1 ## Copyright (C) 2001-2012 Paul Kienzle
3 ## This file is part of Octave.
5 ## Octave is free software; you can redistribute it and/or modify it
6 ## under the terms of the GNU General Public License as published by
7 ## the Free Software Foundation; either version 3 of the License, or (at
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13 ## General Public License for more details.
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20 ## @deftypefn {Function File} {@var{s} =} rat (@var{x}, @var{tol})
21 ## @deftypefnx {Function File} {[@var{n}, @var{d}] =} rat (@var{x}, @var{tol})
23 ## Find a rational approximation to @var{x} within the tolerance defined
24 ## by @var{tol} using a continued fraction expansion. For example:
28 ## rat (pi) = 3 + 1/(7 + 1/16) = 355/113
29 ## rat (e) = 3 + 1/(-4 + 1/(2 + 1/(5 + 1/(-2 + 1/(-7)))))
34 ## Called with two arguments returns the numerator and denominator separately
39 function [n,d] = rat(x,tol)
41 if (nargin != [1,2] || nargout > 2)
47 ## Replace Inf with 0 while calculating ratios.
52 tol = 1e-6 * norm(y,1);
55 ## First step in the approximation is the integer portion
57 ## First element in the continued fraction.
61 lastn = ones(size(y));
62 lastd = zeros(size(y));
66 steps = zeros([nsz, 0]);
68 ## Grab new factors until all continued fractions converge.
70 ## Determine which fractions have not yet converged.
71 idx = find(abs (y-n./d) >= tol);
79 ## Grab the next step in the continued fraction.
81 ## Next element in the continued fraction.
85 tsteps = NaN (nsz, 1);
87 steps = [steps, tsteps];
90 frac(idx) = flip-step;
92 ## Update the numerator/denominator.
95 n(idx) = n(idx).*step + lastn(idx);
96 d(idx) = d(idx).*step + lastd(idx);
102 ## Move the minus sign to the top.
106 ## Return the same shape as you receive.
107 n = reshape(n, size(x));
108 d = reshape(d, size(x));
111 n(isinf(x)) = sign(x(isinf(x)));
114 ## Reshape the output.
115 n = reshape (n, size (x));
116 d = reshape (d, size (x));
119 nsteps = size(steps, 2);
121 s = [int2str(y(i))," "];
125 step = steps(i, j++);
129 if (j > nsteps || isnan (steps(i, j)))
131 s = [s(1:end-1), " + 1/(", int2str(step), ")"];
133 s = [s(1:end-1), " + 1/", int2str(step)];
137 s = [s(1:end-1), " + 1/(", int2str(step), ")"];
140 s = [s, repmat(")", 1, j-2)];
144 s(:,s_nc+1:n_nc) = " ";
146 n(:,n_nc+1:s_nc) = " ";
155 %!error rat (1, 2, 3);
158 %! [n, d] = rat ([0.5, 0.3, 1/3]);
159 %! assert (n, [1, 3, 1]);
160 %! assert (d, [2, 10, 3]);