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src/boost/math/special_functions/detail/bessel_ik.hpp  view on Meta::CPAN

   need_i = 1,
   need_k = 2
};

// Compute I(v, x) and K(v, x) simultaneously by Temme's method, see
// Temme, Journal of Computational Physics, vol 19, 324 (1975)
template <typename T, typename Policy>
int bessel_ik(T v, T x, T* I, T* K, int kind, const Policy& pol)
{
    // Kv1 = K_(v+1), fv = I_(v+1) / I_v
    // Ku1 = K_(u+1), fu = I_(u+1) / I_u
    T u, Iv, Kv, Kv1, Ku, Ku1, fv;
    T W, current, prev, next;
    bool reflect = false;
    unsigned n, k;
    int org_kind = kind;
    BOOST_MATH_INSTRUMENT_VARIABLE(v);
    BOOST_MATH_INSTRUMENT_VARIABLE(x);
    BOOST_MATH_INSTRUMENT_VARIABLE(kind);

    BOOST_MATH_STD_USING
    using namespace boost::math::tools;
    using namespace boost::math::constants;

    static const char* function = "boost::math::bessel_ik<%1%>(%1%,%1%)";

    if (v < 0)
    {
        reflect = true;
        v = -v;                             // v is non-negative from here
        kind |= need_k;
    }
    n = iround(v, pol);
    u = v - n;                              // -1/2 <= u < 1/2
    BOOST_MATH_INSTRUMENT_VARIABLE(n);
    BOOST_MATH_INSTRUMENT_VARIABLE(u);

    if (x < 0)
    {
       *I = *K = policies::raise_domain_error<T>(function,
            "Got x = %1% but real argument x must be non-negative, complex number result not supported.", x, pol);
        return 1;
    }
    if (x == 0)
    {
       Iv = (v == 0) ? static_cast<T>(1) : static_cast<T>(0);
       if(kind & need_k)
       {
         Kv = policies::raise_overflow_error<T>(function, 0, pol);
       }
       else
       {
          Kv = std::numeric_limits<T>::quiet_NaN(); // any value will do
       }

       if(reflect && (kind & need_i))
       {
           T z = (u + n % 2);
           Iv = boost::math::sin_pi(z, pol) == 0 ? 
               Iv : 
               policies::raise_overflow_error<T>(function, 0, pol);   // reflection formula
       }

       *I = Iv;
       *K = Kv;
       return 0;
    }

    // x is positive until reflection
    W = 1 / x;                                 // Wronskian
    if (x <= 2)                                // x in (0, 2]
    {
        temme_ik(u, x, &Ku, &Ku1, pol);             // Temme series
    }
    else                                       // x in (2, \infty)
    {
        CF2_ik(u, x, &Ku, &Ku1, pol);               // continued fraction CF2_ik
    }
    BOOST_MATH_INSTRUMENT_VARIABLE(Ku);
    BOOST_MATH_INSTRUMENT_VARIABLE(Ku1);
    prev = Ku;
    current = Ku1;
    T scale = 1;
    for (k = 1; k <= n; k++)                   // forward recurrence for K
    {
        T fact = 2 * (u + k) / x;
        if((tools::max_value<T>() - fabs(prev)) / fact < fabs(current))
        {
           prev /= current;
           scale /= current;
           current = 1;
        }
        next = fact * current + prev;
        prev = current;
        current = next;
    }
    Kv = prev;
    Kv1 = current;
    BOOST_MATH_INSTRUMENT_VARIABLE(Kv);
    BOOST_MATH_INSTRUMENT_VARIABLE(Kv1);
    if(kind & need_i)
    {
       T lim = (4 * v * v + 10) / (8 * x);
       lim *= lim;
       lim *= lim;
       lim /= 24;
       if((lim < tools::epsilon<T>() * 10) && (x > 100))
       {
          // x is huge compared to v, CF1 may be very slow
          // to converge so use asymptotic expansion for large
          // x case instead.  Note that the asymptotic expansion
          // isn't very accurate - so it's deliberately very hard 
          // to get here - probably we're going to overflow:
          Iv = asymptotic_bessel_i_large_x(v, x, pol);
       }
       else if((v > 0) && (x / v < 0.25))
       {
          Iv = bessel_i_small_z_series(v, x, pol);
       }
       else
       {
          CF1_ik(v, x, &fv, pol);                         // continued fraction CF1_ik
          Iv = scale * W / (Kv * fv + Kv1);                  // Wronskian relation
       }
    }
    else
       Iv = std::numeric_limits<T>::quiet_NaN(); // any value will do

    if (reflect)
    {
        T z = (u + n % 2);
        T fact = (2 / pi<T>()) * (boost::math::sin_pi(z) * Kv);
        if(fact == 0)
           *I = Iv;
        else if(tools::max_value<T>() * scale < fact)
           *I = (org_kind & need_i) ? T(sign(fact) * sign(scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
        else
         *I = Iv + fact / scale;   // reflection formula
    }
    else
    {
        *I = Iv;
    }
    if(tools::max_value<T>() * scale < Kv)
      *K = (org_kind & need_k) ? T(sign(Kv) * sign(scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
    else
      *K = Kv / scale;
    BOOST_MATH_INSTRUMENT_VARIABLE(*I);
    BOOST_MATH_INSTRUMENT_VARIABLE(*K);
    return 0;
}

}}} // namespaces

#endif // BOOST_MATH_BESSEL_IK_HPP