/// <summary>
/// Performs adaptive Gauss-Kronrod quadrature on function f over the range (a,b)
/// </summary>
/// <param name="f">The analytic smooth function to integrate</param>
/// <param name="intervalBegin">Where the interval starts</param>
/// <param name="intervalEnd">Where the interval stops</param>
/// <param name="error">The difference between the (N-1)/2 point Gauss approximation and the N-point Gauss-Kronrod approximation</param>
/// <param name="L1Norm">The L1 norm of the result, if there is a significant difference between this and the returned value, then the result is likely to be ill-conditioned.</param>
/// <param name="targetRelativeError">The maximum relative error in the result</param>
/// <param name="maximumDepth">The maximum number of interval splittings permitted before stopping</param>
/// <param name="order">The number of Gauss-Kronrod points. Pre-computed for 15, 21, 31, 41, 51 and 61 points</param>
public static double Integrate(Func <double, double> f, double intervalBegin, double intervalEnd, out double error, out double L1Norm, double targetRelativeError = 1E-10, int maximumDepth = 15, int order = 15)
{
// Formula used for variable subsitution from
// 1. Shampine, L. F. (2008). Vectorized adaptive quadrature in MATLAB. Journal of Computational and Applied Mathematics, 211(2), 131-140.
// 2. quadgk.m, GNU Octave
if (f == null)
{
throw new ArgumentNullException(nameof(f));
}
if (intervalBegin > intervalEnd)
{
return(-Integrate(f, intervalEnd, intervalBegin, out error, out L1Norm, targetRelativeError, maximumDepth, order));
}
GaussPointPair gaussKronrodPoint = GaussKronrodPointFactory.GetGaussPoint(order);
// (-oo, oo) => [-1, 1]
//
// integral_(-oo)^(oo) f(x) dx = integral_(-1)^(1) f(g(t)) g'(t) dt
// g(t) = t / (1 - t^2)
// g'(t) = (1 + t^2) / (1 - t^2)^2
if ((intervalBegin < double.MinValue) && (intervalEnd > double.MaxValue))
{
Func <double, double> u = (t) => f(t / (1 - t * t)) * (1 + t * t) / ((1 - t * t) * (1 - t * t));
return(recursive_adaptive_integrate(u, -1, 1, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint));
}
// [a, oo) => [0, 1]
//
// integral_(a)^(oo) f(x) dx = integral_(0)^(oo) f(a + t^2) 2 t dt
// = integral_(0)^(1) f(a + g(s)^2) 2 g(s) g'(s) ds
// g(s) = s / (1 - s)
// g'(s) = 1 / (1 - s)^2
else if (intervalEnd > double.MaxValue)
{
Func <double, double> u = (s) => 2 * s * f(intervalBegin + (s / (1 - s)) * (s / (1 - s))) / ((1 - s) * (1 - s) * (1 - s));
return(recursive_adaptive_integrate(u, 0, 1, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint));
}
// (-oo, b] => [-1, 0]
//
// integral_(-oo)^(b) f(x) dx = -integral_(-oo)^(0) f(b - t^2) 2 t dt
// = -integral_(-1)^(0) f(b - g(s)^2) 2 g(s) g'(s) ds
// g(s) = s / (1 + s)
// g'(s) = 1 / (1 + s)^2
else if (intervalBegin < double.MinValue)
{
Func <double, double> u = (s) => - 2 * s * f(intervalEnd - s / (1 + s) * (s / (1 + s))) / ((1 + s) * (1 + s) * (1 + s));
return(recursive_adaptive_integrate(u, -1, 0, maximumDepth, targetRelativeError, 0, out error, out L1Norm, gaussKronrodPoint));
}
// [a, b] => [-1, 1]
//
// integral_(a)^(b) f(x) dx = integral_(-1)^(1) f(g(t)) g'(t) dt
// g(t) = (b - a) * t * (3 - t^2) / 4 + (b + a) / 2
// g'(t) = 3 / 4 * (b - a) * (1 - t^2)
else
{
Func <double, double> u = (t) => f((intervalEnd - intervalBegin) / 4 * t * (3 - t * t) + (intervalEnd + intervalBegin) / 2) * 3 * (intervalEnd - intervalBegin) / 4 * (1 - t * t);
return(recursive_adaptive_integrate(u, -1, 1, maximumDepth, targetRelativeError, 0d, out error, out L1Norm, gaussKronrodPoint));
}
}
public GaussKronrodRule(int order)
{
_gaussKronrodPoint = GaussKronrodPointFactory.GetGaussPoint(order);
}
static Complex contour_recursive_adaptive_integrate(Func <double, Complex> f, double a, double b, int maxLevels, double relTol, double absTol, out double error, out double L1, GaussPointPair gaussKronrodPoint)
{
double error_local;
double mean = (b + a) / 2;
double scale = (b - a) / 2;
var r1 = contour_integrate_non_adaptive_m1_1((x) => f(scale * x + mean), out error_local, out L1, gaussKronrodPoint);
var estimate = scale * r1;
var tmp = estimate * relTol;
var absTol1 = Complex.Abs(tmp);
if (absTol == 0)
{
absTol = absTol1;
}
if (maxLevels > 0 && (absTol1 < error_local) && (absTol < error_local))
{
double mid = (a + b) / 2d;
double L1_local;
estimate = contour_recursive_adaptive_integrate(f, a, mid, maxLevels - 1, relTol, absTol / 2, out error, out L1, gaussKronrodPoint);
estimate += contour_recursive_adaptive_integrate(f, mid, b, maxLevels - 1, relTol, absTol / 2, out error_local, out L1_local, gaussKronrodPoint);
error += error_local;
L1 += L1_local;
return(estimate);
}
L1 *= scale;
error = error_local;
return(estimate);
}
static Complex contour_integrate_non_adaptive_m1_1(Func <double, Complex> f, out double error, out double pL1, GaussPointPair gaussKronrodPoint)
{
int gaussStart = 2;
int kronrodStart = 1;
int gaussOrder = (gaussKronrodPoint.Order - 1) / 2;
Complex kronrodResult;
Complex gaussResult = new Complex();
Complex fp, fm;
var KAbscissa = gaussKronrodPoint.Abscissas;
var KWeights = gaussKronrodPoint.Weights;
var GWeights = gaussKronrodPoint.SecondWeights;
if (gaussOrder.IsOdd())
{
fp = f(0);
kronrodResult = fp * KWeights[0];
gaussResult += fp * GWeights[0];
}
else
{
fp = f(0);
kronrodResult = fp * KWeights[0];
gaussStart = 1;
kronrodStart = 2;
}
double L1 = Complex.Abs(kronrodResult);
for (int i = gaussStart; i < KAbscissa.Length; i += 2)
{
fp = f(KAbscissa[i]);
fm = f(-KAbscissa[i]);
kronrodResult += (fp + fm) * KWeights[i];
L1 += (Complex.Abs(fp) + Complex.Abs(fm)) * KWeights[i];
gaussResult += (fp + fm) * GWeights[i / 2];
}
for (int i = kronrodStart; i < KAbscissa.Length; i += 2)
{
fp = f(KAbscissa[i]);
fm = f(-KAbscissa[i]);
kronrodResult += (fp + fm) * KWeights[i];
L1 += (Complex.Abs(fp) + Complex.Abs(fm)) * KWeights[i];
}
pL1 = L1;
error = Math.Max(Complex.Abs(kronrodResult - gaussResult), Complex.Abs(kronrodResult * Precision.MachineEpsilon * 2d));
return(kronrodResult);
}
static double integrate_non_adaptive_m1_1(Func <double, double> f, out double error, out double pL1, GaussPointPair gaussKronrodPoint)
{
int gaussStart = 2;
int kronrodStart = 1;
int gaussOrder = (gaussKronrodPoint.Order - 1) / 2;
double kronrodResult = 0d;
double gaussResult = 0d;
double fp, fm;
var KAbscissa = gaussKronrodPoint.Abscissas;
var KWeights = gaussKronrodPoint.Weights;
var GWeights = gaussKronrodPoint.SecondWeights;
if ((gaussOrder & 1) == 1)
{
fp = f(0);
kronrodResult = fp * KWeights[0];
gaussResult += fp * GWeights[0];
}
else
{
fp = f(0);
kronrodResult = fp * KWeights[0];
gaussStart = 1;
kronrodStart = 2;
}
double L1 = Math.Abs(kronrodResult);
for (int i = gaussStart; i < KAbscissa.Length; i += 2)
{
fp = f(KAbscissa[i]);
fm = f(-KAbscissa[i]);
kronrodResult += (fp + fm) * KWeights[i];
L1 += (Math.Abs(fp) + Math.Abs(fm)) * KWeights[i];
gaussResult += (fp + fm) * GWeights[i / 2];
}
for (int i = kronrodStart; i < KAbscissa.Length; i += 2)
{
fp = f(KAbscissa[i]);
fm = f(-KAbscissa[i]);
kronrodResult += (fp + fm) * KWeights[i];
L1 += (Math.Abs(fp) + Math.Abs(fm)) * KWeights[i];
}
pL1 = L1;
error = Math.Max(Math.Abs(kronrodResult - gaussResult), Math.Abs(kronrodResult * Precision.MachineEpsilon * 2d));
return(kronrodResult);
}
private static double recursive_adaptive_integrate(Func <double, double> f, double a, double b, int max_levels, double rel_tol, double abs_tol, out double error, out double L1, GaussPointPair gaussKronrodPoint)
{
double error_local;
double mean = (b + a) / 2;
double scale = (b - a) / 2;
var r1 = integrate_non_adaptive_m1_1((x) => f((scale * x) + mean), out error_local, out L1, gaussKronrodPoint);
var estimate = scale * r1;
var tmp = estimate * rel_tol;
var abs_tol1 = Math.Abs(tmp);
if (abs_tol == 0)
{
abs_tol = abs_tol1;
}
if ((max_levels > 0) && (abs_tol1 < error_local) && (abs_tol < error_local))
{
double mid = (a + b) / 2d;
double L1_local;
estimate = recursive_adaptive_integrate(f, a, mid, max_levels - 1, rel_tol, abs_tol / 2, out error, out L1, gaussKronrodPoint);
estimate += recursive_adaptive_integrate(f, mid, b, max_levels - 1, rel_tol, abs_tol / 2, out error_local, out L1_local, gaussKronrodPoint);
error += error_local;
L1 += L1_local;
return(estimate);
}
L1 *= scale;
error = error_local;
return(estimate);
}
