T cyl_bessel_j_imp(T v, T x, const bessel_no_int_tag& t, const Policy& pol)
{
BOOST_MATH_STD_USING
static const char* function = "boost::math::bessel_j<%1%>(%1%,%1%)";
if(x < 0)
{
// better have integer v:
if(floor(v) == v)
{
T r = cyl_bessel_j_imp(v, T(-x), t, pol);
if(iround(v, pol) & 1)
r = -r;
return r;
}
else
return policies::raise_domain_error<T>(
function,
"Got x = %1%, but we need x >= 0", x, pol);
}
if(x == 0)
return (v == 0) ? 1 : (v > 0) ? 0 :
policies::raise_domain_error<T>(
function,
"Got v = %1%, but require v >= 0 or a negative integer: the result would be complex.", v, pol);
if((v >= 0) && ((x < 1) || (v > x * x / 4)))
{
return bessel_j_small_z_series(v, x, pol);
}
T j, y;
bessel_jy(v, x, &j, &y, need_j, pol);
return j;
}
T cyl_bessel_j_imp(T v, T x, const bessel_no_int_tag& t, const Policy& pol)
{
BOOST_MATH_STD_USING
static const char* function = "boost::math::bessel_j<%1%>(%1%,%1%)";
if(x < 0)
{
// better have integer v:
if(floor(v) == v)
{
T r = cyl_bessel_j_imp(v, T(-x), t, pol);
if(iround(v, pol) & 1)
r = -r;
return r;
}
else
return policies::raise_domain_error<T>(
function,
"Got x = %1%, but we need x >= 0", x, pol);
}
if(x == 0)
return (v == 0) ? 1 : (v > 0) ? 0 :
policies::raise_domain_error<T>(
function,
"Got v = %1%, but require v >= 0 or a negative integer: the result would be complex.", v, pol);
if((v >= 0) && ((x < 1) || (v > x * x / 4) || (x < 5)))
{
//
// This series will actually converge rapidly for all small
// x - say up to x < 20 - but the first few terms are large
// and divergent which leads to large errors :-(
//
return bessel_j_small_z_series(v, x, pol);
}
T j, y;
bessel_jy(v, x, &j, &y, need_j, pol);
return j;
}
T bessel_jn(int n, T x, const Policy& pol)
{
T value(0), factor, current, prev, next;
BOOST_MATH_STD_USING
//
// Reflection has to come first:
//
if (n < 0)
{
factor = static_cast<T>((n & 0x1) ? -1 : 1); // J_{-n}(z) = (-1)^n J_n(z)
n = -n;
}
else
{
factor = 1;
}
if(x < 0)
{
factor *= (n & 0x1) ? -1 : 1; // J_{n}(-z) = (-1)^n J_n(z)
x = -x;
}
//
// Special cases:
//
if(asymptotic_bessel_large_x_limit(T(n), x))
return factor * asymptotic_bessel_j_large_x_2<T>(T(n), x);
if (n == 0)
{
return factor * bessel_j0(x);
}
if (n == 1)
{
return factor * bessel_j1(x);
}
if (x == 0) // n >= 2
{
return static_cast<T>(0);
}
BOOST_ASSERT(n > 1);
T scale = 1;
if (n < abs(x)) // forward recurrence
{
prev = bessel_j0(x);
current = bessel_j1(x);
policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
for (int k = 1; k < n; k++)
{
T fact = 2 * k / x;
//
// rescale if we would overflow or underflow:
//
if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
{
scale /= current;
prev /= current;
current = 1;
}
value = fact * current - prev;
prev = current;
current = value;
}
}
else if((x < 1) || (n > x * x / 4) || (x < 5))
{
return factor * bessel_j_small_z_series(T(n), x, pol);
}
else // backward recurrence
{
T fn;
int s; // fn = J_(n+1) / J_n
// |x| <= n, fast convergence for continued fraction CF1
boost::math::detail::CF1_jy(static_cast<T>(n), x, &fn, &s, pol);
prev = fn;
current = 1;
// Check recursion won't go on too far:
policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
for (int k = n; k > 0; k--)
{
T fact = 2 * k / x;
if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
{
prev /= current;
scale /= current;
current = 1;
}
next = fact * current - prev;
prev = current;
current = next;
}
value = bessel_j0(x) / current; // normalization
scale = 1 / scale;
}
value *= factor;
if(tools::max_value<T>() * scale < fabs(value))
return policies::raise_overflow_error<T>("boost::math::bessel_jn<%1%>(%1%,%1%)", 0, pol);
return value / scale;
}