/// <summary>
/// Returns the modified Struve function of order 0.
/// </summary>
/// <param name="x">The value to compute the function of.</param>
public static double StruveL0(double x)
{
//*********************************************************************72
//
//c STRVL0 calculates the modified Struve function of order 0.
//
// DESCRIPTION:
//
// This function calculates the modified Struve function of
// order 0, denoted L0(x), defined as the solution of the
// second-order equation
//
// x*D(Df) + Df - x*f = 2x/pi
//
// This subroutine is set up to work on IEEE machines.
// For other machines, you should retrieve the code
// from the general MISCFUN archive.
//
//
// ERROR RETURNS:
//
// If the value of |XVALUE| is too large, the result
// would cause an floating-pt overflow. An error message
// is printed and the function returns the value of
// sign(XVALUE)*XMAX where XMAX is the largest possible
// floating-pt argument.
//
//
// MACHINE-DEPENDENT PARAMETERS:
//
// NTERM1 - INTEGER - The no. of terms for the array ARL0.
// The recommended value is such that
// ABS(ARL0(NTERM1)) < EPS/100
//
// NTERM2 - INTEGER - The no. of terms for the array ARL0AS.
// The recommended value is such that
// ABS(ARL0AS(NTERM2)) < EPS/100
//
// NTERM3 - INTEGER - The no. of terms for the array AI0ML0.
// The recommended value is such that
// ABS(AI0ML0(NTERM3)) < EPS/100
//
// XLOW - DOUBLE PRECISION - The value of x below which L0(x) = 2*x/pi
// to machine precision. The recommended value is
// 3*SQRT(EPS)
//
// XHIGH1 - DOUBLE PRECISION - The value beyond which the Chebyshev series
// in the asymptotic expansion of I0 - L0 gives
// 1.0 to machine precision. The recommended value
// is SQRT( 30/EPSNEG )
//
// XHIGH2 - DOUBLE PRECISION - The value beyond which the Chebyshev series
// in the asymptotic expansion of I0 gives 1.0
// to machine precision. The recommended value
// is 28 / EPSNEG
//
// XMAX - DOUBLE PRECISION - The value of XMAX, where XMAX is the
// largest possible floating-pt argument.
// This is used to prevent overflow.
//
// For values of EPS, EPSNEG and XMAX the user should refer
// to the file MACHCON.TXT
//
// The machine-arithmetic constants are given in DATA
// statements.
//
//
// INTRINSIC FUNCTIONS USED:
//
// EXP , LOG , SQRT
//
//
// OTHER MISCFUN SUBROUTINES USED:
//
// CHEVAL , ERRPRN
//
//
// AUTHOR:
// DR. ALLAN J. MACLEOD
// DEPT. OF MATHEMATICS AND STATISTICS
// UNIVERSITY OF PAISLEY
// HIGH ST.
// PAISLEY
// SCOTLAND
// PA1 2BE
//
// (e-mail: [email protected] )
//
//
// LATEST REVISION:
// 12 JANUARY, 1996
//
//
if (x < 0.0)
{
return(-StruveL0(-x));
}
double[] ARL0 = new double[28];
ARL0[0] = 0.42127458349979924863;
ARL0[1] = -0.33859536391220612188;
ARL0[2] = 0.21898994812710716064;
ARL0[3] = -0.12349482820713185712;
ARL0[4] = 0.6214209793866958440e-1;
ARL0[5] = -0.2817806028109547545e-1;
ARL0[6] = 0.1157419676638091209e-1;
ARL0[7] = -0.431658574306921179e-2;
ARL0[8] = 0.146142349907298329e-2;
ARL0[9] = -0.44794211805461478e-3;
ARL0[10] = 0.12364746105943761e-3;
ARL0[11] = -0.3049028334797044e-4;
ARL0[12] = 0.663941401521146e-5;
ARL0[13] = -0.125538357703889e-5;
ARL0[14] = 0.20073446451228e-6;
ARL0[15] = -0.2588260170637e-7;
ARL0[16] = 0.241143742758e-8;
ARL0[17] = -0.10159674352e-9;
ARL0[18] = -0.1202430736e-10;
ARL0[19] = 0.262906137e-11;
ARL0[20] = -0.15313190e-12;
ARL0[21] = -0.1574760e-13;
ARL0[22] = 0.315635e-14;
ARL0[23] = -0.4096e-16;
ARL0[24] = -0.3620e-16;
ARL0[25] = 0.239e-17;
ARL0[26] = 0.36e-18;
ARL0[27] = -0.4e-19;
double[] ARL0AS = new double[16];
ARL0AS[0] = 2.00861308235605888600;
ARL0AS[1] = 0.403737966500438470e-2;
ARL0AS[2] = -0.25199480286580267e-3;
ARL0AS[3] = 0.1605736682811176e-4;
ARL0AS[4] = -0.103692182473444e-5;
ARL0AS[5] = 0.6765578876305e-7;
ARL0AS[6] = -0.444999906756e-8;
ARL0AS[7] = 0.29468889228e-9;
ARL0AS[8] = -0.1962180522e-10;
ARL0AS[9] = 0.131330306e-11;
ARL0AS[10] = -0.8819190e-13;
ARL0AS[11] = 0.595376e-14;
ARL0AS[12] = -0.40389e-15;
ARL0AS[13] = 0.2651e-16;
ARL0AS[14] = -0.208e-17;
ARL0AS[15] = 0.11e-18;
double[] AI0ML0 = new double[24];
AI0ML0[0] = 2.00326510241160643125;
AI0ML0[1] = 0.195206851576492081e-2;
AI0ML0[2] = 0.38239523569908328e-3;
AI0ML0[3] = 0.7534280817054436e-4;
AI0ML0[4] = 0.1495957655897078e-4;
AI0ML0[5] = 0.299940531210557e-5;
AI0ML0[6] = 0.60769604822459e-6;
AI0ML0[7] = 0.12399495544506e-6;
AI0ML0[8] = 0.2523262552649e-7;
AI0ML0[9] = 0.504634857332e-8;
AI0ML0[10] = 0.97913236230e-9;
AI0ML0[11] = 0.18389115241e-9;
AI0ML0[12] = 0.3376309278e-10;
AI0ML0[13] = 0.611179703e-11;
AI0ML0[14] = 0.108472972e-11;
AI0ML0[15] = 0.18861271e-12;
AI0ML0[16] = 0.3280345e-13;
AI0ML0[17] = 0.565647e-14;
AI0ML0[18] = 0.93300e-15;
AI0ML0[19] = 0.15881e-15;
AI0ML0[20] = 0.2791e-16;
AI0ML0[21] = 0.389e-17;
AI0ML0[22] = 0.70e-18;
AI0ML0[23] = 0.16e-18;
// MACHINE-DEPENDENT VALUES (Suitable for IEEE-arithmetic machines)
const int nterm1 = 25;
const int nterm2 = 14;
const int nterm3 = 21;
const double xlow = 4.4703484e-8;
const double xmax = 1.797693e308;
const double xhigh1 = 5.1982303e8;
const double xhigh2 = 2.5220158e17;
// Code for |xvalue| <= 16
if (x <= 16.0)
{
if (x < xlow)
{
return(Constants.TwoInvPi * x);
}
double T = ((4.0 * x) - 24.0) / (x + 24.0);
return(Constants.TwoInvPi * x * Evaluate.ChebyshevSum(nterm1, ARL0, T) * Math.Exp(x));
}
// Code for |xvalue| > 16
double ch1;
if (x > xhigh2)
{
ch1 = 1.0;
}
else
{
double T = (x - 28.0) / (4.0 - x);
ch1 = Evaluate.ChebyshevSum(nterm2, ARL0AS, T);
}
double ch2;
if (x > xhigh1)
{
ch2 = 1.0;
}
else
{
double xsq = x * x;
double T = (800.0 - xsq) / (288.0 + xsq);
ch2 = Evaluate.ChebyshevSum(nterm3, AI0ML0, T);
}
double test = (Math.Log(ch1) - Constants.LogSqrt2Pi - (Math.Log(x) / 2.0)) + x;
if (test > Math.Log(xmax))
{
throw new ArithmeticException("ERROR IN MISCFUN FUNCTION STRVL0: ARGUMENT CAUSES OVERFLOW");
}
return(Math.Exp(test) - ((Constants.TwoInvPi * ch2) / x));
}
/// <summary>
/// Least-Squares fitting the points (x,y) to a k-order polynomial y : x -> p0 + p1*x + p2*x^2 + ... + pk*x^k,
/// returning a function y' for the best fitting polynomial.
/// </summary>
public static Func <double, double> PolynomialFunc(double[] x, double[] y, int order)
{
var parameters = Polynomial(x, y, order);
return(z => Evaluate.Polynomial(z, parameters));
}
/// <summary>
/// Returns the modified Struve function of order 1.
/// </summary>
/// <param name="x">The value to compute the function of.</param>
public static double StruveL1(double x)
{
//*********************************************************************72
//
//c STRVL1 calculates the modified Struve function of order 1.
//
// DESCRIPTION:
//
// This function calculates the modified Struve function of
// order 1, denoted L1(x), defined as the solution of
//
// x*x*D(Df) + x*Df - (x*x+1)f = 2*x*x/pi
//
// This subroutine is set up to work on IEEE machines.
// For other machines, you should retrieve the code
// from the general MISCFUN archive.
//
//
// ERROR RETURNS:
//
// If the value of |XVALUE| is too large, the result
// would cause an floating-pt overflow. An error message
// is printed and the function returns the value of
// sign(XVALUE)*XMAX where XMAX is the largest possible
// floating-pt argument.
//
//
// MACHINE-DEPENDENT PARAMETERS:
//
// NTERM1 - INTEGER - The no. of terms for the array ARL1.
// The recommended value is such that
// ABS(ARL1(NTERM1)) < EPS/100
//
// NTERM2 - INTEGER - The no. of terms for the array ARL1AS.
// The recommended value is such that
// ABS(ARL1AS(NTERM2)) < EPS/100
//
// NTERM3 - INTEGER - The no. of terms for the array AI1ML1.
// The recommended value is such that
// ABS(AI1ML1(NTERM3)) < EPS/100
//
// XLOW1 - DOUBLE PRECISION - The value of x below which
// L1(x) = 2*x*x/(3*pi)
// to machine precision. The recommended
// value is SQRT(15*EPS)
//
// XLOW2 - DOUBLE PRECISION - The value of x below which L1(x) set to 0.0.
// This is used to prevent underflow. The
// recommended value is
// SQRT(5*XMIN)
//
// XHIGH1 - DOUBLE PRECISION - The value of |x| above which the Chebyshev
// series in the asymptotic expansion of I1
// equals 1.0 to machine precision. The
// recommended value is SQRT( 30 / EPSNEG ).
//
// XHIGH2 - DOUBLE PRECISION - The value of |x| above which the Chebyshev
// series in the asymptotic expansion of I1 - L1
// equals 1.0 to machine precision. The recommended
// value is 30 / EPSNEG.
//
// XMAX - DOUBLE PRECISION - The value of XMAX, where XMAX is the
// largest possible floating-pt argument.
// This is used to prevent overflow.
//
// For values of EPS, EPSNEG, XMIN, and XMAX the user should refer
// to the file MACHCON.TXT
//
// The machine-arithmetic constants are given in DATA
// statements.
//
//
// INTRINSIC FUNCTIONS USED:
//
// EXP , LOG , SQRT
//
//
// OTHER MISCFUN SUBROUTINES USED:
//
// CHEVAL , ERRPRN
//
//
// AUTHOR:
// DR. ALLAN J. MACLEOD
// DEPT. OF MATHEMATICS AND STATISTICS
// UNIVERSITY OF PAISLEY
// HIGH ST.
// PAISLEY
// SCOTLAND
// PA1 2BE
//
// (e-mail: [email protected] )
//
//
// LATEST UPDATE:
// 12 JANUARY, 1996
//
//
if (x < 0.0)
{
return(StruveL1(-x));
}
double[] ARL1 = new double[27];
ARL1[0] = 0.38996027351229538208;
ARL1[1] = -0.33658096101975749366;
ARL1[2] = 0.23012467912501645616;
ARL1[3] = -0.13121594007960832327;
ARL1[4] = 0.6425922289912846518e-1;
ARL1[5] = -0.2750032950616635833e-1;
ARL1[6] = 0.1040234148637208871e-1;
ARL1[7] = -0.350532294936388080e-2;
ARL1[8] = 0.105748498421439717e-2;
ARL1[9] = -0.28609426403666558e-3;
ARL1[10] = 0.6925708785942208e-4;
ARL1[11] = -0.1489693951122717e-4;
ARL1[12] = 0.281035582597128e-5;
ARL1[13] = -0.45503879297776e-6;
ARL1[14] = 0.6090171561770e-7;
ARL1[15] = -0.623543724808e-8;
ARL1[16] = 0.38430012067e-9;
ARL1[17] = 0.790543916e-11;
ARL1[18] = -0.489824083e-11;
ARL1[19] = 0.46356884e-12;
ARL1[20] = 0.684205e-14;
ARL1[21] = -0.569748e-14;
ARL1[22] = 0.35324e-15;
ARL1[23] = 0.4244e-16;
ARL1[24] = -0.644e-17;
ARL1[25] = -0.21e-18;
ARL1[26] = 0.9e-19;
double[] ARL1AS = new double[17];
ARL1AS[0] = 1.97540378441652356868;
ARL1AS[1] = -0.1195130555088294181e-1;
ARL1AS[2] = 0.33639485269196046e-3;
ARL1AS[3] = -0.1009115655481549e-4;
ARL1AS[4] = 0.30638951321998e-6;
ARL1AS[5] = -0.953704370396e-8;
ARL1AS[6] = 0.29524735558e-9;
ARL1AS[7] = -0.951078318e-11;
ARL1AS[8] = 0.28203667e-12;
ARL1AS[9] = -0.1134175e-13;
ARL1AS[10] = 0.147e-17;
ARL1AS[11] = -0.6232e-16;
ARL1AS[12] = -0.751e-17;
ARL1AS[13] = -0.17e-18;
ARL1AS[14] = 0.51e-18;
ARL1AS[15] = 0.23e-18;
ARL1AS[16] = 0.5e-19;
double[] AI1ML1 = new double[26];
AI1ML1[0] = 1.99679361896789136501;
AI1ML1[1] = -0.190663261409686132e-2;
AI1ML1[2] = -0.36094622410174481e-3;
AI1ML1[3] = -0.6841847304599820e-4;
AI1ML1[4] = -0.1299008228509426e-4;
AI1ML1[5] = -0.247152188705765e-5;
AI1ML1[6] = -0.47147839691972e-6;
AI1ML1[7] = -0.9020819982592e-7;
AI1ML1[8] = -0.1730458637504e-7;
AI1ML1[9] = -0.332323670159e-8;
AI1ML1[10] = -0.63736421735e-9;
AI1ML1[11] = -0.12180239756e-9;
AI1ML1[12] = -0.2317346832e-10;
AI1ML1[13] = -0.439068833e-11;
AI1ML1[14] = -0.82847110e-12;
AI1ML1[15] = -0.15562249e-12;
AI1ML1[16] = -0.2913112e-13;
AI1ML1[17] = -0.543965e-14;
AI1ML1[18] = -0.101177e-14;
AI1ML1[19] = -0.18767e-15;
AI1ML1[20] = -0.3484e-16;
AI1ML1[21] = -0.643e-17;
AI1ML1[22] = -0.118e-17;
AI1ML1[23] = -0.22e-18;
AI1ML1[24] = -0.4e-19;
AI1ML1[25] = -0.1e-19;
// MACHINE-DEPENDENT VALUES (Suitable for IEEE-arithmetic machines)
const int nterm1 = 24;
const int nterm2 = 13;
const int nterm3 = 22;
const double xlow1 = 5.7711949e-8;
const double xlow2 = 3.3354714e-154;
const double xmax = 1.797693e308;
const double xhigh1 = 5.19823025e8;
const double xhigh2 = 2.7021597e17;
// CODE FOR |x| <= 16
if (x <= 16.0)
{
if (x <= xlow2)
{
return(0.0);
}
double xsq = x * x;
if (x < xlow1)
{
return(xsq / Constants.Pi3Over2);
}
double t = ((4.0 * x) - 24.0) / (x + 24.0);
return((xsq * Evaluate.ChebyshevSum(nterm1, ARL1, t) * Math.Exp(x)) / Constants.Pi3Over2);
}
// CODE FOR |x| > 16
double ch1;
if (x > xhigh2)
{
ch1 = 1.0;
}
else
{
double t = (x - 30.0) / (2.0 - x);
ch1 = Evaluate.ChebyshevSum(nterm2, ARL1AS, t);
}
double ch2;
if (x > xhigh1)
{
ch2 = 1.0;
}
else
{
double xsq = x * x;
double t = (800.0 - xsq) / (288.0 + xsq);
ch2 = Evaluate.ChebyshevSum(nterm3, AI1ML1, t);
}
double test = (Math.Log(ch1) - Constants.LogSqrt2Pi - (Math.Log(x) / 2.0)) + x;
if (test > Math.Log(xmax))
{
throw new ArithmeticException("ERROR IN MISCFUN FUNCTION STRVL1: ARGUMENT CAUSES OVERFLOW");
}
return(Math.Exp(test) - (Constants.TwoInvPi * ch2));
}
/// <summary>
/// The implementation of the inverse error function.
/// </summary>
/// <param name="p">First intermediate parameter.</param>
/// <param name="q">Second intermediate parameter.</param>
/// <param name="s">Third intermediate parameter.</param>
/// <returns>the inverse error function.</returns>
static double ErfInvImpl(double p, double q, double s)
{
double result;
if (p <= 0.5)
{
// Evaluate inverse erf using the rational approximation:
//
// x = p(p+10)(Y+R(p))
//
// Where Y is a constant, and R(p) is optimized for a low
// absolute error compared to |Y|.
//
// double: Max error found: 2.001849e-18
// long double: Max error found: 1.017064e-20
// Maximum Deviation Found (actual error term at infinite precision) 8.030e-21
const float y = 0.0891314744949340820313f;
double g = p * (p + 10);
double r = Evaluate.Polynomial(p, ErvInvImpAn) / Evaluate.Polynomial(p, ErvInvImpAd);
result = (g * y) + (g * r);
}
else if (q >= 0.25)
{
// Rational approximation for 0.5 > q >= 0.25
//
// x = sqrt(-2*log(q)) / (Y + R(q))
//
// Where Y is a constant, and R(q) is optimized for a low
// absolute error compared to Y.
//
// double : Max error found: 7.403372e-17
// long double : Max error found: 6.084616e-20
// Maximum Deviation Found (error term) 4.811e-20
const float y = 2.249481201171875f;
double g = Math.Sqrt(-2 * Math.Log(q));
double xs = q - 0.25;
double r = Evaluate.Polynomial(xs, ErvInvImpBn) / Evaluate.Polynomial(xs, ErvInvImpBd);
result = g / (y + r);
}
else
{
// For q < 0.25 we have a series of rational approximations all
// of the general form:
//
// let: x = sqrt(-log(q))
//
// Then the result is given by:
//
// x(Y+R(x-B))
//
// where Y is a constant, B is the lowest value of x for which
// the approximation is valid, and R(x-B) is optimized for a low
// absolute error compared to Y.
//
// Note that almost all code will really go through the first
// or maybe second approximation. After than we're dealing with very
// small input values indeed: 80 and 128 bit long double's go all the
// way down to ~ 1e-5000 so the "tail" is rather long...
double x = Math.Sqrt(-Math.Log(q));
if (x < 3)
{
// Max error found: 1.089051e-20
const float y = 0.807220458984375f;
double xs = x - 1.125;
double r = Evaluate.Polynomial(xs, ErvInvImpCn) / Evaluate.Polynomial(xs, ErvInvImpCd);
result = (y * x) + (r * x);
}
else if (x < 6)
{
// Max error found: 8.389174e-21
const float y = 0.93995571136474609375f;
double xs = x - 3;
double r = Evaluate.Polynomial(xs, ErvInvImpDn) / Evaluate.Polynomial(xs, ErvInvImpDd);
result = (y * x) + (r * x);
}
else if (x < 18)
{
// Max error found: 1.481312e-19
const float y = 0.98362827301025390625f;
double xs = x - 6;
double r = Evaluate.Polynomial(xs, ErvInvImpEn) / Evaluate.Polynomial(xs, ErvInvImpEd);
result = (y * x) + (r * x);
}
else if (x < 44)
{
// Max error found: 5.697761e-20
const float y = 0.99714565277099609375f;
double xs = x - 18;
double r = Evaluate.Polynomial(xs, ErvInvImpFn) / Evaluate.Polynomial(xs, ErvInvImpFd);
result = (y * x) + (r * x);
}
else
{
// Max error found: 1.279746e-20
const float y = 0.99941349029541015625f;
double xs = x - 44;
double r = Evaluate.Polynomial(xs, ErvInvImpGn) / Evaluate.Polynomial(xs, ErvInvImpGd);
result = (y * x) + (r * x);
}
}
return(s * result);
}
/// <summary>
/// Implementation of the error function.
/// </summary>
/// <param name="z">Where to evaluate the error function.</param>
/// <param name="invert">Whether to compute 1 - the error function.</param>
/// <returns>the error function.</returns>
static double ErfImp(double z, bool invert)
{
if (z < 0)
{
if (!invert)
{
return(-ErfImp(-z, false));
}
if (z < -0.5)
{
return(2 - ErfImp(-z, true));
}
return(1 + ErfImp(-z, false));
}
double result;
// Big bunch of selection statements now to pick which
// implementation to use, try to put most likely options
// first:
if (z < 0.5)
{
// We're going to calculate erf:
if (z < 1e-10)
{
result = (z * 1.125) + (z * 0.003379167095512573896158903121545171688);
}
else
{
// Worst case absolute error found: 6.688618532e-21
result = (z * 1.125) + (z * Evaluate.Polynomial(z, ErfImpAn) / Evaluate.Polynomial(z, ErfImpAd));
}
}
else if ((z < 110) || ((z < 110) && invert))
{
// We'll be calculating erfc:
invert = !invert;
double r, b;
if (z < 0.75)
{
// Worst case absolute error found: 5.582813374e-21
r = Evaluate.Polynomial(z - 0.5, ErfImpBn) / Evaluate.Polynomial(z - 0.5, ErfImpBd);
b = 0.3440242112F;
}
else if (z < 1.25)
{
// Worst case absolute error found: 4.01854729e-21
r = Evaluate.Polynomial(z - 0.75, ErfImpCn) / Evaluate.Polynomial(z - 0.75, ErfImpCd);
b = 0.419990927F;
}
else if (z < 2.25)
{
// Worst case absolute error found: 2.866005373e-21
r = Evaluate.Polynomial(z - 1.25, ErfImpDn) / Evaluate.Polynomial(z - 1.25, ErfImpDd);
b = 0.4898625016F;
}
else if (z < 3.5)
{
// Worst case absolute error found: 1.045355789e-21
r = Evaluate.Polynomial(z - 2.25, ErfImpEn) / Evaluate.Polynomial(z - 2.25, ErfImpEd);
b = 0.5317370892F;
}
else if (z < 5.25)
{
// Worst case absolute error found: 8.300028706e-22
r = Evaluate.Polynomial(z - 3.5, ErfImpFn) / Evaluate.Polynomial(z - 3.5, ErfImpFd);
b = 0.5489973426F;
}
else if (z < 8)
{
// Worst case absolute error found: 1.700157534e-21
r = Evaluate.Polynomial(z - 5.25, ErfImpGn) / Evaluate.Polynomial(z - 5.25, ErfImpGd);
b = 0.5571740866F;
}
else if (z < 11.5)
{
// Worst case absolute error found: 3.002278011e-22
r = Evaluate.Polynomial(z - 8, ErfImpHn) / Evaluate.Polynomial(z - 8, ErfImpHd);
b = 0.5609807968F;
}
else if (z < 17)
{
// Worst case absolute error found: 6.741114695e-21
r = Evaluate.Polynomial(z - 11.5, ErfImpIn) / Evaluate.Polynomial(z - 11.5, ErfImpId);
b = 0.5626493692F;
}
else if (z < 24)
{
// Worst case absolute error found: 7.802346984e-22
r = Evaluate.Polynomial(z - 17, ErfImpJn) / Evaluate.Polynomial(z - 17, ErfImpJd);
b = 0.5634598136F;
}
else if (z < 38)
{
// Worst case absolute error found: 2.414228989e-22
r = Evaluate.Polynomial(z - 24, ErfImpKn) / Evaluate.Polynomial(z - 24, ErfImpKd);
b = 0.5638477802F;
}
else if (z < 60)
{
// Worst case absolute error found: 5.896543869e-24
r = Evaluate.Polynomial(z - 38, ErfImpLn) / Evaluate.Polynomial(z - 38, ErfImpLd);
b = 0.5640528202F;
}
else if (z < 85)
{
// Worst case absolute error found: 3.080612264e-21
r = Evaluate.Polynomial(z - 60, ErfImpMn) / Evaluate.Polynomial(z - 60, ErfImpMd);
b = 0.5641309023F;
}
else
{
// Worst case absolute error found: 8.094633491e-22
r = Evaluate.Polynomial(z - 85, ErfImpNn) / Evaluate.Polynomial(z - 85, ErfImpNd);
b = 0.5641584396F;
}
double g = Math.Exp(-z * z) / z;
result = (g * b) + (g * r);
}
else
{
// Any value of z larger than 28 will underflow to zero:
result = 0;
invert = !invert;
}
if (invert)
{
result = 1 - result;
}
return(result);
}
/// <summary>
/// The implementation of the inverse error function.
/// </summary>
/// <param name="p">First intermediate parameter.</param>
/// <param name="q">Second intermediate parameter.</param>
/// <param name="s">Third intermediate parameter.</param>
/// <returns>the inverse error function.</returns>
private static double ErfInvImpl(double p, double q, double s)
{
double result;
if (p <= 0.5)
{
//
// Evaluate inverse erf using the rational approximation:
//
// x = p(p+10)(Y+R(p))
//
// Where Y is a constant, and R(p) is optimized for a low
// absolute error compared to |Y|.
//
// double: Max error found: 2.001849e-18
// long double: Max error found: 1.017064e-20
// Maximum Deviation Found (actual error term at infinite precision) 8.030e-21
//
const float Y = 0.0891314744949340820313f;
double[] P = new[] { -0.000508781949658280665617, -0.00836874819741736770379, 0.0334806625409744615033, -0.0126926147662974029034, -0.0365637971411762664006, 0.0219878681111168899165, 0.00822687874676915743155, -0.00538772965071242932965 };
double[] Q = new[] { 1, -0.970005043303290640362, -1.56574558234175846809, 1.56221558398423026363, 0.662328840472002992063, -0.71228902341542847553, -0.0527396382340099713954, 0.0795283687341571680018, -0.00233393759374190016776, 0.000886216390456424707504 };
double g = p * (p + 10);
double r = Evaluate.Polynomial(p, P) / Evaluate.Polynomial(p, Q);
result = (g * Y) + (g * r);
}
else if (q >= 0.25)
{
//
// Rational approximation for 0.5 > q >= 0.25
//
// x = sqrt(-2*log(q)) / (Y + R(q))
//
// Where Y is a constant, and R(q) is optimized for a low
// absolute error compared to Y.
//
// double : Max error found: 7.403372e-17
// long double : Max error found: 6.084616e-20
// Maximum Deviation Found (error term) 4.811e-20
//
const float Y = 2.249481201171875f;
double[] P = new[] { -0.202433508355938759655, 0.105264680699391713268, 8.37050328343119927838, 17.6447298408374015486, -18.8510648058714251895, -44.6382324441786960818, 17.445385985570866523, 21.1294655448340526258, -3.67192254707729348546 };
double[] Q = new[] { 1, 6.24264124854247537712, 3.9713437953343869095, -28.6608180499800029974, -20.1432634680485188801, 48.5609213108739935468, 10.8268667355460159008, -22.6436933413139721736, 1.72114765761200282724 };
double g = Math.Sqrt(-2 * Math.Log(q));
double xs = q - 0.25;
double r = Evaluate.Polynomial(xs, P) / Evaluate.Polynomial(xs, Q);
result = g / (Y + r);
}
else
{
//
// For q < 0.25 we have a series of rational approximations all
// of the general form:
//
// let: x = sqrt(-log(q))
//
// Then the result is given by:
//
// x(Y+R(x-B))
//
// where Y is a constant, B is the lowest value of x for which
// the approximation is valid, and R(x-B) is optimized for a low
// absolute error compared to Y.
//
// Note that almost all code will really go through the first
// or maybe second approximation. After than we're dealing with very
// small input values indeed: 80 and 128 bit long double's go all the
// way down to ~ 1e-5000 so the "tail" is rather long...
//
double x = Math.Sqrt(-Math.Log(q));
if (x < 3)
{
// Max error found: 1.089051e-20
const float Y = 0.807220458984375f;
double[] P = new[] { -0.131102781679951906451, -0.163794047193317060787, 0.117030156341995252019, 0.387079738972604337464, 0.337785538912035898924, 0.142869534408157156766, 0.0290157910005329060432, 0.00214558995388805277169, -0.679465575181126350155e-6, 0.285225331782217055858e-7, -0.681149956853776992068e-9 };
double[] Q = new[] { 1, 3.46625407242567245975, 5.38168345707006855425, 4.77846592945843778382, 2.59301921623620271374, 0.848854343457902036425, 0.152264338295331783612, 0.01105924229346489121 };
double xs = x - 1.125;
double R = Evaluate.Polynomial(xs, P) / Evaluate.Polynomial(xs, Q);
result = (Y * x) + (R * x);
}
else if (x < 6)
{
// Max error found: 8.389174e-21
const float Y = 0.93995571136474609375f;
double[] P = new[] { -0.0350353787183177984712, -0.00222426529213447927281, 0.0185573306514231072324, 0.00950804701325919603619, 0.00187123492819559223345, 0.000157544617424960554631, 0.460469890584317994083e-5, -0.230404776911882601748e-9, 0.266339227425782031962e-11 };
double[] Q = new[] { 1, 1.3653349817554063097, 0.762059164553623404043, 0.220091105764131249824, 0.0341589143670947727934, 0.00263861676657015992959, 0.764675292302794483503e-4 };
double xs = x - 3;
double R = Evaluate.Polynomial(xs, P) / Evaluate.Polynomial(xs, Q);
result = (Y * x) + (R * x);
}
else if (x < 18)
{
// Max error found: 1.481312e-19
const float Y = 0.98362827301025390625f;
double[] P = new[] { -0.0167431005076633737133, -0.00112951438745580278863, 0.00105628862152492910091, 0.000209386317487588078668, 0.149624783758342370182e-4, 0.449696789927706453732e-6, 0.462596163522878599135e-8, -0.281128735628831791805e-13, 0.99055709973310326855e-16 };
double[] Q = new[] { 1, 0.591429344886417493481, 0.138151865749083321638, 0.0160746087093676504695, 0.000964011807005165528527, 0.275335474764726041141e-4, 0.282243172016108031869e-6 };
double xs = x - 6;
double R = Evaluate.Polynomial(xs, P) / Evaluate.Polynomial(xs, Q);
result = (Y * x) + (R * x);
}
else if (x < 44)
{
// Max error found: 5.697761e-20
const float Y = 0.99714565277099609375f;
double[] P = new[] { -0.0024978212791898131227, -0.779190719229053954292e-5, 0.254723037413027451751e-4, 0.162397777342510920873e-5, 0.396341011304801168516e-7, 0.411632831190944208473e-9, 0.145596286718675035587e-11, -0.116765012397184275695e-17 };
double[] Q = new[] { 1, 0.207123112214422517181, 0.0169410838120975906478, 0.000690538265622684595676, 0.145007359818232637924e-4, 0.144437756628144157666e-6, 0.509761276599778486139e-9 };
double xs = x - 18;
double R = Evaluate.Polynomial(xs, P) / Evaluate.Polynomial(xs, Q);
result = (Y * x) + (R * x);
}
else
{
// Max error found: 1.279746e-20
const float Y = 0.99941349029541015625f;
double[] P = new[] { -0.000539042911019078575891, -0.28398759004727721098e-6, 0.899465114892291446442e-6, 0.229345859265920864296e-7, 0.225561444863500149219e-9, 0.947846627503022684216e-12, 0.135880130108924861008e-14, -0.348890393399948882918e-21 };
double[] Q = new[] { 1, 0.0845746234001899436914, 0.00282092984726264681981, 0.468292921940894236786e-4, 0.399968812193862100054e-6, 0.161809290887904476097e-8, 0.231558608310259605225e-11 };
double xs = x - 44;
double R = Evaluate.Polynomial(xs, P) / Evaluate.Polynomial(xs, Q);
result = (Y * x) + (R * x);
}
}
return(s * result);
}
/// <summary>
/// Implementation of the error function.
/// </summary>
/// <param name="z">Where to evaluate the error function.</param>
/// <param name="invert">Whether to compute 1 - the error function.</param>
/// <returns>the error function.</returns>
private static double ErfImp(double z, bool invert)
{
if (z < 0)
{
if (!invert)
{
return(-ErfImp(-z, false));
}
if (z < -0.5)
{
return(2 - ErfImp((-z), true));
}
return(1 + ErfImp(-z, false));
}
double result;
//
// Big bunch of selection statements now to pick which
// implementation to use, try to put most likely options
// first:
//
if (z < 0.5)
{
//
// We're going to calculate erf:
//
if (z < 1e-10)
{
result = (z * 1.125) + (z * 0.003379167095512573896158903121545171688);
}
else
{
// Worst case absolute error found: 6.688618532e-21
double[] n = new[] { 0.00337916709551257388990745, -0.00073695653048167948530905, -0.374732337392919607868241, 0.0817442448733587196071743, -0.0421089319936548595203468, 0.0070165709512095756344528, -0.00495091255982435110337458, 0.000871646599037922480317225 };
double[] d = new[] { 1, -0.218088218087924645390535, 0.412542972725442099083918, -0.0841891147873106755410271, 0.0655338856400241519690695, -0.0120019604454941768171266, 0.00408165558926174048329689, -0.000615900721557769691924509 };
result = (z * 1.125) + (z * Evaluate.Polynomial(z, n) / Evaluate.Polynomial(z, d));
}
}
else if ((z < 110) || ((z < 110) && invert))
{
//
// We'll be calculating erfc:
//
invert = !invert;
double r, b;
if (z < 0.75)
{
// Worst case absolute error found: 5.582813374e-21
double[] n = new[] { -0.0361790390718262471360258, 0.292251883444882683221149, 0.281447041797604512774415, 0.125610208862766947294894, 0.0274135028268930549240776, 0.00250839672168065762786937 };
double[] d = new[] { 1, 1.8545005897903486499845, 1.43575803037831418074962, 0.582827658753036572454135, 0.124810476932949746447682, 0.0113724176546353285778481 };
r = Evaluate.Polynomial(z - 0.5, n) / Evaluate.Polynomial(z - 0.5, d);
b = 0.3440242112F;
}
else if (z < 1.25)
{
// Worst case absolute error found: 4.01854729e-21
double[] n = new[] { -0.0397876892611136856954425, 0.153165212467878293257683, 0.191260295600936245503129, 0.10276327061989304213645, 0.029637090615738836726027, 0.0046093486780275489468812, 0.000307607820348680180548455 };
double[] d = new[] { 1, 1.95520072987627704987886, 1.64762317199384860109595, 0.768238607022126250082483, 0.209793185936509782784315, 0.0319569316899913392596356, 0.00213363160895785378615014 };
r = Evaluate.Polynomial(z - 0.75, n) / Evaluate.Polynomial(z - 0.75, d);
b = 0.419990927F;
}
else if (z < 2.25)
{
// Worst case absolute error found: 2.866005373e-21
double[] n = new[] { -0.0300838560557949717328341, 0.0538578829844454508530552, 0.0726211541651914182692959, 0.0367628469888049348429018, 0.00964629015572527529605267, 0.00133453480075291076745275, 0.778087599782504251917881e-4 };
double[] d = new[] { 1, 1.75967098147167528287343, 1.32883571437961120556307, 0.552528596508757581287907, 0.133793056941332861912279, 0.0179509645176280768640766, 0.00104712440019937356634038, -0.106640381820357337177643e-7 };
r = Evaluate.Polynomial(z - 1.25, n) / Evaluate.Polynomial(z - 1.25, d);
b = 0.4898625016F;
}
else if (z < 3.5)
{
// Worst case absolute error found: 1.045355789e-21
double[] n = new[] { -0.0117907570137227847827732, 0.014262132090538809896674, 0.0202234435902960820020765, 0.00930668299990432009042239, 0.00213357802422065994322516, 0.00025022987386460102395382, 0.120534912219588189822126e-4 };
double[] d = new[] { 1, 1.50376225203620482047419, 0.965397786204462896346934, 0.339265230476796681555511, 0.0689740649541569716897427, 0.00771060262491768307365526, 0.000371421101531069302990367 };
r = Evaluate.Polynomial(z - 2.25, n) / Evaluate.Polynomial(z - 2.25, d);
b = 0.5317370892F;
}
else if (z < 5.25)
{
// Worst case absolute error found: 8.300028706e-22
double[] n = new[] { -0.00546954795538729307482955, 0.00404190278731707110245394, 0.0054963369553161170521356, 0.00212616472603945399437862, 0.000394984014495083900689956, 0.365565477064442377259271e-4, 0.135485897109932323253786e-5 };
double[] d = new[] { 1, 1.21019697773630784832251, 0.620914668221143886601045, 0.173038430661142762569515, 0.0276550813773432047594539, 0.00240625974424309709745382, 0.891811817251336577241006e-4, -0.465528836283382684461025e-11 };
r = Evaluate.Polynomial(z - 3.5, n) / Evaluate.Polynomial(z - 3.5, d);
b = 0.5489973426F;
}
else if (z < 8)
{
// Worst case absolute error found: 1.700157534e-21
double[] n = new[] { -0.00270722535905778347999196, 0.0013187563425029400461378, 0.00119925933261002333923989, 0.00027849619811344664248235, 0.267822988218331849989363e-4, 0.923043672315028197865066e-6 };
double[] d = new[] { 1, 0.814632808543141591118279, 0.268901665856299542168425, 0.0449877216103041118694989, 0.00381759663320248459168994, 0.000131571897888596914350697, 0.404815359675764138445257e-11 };
r = Evaluate.Polynomial(z - 5.25, n) / Evaluate.Polynomial(z - 5.25, d);
b = 0.5571740866F;
}
else if (z < 11.5)
{
// Worst case absolute error found: 3.002278011e-22
double[] n = new[] { -0.00109946720691742196814323, 0.000406425442750422675169153, 0.000274499489416900707787024, 0.465293770646659383436343e-4, 0.320955425395767463401993e-5, 0.778286018145020892261936e-7 };
double[] d = new[] { 1, 0.588173710611846046373373, 0.139363331289409746077541, 0.0166329340417083678763028, 0.00100023921310234908642639, 0.24254837521587225125068e-4 };
r = Evaluate.Polynomial(z - 8, n) / Evaluate.Polynomial(z - 8, d);
b = 0.5609807968F;
}
else if (z < 17)
{
// Worst case absolute error found: 6.741114695e-21
double[] n = new[] { -0.00056907993601094962855594, 0.000169498540373762264416984, 0.518472354581100890120501e-4, 0.382819312231928859704678e-5, 0.824989931281894431781794e-7 };
double[] d = new[] { 1, 0.339637250051139347430323, 0.043472647870310663055044, 0.00248549335224637114641629, 0.535633305337152900549536e-4, -0.117490944405459578783846e-12 };
r = Evaluate.Polynomial(z - 11.5, n) / Evaluate.Polynomial(z - 11.5, d);
b = 0.5626493692F;
}
else if (z < 24)
{
// Worst case absolute error found: 7.802346984e-22
double[] n = new[] { -0.000241313599483991337479091, 0.574224975202501512365975e-4, 0.115998962927383778460557e-4, 0.581762134402593739370875e-6, 0.853971555085673614607418e-8 };
double[] d = new[] { 1, 0.233044138299687841018015, 0.0204186940546440312625597, 0.000797185647564398289151125, 0.117019281670172327758019e-4 };
r = Evaluate.Polynomial(z - 17, n) / Evaluate.Polynomial(z - 17, d);
b = 0.5634598136F;
}
else if (z < 38)
{
// Worst case absolute error found: 2.414228989e-22
double[] n = new[] { -0.000146674699277760365803642, 0.162666552112280519955647e-4, 0.269116248509165239294897e-5, 0.979584479468091935086972e-7, 0.101994647625723465722285e-8 };
double[] d = new[] { 1, 0.165907812944847226546036, 0.0103361716191505884359634, 0.000286593026373868366935721, 0.298401570840900340874568e-5 };
r = Evaluate.Polynomial(z - 24, n) / Evaluate.Polynomial(z - 24, d);
b = 0.5638477802F;
}
else if (z < 60)
{
// Worst case absolute error found: 5.896543869e-24
double[] n = new[] { -0.583905797629771786720406e-4, 0.412510325105496173512992e-5, 0.431790922420250949096906e-6, 0.993365155590013193345569e-8, 0.653480510020104699270084e-10 };
double[] d = new[] { 1, 0.105077086072039915406159, 0.00414278428675475620830226, 0.726338754644523769144108e-4, 0.477818471047398785369849e-6 };
r = Evaluate.Polynomial(z - 38, n) / Evaluate.Polynomial(z - 38, d);
b = 0.5640528202F;
}
else if (z < 85)
{
// Worst case absolute error found: 3.080612264e-21
double[] n = new[] { -0.196457797609229579459841e-4, 0.157243887666800692441195e-5, 0.543902511192700878690335e-7, 0.317472492369117710852685e-9 };
double[] d = new[] { 1, 0.052803989240957632204885, 0.000926876069151753290378112, 0.541011723226630257077328e-5, 0.535093845803642394908747e-15 };
r = Evaluate.Polynomial(z - 60, n) / Evaluate.Polynomial(z - 60, d);
b = 0.5641309023F;
}
else
{
// Worst case absolute error found: 8.094633491e-22
double[] n = new[] { -0.789224703978722689089794e-5, 0.622088451660986955124162e-6, 0.145728445676882396797184e-7, 0.603715505542715364529243e-10 };
double[] d = new[] { 1, 0.0375328846356293715248719, 0.000467919535974625308126054, 0.193847039275845656900547e-5 };
r = Evaluate.Polynomial(z - 85, n) / Evaluate.Polynomial(z - 85, d);
b = 0.5641584396F;
}
double g = Math.Exp(-z * z) / z;
result = (g * b) + (g * r);
}
else
{
//
// Any value of z larger than 28 will underflow to zero:
//
result = 0;
invert = !invert;
}
if (invert)
{
result = 1 - result;
}
return(result);
}
/// <summary>
/// Least-Squares fitting the points (x,y) to a k-order polynomial y : x -> p0 + p1*x + p2*x^2 + ... + pk*x^k,
/// returning a function y' for the best fitting polynomial.
/// A polynomial with order/degree k has (k+1) coefficients and thus requires at least (k+1) samples.
/// </summary>
public static Func <double, double> PolynomialFunc(double[] x, double[] y, int order, DirectRegressionMethod method = DirectRegressionMethod.QR)
{
var parameters = Polynomial(x, y, order, method);
return(z => Evaluate.Polynomial(z, parameters));
}
