/// <summary>
/// The Faddeeva function or Kramp function is a scaled complex complementary error function.
/// </summary>
/// <param name="z">The argument z.</param>
/// <returns>The evaluated value.</returns>
public static ScalarValue Faddeeva(ScalarValue z)
{
if (z.Im < 0.0)
{
return(2.0 * (-z * z).Exp() - Faddeeva(-z));
}
if (z.Re < 0.0)
{
return(Faddeeva(-z.Conjugate()).Conjugate());
}
var r = z.Abs();
if (r < 2.0)
{
return((-z * z).Exp() * (1.0 - Erf_Series(-ScalarValue.I * z)));
}
else if ((z.Im < 0.1) && (z.Re < 30.0))
{
return(Taylor(new ScalarValue(z.Re), Math.Exp(-z.Re * z.Re) + 2.0 * Dawson.DawsonIntegral(z.Re) / Helpers.SqrtPI * ScalarValue.I, new ScalarValue(0.0, z.Im)));
}
else if (r > 7.0)
{
return(ContinuedFraction(z));
}
return(Weideman(z));
}
static ScalarValue LambertW(ScalarValue x)
{
var EI = 1.0 / Math.E;
// use an initial approximation
var W = ScalarValue.Zero;
var abs = x.Abs();
if (abs < EI / 2.0)
{
W = SeriesSmall(x);
if ((x + EI).Abs() < 1e-6)
{
return(W);
}
}
else if (abs < EI)
{
W = SeriesZero(x);
}
else if (abs > Math.E)
{
W = SeriesLarge(x);
}
else
{
W = new ScalarValue(0.5);
}
return(Halley(x, W));
}
/// <summary>
/// Computes the complex Riemann Zeta function.
/// </summary>
/// <param name="s">The complex argument.</param>
/// <returns>The (in general complex) value.</returns>
public static ScalarValue RiemannZeta(ScalarValue s)
{
if (s == 1.0)
{
return(new ScalarValue(Double.PositiveInfinity));
}
else if (s.Re >= 0.0)
{
return(RiemannZetaGt0(s));
}
//See real zeta function for more information
var zeta_one_minus_s = RiemannZeta1msLt0(s);
var sin_term = (0.5 * Math.PI * s).Sin() / Math.PI;
var sabs = s.Abs();
if (sin_term == 0.0)
{
return(ScalarValue.Zero);
}
else if (sabs < 170)
{
//See below
int n = (int)Math.Floor(sabs / 10.0);
var p = new ScalarValue(2.0 * Math.PI).Pow(s + 10.0 * n) / Helpers.TwoPIpow[n];
var g = Gamma.LinearGamma(1.0 - s);
return(p * g * sin_term * zeta_one_minus_s);
}
throw new YAMPNumericOverflowException("Zeta");
}
public static ScalarValue Gegenbauer(Int32 n, Double alpha, ScalarValue x)
{
if (n < 0)
throw new YAMPArgumentRangeException("n");
if (alpha <= 0)
throw new YAMPArgumentRangeException("alpha", 0);
if (x.Abs() > 1.0)
throw new YAMPArgumentRangeException("x", -1, 1);
var C0 = ScalarValue.One;
if (n == 0)
{
return C0;
}
var C1 = 2.0 * alpha * x;
if (n == 1)
{
return C1;
}
for (var k = 2; k <= n; k++)
{
var Ck = (2 * x * (k + alpha - 1) * C1 - (k + 2 * alpha - 2) * C0) / k;
C0 = C1;
C1 = Ck;
}
return C1;
}
static ScalarValue LambertW(ScalarValue x)
{
var EI = 1.0 / Math.E;
// use an initial approximation
var W = ScalarValue.Zero;
var abs = x.Abs();
if (abs < EI / 2.0)
{
W = SeriesSmall(x);
if ((x + EI).Abs() < 1e-6)
{
return W;
}
}
else if (abs < EI)
{
W = SeriesZero(x);
}
else if (abs > Math.E)
{
W = SeriesLarge(x);
}
else
{
W = new ScalarValue(0.5);
}
return Halley(x, W);
}
/// <summary>
/// Computes the complex error function erf(z).
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The value of erf(z).</returns>
public static ScalarValue Erf(ScalarValue z)
{
if (z.Abs() < 4.0)
{
return(Erf_Series(z));
}
else if (z.Re < 0.0)
{
return((-z * z).Exp() * Faddeeva(-ScalarValue.I * z) - 1.0);
}
return(1.0 - (-z * z).Exp() * Faddeeva(ScalarValue.I * z));
}
/// <summary>
/// Computes the complex (log) gamma function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The evaluated value.</returns>
public static ScalarValue LogGamma(ScalarValue z)
{
if (z.Re < 0.0)
{
return(new ScalarValue(double.PositiveInfinity));
}
if (z.Abs() > 15.0)
{
return(LogGamma_Stirling(z));
}
return(LanczosLogGamma(z));
}
public static ScalarValue Gegenbauer(Int32 n, Double alpha, ScalarValue x)
{
if (n < 0)
{
throw new YAMPArgumentRangeException("n");
}
if (alpha <= 0)
{
throw new YAMPArgumentRangeException("alpha", 0);
}
if (x.Abs() > 1.0)
{
throw new YAMPArgumentRangeException("x", -1, 1);
}
var C0 = ScalarValue.One;
if (n == 0)
{
return(C0);
}
var C1 = 2.0 * alpha * x;
if (n == 1)
{
return(C1);
}
for (var k = 2; k <= n; k++)
{
var Ck = (2 * x * (k + alpha - 1) * C1 - (k + 2 * alpha - 2) * C0) / k;
C0 = C1;
C1 = Ck;
}
return(C1);
}
/// <summary>
/// Computes the complex dilogarithm function, also called Spence's function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The value Li<sub>2</sub>(z).</returns>
public static ScalarValue DiLog(ScalarValue z)
{
ScalarValue f;
double a0 = z.Abs();
if (a0 > 1.0)
{
// outside the unit disk, reflect into the unit disk
var ln = (-z).Ln();
f = -Math.PI * Math.PI / 6.0 - ln * ln / 2.0 - DiLog(1.0 / z);
}
else
{
// inside the unit disk...
if (a0 < 0.75)
{
f = DiLog0(z);
}
else if (z.Re < 0.0)
{
f = DiLog(z * z) / 2.0 - DiLog(-z);
}
else
{
var e = 1.0 - z;
f = e.Abs() < 0.5 ? DiLog1(e) : DiLog_Log_Series(z);
}
}
if ((z.Re > 1.0) && (Math.Sign(f.Im) != Math.Sign(z.Im)))
{
f = f.Conjugate();
}
return(f);
}
/// <summary>
/// Computes the complex dilogarithm function, also called Spence's function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The value Li<sub>2</sub>(z).</returns>
public static ScalarValue DiLog(ScalarValue z)
{
var f = default(ScalarValue);
var a0 = z.Abs();
if (a0 > 1.0)
{
// outside the unit disk, reflect into the unit disk
var ln = (-z).Ln();
f = -Math.PI * Math.PI / 6.0 - ln * ln / 2.0 - DiLog(1.0 / z);
}
else
{
// inside the unit disk...
if (a0 < 0.75)
{
f = DiLog0(z);
}
else if (z.Re < 0.0)
{
f = DiLog(z * z) / 2.0 - DiLog(-z);
}
else
{
var e = 1.0 - z;
f = e.Abs() < 0.5 ? DiLog1(e) : DiLog_Log_Series(z);
}
}
if ((z.Re > 1.0) && (Math.Sign(f.Im) != Math.Sign(z.Im)))
{
f = f.Conjugate();
}
return f;
}
/// <summary>
/// Computes the complex Riemann Zeta function.
/// </summary>
/// <param name="s">The complex argument.</param>
/// <returns>The (in general complex) value.</returns>
public static ScalarValue RiemannZeta(ScalarValue s)
{
if (s == 1.0)
{
return new ScalarValue(Double.PositiveInfinity);
}
else if (s.Re >= 0.0)
{
return RiemannZetaGt0(s);
}
//See real zeta function for more information
var zeta_one_minus_s = RiemannZeta1msLt0(s);
var sin_term = (0.5 * Math.PI * s).Sin() / Math.PI;
var sabs = s.Abs();
if (sin_term == 0.0)
{
return ScalarValue.Zero;
}
else if (sabs < 170)
{
//See below
int n = (int)Math.Floor(sabs / 10.0);
var p = new ScalarValue(2.0 * Math.PI).Pow(s + 10.0 * n) / Helpers.TwoPIpow[n];
var g = Gamma.LinearGamma(1.0 - s);
return p * g * sin_term * zeta_one_minus_s;
}
throw new YAMPNumericOverflowException("Zeta");
}
public static ScalarValue HurwitzZeta(ScalarValue s, ScalarValue q)
{
if (s.Re <= 1.0)
{
throw new YAMPArgumentRangeException("s", 1.0);
}
if (q.Re <= 0.0)
{
throw new YAMPArgumentRangeException("q", 0.0);
}
var max_bits = 54.0;
var ln_term0 = -s *q.Log();
var qabs = q.Abs();
var sabs = s.Abs();
var ss = s;
if ((sabs > max_bits && qabs < 1.0) || (sabs > 0.5 * max_bits && qabs < 0.25))
{
return(q.Pow(-ss));
}
else if (sabs > 0.5 * max_bits && qabs < 1.0)
{
var p1 = q.Pow(-ss);
var p2 = (q / (1.0 + q)).Pow(ss);
var p3 = (q / (2.0 + q)).Pow(ss);
return(p1 * (1.0 + p2 + p3));
}
/* Euler-Maclaurin summation formula
* [Moshier, p. 400, with several typo corrections]
*/
const int jmax = 12;
const int kmax = 10;
var pmax = (kmax + q).Pow(-ss);
var scp = s;
var pcp = pmax / (kmax + q);
var ans = pmax * ((kmax + q) / (s - 1.0) + 0.5);
for (var k = 0; k < kmax; k++)
{
ans += (k + q).Pow(-ss);
}
for (var j = 0; j <= jmax; j++)
{
var delta = COEFFICIENTS[j + 1] * scp * pcp;
ans += delta;
if ((delta / ans).Abs() < 0.5 * Double.Epsilon)
{
break;
}
scp *= (s + 2 * j + 1) * (s + 2 * j + 2);
pcp /= (kmax + q) * (kmax + q);
}
return(ans);
}
/// <summary>
/// The Faddeeva function or Kramp function is a scaled complex complementary error function.
/// </summary>
/// <param name="z">The argument z.</param>
/// <returns>The evaluated value.</returns>
public static ScalarValue Faddeeva(ScalarValue z)
{
if (z.Im < 0.0)
return 2.0 * (-z * z).Exp() - Faddeeva(-z);
if (z.Re < 0.0)
return Faddeeva(-z.Conjugate()).Conjugate();
var r = z.Abs();
if (r < 2.0)
return (-z * z).Exp() * (1.0 - Erf_Series(-ScalarValue.I * z));
else if ((z.Im < 0.1) && (z.Re < 30.0))
return Taylor(new ScalarValue(z.Re), Math.Exp(-z.Re * z.Re) + 2.0 * Dawson.DawsonIntegral(z.Re) / Helpers.SqrtPI * ScalarValue.I, new ScalarValue(0.0, z.Im));
else if (r > 7.0)
return ContinuedFraction(z);
return Weideman(z);
}
static ScalarValue Zernike(int n, int m, ScalarValue rho)
{
if (n < 0)
{
throw new YAMPArgumentRangeException("n");
}
if ((m < 0) || (m > n))
{
throw new YAMPArgumentRangeException("m", "n >= m >= 0");
}
// n and m have the same parity
if ((n - m) % 2 != 0)
{
return(ScalarValue.Zero);
}
// R00
if (n == 0)
{
return(ScalarValue.One);
}
var absrho = rho.Abs();
if ((absrho < 0.0) || (absrho > 1.0))
{
throw new YAMPNotConvergedException("zernike");
}
// R^{m}_m
var r2 = rho.Pow(new ScalarValue(m));
if (n == m)
{
return(r2);
}
// R^{m+1}_{m+1}
var k = m;
var r1 = r2 * rho;
while (true)
{
k += 2;
// *
// \
// * recurrence involving two lesser m's
// /
// *
// 2n R^{m+1}_{n-1} = (n+m) R^{m}_{n-2} + (n-m) R^{m}_{n}
var r0 = ((2 * k) * rho * r1 - (k + m) * r2) / (k - m);
if (k == n)
{
return(r0);
}
// *
// /
// * recurrence involving two greater m's
// \
// *
//
var rp = (2 * (k + 1) * rho * r0 - (k - m) * r1) / (k + m + 2);
r2 = r0;
r1 = rp;
}
}
/// <summary>
/// Computes the complex (log) gamma function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The evaluated value.</returns>
public static ScalarValue LogGamma(ScalarValue z)
{
if (z.Re < 0.0)
return new ScalarValue(double.PositiveInfinity);
if (z.Abs() > 15.0)
return LogGamma_Stirling(z);
return LanczosLogGamma(z);
}
static ScalarValue Zernike(int n, int m, ScalarValue rho)
{
if (n < 0)
throw new YAMPArgumentRangeException("n");
if ((m < 0) || (m > n))
throw new YAMPArgumentRangeException("m", "n >= m >= 0");
// n and m have the same parity
if ((n - m) % 2 != 0)
{
return ScalarValue.Zero;
}
// R00
if (n == 0)
{
return ScalarValue.One;
}
var absrho = rho.Abs();
if ((absrho < 0.0) || (absrho > 1.0))
throw new YAMPNotConvergedException("zernike");
// R^{m}_m
var r2 = rho.Pow(new ScalarValue(m));
if (n == m)
{
return r2;
}
// R^{m+1}_{m+1}
var k = m;
var r1 = r2 * rho;
while (true)
{
k += 2;
// *
// \
// * recurrence involving two lesser m's
// /
// *
// 2n R^{m+1}_{n-1} = (n+m) R^{m}_{n-2} + (n-m) R^{m}_{n}
var r0 = ((2 * k) * rho * r1 - (k + m) * r2) / (k - m);
if (k == n)
{
return r0;
}
// *
// /
// * recurrence involving two greater m's
// \
// *
//
var rp = (2 * (k + 1) * rho * r0 - (k - m) * r1) / (k + m + 2);
r2 = r0;
r1 = rp;
}
}
public static ScalarValue HurwitzZeta(ScalarValue s, ScalarValue q)
{
if (s.Re <= 1.0)
throw new YAMPArgumentRangeException("s", 1.0);
if (q.Re <= 0.0)
throw new YAMPArgumentRangeException("q", 0.0);
var max_bits = 54.0;
var ln_term0 = -s * q.Log();
var qabs = q.Abs();
var sabs = s.Abs();
var ss = s;
if ((sabs > max_bits && qabs < 1.0) || (sabs > 0.5 * max_bits && qabs < 0.25))
{
return q.Pow(-ss);
}
else if (sabs > 0.5 * max_bits && qabs < 1.0)
{
var p1 = q.Pow(-ss);
var p2 = (q / (1.0 + q)).Pow(ss);
var p3 = (q / (2.0 + q)).Pow(ss);
return p1 * (1.0 + p2 + p3);
}
/* Euler-Maclaurin summation formula
* [Moshier, p. 400, with several typo corrections]
*/
const int jmax = 12;
const int kmax = 10;
var pmax = (kmax + q).Pow(-ss);
var scp = s;
var pcp = pmax / (kmax + q);
var ans = pmax * ((kmax + q) / (s - 1.0) + 0.5);
for (var k = 0; k < kmax; k++)
{
ans += (k + q).Pow(-ss);
}
for (var j = 0; j <= jmax; j++)
{
var delta = COEFFICIENTS[j + 1] * scp * pcp;
ans += delta;
if ((delta / ans).Abs() < 0.5 * Double.Epsilon)
{
break;
}
scp *= (s + 2 * j + 1) * (s + 2 * j + 2);
pcp /= (kmax + q) * (kmax + q);
}
return ans;
}
/// <summary>
/// Computes the complex error function erf(z).
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The value of erf(z).</returns>
public static ScalarValue Erf(ScalarValue z)
{
if (z.Abs() < 4.0)
return Erf_Series(z);
else if (z.Re < 0.0)
return (-z * z).Exp() * Faddeeva(-ScalarValue.I * z) - 1.0;
return 1.0 - (-z * z).Exp() * Faddeeva(ScalarValue.I * z);
}
