psi.cc

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#if defined (HAVE_CONFIG_H)
#  include "config.h"
#endif
 
#include "ov.h"
#include "defun.h"
#include "error.h"
#include "dNDArray.h"
#include "fNDArray.h"
 
#include "lo-specfun.h"
 
DEFUN (psi, args, ,
       doc: /* -*- texinfo -*-
@deftypefn  {} {} psi (@var{z})
@deftypefnx {} {} psi (@var{k}, @var{z})
Compute the psi (polygamma) function.
 
The polygamma functions are the @var{k}th derivative of the logarithm
of the gamma function.  If unspecified, @var{k} defaults to zero.  A value
of zero computes the digamma function, a value of 1, the trigamma function,
and so on.
 
The digamma function is defined:
 
@tex
$$
\Psi (z) = {d (log (\Gamma (z))) \over dx}
$$
@end tex
@ifnottex
 
@example
@group
psi (z) = d (log (gamma (z))) / dx
@end group
@end example
 
@end ifnottex
 
When computing the digamma function (when @var{k} equals zero), @var{z}
can have any value real or complex value.  However, for polygamma functions
(@var{k} higher than 0), @var{z} must be real and non-negative.
 
@seealso{gamma, gammainc, gammaln}
@end deftypefn */)
{
  int nargin = args.length ();
 
  if (nargin < 1 || nargin > 2)
    print_usage ();
 
  const octave_value oct_z = (nargin == 1) ? args(0) : args(1);
  const octave_idx_type k = (nargin == 1) ? 0 : args(0).xidx_type_value ("psi: K must be an integer");
  if (k < 0)
    error ("psi: K must be non-negative");
 
  octave_value retval;
 
  if (k == 0)
    {
#define FLOAT_BRANCH(T, A, M, E)                                \
      if (oct_z.is_ ## T ##_type ())                            \
        {                                                       \
          const A ## NDArray z = oct_z.M ## array_value ();     \
          A ## NDArray psi_z (z.dims ());                       \
                                                                \
          const E *zv = z.data ();                              \
          E *psi_zv = psi_z.fortran_vec ();                     \
          const octave_idx_type n = z.numel ();                 \
          for (octave_idx_type i = 0; i < n; i++)               \
            *psi_zv++ = octave::math::psi (*zv++);              \
                                                                \
          retval = psi_z;                                       \
        }
 
      if (oct_z.iscomplex ())
        {
          FLOAT_BRANCH(double, Complex, complex_, Complex)
          else FLOAT_BRANCH(single, FloatComplex, float_complex_, FloatComplex)
          else
            error ("psi: Z must be a floating point");
        }
      else
        {
          FLOAT_BRANCH(double, , , double)
          else FLOAT_BRANCH(single, Float, float_, float)
          else
            error ("psi: Z must be a floating point");
        }
 
#undef FLOAT_BRANCH
    }
  else
    {
      if (! oct_z.isreal ())
        error ("psi: Z must be real value for polygamma (K > 0)");
 
#define FLOAT_BRANCH(T, A, M, E)                                        \
      if (oct_z.is_ ## T ##_type ())                                    \
        {                                                               \
          const A ## NDArray z = oct_z.M ## array_value ();             \
          A ## NDArray psi_z (z.dims ());                               \
                                                                        \
          const E *zv = z.data ();                                      \
          E *psi_zv = psi_z.fortran_vec ();                             \
          const octave_idx_type n = z.numel ();                         \
          for (octave_idx_type i = 0; i < n; i++)                       \
            {                                                           \
              if (*zv < 0)                                              \
                error ("psi: Z must be non-negative for polygamma (K > 0)"); \
                                                                        \
              *psi_zv++ = octave::math::psi (k, *zv++);                 \
            }                                                           \
          retval = psi_z;                                               \
        }
 
      FLOAT_BRANCH(double, , , double)
      else FLOAT_BRANCH(single, Float, float_, float)
      else
        error ("psi: Z must be a floating point for polygamma (K > 0)");
 
#undef FLOAT_BRANCH
    }
 
  return retval;
}
 
/*
%!shared em
%! em = 0.577215664901532860606512090082402431042; # Euler-Mascheroni Constant
 
%!assert (psi (ones (7, 3, 5)), repmat (-em, [7 3 5]))
%!assert (psi ([0 1]), [-Inf -em])
%!assert (psi ([-20:1]), [repmat(-Inf, [1 21]) -em])
%!assert (psi (single ([0 1])), single ([-Inf -em]))
 
## Abramowitz and Stegun, page 258, eq 6.3.5
%!test
%! z = [-100:-1 1:200] ./ 10; # drop the 0
%! assert (psi (z + 1), psi (z) + 1 ./ z, eps*1000);
 
## Abramowitz and Stegun, page 258, eq 6.3.2
%!assert (psi (1), -em)
 
## Abramowitz and Stegun, page 258, eq 6.3.3
%!assert (psi (1/2), -em - 2 * log (2))
 
## The following tests are from Pascal Sebah and Xavier Gourdon (2002)
## "Introduction to the Gamma Function"
 
## Interesting identities of the digamma function, in section of 5.1.3
%!assert (psi (1/3), - em - (3/2) * log(3) - ((sqrt (3) / 6) * pi), eps*10)
%!assert (psi (1/4), - em -3 * log (2) - pi/2, eps*10)
%!assert (psi (1/6), - em -2 * log (2) - (3/2) * log (3) - ((sqrt (3) / 2) * pi), eps*10)
 
## First 6 zeros of the digamma function, in section of 5.1.5 (and also on
## Abramowitz and Stegun, page 258, eq 6.3.19)
%!assert (psi ( 1.46163214496836234126265954232572132846819620400644), 0, eps)
%!assert (psi (-0.504083008264455409258269304533302498955385182368579), 0, eps*2)
%!assert (psi (-1.573498473162390458778286043690434612655040859116846), 0, eps*2)
%!assert (psi (-2.610720868444144650001537715718724207951074010873480), 0, eps*10)
%!assert (psi (-3.635293366436901097839181566946017713948423861193530), 0, eps*10)
%!assert (psi (-4.653237761743142441714598151148207363719069416133868), 0, eps*100)
 
## Tests for complex values
%!shared z
%! z = [-100:-1 1:200] ./ 10; # drop the 0
 
## Abramowitz and Stegun, page 259 eq 6.3.10
%!assert (real (psi (i*z)), real (psi (1 - i*z)))
 
## Abramowitz and Stegun, page 259 eq 6.3.11
%!assert (imag (psi (i*z)), 1/2 .* 1./z + 1/2 * pi * coth (pi * z), eps *10)
 
## Abramowitz and Stegun, page 259 eq 6.3.12
%!assert (imag (psi (1/2 + i*z)), 1/2 * pi * tanh (pi * z), eps*10)
 
## Abramowitz and Stegun, page 259 eq 6.3.13
%!assert (imag (psi (1 + i*z)), - 1./(2*z) + 1/2 * pi * coth (pi * z), eps*10)
 
## Abramowitz and Stegun, page 260 eq 6.4.5
%!test
%! for z = 0:20
%!   assert (psi (1, z + 0.5),
%!           0.5 * (pi^2) - 4 * sum ((2*(1:z) -1) .^(-2)),
%!           eps*10);
%! endfor
 
## Abramowitz and Stegun, page 260 eq 6.4.6
%!test
%! z = 0.1:0.1:20;
%! for n = 0:8
%!   ## our precision goes down really quick when computing n is too high.
%!   assert (psi (n, z+1),
%!           psi (n, z) + ((-1)^n) * factorial (n) * (z.^(-n-1)), 0.1);
%! endfor
 
## Test input validation
%!error psi ()
%!error psi (1, 2, 3)
%!error <Z must be> psi ("non numeric")
%!error <K must be an integer> psi ({5.3}, 1)
%!error <K must be non-negative> psi (-5, 1)
%!error <Z must be non-negative for polygamma> psi (5, -1)
%!error <Z must be a floating point> psi (5, uint8 (-1))
%!error <Z must be real value for polygamma> psi (5, 5i)
 
*/