## Diff of /rat.m [000000] .. [67a5bd]  Maximize  Restore

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```--- a
+++ b/rat.m
@@ -0,0 +1,70 @@
+## Copyright (C) 2001 Paul Kienzle
+##
+## This program is free software; you can redistribute it and/or modify
+## the Free Software Foundation; either version 2 of the License, or
+## (at your option) any later version.
+##
+## This program 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 this program; if not, write to the Free Software
+## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+
+## [n,d] = rat(x,tol)
+## Find a rational approximation to x within tolerance using a continued
+## fraction expansion. E.g,
+##
+##    rat(pi) = 3 + 1/(7 + 1/16) = 355/113
+##    rat(e) = 3 + 1/(-4 + 1/(2 + 1/(5 + 1/(-2 + 1/(-7))))) = 1457/536
+
+function [n,d] = rat(x,tol)
+
+  if (nargin != [1,2] || nargout != 2)
+    usage("[n,d] = rat(x,tol)");
+  endif
+
+  y = x(:);
+  if (nargin < 2)
+    tol = 1e-6 * norm(y,1);
+  endif
+
+  ## First step in the approximation is the integer portion
+  n = round(y);  # first element in the continued fraction
+  d = ones(size(y));
+  frac = y-n;
+  lastn = ones(size(y));
+  lastd = zeros(size(y));
+
+  ## 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)) break; endif
+
+    ## grab the next step in the continued fraction
+    flip = 1./frac(idx);
+    step = round(flip); # next element in the continued fraction
+    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
+
+  ## 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));
+
+endfunction
```