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Optimisations for backprop
author Jordi Gutiérrez Hermoso <jordigh@octave.org>
date Fri, 11 Nov 2011 20:36:02 -0500
parents 395fc40248c3
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1 function [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
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2 % Minimize a continuous differentialble multivariate function. Starting point
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3 % is given by "X" (D by 1), and the function named in the string "f", must
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4 % return a function value and a vector of partial derivatives. The Polack-
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5 % Ribiere flavour of conjugate gradients is used to compute search directions,
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6 % and a line search using quadratic and cubic polynomial approximations and the
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7 % Wolfe-Powell stopping criteria is used together with the slope ratio method
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8 % for guessing initial step sizes. Additionally a bunch of checks are made to
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9 % make sure that exploration is taking place and that extrapolation will not
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10 % be unboundedly large. The "length" gives the length of the run: if it is
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11 % positive, it gives the maximum number of line searches, if negative its
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12 % absolute gives the maximum allowed number of function evaluations. You can
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13 % (optionally) give "length" a second component, which will indicate the
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14 % reduction in function value to be expected in the first line-search (defaults
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15 % to 1.0). The function returns when either its length is up, or if no further
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16 % progress can be made (ie, we are at a minimum, or so close that due to
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17 % numerical problems, we cannot get any closer). If the function terminates
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18 % within a few iterations, it could be an indication that the function value
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19 % and derivatives are not consistent (ie, there may be a bug in the
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20 % implementation of your "f" function). The function returns the found
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21 % solution "X", a vector of function values "fX" indicating the progress made
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22 % and "i" the number of iterations (line searches or function evaluations,
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23 % depending on the sign of "length") used.
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24 %
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25 % Usage: [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
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26 %
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27 % See also: checkgrad
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28 %
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29 % Copyright (C) 2001 and 2002 by Carl Edward Rasmussen. Date 2002-02-13
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30 %
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31 %
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32 % (C) Copyright 1999, 2000 & 2001, Carl Edward Rasmussen
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33 %
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34 % Permission is granted for anyone to copy, use, or modify these
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35 % programs and accompanying documents for purposes of research or
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36 % education, provided this copyright notice is retained, and note is
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37 % made of any changes that have been made.
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38 %
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39 % These programs and documents are distributed without any warranty,
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40 % express or implied. As the programs were written for research
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41 % purposes only, they have not been tested to the degree that would be
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42 % advisable in any important application. All use of these programs is
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43 % entirely at the user's own risk.
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44 %
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45 % [ml-class] Changes Made:
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46 % 1) Function name and argument specifications
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47 % 2) Output display
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48 %
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49
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50 % Read options
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51 if exist('options', 'var') && ~isempty(options) && isfield(options, 'MaxIter')
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52 length = options.MaxIter;
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53 else
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54 length = 100;
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55 end
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56
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57
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58 RHO = 0.01; % a bunch of constants for line searches
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59 SIG = 0.5; % RHO and SIG are the constants in the Wolfe-Powell conditions
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60 INT = 0.1; % don't reevaluate within 0.1 of the limit of the current bracket
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61 EXT = 3.0; % extrapolate maximum 3 times the current bracket
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62 MAX = 20; % max 20 function evaluations per line search
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63 RATIO = 100; % maximum allowed slope ratio
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64
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65 argstr = ['feval(f, X']; % compose string used to call function
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66 for i = 1:(nargin - 3)
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67 argstr = [argstr, ',P', int2str(i)];
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68 end
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69 argstr = [argstr, ')'];
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70
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71 if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end
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72 S=['Iteration '];
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73
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74 i = 0; % zero the run length counter
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75 ls_failed = 0; % no previous line search has failed
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76 fX = [];
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77 [f1 df1] = eval(argstr); % get function value and gradient
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78 i = i + (length<0); % count epochs?!
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79 s = -df1; % search direction is steepest
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80 d1 = -s'*s; % this is the slope
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81 z1 = red/(1-d1); % initial step is red/(|s|+1)
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82
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83 while i < abs(length) % while not finished
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84 i = i + (length>0); % count iterations?!
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85
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86 X0 = X; f0 = f1; df0 = df1; % make a copy of current values
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87 X = X + z1*s; % begin line search
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88 [f2 df2] = eval(argstr);
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89 i = i + (length<0); % count epochs?!
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90 d2 = df2'*s;
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91 f3 = f1; d3 = d1; z3 = -z1; % initialize point 3 equal to point 1
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92 if length>0, M = MAX; else M = min(MAX, -length-i); end
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93 success = 0; limit = -1; % initialize quanteties
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94 while 1
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95 while ((f2 > f1+z1*RHO*d1) | (d2 > -SIG*d1)) & (M > 0)
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96 limit = z1; % tighten the bracket
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97 if f2 > f1
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98 z2 = z3 - (0.5*d3*z3*z3)/(d3*z3+f2-f3); % quadratic fit
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99 else
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100 A = 6*(f2-f3)/z3+3*(d2+d3); % cubic fit
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101 B = 3*(f3-f2)-z3*(d3+2*d2);
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102 z2 = (sqrt(B*B-A*d2*z3*z3)-B)/A; % numerical error possible - ok!
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103 end
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104 if isnan(z2) | isinf(z2)
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105 z2 = z3/2; % if we had a numerical problem then bisect
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106 end
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107 z2 = max(min(z2, INT*z3),(1-INT)*z3); % don't accept too close to limits
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108 z1 = z1 + z2; % update the step
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109 X = X + z2*s;
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110 [f2 df2] = eval(argstr);
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111 M = M - 1; i = i + (length<0); % count epochs?!
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112 d2 = df2'*s;
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113 z3 = z3-z2; % z3 is now relative to the location of z2
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114 end
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115 if f2 > f1+z1*RHO*d1 | d2 > -SIG*d1
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116 break; % this is a failure
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117 elseif d2 > SIG*d1
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118 success = 1; break; % success
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119 elseif M == 0
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120 break; % failure
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121 end
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122 A = 6*(f2-f3)/z3+3*(d2+d3); % make cubic extrapolation
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123 B = 3*(f3-f2)-z3*(d3+2*d2);
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124 z2 = -d2*z3*z3/(B+sqrt(B*B-A*d2*z3*z3)); % num. error possible - ok!
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125 if ~isreal(z2) | isnan(z2) | isinf(z2) | z2 < 0 % num prob or wrong sign?
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126 if limit < -0.5 % if we have no upper limit
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127 z2 = z1 * (EXT-1); % the extrapolate the maximum amount
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128 else
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129 z2 = (limit-z1)/2; % otherwise bisect
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130 end
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131 elseif (limit > -0.5) & (z2+z1 > limit) % extraplation beyond max?
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132 z2 = (limit-z1)/2; % bisect
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133 elseif (limit < -0.5) & (z2+z1 > z1*EXT) % extrapolation beyond limit
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134 z2 = z1*(EXT-1.0); % set to extrapolation limit
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135 elseif z2 < -z3*INT
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136 z2 = -z3*INT;
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137 elseif (limit > -0.5) & (z2 < (limit-z1)*(1.0-INT)) % too close to limit?
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138 z2 = (limit-z1)*(1.0-INT);
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139 end
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140 f3 = f2; d3 = d2; z3 = -z2; % set point 3 equal to point 2
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141 z1 = z1 + z2; X = X + z2*s; % update current estimates
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142 [f2 df2] = eval(argstr);
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143 M = M - 1; i = i + (length<0); % count epochs?!
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144 d2 = df2'*s;
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145 end % end of line search
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146
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147 if success % if line search succeeded
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148 f1 = f2; fX = [fX' f1]';
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149 fprintf('%s %4i | Cost: %4.6e\r', S, i, f1);
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150 s = (df2'*df2-df1'*df2)/(df1'*df1)*s - df2; % Polack-Ribiere direction
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151 tmp = df1; df1 = df2; df2 = tmp; % swap derivatives
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152 d2 = df1'*s;
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153 if d2 > 0 % new slope must be negative
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154 s = -df1; % otherwise use steepest direction
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155 d2 = -s'*s;
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156 end
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157 z1 = z1 * min(RATIO, d1/(d2-realmin)); % slope ratio but max RATIO
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158 d1 = d2;
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159 ls_failed = 0; % this line search did not fail
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160 else
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161 X = X0; f1 = f0; df1 = df0; % restore point from before failed line search
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162 if ls_failed | i > abs(length) % line search failed twice in a row
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163 break; % or we ran out of time, so we give up
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164 end
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165 tmp = df1; df1 = df2; df2 = tmp; % swap derivatives
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166 s = -df1; % try steepest
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167 d1 = -s'*s;
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168 z1 = 1/(1-d1);
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169 ls_failed = 1; % this line search failed
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170 end
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171 if exist('OCTAVE_VERSION')
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172 fflush(stdout);
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173 end
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174 end
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175 fprintf('\n');