/// <summary>
/// Computes the complex Faddeeva function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The complex value of w(z).</returns>
/// <remarks>
/// <para>The Faddeeva function w(z) is related to the error function with a complex argument.</para>
/// <img src="../images/FaddeevaErfcRelation.png" />
/// <para>It also has an integral representation.</para>
/// <img src="../images/FaddeevaIntegral.png" />
/// <para>For purely imaginary values, it reduces to the complementary error function (<see cref="AdvancedMath.Erfc"/>).
/// For purely real values, it reduces to Dawson's integral (<see cref="AdvancedMath.Dawson"/>).</para>
/// <para>It appears in the computation of the Voigt line profile function V(x;σ,γ).</para>
/// <img src="../images/Voigt.png" />
/// <para>Near the origin, w(z) ≈ 1. To accurately determine w(z) - 1 in this region, use the <see cref="Erf"/>
/// function. Away from the origin near the large negative imaginary axis, the magnitude w(z) increases rapidly and
/// may overflow.</para>
/// <para>The image below shows the complex Faddeeva function near the origin, using domain coloring.</para>
/// <img src="../images/ComplexFaddeevaPlot.png" />
/// </remarks>
/// <seealso cref="AdvancedComplexMath.Erf"/>
/// <seealso cref="AdvancedMath.Erf" />
/// <seealso cref="AdvancedMath.Erfc" />
/// <seealso cref="AdvancedMath.Dawson"/>
/// <seealso href="http://en.wikipedia.org/wiki/Voigt_profile" />
public static Complex Faddeeva(Complex z)
{
// use reflection formulae to ensure that we are in the first quadrant
if (z.Im < 0.0)
{
return(2.0 * ComplexMath.Exp(-ComplexMath.Sqr(z)) - Faddeeva(-z));
}
if (z.Re < 0.0)
{
return(Faddeeva(-z.Conjugate).Conjugate);
}
double r = ComplexMath.Abs(z);
if (r < 2.0)
{
// use series for small z
return(ComplexMath.Exp(-ComplexMath.Sqr(z)) * (1.0 - Erf_Series(-ComplexMath.I * z)));
//return (Faddeeva_Series(z));
}
else if ((z.Im < 0.1) && (z.Re < 30.0))
{
// this is a special, awkward region
// along the real axis, Re{w(x)} ~ e^{-x^2}; the Weideman algorithm doesn't compute this small number
// well and the Laplace continued fraction misses it entirely; therefore very close to the real axis
// we will use an analytic result on the real axis and Taylor expand to where we need to go.
// unfortunately the Taylor expansion converges poorly for large x, so we drop this work-around near x~30,
// when this real part becomes too small to represent as a double anyway
double x = z.Re;
double y = z.Im;
return(Faddeeva_Taylor(new Complex(x, 0.0),
Math.Exp(-x * x) + 2.0 * AdvancedMath.Dawson(x) / Global.SqrtPI * ComplexMath.I,
new Complex(0.0, y)));
}
else if (r > 7.0)
{
// use Laplace continued fraction for large z
return(Faddeeva_ContinuedFraction(z));
}
else
{
// use Weideman algorithm for intermediate region
return(Faddeeva_Weideman(z));
}
}
/*
* private static double Fadeeva_Weideman_L = Math.Sqrt(20.0 / Math.Sqrt(2.0));
* private static double[] Faddeeva_Weideman_Coefficients = {
* 0.145204167876254758882, // 0
* 0.136392115479649636711,
* 0.112850364752776356308,
* 0.081814663814142472504,
* 0.051454522270816839073,
* 0.027588397192576112169, // 5
* 0.012224615682792673110,
* 0.004202799708104555100,
* 0.00094232196590738341762,
* 0.000024250951611669019436,
* -0.000075616164647923403908, // 10
* -0.000025048115578198088771,
* 1.35444636058911259445e-6,
* 3.1103803514939499680e-6,
* 4.3812443485181690079e-7,
* -3.1958626141517655223e-7, // 15
* -9.9512459304812101824e-8,
* 3.38225585699344840075e-8,
* 1.68263253379413070440e-8,
* -4.206345505308078461584e-9,
* -2.7027917878529318399463e-9 // 20
* };
*/
// The power series for the error function (DLMF 7.6.1)
// erf(z) = \frac{2}{\sqrt{\pi}} \sum_{k=0}^{\infty} \frac{ (-1)^k z^{2k + 1} }{(2k+1) k!}
// = \frac{2}{\sqrt{\pi}} \left[ z - z^3 / 3 + z^5 / 10 - \cdots \right]
// requires about 15 terms at |z| ~ 1, 30 terms at |z| ~ 2, 45 terms at |z| ~ 3
private static Complex Erf_Series(Complex z)
{
Complex zp = 2.0 / Global.SqrtPI * z;
Complex zz = -ComplexMath.Sqr(z);
Complex f = zp;
for (int k = 1; k < Global.SeriesMax; k++)
{
Complex f_old = f;
zp *= zz / k;
f += zp / (2 * k + 1);
if (f == f_old)
{
return(f);
}
}
throw new NonconvergenceException();
}
private static Complex Sum(Complex z)
{
Complex rzPower = 1.0 / z;
Complex rzSquared = ComplexMath.Sqr(rzPower);
Complex f = 0.5 * AdvancedIntegerMath.Bernoulli[1] * rzPower;
for (int k = 2; k < AdvancedIntegerMath.Bernoulli.Length; k++)
{
Complex f_old = f;
rzPower *= rzSquared;
f += AdvancedIntegerMath.Bernoulli[k] / ((2 * k) * (2 * k - 1)) * rzPower;
if (f == f_old)
{
return(f);
}
}
throw new NonconvergenceException();
}
public static Complex AiryAi_Series(Complex z)
{
Complex p = AdvancedMath.Ai0;
Complex q = AdvancedMath.AiPrime0 * z;
Complex f = p + q;
Complex z3 = ComplexMath.Sqr(z) * z;
for (int k = 0; k < Global.SeriesMax; k += 3)
{
Complex f_old = f;
p *= z3 / ((k + 2) * (k + 3));
q *= z3 / ((k + 3) * (k + 4));
f += p + q;
if (f == f_old)
{
return(f);
}
}
throw new NonconvergenceException();
}
/// <summary>
/// Computes the complex error function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The value of erf(z).</returns>
/// <remarks>
/// <para>This function is the analytic continuation of the error function (<see cref="AdvancedMath.Erf"/>) to the complex plane.</para>
/// <para>The image below shows the complex error function near the origin, using domain coloring.</para>
/// <img src="../images/ComplexErfPlot.png" />
/// <para>The complex error function is entire: it has no poles, cuts, or discontinuities anywhere in the complex plane.</para>
/// <para>For pure imaginary arguments, erf(z) reduces to the Dawson integral (<see cref="AdvancedMath.Dawson"/>).</para>
/// <para>Away from the origin near the real axis, the real part of erf(z) quickly approaches ±1. To accurately determine
/// the small difference erf(z) ∓ 1 in this region, use the <see cref="Faddeeva"/> function. Away from the origin near
/// the imaginary axis, the magnitude of erf(z) increases very quickly. Although erf(z) may overflow in this region, you
/// can still accurately determine the value of the product erf(z) exp(z<sup>2</sup>) using the <see cref="Faddeeva"/>
/// function.</para>
/// </remarks>
/// <seealso cref="AdvancedMath.Erf"/>
/// <seealso cref="AdvancedMath.Dawson"/>
/// <seealso cref="AdvancedComplexMath.Faddeeva"/>
public static Complex Erf(Complex z)
{
double r = ComplexMath.Abs(z);
if (r < 4.0)
{
// near the origin, use the series
return(Erf_Series(z));
}
else
{
// otherwise, just compute from Faddeva
if (z.Re < 0.0)
{
// since Fadddeeva blows up for negative z.Re, use erf(z) = -erf(-z)
return(ComplexMath.Exp(-ComplexMath.Sqr(z)) * Faddeeva(-ComplexMath.I * z) - 1.0);
}
else
{
return(1.0 - ComplexMath.Exp(-ComplexMath.Sqr(z)) * Faddeeva(ComplexMath.I * z));
}
// we don't do this near the origin beause we would loose accuracy in the very small real parts there by subtracting from 1
}
}
