invIncGamma.C

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    Copyright (C) 2016 OpenFOAM Foundation
    Copyright (C) 2021 OpenCFD Ltd.
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License
    This file is part of OpenFOAM.
 
    OpenFOAM is free software: you can redistribute it and/or modify it
    under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.
 
    OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
    ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
    FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
    for more details.
 
    You should have received a copy of the GNU General Public License
    along with OpenFOAM.  If not, see <http://www.gnu.org/licenses/>.
 
Global
    Foam::Math::invIncGamma
 
Description
    Implementation of the inverse incomplete gamma function.
 
\*---------------------------------------------------------------------------*/
 
#include "MathFunctions.H"
#include "mathematicalConstants.H"
#include "error.H"
#include <cmath>
#include <limits>
 
using namespace Foam::constant::mathematical;
 
// * * * * * * * * * * * * * * * Local Functions * * * * * * * * * * * * * * //
 
namespace Foam
{
 
static scalar minimaxs(const scalar P)
{
    // (DM:Eq. 32)
 
    constexpr scalar a_0 = 3.31125922108741;
    constexpr scalar a_1 = 11.6616720288968;
    constexpr scalar a_2 = 4.28342155967104;
    constexpr scalar a_3 = 0.213623493715853;
 
    constexpr scalar b_0 = 6.61053765625462;
    constexpr scalar b_1 = 6.40691597760039;
    constexpr scalar b_2 = 1.27364489782223;
    constexpr scalar b_3 = 0.03611708101884203;
 
    const scalar t = P < 0.5 ? sqrt(-2*log(P)) : sqrt(-2*log(1 - P));
 
    const scalar s =
        t
      - (a_0 + t*(a_1 + t*(a_2 + t*a_3)))
       /(1 + t*(b_0 + t*(b_1 + t*(b_2 + t*b_3))));
 
    return P < 0.5 ? -s : s;
}
 
 
static scalar Sn(const scalar a, const scalar x)
{
    // (DM:Eq. 34)
 
    scalar Sn = 1;
    scalar Si = 1;
 
    for (int i=1; i<100; ++i)
    {
        Si *= x/(a + i);
        Sn += Si;
 
        if (Si < 1e-4) break;
    }
 
    return Sn;
}
 
} // End namespace Foam
 
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
Foam::scalar Foam::Math::invIncGamma(const scalar a, const scalar P)
{
    #ifdef FULLDEBUG
    if (a <= 0)
    {
        WarningInFunction
            << "The parameter (i.e. a) cannot be negative or zero"
            << "    a = " << a
            << endl;
 
        return std::numeric_limits<scalar>::infinity();
    }
 
    if (P < 0 || P > 1)
    {
        WarningInFunction
            << "The domain of the parameter (i.e. P) should be limited to [0,1]"
            << "    P = " << P
            << endl;
 
        return std::numeric_limits<scalar>::infinity();
    }
    #endif
 
    const scalar Q = 1 - P;
 
    if (a == 1)
    {
        return -log(Q);
    }
    else if (a < 1)
    {
        const scalar Ga = tgamma(a);
        const scalar B = Q*Ga;
 
        if (B > 0.6 || (B >= 0.45 && a >= 0.3))
        {
            // (DM:Eq. 21)
            const scalar u =
                (B*Q > 1e-8) ? pow(P*Ga*a, 1/a) : exp((-Q/a) - Eu);
 
            return u/(1 - (u/(a + 1)));
        }
        else if (a < 0.3 && B >= 0.35)
        {
            // (DM:Eq. 22)
            const scalar t = exp(-Eu - B);
            const scalar u = t*exp(t);
 
            return t*exp(u);
        }
        else if (B > 0.15 || a >= 0.3)
        {
            // (DM:Eq. 23)
            const scalar y = -log(B);
            const scalar u = y - (1 - a)*log(y);
 
            return y - (1 - a)*log(u) - log(1 + (1 - a)/(1 + u));
        }
        else if (B > 0.1)
        {
            // (DM:Eq. 24)
            const scalar y = -log(B);
            const scalar u = y - (1 - a)*log(y);
 
            return y
              - (1 - a)*log(u)
              - log
                (
                    (sqr(u) + 2*(3 - a)*u + (2 - a)*(3 - a))
                   /(sqr(u) + (5 - a)*u + 2)
                );
        }
        else
        {
            // (DM:Eq. 25)
            const scalar y = -log(B);
            const scalar c1 = (a - 1)*log(y);
            const scalar c12 = c1*c1;
            const scalar c13 = c12*c1;
            const scalar c14 = c12*c12;
            const scalar a2 = a*a;
            const scalar a3 = a2*a;
            const scalar c2 = (a - 1)*(1 + c1);
            const scalar c3 = (a - 1)*(-(c12/2) + (a - 2)*c1 + (3*a - 5)/2);
            const scalar c4 =
                (a - 1)
               *(
                    (c13/3)
                  - (3*a - 5)*c12/2
                  + (a2 - 6*a + 7)*c1
                  + (11*a2 - 46*a + 47)/6
                );
            const scalar c5 =
                (a - 1)*(-(c14/4)
              + (11*a - 17)*c13/6
              + (-3*a2 + 13*a - 13)*c12
              + (2*a3 - 25*a2 + 72*a - 61)*c1/2
              + (25*a3 - 195*a2 + 477*a - 379)/12);
            const scalar y2 = y*y;
            const scalar y3 = y2*y;
            const scalar y4 = y2*y2;
 
            return y + c1 + (c2/y) + (c3/y2) + (c4/y3) + (c5/y4);
        }
    }
    else
    {
        // (DM:Eq. 31)
        scalar s = minimaxs(P);
 
        const scalar s2 = sqr(s);
        const scalar s3 = s*s2;
        const scalar s4 = s2*s2;
        const scalar s5 = s*s4;
        const scalar sqrta = sqrt(a);
 
        const scalar w =
            a + s*sqrta + (s2 - 1)/3
          + (s3 - 7*s)/(36*sqrta)
          - (3*s4 + 7*s2 - 16)/(810*a)
          + (9*s5 + 256*s3 - 433*s)/(38880*a*sqrta);
 
        if (a >= 500 && mag(1 - w/a) < 1e-6)
        {
            return w;
        }
        else if (P > 0.5)
        {
            if (w < 3*a)
            {
                return w;
            }
            else
            {
                const scalar D = max(scalar(2), scalar(a*(a - 1)));
                const scalar lnGa = lgamma(a);
                const scalar lnB = log(Q) + lnGa;
 
                if (lnB < -2.3*D)
                {
                    // (DM:Eq. 25)
                    const scalar y = -lnB;
                    const scalar c1 = (a - 1)*log(y);
                    const scalar c12 = c1*c1;
                    const scalar c13 = c12*c1;
                    const scalar c14 = c12*c12;
                    const scalar a2 = a*a;
                    const scalar a3 = a2*a;
 
                    const scalar c2 = (a - 1)*(1 + c1);
                    const scalar c3 =
                        (a - 1)
                       *(
                          - (c12/2)
                          + (a - 2)*c1
                          + (3*a - 5)/2
                        );
                    const scalar c4 =
                        (a - 1)
                       *(
                            (c13/3)
                          - (3*a - 5)*c12/2
                          + (a2 - 6*a + 7)*c1
                          + (11*a2 - 46*a + 47)/6
                        );
                    const scalar c5 =
                        (a - 1)
                       *(
                          - (c14/4)
                          + (11*a - 17)*c13/6
                          + (-3*a2 + 13*a - 13)*c12
                          + (2*a3 - 25*a2 + 72*a - 61)*c1/2
                          + (25*a3 - 195*a2 + 477*a - 379)/12
                        );
 
                    const scalar y2 = y*y;
                    const scalar y3 = y2*y;
                    const scalar y4 = y2*y2;
 
                    return y + c1 + (c2/y) + (c3/y2) + (c4/y3) + (c5/y4);
                }
                else
                {
                    // (DM:Eq. 33)
                    const scalar u =
                        -lnB + (a - 1)*log(w) - log(1 + (1 - a)/(1 + w));
 
                    return -lnB + (a - 1)*log(u) - log(1 + (1 - a)/(1 + u));
                }
            }
        }
        else
        {
            scalar z = w;
            const scalar ap1 = a + 1;
 
            if (w < 0.15*ap1)
            {
                // (DM:Eq. 35)
                const scalar ap2 = a + 2;
                const scalar v = log(P) + lgamma(ap1);
                z = exp((v + w)/a);
                s = log1p(z/ap1*(1 + z/ap2));
                z = exp((v + z - s)/a);
                s = log1p(z/ap1*(1 + z/ap2));
                z = exp((v + z - s)/a);
                s = log1p(z/ap1*(1 + z/ap2*(1 + z/(a + 3))));
                z = exp((v + z - s)/a);
            }
 
            if (z <= 0.01*ap1 || z > 0.7*ap1)
            {
                return z;
            }
            else
            {
                // (DM:Eq. 36)
                const scalar lnSn = log(Sn(a, z));
                const scalar v = log(P) + lgamma(ap1);
                z = exp((v + z - lnSn)/a);
 
                return z*(1 - (a*log(z) - z - v + lnSn)/(a - z));
            }
        }
    }
}
 
 
// ************************************************************************* //