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1 ## Copyright (C) 2000-2006 Paul Kienzle |
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2 ## |
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3 ## This program is free software; you can redistribute it and/or modify |
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4 ## it under the terms of the GNU General Public License as published by |
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5 ## the Free Software Foundation; either version 2 of the License, or |
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6 ## (at your option) any later version. |
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7 ## |
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8 ## This program is distributed in the hope that it will be useful, |
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9 ## but WITHOUT ANY WARRANTY; without even the implied warranty of |
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10 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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11 ## GNU General Public License for more details. |
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12 ## |
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13 ## You should have received a copy of the GNU General Public License |
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14 ## along with this program; if not, write to the Free Software |
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15 ## Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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16 ## 02110-1301 USA |
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17 |
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18 ## -*- texinfo -*- |
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19 ## @deftypefn {Function File} {} speed (@var{f}, @var{init}, @var{max_n}, @var{f2}, @var{tol}) |
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20 ## @deftypefnx {Function File} {[@var{order}, @var{n}, @var{T_f}, @var{T_f2}] =} speed (@dots{}) |
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21 ## |
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22 ## Determine the execution time of an expression for various @var{n}. |
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23 ## The @var{n} are log-spaced from 1 to @var{max_n}. For each @var{n}, |
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24 ## an initialization expression is computed to create whatever data |
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25 ## are needed for the test. If a second expression is given, the |
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26 ## execution times of the two expressions will be compared. Called |
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27 ## without output arguments the results are presented graphically. |
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28 ## |
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29 ## @table @code |
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30 ## @item @var{f} |
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31 ## The expression to evaluate. |
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32 ## |
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33 ## @item @var{max_n} |
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34 ## The maximum test length to run. Default value is 100. Alternatively, |
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35 ## use @code{[min_n,max_n]} or for complete control, @code{[n1,n2,@dots{},nk]}. |
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36 ## |
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37 ## @item @var{init} |
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38 ## Initialization expression for function argument values. Use @var{k} |
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39 ## for the test number and @var{n} for the size of the test. This should |
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40 ## compute values for all variables listed in args. Note that init will |
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41 ## be evaluated first for k=0, so things which are constant throughout |
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42 ## the test can be computed then. The default value is @code{@var{x} = |
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43 ## randn (@var{n}, 1);}. |
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44 ## |
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45 ## @item @var{f2} |
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46 ## An alternative expression to evaluate, so the speed of the two |
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47 ## can be compared. Default is @code{[]}. |
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48 ## |
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49 ## @item @var{tol} |
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50 ## If @var{tol} is @code{Inf}, then no comparison will be made between the |
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51 ## results of expression @var{f} and expression @var{f2}. Otherwise, |
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52 ## expression @var{f} should produce a value @var{v} and expression @var{f2} |
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53 ## should produce a value @var{v2}, and these shall be compared using |
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54 ## @code{assert(@var{v},@var{v2},@var{tol})}. The default is |
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55 ## @code{eps}. |
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56 ## |
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57 ## @item @var{order} |
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58 ## The time complexity of the expression @code{O(a n^p)}. This |
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59 ## is a structure with fields @code{a} and @code{p}. |
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60 ## |
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61 ## @item @var{n} |
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62 ## The values @var{n} for which the expression was calculated and the |
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63 ## the execution time was greater than zero. |
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64 ## |
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65 ## @item @var{T_f} |
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66 ## The nonzero execution times recorded for the expression @var{f} in seconds. |
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67 ## |
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68 ## @item @var{T_f2} |
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69 ## The nonzero execution times recorded for the expression @var{f2} in seconds. |
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70 ## If it is needed, the mean time ratio is just @code{mean(T_f./T_f2)}. |
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71 ## |
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72 ## @end table |
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73 ## |
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74 ## The slope of the execution time graph shows the approximate |
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75 ## power of the asymptotic running time @code{O(n^p)}. This |
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76 ## power is plotted for the region over which it is approximated |
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77 ## (the latter half of the graph). The estimated power is not |
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78 ## very accurate, but should be sufficient to determine the |
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79 ## general order of your algorithm. It should indicate if for |
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80 ## example your implementation is unexpectedly @code{O(n^2)} |
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81 ## rather than @code{O(n)} because it extends a vector each |
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82 ## time through the loop rather than preallocating one which is |
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83 ## big enough. For example, in the current version of Octave, |
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84 ## the following is not the expected @code{O(n)}: |
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85 ## |
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86 ## @example |
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87 ## speed("for i=1:n,y@{i@}=x(i); end", "", [1000,10000]) |
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88 ## @end example |
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89 ## |
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90 ## but it is if you preallocate the cell array @code{y}: |
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91 ## |
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92 ## @example |
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93 ## speed("for i=1:n,y@{i@}=x(i);end", ... |
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94 ## "x=rand(n,1);y=cell(size(x));", [1000,10000]) |
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95 ## @end example |
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96 ## |
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97 ## An attempt is made to approximate the cost of the individual |
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98 ## operations, but it is wildly inaccurate. You can improve the |
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99 ## stability somewhat by doing more work for each @code{n}. For |
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100 ## example: |
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101 ## |
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102 ## @example |
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103 ## speed("airy(x)", "x=rand(n,10)", [10000,100000]) |
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104 ## @end example |
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105 ## |
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106 ## When comparing a new and original expression, the line on the |
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107 ## speedup ratio graph should be larger than 1 if the new expression |
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108 ## is faster. Better algorithms have a shallow slope. Generally, |
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109 ## vectorizing an algorithm will not change the slope of the execution |
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110 ## time graph, but it will shift it relative to the original. For |
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111 ## example: |
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112 ## |
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113 ## @example |
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114 ## speed("v=sum(x)", "", [10000,100000], ... |
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115 ## "v=0;for i=1:length(x),v+=x(i);end") |
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116 ## @end example |
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117 ## |
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118 ## A more complex example, if you had an original version of @code{xcorr} |
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119 ## using for loops and another version using an FFT, you could compare the |
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120 ## run speed for various lags as follows, or for a fixed lag with varying |
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121 ## vector lengths as follows: |
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122 ## |
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123 ## @example |
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124 ## speed("v=xcorr(x,n)", "x=rand(128,1);", 100, ... |
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125 ## "v2=xcorr_orig(x,n)", 100*eps,'rel') |
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126 ## speed("v=xcorr(x,15)", "x=rand(20+n,1);", 100, ... |
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127 ## "v2=xcorr_orig(x,n)", 100*eps,'rel') |
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128 ## @end example |
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129 ## |
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130 ## Assuming one of the two versions is in @var{xcorr_orig}, this would |
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131 ## would compare their speed and their output values. Note that the |
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132 ## FFT version is not exact, so we specify an acceptable tolerance on |
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133 ## the comparison @code{100*eps}, and the errors should be computed |
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134 ## relatively, as @code{abs((@var{x} - @var{y})./@var{y})} rather than |
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135 ## absolutely as @code{abs(@var{x} - @var{y})}. |
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136 ## |
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137 ## Type @code{example('speed')} to see some real examples. Note for |
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138 ## obscure reasons, you can't run examples 1 and 2 directly using |
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139 ## @code{demo('speed')}. Instead use, @code{eval(example('speed',1))} |
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140 ## and @code{eval(example('speed',2))}. |
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141 ## @end deftypefn |
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142 |
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143 ## TODO: consider two dimensional speedup surfaces for functions like kron. |
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144 function [__order, __test_n, __tnew, __torig] ... |
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145 = speed (__f1, __init, __max_n, __f2, __tol) |
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146 if nargin < 1 || nargin > 6, |
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147 print_usage (); |
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148 endif |
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149 if nargin < 2 || isempty(__init), |
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150 __init = "x = randn(n, 1);"; |
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151 endif |
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152 if nargin < 3 || isempty(__max_n), __max_n = 100; endif |
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153 if nargin < 4, __f2 = []; endif |
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154 if nargin < 5 || isempty(__tol), __tol = eps; endif |
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155 |
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156 __numtests = 15; |
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157 |
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158 ## Let user specify range of n |
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159 if isscalar(__max_n) |
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160 __min_n = 1; |
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161 assert(__max_n > __min_n); |
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162 __test_n = logspace(0,log10(__max_n),__numtests); |
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163 elseif length(__max_n) == 2 |
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164 __min_n = __max_n(1); |
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165 __max_n = __max_n(2); |
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166 assert(__min_n >= 1); |
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167 __test_n = logspace(log10(__min_n),log10(__max_n),__numtests); |
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168 else |
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169 __test_n = __max_n; |
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170 endif |
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171 __test_n = unique(round(__test_n)); # Force n to be an integer |
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172 assert(__test_n >= 1); |
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173 |
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174 __torig = __tnew = zeros (size(__test_n)) ; |
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175 |
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176 disp (["testing ", __f1, "\ninit: ", __init]); |
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177 |
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178 ## make sure the functions are freshly loaded by evaluating them at |
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179 ## test_n(1); first have to initialize the args though. |
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180 n=1; k=0; |
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181 eval ([__init, ";"]); |
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182 if !isempty(__f2), eval ([__f2, ";"]); endif |
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183 eval ([__f1, ";"]); |
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184 |
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185 ## run the tests |
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186 for k=1:length(__test_n) |
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187 n=__test_n(k); |
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188 eval ([__init, ";"]); |
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189 |
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190 printf ("n%i=%i ",k, n) ; fflush(1); |
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191 eval (["__t=time();", __f1, "; __v1=ans; __t = time()-__t;"]); |
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192 if (__t < 0.25) |
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193 eval (["__t2=time();", __f1, "; __t2 = time()-__t2;"]); |
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194 eval (["__t3=time();", __f1, "; __t3 = time()-__t3;"]); |
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195 __t = min([__t,__t2,__t3]); |
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196 endif |
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197 __tnew(k) = __t; |
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198 |
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199 if !isempty(__f2) |
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200 eval (["__t=time();", __f2, "; __v2=ans; __t = time()-__t;"]); |
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201 if (__t < 0.25) |
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202 eval (["__t2=time();", __f2, "; __t2 = time()-__t2;"]); |
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203 eval (["__t3=time();", __f2, "; __t3 = time()-__t3;"]); |
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204 endif |
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205 __torig(k) = __t; |
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206 if !isinf(__tol) |
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207 assert(__v1,__v2,__tol); |
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208 endif |
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209 endif |
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210 |
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211 endfor |
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212 |
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213 ## Drop times of zero |
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214 if !isempty(__f2) |
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215 zidx = ( __tnew < 100*eps | __torig < 100*eps ) ; |
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216 __test_n(zidx) = []; |
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217 __tnew(zidx) = []; |
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218 __torig(zidx) = []; |
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219 else |
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220 zidx = ( __tnew < 100*eps ) ; |
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221 __test_n(zidx) = []; |
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222 __tnew(zidx) = []; |
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223 endif |
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224 |
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225 ## Approximate time complexity and return it if requested |
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226 tailidx = [ceil(length(__test_n)/2):length(__test_n)]; |
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227 p = polyfit(log(__test_n(tailidx)),log(__tnew(tailidx)), 1); |
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228 if nargout > 0, |
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229 __order.p = p(1); |
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230 __order.a = exp(p(2)); |
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231 endif |
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232 |
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233 |
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234 ## Plot the data if no output is requested. |
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235 doplot = (nargout == 0); |
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236 |
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237 if doplot && !isempty(__f2) |
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238 |
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239 |
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240 subplot(121); |
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241 xlabel("test length"); |
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242 title (__f1); |
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243 ylabel("speedup ratio"); |
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244 semilogx ( __test_n, __torig./__tnew, |
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245 ["-*r;", strrep(__f1,";","."), "/", strrep(__f2,";","."), ";"], |
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246 __test_n, __tnew./__torig, |
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247 ["-*g;", strrep(__f2,";","."), "/", strrep(__f1,";","."), ";"]); |
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248 subplot (122); |
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249 |
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250 ylabel ("best execution time (ms)"); |
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251 title (["init: ", __init]); |
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252 loglog ( __test_n, __tnew*1000, ["*-g;", strrep(__f1,";","."), ";" ], |
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253 __test_n, __torig*1000, ["*-r;", strrep(__f2,";","."), ";"]) |
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254 |
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255 ratio = mean (__torig ./ __tnew); |
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256 printf ("\n\nMean runtime ratio = %.3g for '%s' vs '%s'\n", ... |
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257 ratio, __f2, __f1); |
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258 |
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259 elseif doplot |
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260 |
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261 subplot(111); |
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262 xlabel("test length"); |
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263 ylabel ("best execution time (ms)"); |
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264 title ([__f1, " init: ", __init]); |
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265 loglog ( __test_n, __tnew*1000, "*-g;execution time;"); |
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266 |
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267 endif |
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268 |
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269 if doplot |
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270 |
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271 ## Plot time complexity approximation (using milliseconds). |
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272 order = sprintf("O(n^%g)",round(10*p(1))/10); |
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273 v = polyval(p,log(__test_n(tailidx))); |
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274 hold on; |
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275 loglog(__test_n(tailidx), exp(v)*1000, sprintf("b;%s;",order)); |
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276 hold off; |
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277 |
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278 ## Get base time to 1 digit of accuracy |
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279 dt = exp(p(2)); |
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280 dt = floor(dt/10^floor(log10(dt)))*10^floor(log10(dt)); |
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281 if log10(dt) >= -0.5, time = sprintf("%g s", dt); |
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282 elseif log10(dt) >= -3.5, time = sprintf("%g ms", dt*1e3); |
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283 elseif log10(dt) >= -6.5, time = sprintf("%g us", dt*1e6); |
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284 else time = sprintf("%g ns", dt*1e9); |
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285 endif |
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286 |
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287 ## Display nicely formatted complexity. |
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288 printf ("\nFor %s:\n",__f1); |
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289 printf (" asymptotic power: %s\n", order); |
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290 printf (" approximate time per operation: %s\n", time); |
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291 |
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292 endif |
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293 |
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294 endfunction |
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295 |
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296 %!demo if 1 |
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297 %! function x = build_orig(n) |
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298 %! ## extend the target vector on the fly |
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299 %! for i=0:n-1, x([1:10]+i*10) = 1:10; endfor |
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300 %! endfunction |
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301 %! function x = build(n) |
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302 %! ## preallocate the target vector |
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303 %! x = zeros(1, n*10); |
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304 %! try |
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305 %! if (prefer_column_vectors), x = x.'; endif |
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306 %! catch |
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307 %! end |
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308 %! for i=0:n-1, x([1:10]+i*10) = 1:10; endfor |
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309 %! endfunction |
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310 %! |
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311 %! disp("-----------------------"); |
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312 %! type build_orig; |
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313 %! disp("-----------------------"); |
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314 %! type build; |
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315 %! disp("-----------------------"); |
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316 %! |
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317 %! disp("Preallocated vector test.\nThis takes a little while..."); |
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318 %! speed('build', 'build_orig', 1000, 'v=n;'); |
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319 %! clear build build_orig |
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320 %! disp("Note how much faster it is to pre-allocate a vector."); |
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321 %! disp("Notice the peak speedup ratio."); |
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322 %! clear build build_orig |
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323 %! endif |
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324 |
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325 %!demo if 1 |
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326 %! function x = build_orig(n) |
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327 %! for i=0:n-1, x([1:10]+i*10) = 1:10; endfor |
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328 %! endfunction |
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329 %! function x = build(n) |
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330 %! idx = [1:10]'; |
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331 %! x = idx(:,ones(1,n)); |
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332 %! x = reshape(x, 1, n*10); |
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333 %! try |
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334 %! if (prefer_column_vectors), x = x.'; endif |
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335 %! catch |
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336 %! end |
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337 %! endfunction |
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338 %! |
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339 %! disp("-----------------------"); |
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340 %! type build_orig; |
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341 %! disp("-----------------------"); |
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342 %! type build; |
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343 %! disp("-----------------------"); |
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344 %! |
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345 %! disp("Vectorized test. This takes a little while..."); |
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346 %! speed('build', 'build_orig', 1000, 'v=n;'); |
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347 %! clear build build_orig |
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348 %! disp("-----------------------"); |
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349 %! disp("This time, the for loop is done away with entirely."); |
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350 %! disp("Notice how much bigger the speedup is then in example 1."); |
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351 %! endif |
