gcd.cc

Go to the documentation of this file.

////////////////////////////////////////////////////////////////////////
//
// Copyright (C) 2004-2024 The Octave Project Developers
//
// See the file COPYRIGHT.md in the top-level directory of this
// distribution or <https://octave.org/copyright/>.
//
// 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
// <https://www.gnu.org/licenses/>.
//
////////////////////////////////////////////////////////////////////////
 
#if defined (HAVE_CONFIG_H)
#  include "config.h"
#endif
 
#include "dNDArray.h"
#include "CNDArray.h"
#include "fNDArray.h"
#include "fCNDArray.h"
#include "lo-mappers.h"
#include "oct-binmap.h"
 
#include "defun.h"
#include "error.h"
#include "ovl.h"
 
OCTAVE_BEGIN_NAMESPACE(octave)
 
static double
simple_gcd (double a, double b)
{
  if (! math::isinteger (a) || ! math::isinteger (b))
    error ("gcd: all values must be integers");
 
  double aa = fabs (a);
  double bb = fabs (b);
 
  while (bb != 0)
    {
      double tt = fmod (aa, bb);
      aa = bb;
      bb = tt;
    }
 
  return aa;
}
 
template <typename FP>
static void
divide (const std::complex<FP>& a, const std::complex<FP>& b,
        std::complex<FP>& q, std::complex<FP>& r)
{
  FP qr = std::floor ((a/b).real () + 0.5);
  FP qi = std::floor ((a/b).imag () + 0.5);
 
  q = std::complex<FP> (qr, qi);
 
  r = a - q*b;
}
 
template <typename FP>
static std::complex<FP>
simple_gcd (const std::complex<FP>& a, const std::complex<FP>& b)
{
  if (! math::isinteger (a.real ())
      || ! math::isinteger (a.imag ())
      || ! math::isinteger (b.real ())
      || ! math::isinteger (b.imag ()))
    error ("gcd: all complex parts must be integers");
 
  std::complex<FP> aa = a;
  std::complex<FP> bb = b;
 
  if (abs (aa) < abs (bb))
    std::swap (aa, bb);
 
  while (abs (bb) != 0)
    {
      std::complex<FP> qq, rr;
      divide (aa, bb, qq, rr);
      aa = bb;
      bb = rr;
    }
 
  return aa;
}
 
template <typename T>
static octave_int<T>
simple_gcd (const octave_int<T>& a, const octave_int<T>& b)
{
  T aa = a.abs ().value ();
  T bb = b.abs ().value ();
 
  while (bb != 0)
    {
      T tt = aa % bb;
      aa = bb;
      bb = tt;
    }
 
  return aa;
}
 
static double
extended_gcd (double a, double b, double& x, double& y)
{
  if (! math::isinteger (a) || ! math::isinteger (b))
    error ("gcd: all values must be integers");
 
  double aa = fabs (a);
  double bb = fabs (b);
 
  double xx, lx, yy, ly;
  xx = 0, lx = 1;
  yy = 1, ly = 0;
 
  while (bb != 0)
    {
      double qq = std::floor (aa / bb);
      double tt = fmod (aa, bb);
 
      aa = bb;
      bb = tt;
 
      double tx = lx - qq*xx;
      lx = xx;
      xx = tx;
 
      double ty = ly - qq*yy;
      ly = yy;
      yy = ty;
    }
 
  x = (a >= 0 ? lx : -lx);
  y = (b >= 0 ? ly : -ly);
 
  return aa;
}
 
template <typename FP>
static std::complex<FP>
extended_gcd (const std::complex<FP>& a, const std::complex<FP>& b,
              std::complex<FP>& x, std::complex<FP>& y)
{
  if (! math::isinteger (a.real ())
      || ! math::isinteger (a.imag ())
      || ! math::isinteger (b.real ())
      || ! math::isinteger (b.imag ()))
    error ("gcd: all complex parts must be integers");
 
  std::complex<FP> aa = a;
  std::complex<FP> bb = b;
  bool swapped = false;
  if (abs (aa) < abs (bb))
    {
      std::swap (aa, bb);
      swapped = true;
    }
 
  std::complex<FP> xx, lx, yy, ly;
  xx = 0, lx = 1;
  yy = 1, ly = 0;
 
  while (abs(bb) != 0)
    {
      std::complex<FP> qq, rr;
      divide (aa, bb, qq, rr);
      aa = bb;
      bb = rr;
 
      std::complex<FP> tx = lx - qq*xx;
      lx = xx;
      xx = tx;
 
      std::complex<FP> ty = ly - qq*yy;
      ly = yy;
      yy = ty;
    }
 
  x = lx;
  y = ly;
 
  if (swapped)
    std::swap (x, y);
 
  return aa;
}
 
template <typename T>
static octave_int<T>
extended_gcd (const octave_int<T>& a, const octave_int<T>& b,
              octave_int<T>& x, octave_int<T>& y)
{
  T aa = a.abs ().value ();
  T bb = b.abs ().value ();
  T xx, lx, yy, ly;
  xx = 0, lx = 1;
  yy = 1, ly = 0;
 
  while (bb != 0)
    {
      T qq = aa / bb;
      T tt = aa % bb;
      aa = bb;
      bb = tt;
 
      T tx = lx - qq*xx;
      lx = xx;
      xx = tx;
 
      T ty = ly - qq*yy;
      ly = yy;
      yy = ty;
    }
 
  x = octave_int<T> (lx) * a.signum ();
  y = octave_int<T> (ly) * b.signum ();
 
  return aa;
}
 
template <typename NDA>
static octave_value
do_simple_gcd (const octave_value& a, const octave_value& b)
{
  typedef typename NDA::element_type T;
  octave_value retval;
 
  if (a.is_scalar_type () && b.is_scalar_type ())
    {
      // Optimize scalar case.
      T aa = octave_value_extract<T> (a);
      T bb = octave_value_extract<T> (b);
      retval = simple_gcd (aa, bb);
    }
  else
    {
      NDA aa = octave_value_extract<NDA> (a);
      NDA bb = octave_value_extract<NDA> (b);
      retval = binmap<T> (aa, bb, simple_gcd, "gcd");
    }
 
  return retval;
}
 
// Dispatcher
static octave_value
do_simple_gcd (const octave_value& a, const octave_value& b)
{
  octave_value retval;
  builtin_type_t btyp = btyp_mixed_numeric (a.builtin_type (),
                        b.builtin_type ());
  switch (btyp)
    {
    case btyp_double:
      if (a.issparse () && b.issparse ())
        {
          retval = do_simple_gcd<SparseMatrix> (a, b);
          break;
        }
      OCTAVE_FALLTHROUGH;
 
    case btyp_float:
      retval = do_simple_gcd<NDArray> (a, b);
      break;
 
#define MAKE_INT_BRANCH(X)                            \
    case btyp_ ## X:                                  \
      retval = do_simple_gcd<X ## NDArray> (a, b);    \
      break
 
      MAKE_INT_BRANCH (int8);
      MAKE_INT_BRANCH (int16);
      MAKE_INT_BRANCH (int32);
      MAKE_INT_BRANCH (int64);
      MAKE_INT_BRANCH (uint8);
      MAKE_INT_BRANCH (uint16);
      MAKE_INT_BRANCH (uint32);
      MAKE_INT_BRANCH (uint64);
 
#undef MAKE_INT_BRANCH
 
    case btyp_complex:
      retval = do_simple_gcd<ComplexNDArray> (a, b);
      break;
 
    case btyp_float_complex:
      retval = do_simple_gcd<FloatComplexNDArray> (a, b);
      break;
 
    default:
      error ("gcd: invalid class combination for gcd: %s and %s\n",
             a.class_name ().c_str (), b.class_name ().c_str ());
    }
 
  if (btyp == btyp_float)
    retval = retval.float_array_value ();
 
  return retval;
}
 
template <typename NDA>
static octave_value
do_extended_gcd (const octave_value& a, const octave_value& b,
                 octave_value& x, octave_value& y)
{
  typedef typename NDA::element_type T;
  octave_value retval;
 
  if (a.is_scalar_type () && b.is_scalar_type ())
    {
      // Optimize scalar case.
      T aa = octave_value_extract<T> (a);
      T bb = octave_value_extract<T> (b);
      T xx, yy;
      retval = extended_gcd (aa, bb, xx, yy);
      x = xx;
      y = yy;
    }
  else
    {
      NDA aa = octave_value_extract<NDA> (a);
      NDA bb = octave_value_extract<NDA> (b);
 
      dim_vector dv = aa.dims ();
      if (aa.numel () == 1)
        dv = bb.dims ();
      else if (bb.numel () != 1 && bb.dims () != dv)
        err_nonconformant ("gcd", a.dims (), b.dims ());
 
      NDA gg (dv), xx (dv), yy (dv);
 
      const T *aptr = aa.data ();
      const T *bptr = bb.data ();
 
      bool inca = aa.numel () != 1;
      bool incb = bb.numel () != 1;
 
      T *gptr = gg.fortran_vec ();
      T *xptr = xx.fortran_vec ();
      T *yptr = yy.fortran_vec ();
 
      octave_idx_type n = gg.numel ();
      for (octave_idx_type i = 0; i < n; i++)
        {
          octave_quit ();
 
          *gptr++ = extended_gcd (*aptr, *bptr, *xptr++, *yptr++);
 
          aptr += inca;
          bptr += incb;
        }
 
      x = xx;
      y = yy;
 
      retval = gg;
    }
 
  return retval;
}
 
// Dispatcher
static octave_value
do_extended_gcd (const octave_value& a, const octave_value& b,
                 octave_value& x, octave_value& y)
{
  octave_value retval;
 
  builtin_type_t btyp = btyp_mixed_numeric (a.builtin_type (),
                        b.builtin_type ());
  switch (btyp)
    {
    case btyp_double:
    case btyp_float:
      retval = do_extended_gcd<NDArray> (a, b, x, y);
      break;
 
#define MAKE_INT_BRANCH(X)                                    \
    case btyp_ ## X:                                          \
      retval = do_extended_gcd<X ## NDArray> (a, b, x, y);    \
      break
 
      MAKE_INT_BRANCH (int8);
      MAKE_INT_BRANCH (int16);
      MAKE_INT_BRANCH (int32);
      MAKE_INT_BRANCH (int64);
      MAKE_INT_BRANCH (uint8);
      MAKE_INT_BRANCH (uint16);
      MAKE_INT_BRANCH (uint32);
      MAKE_INT_BRANCH (uint64);
 
#undef MAKE_INT_BRANCH
 
    case btyp_complex:
      retval = do_extended_gcd<ComplexNDArray> (a, b, x, y);
      break;
 
    case btyp_float_complex:
      retval = do_extended_gcd<FloatComplexNDArray> (a, b, x, y);
      break;
 
    default:
      error ("gcd: invalid class combination for gcd: %s and %s\n",
             a.class_name ().c_str (), b.class_name ().c_str ());
    }
 
  // For consistency.
  if (a.issparse () && b.issparse ())
    {
      retval = retval.sparse_matrix_value ();
      x = x.sparse_matrix_value ();
      y = y.sparse_matrix_value ();
    }
 
  if (btyp == btyp_float)
    {
      retval = retval.float_array_value ();
      x = x.float_array_value ();
      y = y.float_array_value ();
    }
 
  return retval;
}
 
DEFUN (gcd, args, nargout,
       doc: /* -*- texinfo -*-
@deftypefn  {} {@var{g} =} gcd (@var{a1}, @var{a2}, @dots{})
@deftypefnx {} {[@var{g}, @var{v1}, @dots{}] =} gcd (@var{a1}, @var{a2}, @dots{})
Compute the greatest common divisor of @var{a1}, @var{a2}, @dots{}.
 
All arguments must be the same size or scalar.  For arrays, the greatest common
divisor is calculated for each element individually.  All elements must be
ordinary or Gaussian (complex) integers.  Note that for Gaussian integers, the
gcd is only unique up to a phase factor (multiplication by 1, -1, i, or -i), so
an arbitrary greatest common divisor among the four possible is returned.
 
Optional return arguments @var{v1}, @dots{}, contain integer vectors such
that,
 
@tex
$g = v_1 a_1 + v_2 a_2 + \cdots$
@end tex
@ifnottex
 
@example
@var{g} = @var{v1} .* @var{a1} + @var{v2} .* @var{a2} + @dots{}
@end example
 
@end ifnottex
 
Example code:
 
@example
@group
gcd ([15, 9], [20, 18])
   @result{}  5  9
@end group
@end example
 
Programming tip: To find the GCD of all the elements of a single array, use
@code{num2cell} instead of nested calls or a loop:
 
@example
@group
x = [30 42 70 105];    # vector or array of inputs
gcd (num2cell (x) @{:@})
   @result{}     1
@end group
@end example
 
@seealso{lcm, factor, isprime}
@end deftypefn */)
{
  int nargin = args.length ();
 
  if (nargin < 2)
    print_usage ();
 
  octave_value_list retval;
 
  if (nargout > 1)
    {
      retval.resize (nargin + 1);
 
      retval(0) = do_extended_gcd (args(0), args(1), retval(1), retval(2));
 
      for (int j = 2; j < nargin; j++)
        {
          octave_value x;
          retval(0) = do_extended_gcd (retval(0), args(j), x, retval(j+1));
          for (int i = 0; i < j; i++)
            retval(i+1).assign (octave_value::op_el_mul_eq, x);
        }
    }
  else
    {
      retval(0) = do_simple_gcd (args(0), args(1));
 
      for (int j = 2; j < nargin; j++)
        retval(0) = do_simple_gcd (retval(0), args(j));
    }
 
  return retval;
}
 
/*
%!assert (gcd (200, 300, 50, 35), 5)
%!assert (gcd (int16 (200), int16 (300), int16 (50), int16 (35)), int16 (5))
%!assert (gcd (uint64 (200), uint64 (300), uint64 (50), uint64 (35)),
%!        uint64 (5))
%!assert (gcd (18-i, -29+3i), -3-4i)
 
%!test
%! p = [953 967];
%! u = [953 + i*971, 967 + i*977];
%! [d, k(1), k(2)] = gcd (p(1), p(2));
%! [z, w(1), w(2)] = gcd (u(1), u(2));
%! assert (d, 1);
%! assert (sum (p.*k), d);
%! assert (abs (z), sqrt (2));
%! assert (abs (sum (u.*w)), sqrt (2));
 
%!error <all values must be integers> gcd (1/2, 2)
%!error <all complex parts must be integers> gcd (e + i*pi, 1)
 
%!error gcd ()
 
%!test
%! s.a = 1;
%! fail ("gcd (s)");
*/
 
OCTAVE_END_NAMESPACE(octave)