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## Copyright (C) 2001-2013 Paul Kienzle
##
## 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} {@var{s} =} rat (@var{x}, @var{tol})
## @deftypefnx {Function File} {[@var{n}, @var{d}] =} rat (@var{x}, @var{tol})
##
## Find a rational approximation to @var{x} within the tolerance defined
## by @var{tol} using a continued fraction expansion. For example:
##
## @example
## @group
## rat (pi) = 3 + 1/(7 + 1/16) = 355/113
## rat (e) = 3 + 1/(-4 + 1/(2 + 1/(5 + 1/(-2 + 1/(-7)))))
## = 1457/536
## @end group
## @end example
##
## Called with two arguments returns the numerator and denominator separately
## as two matrices.
## @seealso{rats}
## @end deftypefn
function [n,d] = rat (x,tol)
if (nargin != [1,2] || nargout > 2)
print_usage ();
endif
y = x(:);
## Replace Inf with 0 while calculating ratios.
y(isinf(y)) = 0;
## default norm
if (nargin < 2)
tol = 1e-6 * norm (y,1);
endif
## First step in the approximation is the integer portion
## First element in the continued fraction.
n = round (y);
d = ones (size (y));
frac = y-n;
lastn = ones (size (y));
lastd = zeros (size (y));
nd = ndims (y);
nsz = numel (y);
steps = zeros ([nsz, 0]);
## Grab new factors until all continued fractions converge.
while (1)
## Determine which fractions have not yet converged.
idx = find (abs (y-n./d) >= tol);
if (isempty (idx))
if (isempty (steps))
steps = NaN (nsz, 1);
endif
break;
endif
## Grab the next step in the continued fraction.
flip = 1./frac(idx);
## Next element in the continued fraction.
step = round (flip);
if (nargout < 2)
tsteps = NaN (nsz, 1);
tsteps (idx) = step;
steps = [steps, tsteps];
endif
frac(idx) = flip-step;
## Update the numerator/denominator.
nextn = n;
nextd = d;
n(idx) = n(idx).*step + lastn(idx);
d(idx) = d(idx).*step + lastd(idx);
lastn = nextn;
lastd = nextd;
endwhile
if (nargout == 2)
## Move the minus sign to the top.
n = n .* sign (d);
d = abs (d);
## Return the same shape as you receive.
n = reshape (n, size (x));
d = reshape (d, size (x));
## Use 1/0 for Inf.
n(isinf (x)) = sign (x(isinf (x)));
d(isinf (x)) = 0;
## Reshape the output.
n = reshape (n, size (x));
d = reshape (d, size (x));
else
n = "";
nsteps = columns (steps);
for i = 1: nsz
s = [int2str(y(i))," "];
j = 1;
while (true)
step = steps(i, j++);
if (isnan (step))
break;
endif
if (j > nsteps || isnan (steps(i, j)))
if (step < 0)
s = [s(1:end-1), " + 1/(", int2str(step), ")"];
else
s = [s(1:end-1), " + 1/", int2str(step)];
endif
break;
else
s = [s(1:end-1), " + 1/(", int2str(step), ")"];
endif
endwhile
s = [s, repmat(")", 1, j-2)];
n_nc = columns (n);
s_nc = columns (s);
if (n_nc > s_nc)
s(:,s_nc+1:n_nc) = " ";
elseif (s_nc > n_nc)
n(:,n_nc+1:s_nc) = " ";
endif
n = cat (1, n, s);
endfor
endif
endfunction
%!test
%! [n, d] = rat ([0.5, 0.3, 1/3]);
%! assert (n, [1, 3, 1]);
%! assert (d, [2, 10, 3]);
%!error rat ();
%!error rat (1, 2, 3);
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