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1 ## Copyright (C) 2001, 2007 Paul Kienzle |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 3 of the License, or (at |
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8 ## your option) any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, see |
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17 ## <http://www.gnu.org/licenses/>. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {@var{s} =} rat (@var{x}, @var{tol}) |
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21 ## @deftypefnx {Function File} {[@var{n}, @var{d}] =} rat (@var{x}, @var{tol}) |
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22 ## |
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23 ## Find a rational approximation to @var{x} within tolerance defined |
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24 ## by @var{tol} using a continued fraction expansion. E.g, |
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25 ## |
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26 ## @example |
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27 ## rat(pi) = 3 + 1/(7 + 1/16) = 355/113 |
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28 ## rat(e) = 3 + 1/(-4 + 1/(2 + 1/(5 + 1/(-2 + 1/(-7))))) = 1457/536 |
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29 ## @end example |
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30 ## |
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31 ## Called with two arguments returns the numerator and denominator separately |
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32 ## as two matrices. |
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33 ## @end deftypefn |
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34 ## @seealso{rats} |
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35 |
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36 function [n,d] = rat(x,tol) |
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37 |
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38 if (nargin != [1,2] || nargout > 2) |
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39 print_usage (); |
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40 endif |
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41 |
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42 y = x(:); |
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43 |
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44 ## replace inf with 0 while calculating ratios |
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45 y(isinf(y)) = 0; |
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46 |
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47 ## default norm |
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48 if (nargin < 2) |
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49 tol = 1e-6 * norm(y,1); |
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50 endif |
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51 |
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52 ## First step in the approximation is the integer portion |
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53 n = round(y); # first element in the continued fraction |
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54 d = ones(size(y)); |
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55 frac = y-n; |
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56 lastn = ones(size(y)); |
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57 lastd = zeros(size(y)); |
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58 |
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59 nd = ndims(y); |
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60 nsz = numel (y); |
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61 steps = zeros([nsz, 0]); |
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62 |
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63 ## grab new factors until all continued fractions converge |
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64 while (1) |
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65 ## determine which fractions have not yet converged |
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66 idx = find (abs(y-n./d) >= tol); |
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67 if (isempty(idx)) break; endif |
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68 |
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69 ## grab the next step in the continued fraction |
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70 flip = 1./frac(idx); |
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71 step = round(flip); # next element in the continued fraction |
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72 |
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73 if (nargout < 2) |
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74 tsteps = NaN .* ones (nsz, 1); |
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75 tsteps (idx) = step; |
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76 steps = [steps, tsteps]; |
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77 endif |
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78 |
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79 frac(idx) = flip-step; |
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80 |
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81 ## update the numerator/denominator |
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82 nextn = n; |
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83 nextd = d; |
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84 n(idx) = n(idx).*step + lastn(idx); |
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85 d(idx) = d(idx).*step + lastd(idx); |
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86 lastn = nextn; |
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87 lastd = nextd; |
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88 endwhile |
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89 |
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90 if (nargout == 2) |
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91 ## move the minus sign to the top |
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92 n = n.*sign(d); |
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93 d = abs(d); |
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94 |
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95 ## return the same shape as you receive |
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96 n = reshape(n, size(x)); |
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97 d = reshape(d, size(x)); |
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98 |
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99 ## use 1/0 for Inf |
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100 n(isinf(x)) = sign(x(isinf(x))); |
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101 d(isinf(x)) = 0; |
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102 |
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103 ## reshape the output |
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104 n = reshape (n, size (x)); |
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105 d = reshape (d, size (x)); |
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106 else |
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107 n = ""; |
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108 nsteps = size(steps, 2); |
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109 for i = 1: nsz |
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110 s = [int2str(y(i))," "]; |
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111 j = 1; |
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112 |
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113 while (true) |
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114 step = steps(i, j++); |
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115 if (isnan (step)) |
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116 break; |
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117 endif |
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118 if (j > nsteps || isnan (steps(i, j))) |
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119 if (step < 0) |
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120 s = [s(1:end-1), " + 1/(", int2str(step), ")"]; |
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121 else |
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122 s = [s(1:end-1), " + 1/", int2str(step)]; |
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123 endif |
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124 break; |
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125 else |
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126 s = [s(1:end-1), " + 1/(", int2str(step), ")"]; |
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127 endif |
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128 endwhile |
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129 s = [s, repmat(")", 1, j-2)]; |
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130 n = cat (1, n, s); |
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131 endfor |
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132 endif |
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133 |
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134 endfunction |
