/// <summary>
/// Adaptive approximation of the definite integral by the trapezium rule.
/// </summary>
/// <param name="f">The analytic smooth function to integrate.</param>
/// <param name="intervalBegin">Where the interval starts, inclusive and finite.</param>
/// <param name="intervalEnd">Where the interval stops, inclusive and finite.</param>
/// <param name="levelAbscissas">Abscissa vector per level provider.</param>
/// <param name="levelWeights">Weight vector per level provider.</param>
/// <param name="levelOneStep">First Level Step</param>
/// <param name="targetRelativeError">The expected relative accuracy of the approximation.</param>
/// <returns>Approximation of the finite integral in the given interval.</returns>
public static double IntegrateAdaptiveTransformedOdd(
Func <double, double> f,
double intervalBegin, double intervalEnd,
IEnumerable <double[]> levelAbscissas, IEnumerable <double[]> levelWeights,
double levelOneStep, double targetRelativeError)
{
if (f == null)
{
throw new ArgumentNullException(nameof(f));
}
if (levelAbscissas == null)
{
throw new ArgumentNullException(nameof(levelAbscissas));
}
if (levelWeights == null)
{
throw new ArgumentNullException(nameof(levelWeights));
}
double linearSlope = 0.5 * (intervalEnd - intervalBegin);
double linearOffset = 0.5 * (intervalEnd + intervalBegin);
targetRelativeError /= 5 * linearSlope;
using (var abcissasIterator = levelAbscissas.GetEnumerator())
using (var weightsIterator = levelWeights.GetEnumerator())
{
double step = levelOneStep;
// First Level
abcissasIterator.MoveNext();
weightsIterator.MoveNext();
double[] abcissasL1 = abcissasIterator.Current;
double[] weightsL1 = weightsIterator.Current;
double sum = f(linearOffset) * weightsL1[0];
for (int i = 1; i < abcissasL1.Length; i++)
{
sum += weightsL1[i] * (f((linearSlope * abcissasL1[i]) + linearOffset) + f(-(linearSlope * abcissasL1[i]) + linearOffset));
}
sum *= step;
// Additional Levels
double previousDelta = double.MaxValue;
for (int level = 1; abcissasIterator.MoveNext() && weightsIterator.MoveNext(); level++)
{
double[] abcissas = abcissasIterator.Current;
double[] weights = weightsIterator.Current;
double midpointsum = 0;
for (int i = 0; i < abcissas.Length; i++)
{
midpointsum += weights[i] * (f((linearSlope * abcissas[i]) + linearOffset) + f(-(linearSlope * abcissas[i]) + linearOffset));
}
midpointsum *= step;
sum = 0.5 * (sum + midpointsum);
step *= 0.5;
double delta = Math.Abs(sum - midpointsum);
if (level == 1)
{
previousDelta = delta;
continue;
}
double r = Math.Log(delta) / Math.Log(previousDelta);
previousDelta = delta;
if (r > 1.9 && r < 2.1)
{
// convergence region
delta = Math.Sqrt(delta);
}
if (sum.AlmostEqualNormRelative(midpointsum, delta, targetRelativeError))
{
break;
}
}
return(sum * linearSlope);
}
}
/// <summary>Find a solution of the equation f(x)=0.</summary>
/// <param name="f">The function to find roots from.</param>
/// <param name="lowerBound">The low value of the range where the root is supposed to be.</param>
/// <param name="upperBound">The high value of the range where the root is supposed to be.</param>
/// <param name="accuracy">Desired accuracy. The root will be refined until the accuracy or the maximum number of iterations is reached.</param>
/// <param name="maxIterations">Maximum number of iterations. Usually 100.</param>
/// <param name="root">The root that was found, if any. Undefined if the function returns false.</param>
/// <returns>True if a root with the specified accuracy was found, else false.</returns>
public static bool TryFindRoot(Func <double, double> f, double lowerBound, double upperBound, double accuracy, int maxIterations, out double root)
{
double fmin = f(lowerBound);
double fmax = f(upperBound);
double froot = fmax;
double d = 0.0, e = 0.0;
root = upperBound;
double xMid = double.NaN;
// Root must be bracketed.
if (Math.Sign(fmin) == Math.Sign(fmax))
{
return(false);
}
for (int i = 0; i <= maxIterations; i++)
{
// adjust bounds
if (Math.Sign(froot) == Math.Sign(fmax))
{
upperBound = lowerBound;
fmax = fmin;
e = d = root - lowerBound;
}
if (Math.Abs(fmax) < Math.Abs(froot))
{
lowerBound = root;
root = upperBound;
upperBound = lowerBound;
fmin = froot;
froot = fmax;
fmax = fmin;
}
// convergence check
double xAcc1 = 2.0 * Precision.DoublePrecision * Math.Abs(root) + 0.5 * accuracy;
double xMidOld = xMid;
xMid = (upperBound - root) / 2.0;
if (Math.Abs(xMid) <= xAcc1 || froot.AlmostEqualNormRelative(0, froot, accuracy))
{
return(true);
}
if (xMid == xMidOld)
{
// accuracy not sufficient, but cannot be improved further
return(false);
}
if (Math.Abs(e) >= xAcc1 && Math.Abs(fmin) > Math.Abs(froot))
{
// Attempt inverse quadratic interpolation
double s = froot / fmin;
double p;
double q;
if (lowerBound.AlmostEqualRelative(upperBound))
{
p = 2.0 * xMid * s;
q = 1.0 - s;
}
else
{
q = fmin / fmax;
double r = froot / fmax;
p = s * (2.0 * xMid * q * (q - r) - (root - lowerBound) * (r - 1.0));
q = (q - 1.0) * (r - 1.0) * (s - 1.0);
}
if (p > 0.0)
{
// Check whether in bounds
q = -q;
}
p = Math.Abs(p);
if (2.0 * p < Math.Min(3.0 * xMid * q - Math.Abs(xAcc1 * q), Math.Abs(e * q)))
{
// Accept interpolation
e = d;
d = p / q;
}
else
{
// Interpolation failed, use bisection
d = xMid;
e = d;
}
}
else
{
// Bounds decreasing too slowly, use bisection
d = xMid;
e = d;
}
lowerBound = root;
fmin = froot;
if (Math.Abs(d) > xAcc1)
{
root += d;
}
else
{
root += Sign(xAcc1, xMid);
}
froot = f(root);
}
return(false);
}