## Copyright (C) 2013 Julien Bect
## Copyright (C) 2012 Rik Wehbring
## Copyright (C) 1995-2012 Kurt Hornik
##
## This file is part of Octave.
##
## Octave 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.
##
## Octave 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 Octave; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {} tcdf (@var{x}, @var{n})
## For each element of @var{x}, compute the cumulative distribution
## function (CDF) at @var{x} of the t (Student) distribution with
## @var{n} degrees of freedom, i.e., PROB (t(@var{n}) @leq{} @var{x}).
## @end deftypefn

## Author: KH <Kurt.Hornik@wu-wien.ac.at>
## Description: CDF of the t distribution

function cdf = tcdf (x, n)

  if (nargin != 2)
    print_usage ();
  endif

  if (!isscalar (n))
    [retval, x, n] = common_size (x, n);
    if (retval > 0)
      error ("tcdf: X and N must be of common size or scalars");
    endif
  endif

  if (iscomplex (x) || iscomplex (n))
    error ("tcdf: X and N must not be complex");
  endif

  if (isa (x, "single") || isa (n, "single"))
    cdf = zeros (size (x), "single");
  else
    cdf = zeros (size (x));
  endif

  k = !isinf (x) & (n > 0);

  xx = x .^ 2;
  x_big_abs = (xx > n);

  ## deal with the case "abs(x) big"
  kk = k & x_big_abs;
  if (isscalar (n))
    cdf(kk) = betainc (n ./ (n + xx(kk)), n/2, 1/2) / 2;
  else
    cdf(kk) = betainc (n(kk) ./ (n(kk) + xx(kk)), n(kk)/2, 1/2) / 2;
  endif

  ## deal with the case "abs(x) small"
  kk = k & !x_big_abs;
  if (isscalar (n))
    cdf(kk) = 0.5 * (1 - betainc (xx(kk) ./ (n + xx(kk)), 1/2, n/2));
  else
    cdf(kk) = 0.5 * (1 - betainc (xx(kk) ./ (n(kk) + xx(kk)), 1/2, n(kk)/2));
  endif

  k &= (x > 0);
  if (any (k(:)))
    cdf(k) = 1 - cdf(k);
  endif

  k = isnan (x) | !(n > 0);
  cdf(k) = NaN;

  k = (x == Inf) & (n > 0);
  cdf(k) = 1;

endfunction


%!shared x,y
%! x = [-Inf 0 1 Inf];
%! y = [0 1/2 3/4 1];
%!assert (tcdf (x, ones (1,4)), y, eps)
%!assert (tcdf (x, 1), y, eps)
%!assert (tcdf (x, [0 1 NaN 1]), [NaN 1/2 NaN 1], eps)
%!assert (tcdf ([x(1:2) NaN x(4)], 1), [y(1:2) NaN y(4)], eps)

%% Test class of input preserved
%!assert (tcdf ([x, NaN], 1), [y, NaN], eps)
%!assert (tcdf (single ([x, NaN]), 1), single ([y, NaN]), eps ("single"))
%!assert (tcdf ([x, NaN], single (1)), single ([y, NaN]), eps ("single"))

%% Test input validation
%!error tcdf ()
%!error tcdf (1)
%!error tcdf (1,2,3)
%!error tcdf (ones (3), ones (2))
%!error tcdf (ones (2), ones (3))
%!error tcdf (i, 2)
%!error tcdf (2, i)

## Check some reference values

%!shared tol_rel
%! tol_rel = 10 * eps;

## check accuracy for small positive values
%!assert (tcdf (10^(-10), 2.5), 0.50000000003618087, -tol_rel)
%!assert (tcdf (10^(-11), 2.5), 0.50000000000361809, -tol_rel)
%!assert (tcdf (10^(-12), 2.5), 0.50000000000036181, -tol_rel)
%!assert (tcdf (10^(-13), 2.5), 0.50000000000003618, -tol_rel)
%!assert (tcdf (10^(-14), 2.5), 0.50000000000000362, -tol_rel)
%!assert (tcdf (10^(-15), 2.5), 0.50000000000000036, -tol_rel)
%!assert (tcdf (10^(-16), 2.5), 0.50000000000000004, -tol_rel)

## check accuracy for large negative values
%!assert (tcdf (-10^1, 2.5), 2.2207478836537124e-03, -tol_rel)
%!assert (tcdf (-10^2, 2.5), 7.1916492116661878e-06, -tol_rel)
%!assert (tcdf (-10^3, 2.5), 2.2747463948307452e-08, -tol_rel)
%!assert (tcdf (-10^4, 2.5), 7.1933970159922115e-11, -tol_rel)
%!assert (tcdf (-10^5, 2.5), 2.2747519231756221e-13, -tol_rel)

## # Reference values obtained using Python 2.7.4 and mpmath 0.17
##
## from mpmath import *
##
## mp.dps = 100
##
## def F(x_in, nu_in):
##     x = mpf(x_in);
##     nu = mpf(nu_in);
##     t = nu / (nu + x*x)
##     a = nu / 2
##     b = mpf(0.5)
##     F = betainc(a, b, 0, t, regularized=True) / 2
##     if (x > 0):
##         F = 1 - F
##     return F
##
## nu = 2.5
##
## for i in range(1, 6):
##     x = - power(mpf(10), mpf(i))
##     print "%%!assert (tcdf (-10^%d, 2.5), %s, -eps)" \
##         % (i, nstr(F(x, nu), 17))
##
## for i in range(10, 17):
##     x = power(mpf(10), -mpf(i))
##     print "%%!assert (tcdf (10^(-%d), 2.5), %s, -eps)" \
##         % (i, nstr(F(x, nu), 17))