double
proj_auth_inv(double beta, const void *a) {
int max;
double dl, c, phi;
c = sin(beta);
phi = beta;
for (max = MAX_ITER ; max ; --max) {
dl = (c - betaf(phi, a))/betap(phi, a);
phi += dl;
if (fabs(dl) < TOLER)
return(phi);
}
/* it will usually not fall out of the loop, in fact tests
* have never made it fall through. But there are convergence
* limits at argument approaches 90 degrees */
return(phi);
}
T non_central_t2_p(T v, T delta, T x, T y, const Policy& pol, T init_val)
{
BOOST_MATH_STD_USING
//
// Variables come first:
//
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
T errtol = policies::get_epsilon<T, Policy>();
T d2 = delta * delta / 2;
//
// k is the starting point for iteration, and is the
// maximum of the poisson weighting term, we don't
// ever allow k == 0 as this can lead to catastrophic
// cancellation errors later (test case is v = 1621286869049072.3
// delta = 0.16212868690490723, x = 0.86987415482475994).
//
int k = itrunc(d2);
T pois;
if(k == 0) k = 1;
// Starting Poisson weight:
pois = gamma_p_derivative(T(k+1), d2, pol)
* tgamma_delta_ratio(T(k + 1), T(0.5f))
* delta / constants::root_two<T>();
if(pois == 0)
return init_val;
T xterm, beta;
// Recurrance & starting beta terms:
beta = x < y
? detail::ibeta_imp(T(k + 1), T(v / 2), x, pol, false, true, &xterm)
: detail::ibeta_imp(T(v / 2), T(k + 1), y, pol, true, true, &xterm);
xterm *= y / (v / 2 + k);
T poisf(pois), betaf(beta), xtermf(xterm);
T sum = init_val;
if((xterm == 0) && (beta == 0))
return init_val;
//
// Backwards recursion first, this is the stable
// direction for recursion:
//
boost::uintmax_t count = 0;
T last_term = 0;
for(int i = k; i >= 0; --i)
{
T term = beta * pois;
sum += term;
// Don't terminate on first term in case we "fixed" k above:
if((fabs(last_term) > fabs(term)) && fabs(term/sum) < errtol)
break;
last_term = term;
pois *= (i + 0.5f) / d2;
beta += xterm;
xterm *= (i) / (x * (v / 2 + i - 1));
++count;
}
last_term = 0;
for(int i = k + 1; ; ++i)
{
poisf *= d2 / (i + 0.5f);
xtermf *= (x * (v / 2 + i - 1)) / (i);
betaf -= xtermf;
T term = poisf * betaf;
sum += term;
if((fabs(last_term) > fabs(term)) && (fabs(term/sum) < errtol))
break;
last_term = term;
++count;
if(count > max_iter)
{
return policies::raise_evaluation_error(
"cdf(non_central_t_distribution<%1%>, %1%)",
"Series did not converge, closest value was %1%", sum, pol);
}
}
return sum;
}
T non_central_t2_q(T v, T delta, T x, T y, const Policy& pol, T init_val)
{
BOOST_MATH_STD_USING
//
// Variables come first:
//
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
T errtol = boost::math::policies::get_epsilon<T, Policy>();
T d2 = delta * delta / 2;
//
// k is the starting point for iteration, and is the
// maximum of the poisson weighting term, we don't allow
// k == 0 as this can cause catastrophic cancellation errors
// (test case is v = 561908036470413.25, delta = 0.056190803647041321,
// x = 1.6155232703966216):
//
int k = itrunc(d2);
if(k == 0) k = 1;
// Starting Poisson weight:
T pois;
if((k < (int)(max_factorial<T>::value)) && (d2 < tools::log_max_value<T>()) && (log(d2) * k < tools::log_max_value<T>()))
{
//
// For small k we can optimise this calculation by using
// a simpler reduced formula:
//
pois = exp(-d2);
pois *= pow(d2, static_cast<T>(k));
pois /= boost::math::tgamma(T(k + 1 + 0.5), pol);
pois *= delta / constants::root_two<T>();
}
else
{
pois = gamma_p_derivative(T(k+1), d2, pol)
* tgamma_delta_ratio(T(k + 1), T(0.5f))
* delta / constants::root_two<T>();
}
if(pois == 0)
return init_val;
// Recurance term:
T xterm;
T beta;
// Starting beta term:
if(k != 0)
{
beta = x < y
? detail::ibeta_imp(T(k + 1), T(v / 2), x, pol, true, true, &xterm)
: detail::ibeta_imp(T(v / 2), T(k + 1), y, pol, false, true, &xterm);
xterm *= y / (v / 2 + k);
}
else
{
beta = pow(y, v / 2);
xterm = beta;
}
T poisf(pois), betaf(beta), xtermf(xterm);
T sum = init_val;
if((xterm == 0) && (beta == 0))
return init_val;
//
// Fused forward and backwards recursion:
//
boost::uintmax_t count = 0;
T last_term = 0;
for(int i = k + 1, j = k; ; ++i, --j)
{
poisf *= d2 / (i + 0.5f);
xtermf *= (x * (v / 2 + i - 1)) / (i);
betaf += xtermf;
T term = poisf * betaf;
if(j >= 0)
{
term += beta * pois;
pois *= (j + 0.5f) / d2;
beta -= xterm;
xterm *= (j) / (x * (v / 2 + j - 1));
}
sum += term;
// Don't terminate on first term in case we "fixed" the value of k above:
if((fabs(last_term) > fabs(term)) && fabs(term/sum) < errtol)
break;
last_term = term;
if(count > max_iter)
{
return policies::raise_evaluation_error(
"cdf(non_central_t_distribution<%1%>, %1%)",
"Series did not converge, closest value was %1%", sum, pol);
}
++count;
}
return sum;
}
inline float beta(float a, float b)
{ return betaf(a, b); }
double
proj_auth_lat(double phi, const void *a) {
return(asin(betaf(phi, a)));
}