribution) e = Math.Sqrt(Math.Sqrt(e));
if (distribution is KolmogorovDistribution) e = Math.Sqrt(e);
if (distribution is KuiperDistribution) e = Math.Sqrt(Math.Sqrt(e));
if (distribution is TriangularDistribution) e = Math.Sqrt(e);
Assert.IsTrue(TestUtilities.IsNearlyEqual(C, CI, e));
}
} catch (NonconvergenceException) {
Console.WriteLine("{0} {1} {2} {3}", distribution.GetType().Name, n, C, "NC");
// deal with these later; they are integration problems, not distribution problems
}
}
}
}
private static double FindMinimum(
Func<double, double> f,
double a, double b
)
{
// evaluate three points within the bracket
double u = (3.0 * a + b) / 4.0;
double v = (a + b) / 2.0;
double w = (a + 3.0 * b) / 4.0;
double fu = f(u); double fv = f(v); double fw = f(w);
Console.WriteLine("f({0})={1} f({2})={3} f({4})={5}", u, fu, v, fv, w, fw);
// move in the bracket boundaries, if possible
if (fv < fu) { a = u; if (fw < fv) a = v; }
if (fv < fw) { b = w; if (fu < fv) b = v; }
Console.WriteLine("a={0} b={1}", a, b);
// sort u, v, w by fu, fv, fw values
// these three comparisons are the most efficient three-item sort
if (fv < fu) { double t = v; v = u; u = t; t = fv; fv = fu; fu = t; }
if (fw < fu) { double t = w; w = u; u = t; t = fw; fw = fu; fu = t; }
if (fw < fv) { double t = w; w = v; v = t; t = fw; fw = fv; fv = t; }
// An evaluation budget of 32 is sufficient for all our test cases except for |x|, which requires 82 (!) evaluations to converge. Parabolic fitting just does a very poor job
// for this function (at all scales, since it is scale invariant). We should look into cubic fitting.
EvaluationSettings settings = new EvaluationSettings() { EvaluationBudget = 128, AbsolutePrecision = 0.0, RelativePrecision = 0.0 };
return (FindMinimum(f, a, b, u, fu, v, fv, w, fw, settings, 3));
}
private double FindMinimumWithDerivative(
FuncWithDerivative f,
double a, double b,
double u, double fu, double fpu,
double v, double fv, double fpv,
EvaluationSettings settings
)
{
double tol = 0.0;
int count = 0;
while (count < settings.EvaluationBudget) {
Console.WriteLine("n = {0}, tol = {1}", count, tol);
Console.WriteLine("[{0} f({1})={2}({3}) f({4})={5}({6}) {7}]", a, u, fu, fpu, v, fv, fpv, b);
// a, b bracket minimum a < u, v < b and f(u), f(v) <= f(a), f(b)
Debug.Assert(a < b);
Debug.Assert((a <= u) && (u <= b));
//Debug.Assert((a <= v) && (v <= b));
Debug.Assert(fu <= fv);
if ((b - a) <= 4.0 * tol) return (u);
// compute the minimum of the interpolating Hermite cubic
double x, fpp; CubicHermiteMinimum(u, fu, fpu, v, fv, fpv, out x, out fpp);
Console.WriteLine("cubic x = {0}, fpp = {1}", x, fpp);
// if the cubic had no minimum, or the minimum lies outside our bounds, fall back to bisection
if (Double.IsNaN(x) || (x <= a) || (x >= b)) {
// the derivative tells us which side to choose
if (fpu > 0.0) {
x = (a + u) / 2.0;
} else {
x = (u + b) / 2.0;
}
Console.WriteLine("bisection x = {0}", x);
}
// ensure we don't evaluate within tolerance of an existing point
if (Math.Abs(x - u) < tol) { Console.WriteLine("shift from u (x={0})", x); x = (x > u) ? u + tol : u - tol; }
if ((x - a) < tol) { Console.WriteLine("shift from a (x={0})", x); x = a + tol; }
if ((b - x) < tol) { Console.WriteLine("shift from b (x={0})", x); x = b - tol; }
// evaluate the function plus derivative at the predicted minimum
double fx, fpx;
f(x, out fx, out fpx);
count++;
Console.WriteLine("f({0}) = {1}({2})", x, fx, fpx);
// check if we have converged
double df = fu - fx;
Console.WriteLine("df={0}", df);
if ((Math.Abs(df) < settings.AbsolutePrecision) || (2.0 * Math.Abs(df) < settings.RelativePrecision * (Math.Abs(fu) + Math.Abs(fx)))) {
Console.WriteLine("count = {0}", count);
return (x);
}
if (fx < fu) {
// x is the new lowest point: f(x) < f(u) < f(v)
// this is the expected outcome
// move the bracket
if (x < u) {
b = u;
} else {
a = u;
}
// x -> u -> v
v = u; fv = fu; fpv = fpu;
u = x; fu = fx; fpu = fpx;
} else {
// move the bracket
if (x < u) {
a = x;
} else {
b = x;
}
if (fx < fv) {
// x lies between other two known points: f(u) < f(x) < f(v)
// x -> v
v = x; fv = fx; fpv = fpx;
} else {
// x is higher than both other points: f(u) < f(v) < f(x)
// this is a really poor outcome; we expected to get a point lower than our other two and we got a point higher than both
// next time we should bisect
Console.WriteLine("bad point");
//throw new NotImplementedException();
//v = x; fv = fx; fpv = fpx;
}
}
// if the user has specified a tollerance, use it
if ((settings.RelativePrecision > 0.0 || settings.AbsolutePrecision > 0.0)) {
tol = Math.Max(Math.Abs(u) * settings.RelativePrecision, settings.AbsolutePrecision);
} else {
// otherwise, try to get the tollerance from the curvature
if (fpp > 0.0) {
tol = Math.Sqrt(2.0 * 1.0E-14 * (Math.Abs(fu) + 1.0E-14) / fpp);
} else {
// but if we don't have a useable curvature either, wing it
if (tol == 0.0) tol = 1.0E-7;
}
}
}
throw new NonconvergenceException();
}
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 }));
}
}
private static UncertainValue Integrate_MonteCarlo(MultiFunctor f, CoordinateTransform[] map, IList <Interval> box, EvaluationSettings settings)
{
int d = box.Count;
// Use a Sobol quasi-random sequence. This give us 1/N accuracy instead of 1/\sqrt{N} accuracy.
//VectorGenerator g = new RandomVectorGenerator(d, new Random(314159265));
VectorGenerator g = new SobolVectorGenerator(d);
// Start with a trivial Lepage grid.
// We will increase the grid size every few cycles.
// My tests indicate that trying to increase every cycle or even every other cycle is too often.
// This makes sense, because we have no reason to believe our new grid will be better until we
// have substantially more evaluations per grid cell than we did for the previous grid.
LePageGrid grid = new LePageGrid(box, 1);
int refineCount = 0;
// Start with a reasonable number of evaluations per cycle that increases with the dimension.
int cycleCount = 8 * d;
//double lastValue = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
// Each cycle consists of three sets of evaluations.
// At first I did this with just two set and used the difference between the two sets as an error estimate.
// I found that it was pretty common for that difference to be low just by chance, causing error underestimatation.
double value1 = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
double value2 = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
double value3 = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
while (f.EvaluationCount < settings.EvaluationBudget)
{
// Take the largest deviation as the error.
double value = (value1 + value2 + value3) / 3.0;
double error = Math.Max(Math.Abs(value1 - value3), Math.Max(Math.Abs(value1 - value2), Math.Abs(value2 - value3)));
Debug.WriteLine("{0} {1} {2}", f.EvaluationCount, value, error);
// Check for convergence.
if ((error <= settings.AbsolutePrecision) || (error <= Math.Abs(value) * settings.RelativePrecision))
{
return(new UncertainValue(value, error));
}
// Do more cycles. In order for new sets to be equal-sized, one of those must be at the current count and the next at twice that.
double smallValue = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
cycleCount *= 2;
double bigValue = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
// Combine all the cycles into new ones with twice the number of evaluations each.
value1 = (value1 + value2) / 2.0;
value2 = (value3 + smallValue) / 2.0;
value3 = bigValue;
//double currentValue = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
//double error = Math.Abs(currentValue - lastValue);
//double value = (currentValue + lastValue) / 2.0;
//lastValue = value;
// Increase the number of evaluations for the next cycle.
//cycleCount *= 2;
// Refine the grid for the next cycle.
refineCount++;
if (refineCount == 2)
{
Debug.WriteLine("Replacing grid with {0} bins after {1} evaluations", grid.BinCount, grid.EvaluationCount);
grid = grid.ComputeNewGrid(grid.BinCount * 2);
refineCount = 0;
}
}
throw new NonconvergenceException();
}
private double RambleIntegral(int d, int s, EvaluationSettings settings)
{
return (MultiFunctionMath.Integrate((IList<double> x) => {
Complex z = 0.0;
for (int k = 0; k < d; k++) {
z += ComplexMath.Exp(2.0 * Math.PI * Complex.I * x[k]);
}
return (MoreMath.Pow(ComplexMath.Abs(z), s));
}, UnitCube(d), settings).Value);
}
public static ColumnVector SolveConservativeOde(Func <double, IList <double>, IList <double> > rhs, double x0, IList <double> y0, IList <double> yp0, double x1, EvaluationSettings settings)
{
if (rhs == null)
{
throw new ArgumentNullException("rhs");
}
if (y0 == null)
{
throw new ArgumentNullException("y0");
}
MultiOdeStepper stepper = new MultiStoermerStepper(rhs, x0, y0, yp0, settings);
stepper.Integrate(x1);
return(new ColumnVector(stepper.Y));
}
public void SteinmetzVolume()
{
// Steinmetz solid is intersection of unit cylinders along all axes. This is another hard-edged integral. Analytic values are known for d=2-5.
// http://www.math.illinois.edu/~hildebr/ugresearch/cylinder-spring2013report.pdf
// http://www.math.uiuc.edu/~hildebr/igl/nvolumes-fall2012report.pdf
EvaluationSettings settings = new EvaluationSettings() { EvaluationBudget = 1000000, RelativePrecision = 1.0E-2 };
for (int d = 2; d <= 5; d++) {
IntegrationResult v1 = MultiFunctionMath.Integrate((IList<double> x) => {
for (int i = 0; i < d; i++) {
double s = 0.0;
for (int j = 0; j < d; j++) {
if (j != i) s += x[j] * x[j];
}
if (s > 1.0) return (0.0);
}
return (1.0);
}, SymmetricUnitCube(d), settings);
double v2 = 0.0;
switch (d) {
case 2:
// trivial square
v2 = 4.0;
break;
case 3:
v2 = 16.0 - 8.0 * Math.Sqrt(2.0);
break;
case 4:
v2 = 48.0 * (Math.PI / 4.0 - Math.Atan(Math.Sqrt(2.0)) / Math.Sqrt(2.0));
break;
case 5:
v2 = 256.0 * (Math.PI / 12.0 - Math.Atan(1.0 / (2.0 * Math.Sqrt(2.0))) / Math.Sqrt(2.0));
break;
}
Console.WriteLine("{0} {1} {2}", d, v1.Value, v2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(v1.Value, v2, new EvaluationSettings() { AbsolutePrecision = 4.0 * v1.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)
)
);
}
private static void FindExtremum_Amobea(MultiFunctor f, Vertex[] vertexes, EvaluationSettings settings)
{
int d = vertexes.Length - 1;
while (f.EvaluationCount < settings.EvaluationBudget)
{
// Identify the best and worst vertexes.
int minVertex = 0; double minY = vertexes[0].Y;
int maxVertex = 0; double maxY = vertexes[0].Y;
int nextMaxVertex = 0; double nextMaxY = vertexes[0].Y;
for (int i = 1; i < vertexes.Length; i++)
{
double y = vertexes[i].Y;
if (y < minY)
{
minVertex = i; minY = y;
}
if (y > nextMaxY)
{
if (y > maxY)
{
nextMaxVertex = maxVertex; nextMaxY = maxY;
maxVertex = i; maxY = y;
}
else
{
nextMaxVertex = i; nextMaxY = y;
}
}
}
// Terminate based on spread between vertexes.
if ((maxY - minY) <= Math.Abs(maxY) * settings.RelativePrecision)
{
Debug.WriteLine(minY);
return;
}
// Produce a new candidate vertex by reflecting the worst vertex through the opposite face.
double[] centroid = new double[d];
for (int i = 0; i < vertexes.Length; i++)
{
if (i != maxVertex)
{
for (int j = 0; j < d; j++)
{
centroid[j] += vertexes[i].X[j] / d;
}
}
}
double[] newX = new double[d];
for (int j = 0; j < d; j++)
{
newX[j] = centroid[j] + alpha * (centroid[j] - vertexes[maxVertex].X[j]);
}
double newY = f.Evaluate(newX);
if (newY < nextMaxY)
{
// As long as the new point is not terrible, we are going to replace the worst point with it.
vertexes[maxVertex] = new Vertex()
{
X = newX, Y = newY
};
Debug.WriteLine("Reflect");
if (newY < minY)
{
// If the new point was very good, we will try to extend the simplex further in that direction.
double[] extendedX = new double[d];
for (int j = 0; j < d; j++)
{
extendedX[j] = centroid[j] + 2.0 * (centroid[j] - vertexes[maxVertex].X[j]);
}
double extendedY = f.Evaluate(extendedX);
if (extendedY < minY)
{
// If the extension is also very good, we replace the second worst point too.
vertexes[maxVertex] = new Vertex()
{
X = extendedX, Y = extendedY
};
Debug.WriteLine("No, Extend");
}
}
}
else
{
// The reflected point was pretty terrible, so we will try to produce a new candidate
// point by contracting the worst point toward the centroid instead.
for (int j = 0; j < d; j++)
{
newX[j] = centroid[j] + (vertexes[maxVertex].X[j] - centroid[j]) / 2.0;
}
newY = f.Evaluate(newX);
if (newY < nextMaxY)
{
// If that candidate is not terrible, accept it.
vertexes[maxVertex] = new Vertex()
{
X = newX, Y = newY
};
Debug.WriteLine("Contract");
}
else
{
// Otherwise, we give up and simply shrink our simplex down toward the minimum.
for (int i = 0; i < vertexes.Length; i++)
{
if (i != minVertex)
{
double[] shrunkX = new double[d];
for (int j = 0; j < d; j++)
{
shrunkX[j] = vertexes[minVertex].X[j] + (vertexes[i].X[j] - vertexes[minVertex].X[j]) / 2.0;
}
double shrunkY = f.Evaluate(shrunkX);
vertexes[i] = new Vertex()
{
X = shrunkX, Y = shrunkY
};
}
}
Debug.WriteLine("Shrink");
}
}
}
}
public static void FindExtremum_Amobea(Func <IList <double>, double> function, IList <double> x0, EvaluationSettings settings)
{
MultiFunctor f = new MultiFunctor(function);
int d = x0.Count;
Vertex[] vertexes = new Vertex[d + 1];
for (int i = 0; i < d; i++)
{
double[] x = new double[d];
x0.CopyTo(x, 0);
x[i] = x[i] + 1.0 + Math.Abs(x[i]);
double y = f.Evaluate(x);
vertexes[i] = new Vertex()
{
X = x, Y = y
};
}
double[] x00 = new double[d];
x0.CopyTo(x00, 0);
double y00 = f.Evaluate(x00);
vertexes[d] = new Vertex()
{
X = x00, Y = y00
};
FindExtremum_Amobea(f, vertexes, settings);
}
internal Extremum(double x, double f, double f2, int count, EvaluationSettings settings) : base(count, settings)
{
this.x = x;
this.f = f;
this.f2 = f2;
}
public BulrischStoerStepper(Func <double, IList <double>, IList <double> > rhs, double x0, IList <double> y0, EvaluationSettings settings) : base(rhs, x0, y0, settings)
{
YPrime = Evaluate(x0, y0);
//DeltaX = 1.0;
DeltaX = 0.1;
//ComputeInitialSetp();
extrapolators = new NevilleExtrapolator[Dimension];
for (int i = 0; i < Dimension; i++)
{
extrapolators[i] = new NevilleExtrapolator(N.Length);
}
errorExtrapolator = new NevilleExtrapolator(N.Length);
}
public MultiOdeStepper(Func <double, IList <double>, IList <double> > rhs, double x0, IList <double> y0, EvaluationSettings settings) : base(rhs, x0, y0, settings)
{
this.Dimension = y0.Count;
}
private static UncertainValue Integrate_Adaptive(MultiFunctor f, CoordinateTransform[] map, IntegrationRegion r, EvaluationSettings settings)
{
// Create an evaluation rule
GenzMalik75Rule rule = new GenzMalik75Rule(r.Dimension);
// Use it on the whole region and put the answer in a linked list
rule.Evaluate(f, map, r);
LinkedList <IntegrationRegion> regionList = new LinkedList <IntegrationRegion>();
regionList.AddFirst(r);
// Iterate until convergence
while (f.EvaluationCount < settings.EvaluationBudget)
{
// Add up value and errors in all regions.
// While we are at it, take note of the region with the largest error.
double value = 0.0;
double error = 0.0;
LinkedListNode <IntegrationRegion> regionNode = regionList.First;
double maxError = 0.0;
LinkedListNode <IntegrationRegion> maxErrorNode = null;
while (regionNode != null)
{
IntegrationRegion region = regionNode.Value;
value += region.Value;
error += region.Error;
//error += MoreMath.Sqr(region.Error);
if (region.Error > maxError)
{
maxError = region.Error;
maxErrorNode = regionNode;
}
regionNode = regionNode.Next;
}
// Check for convergence.
if ((error <= settings.AbsolutePrecision) || (error <= settings.RelativePrecision * Math.Abs(value)))
{
return(new UncertainValue(value, error));
}
// Split the region with the largest error, and evaluate each subregion.
IntegrationRegion maxErrorRegion = maxErrorNode.Value;
regionList.Remove(maxErrorNode);
IList <IntegrationRegion> subRegions = maxErrorRegion.Split(maxErrorRegion.SplitIndex);
/*
* Countdown cnt = new Countdown(2);
* ThreadPool.QueueUserWorkItem((object state) => { rule.Evaluate(f, subRegions[0]); cnt.Signal(); });
* ThreadPool.QueueUserWorkItem((object state) => { rule.Evaluate(f, subRegions[1]); cnt.Signal(); });
* cnt.Wait();
* foreach (IntegrationRegion subRegion in subRegions) {
* regionList.AddLast(subRegion);
* }
*/
foreach (IntegrationRegion subRegion in subRegions)
{
rule.Evaluate(f, map, subRegion);
regionList.AddLast(subRegion);
}
}
throw new NonconvergenceException();
}
public static double SolveOde(Func <double, double, double> rhs, double x0, double y0, double x1, EvaluationSettings settings)
{
if (rhs == null)
{
throw new ArgumentNullException("rhs");
}
MultiOdeStepper stepper = new BulrischStoerStepper((double x, IList <double> y) => new double[] { rhs(x, y[0]) }, x0, new double[] { y0 }, settings);
//MultiOdeStepper stepper = new RungeKutta54Stepper((double x, IList<double> y) => new double[] { rhs(x, y[0]) }, range.LeftEndpoint, new double[] { start }, settings);
stepper.Integrate(x1);
return(stepper.Y[0]);
}
public void AssociatedLaguerreOrthonormality()
{
// don't let orders get too big, or (1) the Gamma function will overflow and (2) our integral will become highly oscilatory
foreach (int n in TestUtilities.GenerateIntegerValues(1, 10, 3)) {
foreach (int m in TestUtilities.GenerateIntegerValues(1, 10, 3)) {
foreach (double a in TestUtilities.GenerateRealValues(0.1, 10.0, 5)) {
//int n = 2;
//int m = 4;
//double a = 3.5;
Console.WriteLine("n={0} m={1} a={2}", n, m, a);
// evaluate the orthonormal integral
Func<double, double> f = delegate(double x) {
return (Math.Pow(x, a) * Math.Exp(-x) *
OrthogonalPolynomials.LaguerreL(m, a, x) *
OrthogonalPolynomials.LaguerreL(n, a, x)
);
};
Interval r = Interval.FromEndpoints(0.0, Double.PositiveInfinity);
// need to loosen default evaluation settings in order to get convergence in some of these cases
// seems to have most convergence problems for large a
EvaluationSettings e = new EvaluationSettings();
e.AbsolutePrecision = TestUtilities.TargetPrecision;
e.RelativePrecision = TestUtilities.TargetPrecision;
double I = FunctionMath.Integrate(f, r, e).Value;
Console.WriteLine(I);
// test for orthonormality
if (n == m) {
Assert.IsTrue(TestUtilities.IsNearlyEqual(
I, AdvancedMath.Gamma(n + a + 1) / AdvancedIntegerMath.Factorial(n)
));
} else {
Assert.IsTrue(Math.Abs(I) < TestUtilities.TargetPrecision);
}
}
}
}
}
internal MultiOdeResult(double x, double[] y, double[] yPrime, int count, EvaluationSettings settings) : base(count, settings)
{
this.x = x;
this.y = y;
this.yPrime = yPrime;
}
// This method is due to Powell (http://en.wikipedia.org/wiki/Michael_J._D._Powell), but it is not what
// is usually called Powell's Method (http://en.wikipedia.org/wiki/Powell%27s_method); Powell
// developed that method in the 1960s, it was included in Numerical Recipies and is very popular.
// This is a model trust algorithm developed by Powell in the 2000s. It typically uses many
// fewer function evaluations, but does more intensive calcuations between each evaluation.
// This is basically the UOBYQA variant of Powell's new methods. It maintains a quadratic model
// that interpolates between (d + 1) (d + 2) / 2 points. The model is trusted
// within a given radius. At each step, it moves to the minimum of the model (or the boundary of
// the trust region in that direction) and evaluates the function. The new value is incorporated
// into the model and the trust region expanded or contracted depending on how accurate its
// prediction of the function value was.
// Papers on these methods are collected at http://mat.uc.pt/~zhang/software.html#powell_software.
// The UOBYQA paper is here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.1756.
// The NEWUOA paper is here: http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2004_08.pdf.
// The CONDOR system (http://www.applied-mathematics.net/optimization/CONDORdownload.html) is based on these same ideas.
// The thesis of CONDOR's author (http://www.applied-mathematics.net/mythesis/index.html) was also helpful.
// It should be very easy to extend this method to constrained optimization, either by incorporating the bounds into
// the step limits or by mapping hyper-space into a hyper-cube.
private static MultiExtremum FindMinimum_ModelTrust(MultiFunctor f, IList <double> x, double s, EvaluationSettings settings)
{
// Construct an initial model.
QuadraticInterpolationModel model = QuadraticInterpolationModel.Construct(f, x, s);
double trustRadius = s;
while (f.EvaluationCount < settings.EvaluationBudget)
{
// Find the minimum point of the model within the trust radius
double[] z = model.FindMinimum(trustRadius);
double expectedValue = model.Evaluate(z);
double deltaExpected = model.MinimumValue - expectedValue;
// Evaluate the function at the suggested minimum
double[] point = model.ConvertPoint(z);
double value = f.Evaluate(point);
double delta = model.MinimumValue - value;
double tol = settings.ComputePrecision(value);
// To terminate, we demand: a reduction, the reduction be small, the reduction be in line with its expected value, that we have run up against trust boundary,
// and that the gradient is small.
// I had wanted to demand delta > 0, but we run into some cases where delta keeps being very slightly negative, typically orders of magnitude less than tol,
// causing the trust radius to shrink in and endless cycle that causes our approximation to ultimately go sour, even though terminating on the original
// very slightly negative delta would have produced an accurate estimate. So we tolerate this case for now.
if ((-tol / 4.0 <= delta) && (delta <= tol))
{
// We demand that the model be decent, i.e. that the expected delta was within tol of the measured delta.
if (Math.Abs(delta - deltaExpected) <= tol)
{
// We demand that the step not just be small because it ran up against the trust radius. If it ran up against the trust radius,
// there is probably more to be hand by continuing.
double zm = Blas1.dNrm2(z, 0, 1, z.Length);
if (zm < trustRadius)
{
// Finally, we demand that the gradient be small. You might think this was obvious since z was small, but if the Hessian is not positive definite
// the interplay of the Hessian and the gradient can produce a small z even if the model looks nothing like a quadratic minimum.
double gm = Blas1.dNrm2(model.GetGradient(), 0, 1, z.Length);
if (gm * zm <= tol)
{
if (f.IsNegated)
{
value = -value;
}
return(new MultiExtremum(f.EvaluationCount, settings, point, value, Math.Max(Math.Abs(delta), 0.75 * tol), model.GetHessian()));
}
}
}
}
// There are now three decisions to be made:
// 1. How to change the trust radius
// 2. Whether to accept the new point
// 3. Which existing point to replace
// If the actual change was very far from the expected change, reduce the trust radius.
// If the expected change did a good job of predicting the actual change, increase the trust radius.
if ((delta < 0.25 * deltaExpected) /*|| (8.0 * deltaExpected < delta)*/)
{
trustRadius = trustRadius / 2.0;
}
else if ((0.75 * deltaExpected <= delta) /*&& (delta <= 2.0 * deltaExpected)*/)
{
trustRadius = 2.0 * trustRadius;
}
// It appears that the limits on delta being too large don't help, and even hurt if made too stringent.
// Replace an old point with the new point.
int iMax = 0; double fMax = model.values[0];
int iBad = 0; double fBad = model.ComputeBadness(0, z, point, value);
for (int i = 1; i < model.values.Length; i++)
{
if (model.values[i] > fMax)
{
iMax = i; fMax = model.values[i];
}
double bad = model.ComputeBadness(i, z, point, value);
if (bad > fBad)
{
iBad = i; fBad = bad;
}
}
if (value < fMax)
{
Debug.WriteLine("iMax={0}, iBad={1}", iMax, iBad);
model.ReplacePoint(iBad, point, z, value);
}
// There is some question about how best to choose which point to replace.
// The largest value? The furthest away? The one closest to new min?
}
throw new NonconvergenceException();
}
/// <summary>
/// Evaluates a definite integral with the given evaluation settings.
/// </summary>
/// <param name="integrand">The function to be integrated.</param>
/// <param name="range">The range of integration.</param>
/// <param name="settings">The settings which control the evaulation of the integal.</param>
/// <returns>The result of the integral, which includes an estimated value and estimated uncertainty of that value.</returns>
public static IntegrationResult Integrate(Func<double,double> integrand, Interval range, EvaluationSettings settings)
{
if (integrand == null) throw new ArgumentNullException("integrand");
// remap infinite integrals to finite integrals
if (Double.IsNegativeInfinity(range.LeftEndpoint) && Double.IsPositiveInfinity(range.RightEndpoint)) {
// -infinity to +infinity
// remap to (-pi/2,pi/2)
Func<double, double> f0 = integrand;
Func<double, double> f1 = delegate (double t) {
double x = Math.Tan(t);
return (f0(x) * (1.0 + x * x));
};
Interval r1 = Interval.FromEndpoints(-Global.HalfPI, Global.HalfPI);
return (Integrate(f1, r1, settings));
} else if (Double.IsPositiveInfinity(range.RightEndpoint)) {
// finite to +infinity
// remap to interval (-1,1)
double a0 = range.LeftEndpoint;
Func<double, double> f0 = integrand;
Func<double, double> f1 = delegate (double t) {
double q = 1.0 - t;
double x = a0 + (1 + t) / q;
return (f0(x) * (2.0 / q / q));
};
Interval r1 = Interval.FromEndpoints(-1.0, 1.0);
return (Integrate(f1, r1, settings));
} else if (Double.IsNegativeInfinity(range.LeftEndpoint)) {
// -infinity to finite
// remap to interval (-1,1)
double b0 = range.RightEndpoint;
Func<double, double> f0 = integrand;
Func<double, double> f1 = delegate (double t) {
double q = t + 1.0;
double x = b0 + (t - 1.0) / q;
return(f0(x) * (2.0 / q / q));
};
Interval r1 = Interval.FromEndpoints(-1.0, 1.0);
return(Integrate(f1, r1, settings));
}
// normal integral over a finitite range
IAdaptiveIntegrator integrator = new GaussKronrodIntegrator(integrand, range);
IntegrationResult result = Integrate_Adaptive(integrator, settings);
return (result);
}
private static double FindMinimum(
Func<double,double> f,
double a, double b,
double u, double fu, double v, double fv, double w, double fw,
EvaluationSettings settings, int count
)
{
double tol = 0.0;
while (count < settings.EvaluationBudget) {
Console.WriteLine("n={0} tol={1}", count, tol);
Console.WriteLine("[{0} f({1})={2} f({3})={4} f({5})={6} {7}]", a, u, fu, v, fv, w, fw, b);
Debug.Assert(a < b);
Debug.Assert((a <= u) && (u <= b));
Debug.Assert((fu <= fv) && (fv <= fw));
// Expected final situation is a<tol><tol>u<tol><tol>b, leaving no point left to evaluate that is not within tol of an existing point.
if ((b - a) <= 4.0 * tol) return (u);
// While a < u < b is guaranteed, a < v, w < b is not guaranteed, since the bracket can sometimes be made tight enough to exclude v or w.
// For example, if u < v < w, then we can set b = v, placing w outside the bracket.
double x, fpp;
ParabolicFit(u, fu, v, fv, w, fw, out x, out fpp);
Console.WriteLine("parabolic x={0} f''={1}", x, fpp);
if (Double.IsNaN(fpp) || (fpp <= 0.0) || (x < a) || (x > b)) {
// the parabolic fit didn't work out, so do a golden section reduction instead
// to get the most reduction of the bracket, pick the larger of au and ub
// for self-similarity, pick a point inside it that divides it into two segments in the golden section ratio,
// i.e. 0.3820 = \frac{1}{\phi + 1} and 0.6180 = \frac{\phi}{\phi+1}
// put the smaller segment closer to u so that x is closer to u, the best minimum so far
double au = u - a;
double ub = b - u;
if (au > ub) {
x = u - au / (AdvancedMath.GoldenRatio + 1.0);
} else {
x = u + ub / (AdvancedMath.GoldenRatio + 1.0);
}
Console.WriteLine("golden section x={0}", x);
}
// ensure we don't evaluate within tolerance of an existing point
if (Math.Abs(x - u) < tol) { Console.WriteLine("shift from u (x={0})", x); x = (x > u) ? u + tol : u - tol; }
if ((x - a) < tol) { Console.WriteLine("shift from a (x={0})", x); x = a + tol; }
if ((b - x) < tol) { Console.WriteLine("shift from b (x={0})", x); x = b - tol; }
count++;
double fx = f(x);
Console.WriteLine("f({0}) = {1}", x, fx);
Console.WriteLine("delta={0}", fu - fx);
if (fx < fu) {
// the new point is lower than all the others; this is success
// u now becomes a bracket point
if (u < x) {
a = u;
} else {
b = u;
}
// x -> u -> v -> w
w = v; fw = fv;
v = u; fv = fu;
u = x; fu = fx;
} else {
// x now becomes a bracket point
if (x < u) {
a = x;
} else {
b = x;
}
if (fx < fv) {
// the new point is higher than u, but still lower than v and w
// this isn't what we expected, but we have lower points that before
// x -> v -> w
w = v; fw = fv;
v = x; fv = fx;
} else if (fx < fw) {
// x -> w
w = x; fw = fx;
} else {
// the new point is higher than all our other points; this is the worst case
// we might still want to replace w with x because
// (i) otherwise a parabolic fit will reproduce the same x and
// (ii) w is quite likely far outside the new bracket and not telling us much about the behavior near u
Console.WriteLine("bad point");
//throw new NotImplementedException();
}
}
// if the user has specified a tollerance, use it
if ((settings.RelativePrecision > 0.0 || settings.AbsolutePrecision > 0.0)) {
tol = Math.Max(Math.Abs(u) * settings.RelativePrecision, settings.AbsolutePrecision);
} else {
// otherwise, try to get the tollerance from the curvature
if (fpp > 0.0) {
tol = Math.Sqrt(2.0 * (Math.Abs(fu) * 1.0E-14 + 1.0E-28) / fpp);
} else {
// but if we don't have a useable curvature either, wing it
}
}
}
throw new NonconvergenceException();
}
// the drivers
private static IntegrationResult Integrate_Adaptive(IAdaptiveIntegrator integrator, EvaluationSettings s)
{
LinkedList<IAdaptiveIntegrator> list = new LinkedList<IAdaptiveIntegrator>();
list.AddFirst(integrator);
int n = integrator.EvaluationCount;
while (true) {
// go through the intervals, adding estimates (and errors)
// and noting which contributes the most error
// keep track of the total value and uncertainty
UncertainValue vTotal = new UncertainValue();
//double v = 0.0;
//double u = 0.0;
// keep track of which node contributes the most error
LinkedListNode<IAdaptiveIntegrator> maxNode = null;
double maxError = 0.0;
LinkedListNode<IAdaptiveIntegrator> node = list.First;
while (node != null) {
IAdaptiveIntegrator i = node.Value;
UncertainValue v = i.Estimate;
vTotal += v;
//UncertainValue uv = i.Estimate;
//v += uv.Value;
//u += uv.Uncertainty;
if (v.Uncertainty > maxError) {
maxNode = node;
maxError = v.Uncertainty;
}
node = node.Next;
}
// if our error is small enough, return
if ((vTotal.Uncertainty <= Math.Abs(vTotal.Value) * s.RelativePrecision) || (vTotal.Uncertainty <= s.AbsolutePrecision)) {
return (new IntegrationResult(vTotal, n, s));
}
//if ((vTotal.Uncertainty <= Math.Abs(vTotal.Value) * s.RelativePrecision) || (vTotal.Uncertainty <= s.AbsolutePrecision)) {
// return (new IntegrationResult(vTotal.Value, n));
//}
// if our evaluation count is too big, throw
if (n > s.EvaluationBudget) throw new NonconvergenceException();
// subdivide the interval with the largest error
IEnumerable<IAdaptiveIntegrator> divisions = maxNode.Value.Divide();
foreach (IAdaptiveIntegrator division in divisions) {
list.AddBefore(maxNode, division);
n += division.EvaluationCount;
//v2 += division.Estimate;
}
list.Remove(maxNode);
}
}
public double FindMinimum(Func<double, double> f, double x, double d, EvaluationSettings settings)
{
// evaluate at x and x + d
double fx = f(x);
double y = x + d;
double fy = f(y);
int count = 2;
// if we stepped uphill, reverse direction of steps and exchange x & y
if (fy > fx) {
double t = x; x = y; y = t;
t = fx; fx = fy; fy = t;
d = -d;
}
// we now know f(x) >= f(y) and we are stepping downhill
// continue stepping until we step uphill
double z, fz;
while (true) {
if (count >= settings.EvaluationBudget) throw new NonconvergenceException();
z = y + d;
fz = f(z);
count++;
Console.WriteLine("f({0})={1} f({2})={3} f({4})={5} d={6}", x, fx, y, fy, z, fz, d);
if (fz > fy) break;
// increase the step size each time
d = AdvancedMath.GoldenRatio * d;
// x <- y <- z
x = y; fx = fy; y = z; fy = fz;
}
// we x and z now bracket a local minimum, with y the lowest point evaluated so far
double a = Math.Min(x, z); double b = Math.Max(x, z);
if (fz < fx) { double t = x; x = z; z = t; t = fx; fx = fz; fz = t; }
return (FindMinimum(f, a, b, y, fy, x, fx, z, fz, settings, count));
}
// the public API
/// <summary>
/// Evaluates a definite integral.
/// </summary>
/// <param name="integrand">The function to be integrated.</param>
/// <param name="range">The range of integration.</param>
/// <returns>A numerical estimate of the given integral.</returns>
/// <remarks>
/// <para>Integral values are accurate to within about a digit of full double precision.</para>
/// <para>To do integrals over infinite regions, simply set the lower bound of the <paramref name="range"/>
/// to <see cref="System.Double.NegativeInfinity"/> or the upper bound to <see cref="System.Double.PositiveInfinity"/>.</para>
/// <para>Our numerical integrator uses a Gauss-Kronrod rule that can integrate efficiently,
/// combined with an adaptive strategy that limits function
/// evaluations to those regions required to achieve the desired accuracy.</para>
/// <para>Our integrator handles smooth functions extremely efficiently. It handles integrands with
/// discontinuities, or discontinuities of derivatives, at the price of slightly more evaluations
/// of the integrand. It handles oscilatory functions, so long as not too many periods contribute
/// significantly to the integral. It can integrate logarithmic and mild power-law singularities.
/// </para>
/// <para>Strong power-law singularities will cause the alrorighm to fail with a NonconvergenceException.
/// This is unavoidable for essentially any double-precision numerical integrator. Consider, for example,
/// the integrable singularity 1/√x. Since
/// ε = ∫<sub>0</sub><sup>δ</sup> x<sup>-1/2</sup> dx = 2 δ<sup>1/2</sup>,
/// points within δ ∼ 10<sup>-16</sup> of the end-points, which as a close as you can get to
/// a point in double precision without being on top of it, contribute at the ε ∼ 10<sup>-8</sup>
/// level to our integral, well beyond limit that nearly-full double precision requires. Said differently,
/// to know the value of the integral to ε ∼ 10<sup>-16</sup> prescision, we would need to
/// evaluate the contributions of points within δ ∼ 10<sup>-32</sup> of the endpoints,
/// far closer than we can get.</para>
/// <para>If you need to evaluate an integral with such a strong singularity, make an analytic
/// change of variable to absorb the singularity before attempting numerical integration. For example,
/// to evaluate I = ∫<sub>0</sub><sup>b</sup> f(x) x<sup>-1/2</sup> dx, substitute y = x<sup>1/2</sup>
/// to obtain I = 2 ∫<sub>0</sub><sup>√b</sup> f(y<sup>2</sup>) dy.</para>
/// </remarks>
public static double Integrate(Func<double, double> integrand, Interval range)
{
EvaluationSettings settings = new EvaluationSettings();
return (Integrate(integrand, range, settings).Value);
}
/// <summary>
/// Minimizes a function on a multi-dimensional space in the vicinity of a given point, subject to the given settings.
/// </summary>
/// <param name="f">The function.</param>
/// <param name="x">The starting point for the search.</param>
/// <param name="settings">The evaluation settings.</param>
/// <returns>The minimum.</returns>
/// <exception cref="ArgumentNullException"><paramref name="f"/>, <paramref name="x"/>, or <paramref name="settings"/> is null.</exception>
/// <exception cref="NonconvergenceException">The minimum was not found to the required precision within the budgeted number of function evaluations.</exception>
internal static SpaceExtremum FindMinimum(Func <double[], double> f, double[] x, EvaluationSettings settings)
{
if (f == null)
{
throw new ArgumentNullException("f");
}
if (x == null)
{
throw new ArgumentNullException("x");
}
if (settings == null)
{
throw new ArgumentNullException("settings");
}
int d = x.Length;
// put the function into a Functor we will use for line searches
LineSearchFunctor fo = new LineSearchFunctor(f);
// keep track of the (conjugate) minimization directions
double[][] Q = new double[d][];
for (int i = 0; i < d; i++)
{
Q[i] = new double[d];
for (int j = 0; j < d; j++)
{
Q[i][j] = 0.0;
}
// pick a step size in each direction that represents a fraction of the input value
Q[i][i] = (1.0 / 16.0) * (Math.Abs(x[i]) + (1.0 / 16.0));
}
// keep track of the curvature in these directions
double[] r = new double[d];
// keep track of the function value
double y = f(x);
// keep track of the total number of evaluations
//int count = 0;
bool skip = false;
// interate until convergence
while (fo.EvaluationCount < settings.EvaluationBudget)
{
// remember our starting position
double[] x0 = new double[d];
Array.Copy(x, x0, d);
double y0 = y;
// keep track of direction of largest decrease
double max_dy = 0.0;
int max_i = 0;
// now minimize in each direction
for (int i = 0; i < d; i++)
{
// if we placed the net direction in the first slot last time,
// we are already at the minimum along that direction, so don't
// minimize along it
if (skip)
{
skip = false;
continue;
}
//if ((n > 0) && (i == 0) && (max_i == 0)) continue;
//Console.WriteLine("i = {0}", i);
//WriteVector(Q[i]);
// form a line function
//LineFunction f1 = new LineFunction(f, x, Q[i]);
fo.Origin = x;
fo.Direction = Q[i];
// minimize it
Extremum m = FindMinimum(fo, 0.0, y, 1.0, new ExtremumSettings()
{
EvaluationBudget = settings.EvaluationBudget, AbsolutePrecision = 0.0, RelativePrecision = 0.0
});
//LineExtremum m = FindMinimum(new Func<double,double>(f1.Evaluate), 0.0, y, 1.0);
// add to the evaluation count
//count += f1.Count;
// update the current position
x = fo.ComputeLocation(m.Location);
//x = f1.Position(m.Location);
//WriteVector(x);
r[i] = m.Curvature;
// keep track of how much the function dropped, and
// if this is the direction of largest decrease
double dy = y - m.Value;
//Console.WriteLine("dy = {0}", dy);
if (dy > max_dy)
{
max_dy = dy;
max_i = i;
}
y = m.Value;
}
//Console.WriteLine("max_i = {0}, max_dy = {1}", max_i, max_dy);
//Console.WriteLine("y0 = {0}, y = {1}", y0, y);
// figure out the net direction we have moved
double[] dx = new double[d];
for (int i = 0; i < d; i++)
{
dx[i] = x[i] - x0[i];
}
//Console.WriteLine("Finish:");
//WriteVector(x);
//Console.WriteLine("Net direction:");
//WriteVector(dx);
// check termination criteria
// we do this before minimizing in the net direction because if dx=0 it loops forever
double Dy = Math.Abs(y0 - y);
if ((Dy < settings.AbsolutePrecision) || (2.0 * Dy < (Math.Abs(y) + Math.Abs(y0)) * settings.RelativePrecision))
{
SymmetricMatrix A = ComputeCurvature(f, x);
return(new SpaceExtremum(x, y, A));
}
// attempt a minimization in the net direction
fo.Origin = x;
fo.Direction = dx;
//LineFunction f2 = new LineFunction(f, x, dx);
//LineExtremum mm = FindMinimum(new Func<double,double>(f2.Evaluate), 0.0, y, 1.0);
Extremum mm = FindMinimum(fo, 0.0, y, 1.0, new ExtremumSettings()
{
EvaluationBudget = settings.EvaluationBudget, RelativePrecision = 0.0, AbsolutePrecision = 0.0
});
//count += f2.Count;
//x = f2.Position(mm.Location);
x = fo.ComputeLocation(mm.Location);
y = mm.Value;
// rotate this direction into the direction set
/*
* for (int i = 0; i < (d - 1); i++) {
* Q[i] = Q[i + 1];
* r[i] = r[i + 1];
* }
* Q[d - 1] = dx;
* r[d - 1] = mm.Curvature;
*/
// this is the basic Powell procedure, and it leads to linear dependence
// replace the direction of largest decrease with the net direction
Q[max_i] = dx;
r[max_i] = mm.Curvature;
if (max_i == 0)
{
skip = true;
}
// this is powell's modification to avoid linear dependence
// reset
}
throw new NonconvergenceException();
}
/// <summary>
/// Finds the maxunyn of a function within the given volume, subject to the given evaluation constraints.
/// </summary>
/// <param name="function">The function.</param>
/// <param name="volume">The volume to search.</param>
/// <param name="settings">The evaluation constraints to apply.</param>
/// <returns>The global maximum.</returns>
public static MultiExtremum FindGlobalMaximum(Func <IList <double>, double> function, IList <Interval> volume, EvaluationSettings settings)
{
return(FindGlobalExtremum(function, volume, settings, true));
}
public void DistributionProbabilityIntegral()
{
Random rng = new Random(4);
// if integral is very small, we still want to get it very accurately
EvaluationSettings settings = new EvaluationSettings();
settings.AbsolutePrecision = 0.0;
foreach (Distribution distribution in distributions) {
if (distribution is TriangularDistribution) continue;
for (int i = 0; i < 3; 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 exponentialy 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;
}
Console.WriteLine("{0} {1}", distribution.GetType().Name, x);
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 innacurate, 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
Console.WriteLine("skip (P={0}, Q={1})", P, Q);
continue;
}
Console.WriteLine(" {0} v. {1}", P, distribution.LeftProbability(x));
Console.WriteLine(" {0} v. {1}", Q, distribution.RightProbability(x));
Assert.IsTrue(TestUtilities.IsNearlyEqual(P, distribution.LeftProbability(x)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(Q, distribution.RightProbability(x)));
}
}
}
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.
}
private static UncertainValue Integrate_MonteCarlo(MultiFunctor f, CoordinateTransform[] map, IList<Interval> box, EvaluationSettings settings)
{
int d = box.Count;
// Use a Sobol quasi-random sequence. This give us 1/N accuracy instead of 1/\sqrt{N} accuracy.
//VectorGenerator g = new RandomVectorGenerator(d, new Random(314159265));
VectorGenerator g = new SobolVectorGenerator(d);
// Start with a trivial Lepage grid.
// We will increase the grid size every few cycles.
// My tests indicate that trying to increase every cycle or even every other cycle is too often.
// This makes sense, because we have no reason to believe our new grid will be better until we
// have substantially more evaluations per grid cell than we did for the previous grid.
LePageGrid grid = new LePageGrid(box, 1);
int refineCount = 0;
// Start with a reasonable number of evaluations per cycle that increases with the dimension.
int cycleCount = 8 * d;
//double lastValue = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
// Each cycle consists of three sets of evaluations.
// At first I did this with just two set and used the difference between the two sets as an error estimate.
// I found that it was pretty common for that difference to be low just by chance, causing error underestimatation.
double value1 = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
double value2 = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
double value3 = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
while (f.EvaluationCount < settings.EvaluationBudget) {
// Take the largest deviation as the error.
double value = (value1 + value2 + value3) / 3.0;
double error = Math.Max(Math.Abs(value1 - value3), Math.Max(Math.Abs(value1 - value2), Math.Abs(value2 - value3)));
Debug.WriteLine("{0} {1} {2}", f.EvaluationCount, value, error);
// Check for convergence.
if ((error <= settings.AbsolutePrecision) || (error <= Math.Abs(value) * settings.RelativePrecision)) {
return (new UncertainValue(value, error));
}
// Do more cycles. In order for new sets to be equal-sized, one of those must be at the current count and the next at twice that.
double smallValue = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
cycleCount *= 2;
double bigValue = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
// Combine all the cycles into new ones with twice the number of evaluations each.
value1 = (value1 + value2) / 2.0;
value2 = (value3 + smallValue) / 2.0;
value3 = bigValue;
//double currentValue = Integrate_MonteCarlo_Cycle(f, map, g, grid, cycleCount);
//double error = Math.Abs(currentValue - lastValue);
//double value = (currentValue + lastValue) / 2.0;
//lastValue = value;
// Increase the number of evaluations for the next cycle.
//cycleCount *= 2;
// Refine the grid for the next cycle.
refineCount++;
if (refineCount == 2) {
Debug.WriteLine("Replacing grid with {0} bins after {1} evaluations", grid.BinCount, grid.EvaluationCount);
grid = grid.ComputeNewGrid(grid.BinCount * 2);
refineCount = 0;
}
}
throw new NonconvergenceException();
}
public RungeKutta54Stepper(Func <double, IList <double>, IList <double> > rhs, double x0, IList <double> y0, EvaluationSettings settings) : base(rhs, x0, y0, settings)
{
YPrime = Evaluate(X, Y);
DeltaX = 1.0;
}
private static MultiExtremum FindGlobalExtremum(Func <IList <double>, double> function, IList <Interval> volume, EvaluationSettings settings, bool negate)
{
if (function == null)
{
throw new ArgumentNullException("function");
}
if (volume == null)
{
throw new ArgumentNullException("volume");
}
MultiFunctor f = new MultiFunctor(function, negate);
DifferentialEvolutionSettings deSettings = GetDefaultSettings(settings, volume.Count);
MultiExtremum extremum = FindGlobalExtremum(f, volume, deSettings);
return(extremum);
}
public void MinimizePerturbedQuadratic2D()
{
EvaluationSettings s = new EvaluationSettings() { EvaluationBudget = 100, RelativePrecision = 1.0E-10 };
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) +
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) }));
}
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.
}
}
/// <summary>
/// Finds a local minimum of a multi-dimensional function in the vincinity of the given starting location, subject to the given evaluation constraints.
/// </summary>
/// <param name="function">The multi-dimensional function to minimize.</param>
/// <param name="start">The starting location for the search.</param>
/// <param name="settings">The evaluation settings that govern the search for the minimum.</param>
/// <returns>The local minimum.</returns>
/// <remarks>
/// <para>The Hessian (matrix of second derivatives) returned with the minimum is an approximation that is constructed in the course of search. It should be
/// considered a crude approximation, and may not even be that if the minimum is highly non-quadratic.</para>
/// <para>If you have a constrained minimization problem, require a high-precision solution, and do not have a good initial guess, consider first feeding
/// your constrained problem into <see cref="FindGlobalMaximum"/>, which supports constraints but gives relatively lower precision solutions, then
/// feeding the result of that method into this method, which finds relatively precision solutions but does not support constraints.</para>
/// </remarks>
/// <exception cref="NonconvergenceException">The number of function evaluations required exceeded the evaluation budget.</exception>
public static MultiExtremum FindLocalMinimum(Func <IList <double>, double> function, IList <double> start, EvaluationSettings settings)
{
if (function == null)
{
throw new ArgumentNullException("function");
}
if (start == null)
{
throw new ArgumentNullException("start");
}
if (settings == null)
{
throw new ArgumentNullException("settings");
}
return(FindLocalExtremum(function, start, settings, false));
}
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 }));
}
}
// the public API
/// <summary>
/// Evaluates a definite integral.
/// </summary>
/// <param name="integrand">The function to be integrated.</param>
/// <param name="range">The range of integration.</param>
/// <returns>A numerical estimate of the given integral.</returns>
/// <remarks>
/// <para>Integral values are accurate to within about a digit of full double precision.</para>
/// <para>To do integrals over infinite regions, simply set the lower bound of the <paramref name="range"/>
/// to <see cref="System.Double.NegativeInfinity"/> or the upper bound to <see cref="System.Double.PositiveInfinity"/>.</para>
/// <para>Our numerical integrator uses a Gauss-Kronrod rule that can integrate efficiently,
/// combined with an adaptive strategy that limits function
/// evaluations to those regions required to achieve the desired accuracy.</para>
/// <para>Our integrator handles smooth functions extremely efficiently. It handles integrands with
/// discontinuities, or discontinuities of derivatives, at the price of slightly more evaluations
/// of the integrand. It handles oscilatory functions, so long as not too many periods contribute
/// significantly to the integral. It can integrate logarithmic and mild power-law singularities.
/// </para>
/// <para>Strong power-law singularities will cause the alrorighm to fail with a NonconvergenceException.
/// This is unavoidable for essentially any double-precision numerical integrator. Consider, for example,
/// the integrable singularity 1/√x. Since
/// ε = ∫<sub>0</sub><sup>δ</sup> x<sup>-1/2</sup> dx = 2 δ<sup>1/2</sup>,
/// points within δ ∼ 10<sup>-16</sup> of the end-points, which as a close as you can get to
/// a point in double precision without being on top of it, contribute at the ε ∼ 10<sup>-8</sup>
/// level to our integral, well beyond limit that nearly-full double precision requires. Said differently,
/// to know the value of the integral to ε ∼ 10<sup>-16</sup> prescision, we would need to
/// evaluate the contributions of points within δ ∼ 10<sup>-32</sup> of the endpoints,
/// far closer than we can get.</para>
/// <para>If you need to evaluate an integral with such a strong singularity, make an analytic
/// change of variable to absorb the singularity before attempting numerical integration. For example,
/// to evaluate I = ∫<sub>0</sub><sup>b</sup> f(x) x<sup>-1/2</sup> dx, substitute y = x<sup>1/2</sup>
/// to obtain I = 2 ∫<sub>0</sub><sup>√b</sup> f(y<sup>2</sup>) dy.</para>
/// </remarks>
public static double Integrate(Func <double, double> integrand, Interval range)
{
EvaluationSettings settings = new EvaluationSettings();
return(Integrate(integrand, range, settings).Value);
}
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 }));
}
}
/// <summary>
/// Evaluates a definite integral with the given evaluation settings.
/// </summary>
/// <param name="integrand">The function to be integrated.</param>
/// <param name="range">The range of integration.</param>
/// <param name="settings">The settings which control the evaulation of the integal.</param>
/// <returns>The result of the integral, which includes an estimated value and estimated uncertainty of that value.</returns>
public static IntegrationResult Integrate(Func <double, double> integrand, Interval range, EvaluationSettings settings)
{
if (integrand == null)
{
throw new ArgumentNullException("integrand");
}
// remap infinite integrals to finite integrals
if (Double.IsNegativeInfinity(range.LeftEndpoint) && Double.IsPositiveInfinity(range.RightEndpoint))
{
// -infinity to +infinity
// remap to (-pi/2,pi/2)
Func <double, double> f0 = integrand;
Func <double, double> f1 = delegate(double t) {
double x = Math.Tan(t);
return(f0(x) * (1.0 + x * x));
};
Interval r1 = Interval.FromEndpoints(-Global.HalfPI, Global.HalfPI);
return(Integrate(f1, r1, settings));
}
else if (Double.IsPositiveInfinity(range.RightEndpoint))
{
// finite to +infinity
// remap to interval (-1,1)
double a0 = range.LeftEndpoint;
Func <double, double> f0 = integrand;
Func <double, double> f1 = delegate(double t) {
double q = 1.0 - t;
double x = a0 + (1 + t) / q;
return(f0(x) * (2.0 / q / q));
};
Interval r1 = Interval.FromEndpoints(-1.0, 1.0);
return(Integrate(f1, r1, settings));
}
else if (Double.IsNegativeInfinity(range.LeftEndpoint))
{
// -infinity to finite
// remap to interval (-1,1)
double b0 = range.RightEndpoint;
Func <double, double> f0 = integrand;
Func <double, double> f1 = delegate(double t) {
double q = t + 1.0;
double x = b0 + (t - 1.0) / q;
return(f0(x) * (2.0 / q / q));
};
Interval r1 = Interval.FromEndpoints(-1.0, 1.0);
return(Integrate(f1, r1, settings));
}
// normal integral over a finitite range
IAdaptiveIntegrator integrator = new GaussKronrodIntegrator(integrand, range);
IntegrationResult result = Integrate_Adaptive(integrator, settings);
return(result);
}
/// <summary>
/// Integrates the given function over the given volume.
/// </summary>
/// <param name="integrand">The function to integrate, which maps R<sup>d</sup> to R.</param>
/// <param name="volume">The box defining the volume over with to integrate.</param>
/// <param name="settings">The evaluation settings to use.</param>
/// <returns>The value of the integral.</returns>
public static double Integrate(Func <IList <double>, double> integrand, IList <Interval> volume, EvaluationSettings settings)
{
if (integrand == null)
{
throw new ArgumentNullException("integrand");
}
if (volume == null)
{
throw new ArgumentNullException("volume");
}
// get the dimension of the problem from the volume
int d = volume.Count;
if ((d < 1) || (d > sobolParameters.Length))
{
throw new InvalidOperationException();
}
// if no settings were provided, use defaults
if (settings == null)
{
settings = new EvaluationSettings()
{
RelativePrecision = 1.0 / (1 << (14 - 3 * d / 2)),
AbsolutePrecision = RelativePrecision / 128.0,
EvaluationBudget = 1 << (18 + 3 * d / 2)
};
}
// compute the volume of the box
double V = 1.0;
for (int j = 0; j < volume.Count; j++)
{
V *= volume[j].Width;
}
// generate the appropriate number of Sobol sequences
SobolSequence[] s = new SobolSequence[d];
for (int j = 0; j < d; j++)
{
SobolSequenceParameters p = sobolParameters[j];
s[j] = new SobolSequence(p.Dimension, p.Coefficients, p.Seeds);
}
// create one vector to store the argument
// we don't want to recreate the vector in the
// heap on each iteration
double[] x = new double[d];
// keep track of the average sampled value
double M = 0.0; double M_old = Double.NaN;
// set the first convergence checkpoint to ~4000 points for d=2, increasing
// for higher d
int i_next = 1 << (10 + d);
for (int i = 1; i <= settings.EvaluationBudget; i++)
{
// move to the next value in each dimension's Sobol sequence
// and construct the argument vector
for (int j = 0; j < d; j++)
{
s[j].MoveNext();
x[j] = volume[j].LeftEndpoint + s[j].Current * volume[j].Width;
}
// evaluate the function
double y = integrand(x);
// update the mean
M += (y - M) / i;
// check for convergence at regular intervals
if (i == i_next)
{
// estimate error as change since last check
double dM = 2.0 * Math.Abs(M - M_old);
if (dM < settings.RelativePrecision * Math.Abs(M) || Math.Abs(V) * dM < settings.AbsolutePrecision)
{
return(V * M);
}
// no convergence, so remember current value and set next checkpoint
M_old = M;
i_next = 2 * i_next;
// Consider how to do this better:
// 1. Is there any way we can do better than doubling? Say increasing by 4/3 and then 3/2, for an average increase of
// 40% instead of 100%? If so, how do we estimate error at each step?
// 2. Can we use Romberg extrapolation? I tried using both M and M_old, assuming an error term of 1/N. This gives
// the extrapolated value 2 M - M_old, but experimentally this is usually a worse value than M. I should try out
// 3-value extrapolation.
}
}
throw new NonconvergenceException();
}
private static Extremum FindMinimum(Functor f, double x, double fx, double d, EvaluationSettings settings)
{
// This function brackets a minimum by starting from x and taking increasing steps downhill until it moves uphill again.
// We write this function assuming f(x) has already been evaluated because when it is called
// to do a line fit for Powell's multi-dimensional minimization routine, that is the case and
// we don't want to do a superfluous evaluation.
Debug.Assert(f != null); Debug.Assert(d > 0.0); Debug.Assert(settings != null);
// evaluate at x + d
double y = x + d;
double fy = f.Evaluate(y);
// if we stepped uphill, reverse direction of steps and exchange x & y
if (fy > fx)
{
Global.Swap(ref x, ref y); Global.Swap(ref fx, ref fy);
d = -d;
}
// we now know f(x) >= f(y) and we are stepping downhill
// continue stepping until we step uphill
double z, fz;
while (true)
{
if (f.EvaluationCount >= settings.EvaluationBudget)
{
throw new NonconvergenceException();
}
z = y + d;
fz = f.Evaluate(z);
Debug.WriteLine(String.Format("f({0})={1} f({2})={3} f({4})={5} d={6}", x, fx, y, fy, z, fz, d));
if (fz > fy)
{
break;
}
// increase the step size each time
d = AdvancedMath.GoldenRatio * d;
// x <- y <- z
x = y; fx = fy; y = z; fy = fz;
}
// we x and z now bracket a local minimum, with y the lowest point evaluated so far
double a = Math.Min(x, z); double b = Math.Max(x, z);
if (fz < fx)
{
Global.Swap(ref x, ref z); Global.Swap(ref fx, ref fz);
}
return(FindMinimum(f, a, b, y, fy, x, fx, z, fz, settings));
}
internal IntegrationResult(UncertainValue estimate, int evaluationCount, EvaluationSettings settings) : base(evaluationCount, settings)
{
this.estimate = estimate;
}
// Brent's algorithm: use 3-point parabolic interpolation,
// switching to golden section if interval does not shrink fast enough
// see Richard Brent, "Algorithms for Minimization Without Derivatives"
// The bracket is [a, b] and the three lowest points are (u,fu), (v,fv), (w, fw)
// Note that a or b may be u, v, or w.
private static Extremum FindMinimum(
Functor f,
double a, double b,
double u, double fu, double v, double fv, double w, double fw,
EvaluationSettings settings
)
{
double tol = 0.0; double fpp = Double.NaN;
while (f.EvaluationCount < settings.EvaluationBudget)
{
Debug.WriteLine(String.Format("n={0} tol={1}", f.EvaluationCount, tol));
Debug.WriteLine(String.Format("[{0} f({1})={2} f({3})={4} f({5})={6} {7}]", a, u, fu, v, fv, w, fw, b));
// While a < u < b is guaranteed, a < v, w < b is not guaranteed, since the bracket can sometimes be made tight enough to exclude v or w.
// For example, if u < v < w, then we can set b = v, placing w outside the bracket.
Debug.Assert(a < b);
Debug.Assert((a <= u) && (u <= b));
Debug.Assert((fu <= fv) && (fv <= fw));
// Expected final situation is a<tol><tol>u<tol><tol>b, leaving no point left to evaluate that is not within tol of an existing point.
if ((b - a) <= 4.0 * tol)
{
return(new Extremum(u, fu, fpp, f.EvaluationCount, settings));
}
double x; ParabolicFit(u, fu, v, fv, w, fw, out x, out fpp);
Debug.WriteLine(String.Format("parabolic x={0} f''={1}", x, fpp));
if (Double.IsNaN(fpp) || (fpp <= 0.0) || (x < a) || (x > b))
{
// the parabolic fit didn't work out, so do a golden section reduction instead
// to get the most reduction of the bracket, pick the larger of au and ub
// for self-similarity, pick a point inside it that divides it into two segments in the golden section ratio,
// i.e. 0.3820 = \frac{1}{\phi + 1} and 0.6180 = \frac{\phi}{\phi+1}
// put the smaller segment closer to u so that x is closer to u, the best minimum so far
double au = u - a;
double ub = b - u;
if (au > ub)
{
x = u - au / (AdvancedMath.GoldenRatio + 1.0);
}
else
{
x = u + ub / (AdvancedMath.GoldenRatio + 1.0);
}
Debug.WriteLine(String.Format("golden section x={0}", x));
}
// ensure we don't evaluate within tolerance of an existing point
if (Math.Abs(x - u) < tol)
{
Debug.WriteLine(String.Format("shift from u (x={0})", x)); x = (x > u) ? u + tol : u - tol;
}
if ((x - a) < tol)
{
Debug.WriteLine(String.Format("shift from a (x={0})", x)); x = a + tol;
}
if ((b - x) < tol)
{
Debug.WriteLine(String.Format("shift from b (x={0})", x)); x = b - tol;
}
// evaluate the function at the new point x
double fx = f.Evaluate(x);
Debug.WriteLine(String.Format("f({0}) = {1}", x, fx));
Debug.WriteLine(String.Format("delta={0}", fu - fx));
// update a, b and u, v, w based on new point x
if (fx < fu)
{
// the new point is lower than all the others; this is success
// u now becomes a bracket point
if (u < x)
{
a = u;
}
else
{
b = u;
}
// x -> u -> v -> w
w = v; fw = fv;
v = u; fv = fu;
u = x; fu = fx;
}
else
{
// x now becomes a bracket point
if (x < u)
{
a = x;
}
else
{
b = x;
}
if (fx < fv)
{
// the new point is higher than u, but still lower than v and w
// this isn't what we expected, but we have lower points that before
// x -> v -> w
w = v; fw = fv;
v = x; fv = fx;
}
else if (fx < fw)
{
// x -> w
w = x; fw = fx;
}
else
{
// the new point is higher than all our other points; this is the worst case
// we might still want to replace w with x because
// (i) otherwise a parabolic fit will reproduce the same x and
// (ii) w is quite likely far outside the new bracket and not telling us much about the behavior near u
// w = x; fw = fx;
// but tests with Rosenbrock function indicate this increases evaluation count
Debug.WriteLine("bad point");
}
}
// if the user has specified a tollerance, use it
if ((settings.RelativePrecision > 0.0 || settings.AbsolutePrecision > 0.0))
{
tol = Math.Max(Math.Abs(u) * settings.RelativePrecision, settings.AbsolutePrecision);
}
else
{
// otherwise, try to get the tollerance from the curvature
if (fpp > 0.0)
{
tol = Math.Sqrt(2.0 * Global.Accuracy * (Math.Abs(fu) + Global.Accuracy) / fpp);
}
else
{
// but if we don't have a useable curvature either, wing it
if (tol == 0.0)
{
tol = Math.Sqrt(Global.Accuracy);
}
}
}
}
throw new NonconvergenceException();
}
// the drivers
private static IntegrationResult Integrate_Adaptive(IAdaptiveIntegrator integrator, EvaluationSettings s)
{
LinkedList <IAdaptiveIntegrator> list = new LinkedList <IAdaptiveIntegrator>();
list.AddFirst(integrator);
int n = integrator.EvaluationCount;
while (true)
{
// go through the intervals, adding estimates (and errors)
// and noting which contributes the most error
// keep track of the total value and uncertainty
UncertainValue vTotal = new UncertainValue();
//double v = 0.0;
//double u = 0.0;
// keep track of which node contributes the most error
LinkedListNode <IAdaptiveIntegrator> maxNode = null;
double maxError = 0.0;
LinkedListNode <IAdaptiveIntegrator> node = list.First;
while (node != null)
{
IAdaptiveIntegrator i = node.Value;
UncertainValue v = i.Estimate;
vTotal += v;
//UncertainValue uv = i.Estimate;
//v += uv.Value;
//u += uv.Uncertainty;
if (v.Uncertainty > maxError)
{
maxNode = node;
maxError = v.Uncertainty;
}
node = node.Next;
}
// if our error is small enough, return
if ((vTotal.Uncertainty <= Math.Abs(vTotal.Value) * s.RelativePrecision) || (vTotal.Uncertainty <= s.AbsolutePrecision))
{
return(new IntegrationResult(vTotal, n, s));
}
//if ((vTotal.Uncertainty <= Math.Abs(vTotal.Value) * s.RelativePrecision) || (vTotal.Uncertainty <= s.AbsolutePrecision)) {
// return (new IntegrationResult(vTotal.Value, n));
//}
// if our evaluation count is too big, throw
if (n > s.EvaluationBudget)
{
throw new NonconvergenceException();
}
// subdivide the interval with the largest error
IEnumerable <IAdaptiveIntegrator> divisions = maxNode.Value.Divide();
foreach (IAdaptiveIntegrator division in divisions)
{
list.AddBefore(maxNode, division);
n += division.EvaluationCount;
//v2 += division.Estimate;
}
list.Remove(maxNode);
}
}
private static MultiExtremum FindLocalExtremum(Func <IList <double>, double> function, IList <double> start, EvaluationSettings settings, bool negate)
{
MultiFunctor f = new MultiFunctor(function, negate);
// Pick an initial radius; we need to do this better.
/*
* double s = Double.MaxValue;
* foreach (double x in start) s = Math.Min((Math.Abs(x) + 1.0 / 8.0) / 8.0, s);
*/
double s = 0.0;
foreach (double x in start)
{
s += (Math.Abs(x) + 1.0 / 4.0) / 4.0;
}
s = s / start.Count;
//double s = 0.2;
Debug.WriteLine("s={0}", s);
return(FindMinimum_ModelTrust(f, start, s, settings));
}
