/// <summary>
/// Computes Owen's T function for arbitrary H and A.
/// </summary>
///
/// <param name="h">Owen's T function argument H.</param>
/// <param name="a">Owen's T function argument A.</param>
///
/// <returns>The value of Owen's T function.</returns>
///
public static double Function(double h, double a)
{
double absa;
double absh;
double ah;
const double cut = 0.67;
double normah;
double normh;
double value;
absh = Math.Abs(h);
absa = Math.Abs(a);
ah = absa * absh;
if (absa <= 1.0)
{
value = Function(absh, absa, ah);
}
else if (absh <= cut)
{
value = 0.25 - (-0.5 + Normal.Function(absh)) * (-0.5 + Normal.Function(ah))
- Function(ah, 1.0 / absa, absh);
}
else
{
normh = Normal.Complemented(absh);
normah = Normal.Complemented(ah);
value = 0.5 * (normh + normah) - normh * normah
- Function(ah, 1.0 / absa, absh);
}
if (a < 0.0)
{
value = -value;
}
return(value);
}
/// <summary>
/// Owen's T function for a restricted range of parameters.
/// </summary>
///
/// <param name="h">Owen's T function argument H (where 0 <= H).</param>
/// <param name="a">Owen's T function argument A (where 0 <= A <= 1).</param>
/// <param name="ah">The value of A*H.</param>
///
/// <returns>The value of Owen's T function.</returns>
///
public static double Function(double h, double a, double ah)
{
double ai;
double aj;
double AS;
double dhs;
double dj;
double gj;
double hs;
int i;
int iaint;
int icode;
int ihint;
int ii;
int j;
int jj;
int m;
int maxii;
double normh;
double r;
const double rrtpi = 0.39894228040143267794;
const double rtwopi = 0.15915494309189533577;
double y;
double yi;
double z;
double zi;
double value = 0;
double vi;
double[] arange =
{
0.025, 0.09, 0.15, 0.36, 0.5,
0.9, 0.99999
};
double[] c2 =
{
0.99999999999999987510,
-0.99999999999988796462, 0.99999999998290743652,
-0.99999999896282500134, 0.99999996660459362918,
-0.99999933986272476760, 0.99999125611136965852,
-0.99991777624463387686, 0.99942835555870132569,
-0.99697311720723000295, 0.98751448037275303682,
-0.95915857980572882813, 0.89246305511006708555,
-0.76893425990463999675, 0.58893528468484693250,
-0.38380345160440256652, 0.20317601701045299653,
-0.82813631607004984866E-01, 0.24167984735759576523E-01,
-0.44676566663971825242E-02, 0.39141169402373836468E-03
};
double[] hrange =
{
0.02, 0.06, 0.09, 0.125, 0.26,
0.4, 0.6, 1.6, 1.7, 2.33,
2.4, 3.36, 3.4, 4.8
};
int[] meth =
{
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6
};
int[] ord =
{
2, 3, 4, 5, 7, 10, 12, 18, 10, 20, 30, 20, 4, 7, 8, 20, 13, 0
};
double[] pts =
{
0.35082039676451715489E-02,
0.31279042338030753740E-01, 0.85266826283219451090E-01,
0.16245071730812277011, 0.25851196049125434828,
0.36807553840697533536, 0.48501092905604697475,
0.60277514152618576821, 0.71477884217753226516,
0.81475510988760098605, 0.89711029755948965867,
0.95723808085944261843, 0.99178832974629703586
};
int[] select =
{
1, 1, 2, 13, 13, 13, 13, 13, 13, 13, 13, 16, 16, 16, 9,
1, 2, 2, 3, 3, 5, 5, 14, 14, 15, 15, 16, 16, 16, 9,
2, 2, 3, 3, 3, 5, 5, 15, 15, 15, 15, 16, 16, 16, 10,
2, 2, 3, 5, 5, 5, 5, 7, 7, 16, 16, 16, 16, 16, 10,
2, 3, 3, 5, 5, 6, 6, 8, 8, 17, 17, 17, 12, 12, 11,
2, 3, 5, 5, 5, 6, 6, 8, 8, 17, 17, 17, 12, 12, 12,
2, 3, 4, 4, 6, 6, 8, 8, 17, 17, 17, 17, 17, 12, 12,
2, 3, 4, 4, 6, 6, 18, 18, 18, 18, 17, 17, 17, 12, 12
};
double[] wts =
{
0.18831438115323502887E-01,
0.18567086243977649478E-01, 0.18042093461223385584E-01,
0.17263829606398753364E-01, 0.16243219975989856730E-01,
0.14994592034116704829E-01, 0.13535474469662088392E-01,
0.11886351605820165233E-01, 0.10070377242777431897E-01,
0.81130545742299586629E-02, 0.60419009528470238773E-02,
0.38862217010742057883E-02, 0.16793031084546090448E-02
};
/*
* Determine appropriate method from t1...t6
*/
ihint = 15;
for (i = 1; i <= 14; i++)
{
if (h <= hrange[i - 1])
{
ihint = i;
break;
}
}
iaint = 8;
for (i = 1; i <= 7; i++)
{
if (a <= arange[i - 1])
{
iaint = i;
break;
}
}
icode = select[ihint - 1 + (iaint - 1) * 15];
m = ord[icode - 1];
/*
* t1(h, a, m) ; m = 2, 3, 4, 5, 7, 10, 12 or 18
* jj = 2j - 1 ; gj = exp(-h*h/2) * (-h*h/2)**j / j
* aj = a**(2j-1) / (2*pi)
*/
if (meth[icode - 1] == 1)
{
hs = -0.5 * h * h;
dhs = Math.Exp(hs);
AS = a * a;
j = 1;
jj = 1;
aj = rtwopi * a;
value = rtwopi * Math.Atan(a);
dj = dhs - 1.0;
gj = hs * dhs;
for (; ;)
{
value = value + dj * aj / (double)(jj);
if (m <= j)
{
return(value);
}
j = j + 1;
jj = jj + 2;
aj = aj * AS;
dj = gj - dj;
gj = gj * hs / (double)(j);
}
}
/*
* t2(h, a, m) ; m = 10, 20 or 30
* z = (-1)^(i-1) * zi ; ii = 2i - 1
* vi = (-1)^(i-1) * a^(2i-1) * exp[-(a*h)^2/2] / sqrt(2*pi)
*/
else if (meth[icode - 1] == 2)
{
maxii = m + m + 1;
ii = 1;
value = 0.0;
hs = h * h;
AS = -a * a;
vi = rrtpi * a * Math.Exp(-0.5 * ah * ah);
z = 0.5 * (-0.5 + Normal.Function(ah)) / h;
y = 1.0 / hs;
for (; ;)
{
value = value + z;
if (maxii <= ii)
{
value = value * rrtpi * Math.Exp(-0.5 * hs);
return(value);
}
z = y * (vi - (double)(ii) * z);
vi = AS * vi;
ii = ii + 2;
}
}
/*
* t3(h, a, m) ; m = 20
* ii = 2i - 1
* vi = a**(2i-1) * exp[-(a*h)**2/2] / sqrt(2*pi)
*/
else if (meth[icode - 1] == 3)
{
i = 1;
ii = 1;
value = 0.0;
hs = h * h;
AS = a * a;
vi = rrtpi * a * Math.Exp(-0.5 * ah * ah);
zi = 0.5 * (-0.5 + Normal.Function(ah)) / h;
y = 1.0 / hs;
for (; ;)
{
value = value + zi * c2[i - 1];
if (m < i)
{
value = value * rrtpi * Math.Exp(-0.5 * hs);
return(value);
}
zi = y * ((double)(ii) * zi - vi);
vi = AS * vi;
i = i + 1;
ii = ii + 2;
}
}
/*
* t4(h, a, m) ; m = 4, 7, 8 or 20; ii = 2i + 1
* ai = a * exp[-h*h*(1+a*a)/2] * (-a*a)**i / (2*pi)
*/
else if (meth[icode - 1] == 4)
{
maxii = m + m + 1;
ii = 1;
hs = h * h;
AS = -a * a;
value = 0.0;
ai = rtwopi * a * Math.Exp(-0.5 * hs * (1.0 - AS));
yi = 1.0;
for (; ;)
{
value = value + ai * yi;
if (maxii <= ii)
{
return(value);
}
ii = ii + 2;
yi = (1.0 - hs * yi) / (double)(ii);
ai = ai * AS;
}
}
/*
* t5(h, a, m) ; m = 13
* 2m - point gaussian quadrature
*/
else if (meth[icode - 1] == 5)
{
value = 0.0;
AS = a * a;
hs = -0.5 * h * h;
for (i = 1; i <= m; i++)
{
r = 1.0 + AS * pts[i - 1];
value = value + wts[i - 1] * Math.Exp(hs * r) / r;
}
value = a * value;
}
/*
* t6(h, a); approximation for a near 1, (a<=1)
*/
else if (meth[icode - 1] == 6)
{
normh = Normal.Complemented(h);
value = 0.5 * normh * (1.0 - normh);
y = 1.0 - a;
r = Math.Atan(y / (1.0 + a));
if (r != 0.0)
{
value = value - rtwopi * r * Math.Exp(-0.5 * y * h * h / r);
}
}
return(value);
}
/// <summary>
/// A function for computing bivariate normal probabilities.
/// BVND calculates the probability that X > DH and Y > DK.
/// </summary>
///
/// <remarks>
/// <para>
/// This method is based on the work done by Alan Genz, Department of
/// Mathematics, Washington State University. Pullman, WA 99164-3113
/// Email: [email protected]. This work was shared under a 3-clause BSD
/// license. Please see source file for more details and the actual
/// license text.</para>
///
/// <para>
/// This function is based on the method described by Drezner, Z and G.O.
/// Wesolowsky, (1989), On the computation of the bivariate normal integral,
/// Journal of Statist. Comput. Simul. 35, pp. 101-107, with major modifications
/// for double precision, and for |R| close to 1.</para>
/// </remarks>
///
private static double BVND(double DH, double DK, double R)
{
// Copyright (C) 2013, Alan Genz, All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided the following conditions are met:
// 1. Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in
// the documentation and/or other materials provided with the
// distribution.
// 3. The contributor name(s) may not be used to endorse or promote
// products derived from this software without specific prior
// written permission.
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
// FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
// COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
// INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
// BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
// OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
// ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR
// TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
double TWOPI = 2.0 * Math.PI;
double[] x;
double[] w;
if (Math.Abs(R) < 0.3)
{
// Gauss Legendre Points and Weights N = 6
x = new double[] { -0.9324695142031522, -0.6612093864662647, -0.2386191860831970 };
w = new double[] { 0.1713244923791705, 0.3607615730481384, 0.4679139345726904 };
}
else if (Math.Abs(R) < 0.75)
{
// Gauss Legendre Points and Weights N = 12
x = new double[]
{
-0.9815606342467191, -0.9041172563704750, -0.7699026741943050,
-0.5873179542866171, -0.3678314989981802, -0.1252334085114692
};
w = new double[]
{
0.04717533638651177, 0.1069393259953183, 0.1600783285433464,
0.2031674267230659, 0.2334925365383547, 0.2491470458134029,
};
}
else
{
// Gauss Legendre Points and Weights N = 20
x = new double[]
{
-0.9931285991850949, -0.9639719272779138,
-0.9122344282513259, -0.8391169718222188,
-0.7463319064601508, -0.6360536807265150,
-0.5108670019508271, -0.3737060887154196,
-0.2277858511416451, -0.07652652113349733
};
w = new double[]
{
0.01761400713915212, 0.04060142980038694,
0.06267204833410906, 0.08327674157670475,
0.1019301198172404, 0.1181945319615184,
0.1316886384491766, 0.1420961093183821,
0.1491729864726037, 0.1527533871307259
};
}
double H = DH;
double K = DK;
double HK = H * K;
double BVN = 0;
if (Math.Abs(R) < 0.925)
{
if (Math.Abs(R) > 0)
{
double HS = (H * H + K * K) / 2;
double ASR = Math.Asin(R);
for (int I = 0; I < x.Length; I++)
{
for (int IS = -1; IS <= 1; IS += 2)
{
double SN = Math.Sin(ASR * (IS * x[I] + 1) / 2);
BVN = BVN + w[I] * Math.Exp((SN * HK - HS) / (1 - SN * SN));
}
}
BVN = BVN * ASR / (2 * TWOPI);
}
return(BVN + Normal.Function(-H) * Normal.Function(-K));
}
if (R < 0)
{
K = -K;
HK = -HK;
}
if (Math.Abs(R) < 1)
{
double AS = (1 - R) * (1 + R);
double A = Math.Sqrt(AS);
double BS = (H - K);
BS = BS * BS;
double C = (4 - HK) / 8;
double D = (12 - HK) / 16;
double ASR = -(BS / AS + HK) / 2;
if (ASR > -100)
{
BVN = A * Math.Exp(ASR) * (1 - C * (BS - AS) * (1 - D * BS / 5) / 3 + C * D * AS * AS / 5);
}
if (-HK < 100)
{
double B = Math.Sqrt(BS);
BVN = BVN - Math.Exp(-HK / 2) * Math.Sqrt(TWOPI) * Normal.Function(-B / A) * B
* (1 - C * BS * (1 - D * BS / 5) / 3);
}
A = A / 2;
for (int I = 0; I < x.Length; I++)
{
for (int IS = -1; IS <= 1; IS += 2)
{
double XS = (A * (IS * x[I] + 1));
XS = XS * XS;
double RS = Math.Sqrt(1 - XS);
ASR = -(BS / XS + HK) / 2;
if (ASR > -100)
{
BVN = BVN + A * w[I] * Math.Exp(ASR)
* (Math.Exp(-HK * XS / (2 * (1 + RS) * (1 + RS))) / RS
- (1 + C * XS * (1 + D * XS)));
}
}
}
BVN = -BVN / TWOPI;
}
if (R > 0)
{
return(BVN + Normal.Function(-Math.Max(H, K)));
}
BVN = -BVN;
if (K <= H)
{
return(BVN);
}
if (H < 0)
{
return(BVN + Normal.Function(K) - Normal.Function(H));
}
return(BVN + Normal.Function(-H) - Normal.Function(-K));
}
/// <summary>
/// Owen's T function for a restricted range of parameters.
/// </summary>
///
/// <param name="h">Owen's T function argument H (where 0 <= H).</param>
/// <param name="a">Owen's T function argument A (where 0 <= A <= 1).</param>
/// <param name="ah">The value of A*H.</param>
///
/// <returns>The value of Owen's T function.</returns>
///
public static double Function(double h, double a, double ah)
{
double ai;
double aj;
double AS;
double dhs;
double dj;
double gj;
double hs;
int i;
int iaint;
int icode;
int ihint;
int ii;
int j;
int jj;
int m;
int maxii;
double normh;
double r;
const double rrtpi = 0.39894228040143267794;
const double rtwopi = 0.15915494309189533577;
double y;
double yi;
double z;
double zi;
double value = 0;
double vi;
/*
* Determine appropriate method from t1...t6
*/
ihint = 15;
for (i = 1; i <= 14; i++)
{
if (h <= hrange[i - 1])
{
ihint = i;
break;
}
}
iaint = 8;
for (i = 1; i <= 7; i++)
{
if (a <= arange[i - 1])
{
iaint = i;
break;
}
}
icode = select[ihint - 1 + (iaint - 1) * 15];
m = ord[icode - 1];
/*
* t1(h, a, m) ; m = 2, 3, 4, 5, 7, 10, 12 or 18
* jj = 2j - 1 ; gj = exp(-h*h/2) * (-h*h/2)**j / j
* aj = a**(2j-1) / (2*pi)
*/
if (meth[icode - 1] == 1)
{
hs = -0.5 * h * h;
dhs = Math.Exp(hs);
AS = a * a;
j = 1;
jj = 1;
aj = rtwopi * a;
value = rtwopi * Math.Atan(a);
dj = dhs - 1.0;
gj = hs * dhs;
for (; ;)
{
value = value + dj * aj / (double)(jj);
if (m <= j)
{
return(value);
}
j = j + 1;
jj = jj + 2;
aj = aj * AS;
dj = gj - dj;
gj = gj * hs / (double)(j);
}
}
/*
* t2(h, a, m) ; m = 10, 20 or 30
* z = (-1)^(i-1) * zi ; ii = 2i - 1
* vi = (-1)^(i-1) * a^(2i-1) * exp[-(a*h)^2/2] / sqrt(2*pi)
*/
else if (meth[icode - 1] == 2)
{
maxii = m + m + 1;
ii = 1;
value = 0.0;
hs = h * h;
AS = -a * a;
vi = rrtpi * a * Math.Exp(-0.5 * ah * ah);
z = 0.5 * (-0.5 + Normal.Function(ah)) / h;
y = 1.0 / hs;
for (; ;)
{
value = value + z;
if (maxii <= ii)
{
value = value * rrtpi * Math.Exp(-0.5 * hs);
return(value);
}
z = y * (vi - (double)(ii) * z);
vi = AS * vi;
ii = ii + 2;
}
}
/*
* t3(h, a, m) ; m = 20
* ii = 2i - 1
* vi = a**(2i-1) * exp[-(a*h)**2/2] / sqrt(2*pi)
*/
else if (meth[icode - 1] == 3)
{
i = 1;
ii = 1;
value = 0.0;
hs = h * h;
AS = a * a;
vi = rrtpi * a * Math.Exp(-0.5 * ah * ah);
zi = 0.5 * (-0.5 + Normal.Function(ah)) / h;
y = 1.0 / hs;
for (; ;)
{
value = value + zi * c2[i - 1];
if (m < i)
{
value = value * rrtpi * Math.Exp(-0.5 * hs);
return(value);
}
zi = y * ((double)(ii) * zi - vi);
vi = AS * vi;
i = i + 1;
ii = ii + 2;
}
}
/*
* t4(h, a, m) ; m = 4, 7, 8 or 20; ii = 2i + 1
* ai = a * exp[-h*h*(1+a*a)/2] * (-a*a)**i / (2*pi)
*/
else if (meth[icode - 1] == 4)
{
maxii = m + m + 1;
ii = 1;
hs = h * h;
AS = -a * a;
value = 0.0;
ai = rtwopi * a * Math.Exp(-0.5 * hs * (1.0 - AS));
yi = 1.0;
for (; ;)
{
value = value + ai * yi;
if (maxii <= ii)
{
return(value);
}
ii = ii + 2;
yi = (1.0 - hs * yi) / (double)(ii);
ai = ai * AS;
}
}
/*
* t5(h, a, m) ; m = 13
* 2m - point gaussian quadrature
*/
else if (meth[icode - 1] == 5)
{
value = 0.0;
AS = a * a;
hs = -0.5 * h * h;
for (i = 1; i <= m; i++)
{
r = 1.0 + AS * pts[i - 1];
value = value + wts[i - 1] * Math.Exp(hs * r) / r;
}
value = a * value;
}
/*
* t6(h, a); approximation for a near 1, (a<=1)
*/
else if (meth[icode - 1] == 6)
{
normh = Normal.Complemented(h);
value = 0.5 * normh * (1.0 - normh);
y = 1.0 - a;
r = Math.Atan(y / (1.0 + a));
if (r != 0.0)
{
value = value - rtwopi * r * Math.Exp(-0.5 * y * h * h / r);
}
}
return(value);
}
/// <summary>
/// A function for computing bivariate normal probabilities.
/// BVND calculates the probability that X > DH and Y > DK.
/// </summary>
///
/// <remarks>
/// <para>
/// This method is based on the work done by Alan Genz, Department of
/// Mathematics, Washington State University. Pullman, WA 99164-3113
/// Email: [email protected]. This work was shared under a 3-clause BSD
/// license. Please see source file for more details and the actual
/// license text.</para>
///
/// <para>
/// This function is based on the method described by Drezner, Z and G.O.
/// Wesolowsky, (1989), On the computation of the bivariate normal integral,
/// Journal of Statist. Comput. Simul. 35, pp. 101-107, with major modifications
/// for double precision, and for |R| close to 1.</para>
/// </remarks>
///
private static double BVND(double dh, double dk, double r)
{
// Copyright (C) 2013, Alan Genz, All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided the following conditions are met:
// 1. Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in
// the documentation and/or other materials provided with the
// distribution.
// 3. The contributor name(s) may not be used to endorse or promote
// products derived from this software without specific prior
// written permission.
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
// FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
// COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
// INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
// BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
// OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
// ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR
// TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
const double TWOPI = 2.0 * Math.PI;
double[] x;
double[] w;
if (Math.Abs(r) < 0.3)
{
// Gauss Legendre Points and Weights N = 6
x = BVND_XN6;
w = BVND_WN6;
}
else if (Math.Abs(r) < 0.75)
{
// Gauss Legendre Points and Weights N = 12
x = BVND_XN12;
w = BVND_WN12;
}
else
{
// Gauss Legendre Points and Weights N = 20
x = BVND_XN20;
w = BVND_WN20;
}
double h = dh;
double k = dk;
double hk = h * k;
double bvn = 0;
if (Math.Abs(r) < 0.925)
{
if (Math.Abs(r) > 0)
{
double sh = (h * h + k * k) / 2;
double asr = Math.Asin(r);
for (int i = 0; i < x.Length; i++)
{
for (int j = -1; j <= 1; j += 2)
{
double sn = Math.Sin(asr * (j * x[i] + 1) / 2);
bvn = bvn + w[i] * Math.Exp((sn * hk - sh) / (1 - sn * sn));
}
}
bvn = bvn * asr / (2 * TWOPI);
}
return(bvn + Normal.Function(-h) * Normal.Function(-k));
}
if (r < 0)
{
k = -k;
hk = -hk;
}
if (Math.Abs(r) < 1)
{
double sa = (1 - r) * (1 + r);
double A = Math.Sqrt(sa);
double sb = (h - k);
sb = sb * sb;
double c = (4 - hk) / 8;
double d = (12 - hk) / 16;
double asr = -(sb / sa + hk) / 2;
if (asr > -100)
{
bvn = A * Math.Exp(asr) * (1 - c * (sb - sa) * (1 - d * sb / 5) / 3 + c * d * sa * sa / 5);
}
if (-hk < 100)
{
double B = Math.Sqrt(sb);
bvn = bvn - Math.Exp(-hk / 2) * Math.Sqrt(TWOPI) * Normal.Function(-B / A) * B
* (1 - c * sb * (1 - d * sb / 5) / 3);
}
A = A / 2;
for (int i = 0; i < x.Length; i++)
{
for (int j = -1; j <= 1; j += 2)
{
double xs = (A * (j * x[i] + 1));
xs = xs * xs;
double rs = Math.Sqrt(1 - xs);
asr = -(sb / xs + hk) / 2;
if (asr > -100)
{
bvn = bvn + A * w[i] * Math.Exp(asr)
* (Math.Exp(-hk * xs / (2 * (1 + rs) * (1 + rs))) / rs
- (1 + c * xs * (1 + d * xs)));
}
}
}
bvn = -bvn / TWOPI;
}
if (r > 0)
{
return(bvn + Normal.Function(-Math.Max(h, k)));
}
bvn = -bvn;
if (k <= h)
{
return(bvn);
}
if (h < 0)
{
return(bvn + Normal.Function(k) - Normal.Function(h));
}
return(bvn + Normal.Function(-h) - Normal.Function(-k));
}