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doc: Expand and edit documentation for memoization (bug #60860)
author Arun Giridhar <arungiridhar@gmail.com>
date Fri, 30 Sep 2022 06:38:59 -0400
parents 43a6be589387
children f957849b2ba5
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796f54d4ddbf update Octave Project Developers copyright for the new year
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1 @c Copyright (C) 2012-2022 The Octave Project Developers
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2 @c
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3 @c This file is part of Octave.
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4 @c
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5 @c Octave is free software: you can redistribute it and/or modify it
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6 @c under the terms of the GNU General Public License as published by
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7 @c the Free Software Foundation, either version 3 of the License, or
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8 @c (at your option) any later version.
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9 @c
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10 @c Octave is distributed in the hope that it will be useful, but
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11 @c WITHOUT ANY WARRANTY; without even the implied warranty of
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12 @c MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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13 @c GNU General Public License for more details.
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14 @c
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15 @c You should have received a copy of the GNU General Public License
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16 @c along with Octave; see the file COPYING. If not, see
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17 @c <https://www.gnu.org/licenses/>.
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18
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19 @node Vectorization and Faster Code Execution
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20 @chapter Vectorization and Faster Code Execution
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21 @cindex vectorization
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22 @cindex vectorize
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23
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24 Vectorization is a programming technique that uses vector operations
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25 instead of element-by-element loop-based operations. Besides frequently
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26 producing more succinct Octave code, vectorization also allows for better
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27 optimization in the subsequent implementation. The optimizations may occur
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28 either in Octave's own Fortran, C, or C++ internal implementation, or even at a
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29 lower level depending on the compiler and external numerical libraries used to
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30 build Octave. The ultimate goal is to make use of your hardware's vector
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31 instructions if possible or to perform other optimizations in software.
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32
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33 Vectorization is not a concept unique to Octave, but it is particularly
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34 important because Octave is a matrix-oriented language. Vectorized
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35 Octave code will see a dramatic speed up (10X--100X) in most cases.
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36
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37 This chapter discusses vectorization and other techniques for writing faster
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38 code.
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39
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40 @menu
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41 * Basic Vectorization:: Basic techniques for code optimization
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42 * Broadcasting:: Broadcasting operations
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43 * Function Application:: Applying functions to arrays, cells, and structs
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44 * Accumulation:: Accumulation functions
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45 * Memoization:: Memoization techniques
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46 * Miscellaneous Techniques:: Other techniques for speeding up code
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47 * Examples::
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48 @end menu
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49
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50 @node Basic Vectorization
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51 @section Basic Vectorization
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52
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53 To a very good first approximation, the goal in vectorization is to
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54 write code that avoids loops and uses whole-array operations. As a
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55 trivial example, consider
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56
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57 @example
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58 @group
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59 for i = 1:n
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60 for j = 1:m
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61 c(i,j) = a(i,j) + b(i,j);
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62 endfor
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63 endfor
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64 @end group
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65 @end example
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66
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67 @noindent
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68 compared to the much simpler
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69
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70 @example
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71 c = a + b;
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72 @end example
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73
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74 @noindent
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75 This isn't merely easier to write; it is also internally much easier to
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76 optimize. Octave delegates this operation to an underlying
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77 implementation which, among other optimizations, may use special vector
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78 hardware instructions or could conceivably even perform the additions in
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79 parallel. In general, if the code is vectorized, the underlying
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80 implementation has more freedom about the assumptions it can make
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81 in order to achieve faster execution.
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82
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83 This is especially important for loops with "cheap" bodies. Often it
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84 suffices to vectorize just the innermost loop to get acceptable
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85 performance. A general rule of thumb is that the "order" of the
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86 vectorized body should be greater or equal to the "order" of the
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87 enclosing loop.
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88
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89 As a less trivial example, instead of
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90
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91 @example
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92 @group
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93 for i = 1:n-1
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94 a(i) = b(i+1) - b(i);
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95 endfor
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96 @end group
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97 @end example
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98
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99 @noindent
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100 write
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101
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102 @example
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103 a = b(2:n) - b(1:n-1);
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104 @end example
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105
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106 This shows an important general concept about using arrays for indexing
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107 instead of looping over an index variable. @xref{Index Expressions}.
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108 Also use boolean indexing generously. If a condition needs to be tested,
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109 this condition can also be written as a boolean index. For instance,
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110 instead of
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111
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112 @example
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113 @group
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114 for i = 1:n
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115 if (a(i) > 5)
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116 a(i) -= 20
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117 endif
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118 endfor
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119 @end group
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120 @end example
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121
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122 @noindent
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123 write
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124
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125 @example
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126 a(a>5) -= 20;
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127 @end example
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128
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129 @noindent
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130 which exploits the fact that @code{a > 5} produces a boolean index.
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131
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132 Use elementwise vector operators whenever possible to avoid looping
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133 (operators like @code{.*} and @code{.^}). @xref{Arithmetic Ops}.
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134
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135 Also exploit broadcasting in these elementwise operators both to avoid
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136 looping and unnecessary intermediate memory allocations.
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137 @xref{Broadcasting}.
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138
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141 are good that it is already optimized, or will be optimized more in a
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142 future release.
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143
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144 For instance, even better than
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145
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146 @example
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147 a = b(2:n) - b(1:n-1);
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148 @end example
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149
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150 @noindent
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151 is
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152
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153 @example
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154 a = diff (b);
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155 @end example
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156
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157 Most Octave functions are written with vector and array arguments in
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158 mind. If you find yourself writing a loop with a very simple operation,
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159 chances are that such a function already exists. The following functions
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160 occur frequently in vectorized code:
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161
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162 @itemize @bullet
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163 @item
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164 Index manipulation
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165
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166 @itemize
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167 @item
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168 find
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169
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170 @item
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171 sub2ind
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172
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173 @item
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174 ind2sub
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175
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176 @item
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177 sort
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178
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179 @item
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180 unique
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181
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182 @item
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183 lookup
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184
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185 @item
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186 ifelse / merge
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187 @end itemize
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188
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189 @item
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190 Repetition
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191
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192 @itemize
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193 @item
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194 repmat
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195
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196 @item
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197 repelems
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198 @end itemize
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199
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200 @item
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201 Vectorized arithmetic
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202
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203 @itemize
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204 @item
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205 sum
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206
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207 @item
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208 prod
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209
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210 @item
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211 cumsum
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212
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213 @item
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214 cumprod
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215
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216 @item
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217 sumsq
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218
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219 @item
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220 diff
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221
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222 @item
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223 dot
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224
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225 @item
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226 cummax
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227
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228 @item
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229 cummin
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230 @end itemize
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231
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232 @item
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233 Shape of higher dimensional arrays
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234
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235 @itemize
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236 @item
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237 reshape
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238
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239 @item
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240 resize
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241
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242 @item
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243 permute
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244
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245 @item
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246 squeeze
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247
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248 @item
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249 deal
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250 @end itemize
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251
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252 @end itemize
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253
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254 @node Broadcasting
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255 @section Broadcasting
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256 @cindex broadcast
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257 @cindex broadcasting
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258 @cindex @nospell{BSX}
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259 @cindex recycling
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260 @cindex SIMD
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261
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262 Broadcasting refers to how Octave binary operators and functions behave
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263 when their matrix or array operands or arguments differ in size. Since
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264 version 3.6.0, Octave now automatically broadcasts vectors, matrices,
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265 and arrays when using elementwise binary operators and functions.
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266 Broadly speaking, smaller arrays are ``broadcast'' across the larger
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267 one, until they have a compatible shape. The rule is that corresponding
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268 array dimensions must either
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269
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270 @enumerate
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271 @item
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272 be equal, or
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273
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274 @item
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275 one of them must be 1.
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276 @end enumerate
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277
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278 @noindent
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279 In case all dimensions are equal, no broadcasting occurs and ordinary
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280 element-by-element arithmetic takes place. For arrays of higher
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281 dimensions, if the number of dimensions isn't the same, then missing
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282 trailing dimensions are treated as 1. When one of the dimensions is 1,
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283 the array with that singleton dimension gets copied along that dimension
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284 until it matches the dimension of the other array. For example, consider
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285
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286 @example
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287 @group
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288 x = [1 2 3;
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289 4 5 6;
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290 7 8 9];
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291
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292 y = [10 20 30];
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293
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294 x + y
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295 @end group
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296 @end example
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297
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298 @noindent
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299 Without broadcasting, @code{x + y} would be an error because the dimensions
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300 do not agree. However, with broadcasting it is as if the following
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301 operation were performed:
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302
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303 @example
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304 @group
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305 x = [1 2 3
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306 4 5 6
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307 7 8 9];
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308
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309 y = [10 20 30
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310 10 20 30
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311 10 20 30];
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312
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313 x + y
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314 @result{} 11 22 33
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315 14 25 36
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316 17 28 39
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317 @end group
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318 @end example
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319
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320 @noindent
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321 That is, the smaller array of size @code{[1 3]} gets copied along the
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322 singleton dimension (the number of rows) until it is @code{[3 3]}. No
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323 actual copying takes place, however. The internal implementation reuses
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324 elements along the necessary dimension in order to achieve the desired
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325 effect without copying in memory.
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326
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327 Both arrays can be broadcast across each other, for example, all
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328 pairwise differences of the elements of a vector with itself:
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329
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330 @example
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331 @group
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332 y - y'
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333 @result{} 0 10 20
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334 -10 0 10
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335 -20 -10 0
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336 @end group
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337 @end example
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338
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339 @noindent
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340 Here the vectors of size @code{[1 3]} and @code{[3 1]} both get
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341 broadcast into matrices of size @code{[3 3]} before ordinary matrix
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342 subtraction takes place.
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343
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344 A special case of broadcasting that may be familiar is when all
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345 dimensions of the array being broadcast are 1, i.e., the array is a
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346 scalar. Thus for example, operations like @code{x - 42} and @code{max
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347 (x, 2)} are basic examples of broadcasting.
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348
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349 For a higher-dimensional example, suppose @code{img} is an RGB image of
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350 size @code{[m n 3]} and we wish to multiply each color by a different
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351 scalar. The following code accomplishes this with broadcasting,
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352
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353 @example
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354 img .*= permute ([0.8, 0.9, 1.2], [1, 3, 2]);
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355 @end example
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356
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357 @noindent
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358 Note the usage of permute to match the dimensions of the
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359 @code{[0.8, 0.9, 1.2]} vector with @code{img}.
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360
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361 For functions that are not written with broadcasting semantics,
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362 @code{bsxfun} can be useful for coercing them to broadcast.
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363
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364 @DOCSTRING(bsxfun)
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365
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366 Broadcasting is only applied if either of the two broadcasting
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367 conditions hold. As usual, however, broadcasting does not apply when two
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368 dimensions differ and neither is 1:
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369
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370 @example
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371 @group
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372 x = [1 2 3
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373 4 5 6];
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374 y = [10 20
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375 30 40];
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376 x + y
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377 @end group
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378 @end example
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379
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380 @noindent
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381 This will produce an error about nonconformant arguments.
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382
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383 Besides common arithmetic operations, several functions of two arguments
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384 also broadcast. The full list of functions and operators that broadcast
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385 is
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386
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387 @example
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388 plus +
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389 minus -
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390 times .*
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391 rdivide ./
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392 ldivide .\
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393 power .^
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394 lt <
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395 le <=
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396 eq ==
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397 gt >
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398 ge >=
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399 ne != ~=
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400 and &
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401 or |
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402 atan2
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403 hypot
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404 max
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405 min
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406 mod
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407 rem
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408 xor
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409
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410 += -= .*= ./= .\= .^= &= |=
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411 @end example
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412
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413 Here is a real example of the power of broadcasting. The Floyd-Warshall
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414 algorithm is used to calculate the shortest path lengths between every pair of
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415 vertices in a graph. A naive implementation for a graph adjacency matrix of
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416 order @var{n} might look like this:
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417
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418 @example
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419 @group
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420 for k = 1:n
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421 for i = 1:n
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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parents: 30564
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422 for j = 1:n
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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diff changeset
423 dist(i,j) = min (dist(i,j), dist(i,k) + dist(k,j));
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
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424 endfor
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
diff changeset
425 endfor
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
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426 endfor
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
diff changeset
427 @end group
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
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diff changeset
428 @end example
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
diff changeset
429
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
diff changeset
430 Upon vectorizing the innermost loop, it might look like this:
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
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431
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
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432 @example
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
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433 @group
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
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434 for k = 1:n
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
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435 for i = 1:n
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
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436 dist(i,:) = min (dist(i,:), dist(i,k) + dist(k,:));
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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437 endfor
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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parents: 30564
diff changeset
438 endfor
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
diff changeset
439 @end group
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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440 @end example
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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parents: 30564
diff changeset
441
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Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
diff changeset
442 Using broadcasting in both directions, it looks like this:
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
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443
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
parents: 30564
diff changeset
444 @example
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
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diff changeset
445 @group
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
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446 for k = 1:n
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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447 dist = min (dist, dist(:,k) + dist(k,:));
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448 endfor
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449 @end group
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450 @end example
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
Arun Giridhar <arungiridhar@gmail.com>
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451
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Arun Giridhar <arungiridhar@gmail.com>
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452 The relative time performance of the three techniques for a given graph with
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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453 100 vertices is 7.3 seconds for the naive code, 87 milliseconds for the
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Arun Giridhar <arungiridhar@gmail.com>
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diff changeset
454 singly vectorized code, and 1.3 milliseconds for the fully broadcast code.
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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parents: 30564
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455 For a graph with 1000 vertices, vectorization takes 11.7 seconds while
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parents: 30564
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456 broadcasting takes only 1.15 seconds. Therefore in general it is worth
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Arun Giridhar <arungiridhar@gmail.com>
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457 writing code with broadcasting semantics for performance.
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458
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459 However, beware of resorting to broadcasting if a simpler operation will
02e4fd516f0a doc: add example about broadcasting and vectorization (bug #61601).
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460 suffice. For matrices @var{a} and @var{b}, consider the following:
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461
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462 @example
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463 @var{c} = sum (permute (@var{a}, [1, 3, 2]) .* permute (@var{b}, [3, 2, 1]), 3);
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464 @end example
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465
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466 @noindent
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diff changeset
467 This operation broadcasts the two matrices with permuted dimensions
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diff changeset
468 across each other during elementwise multiplication in order to obtain a
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469 larger 3-D array, and this array is then summed along the third dimension.
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470 A moment of thought will prove that this operation is simply the much
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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471 faster ordinary matrix multiplication, @code{@var{c} = @var{a}*@var{b};}.
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472
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diff changeset
473 A note on terminology: ``broadcasting'' is the term popularized by the
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474 Numpy numerical environment in the Python programming language. In other
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diff changeset
475 programming languages and environments, broadcasting may also be known
25003
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Rik <rik@octave.org>
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476 as @emph{binary singleton expansion} (@nospell{BSX}, in @sc{matlab}, and the
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diff changeset
477 origin of the name of the @code{bsxfun} function), @emph{recycling} (R
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478 programming language), @emph{single-instruction multiple data} (SIMD),
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diff changeset
479 or @emph{replication}.
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diff changeset
480
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diff changeset
481 @subsection Broadcasting and Legacy Code
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diff changeset
482
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diff changeset
483 The new broadcasting semantics almost never affect code that worked
94e2a76f1e5a doc: Final grammarcheck and spellcheck before 3.6.0 release.
Rik <octave@nomad.inbox5.com>
parents: 14118
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484 in previous versions of Octave. Consequently, all code inherited from
94e2a76f1e5a doc: Final grammarcheck and spellcheck before 3.6.0 release.
Rik <octave@nomad.inbox5.com>
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485 @sc{matlab} that worked in previous versions of Octave should still work
94e2a76f1e5a doc: Final grammarcheck and spellcheck before 3.6.0 release.
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486 without change in Octave. The only exception is code such as
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diff changeset
487
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488 @example
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489 @group
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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490 try
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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491 c = a.*b;
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492 catch
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diff changeset
493 c = a.*a;
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494 end_try_catch
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495 @end group
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496 @end example
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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497
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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498 @noindent
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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499 that may have relied on matrices of different size producing an error.
20491
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Carnë Draug <carandraug@octave.org>
parents: 19836
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500 Because such operation is now valid Octave syntax, this will no longer
6ef177ddc1ee doc: remove section about broadcasting warnings which no longer exists.
Carnë Draug <carandraug@octave.org>
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501 produce an error. Instead, the following code should be used:
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502
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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503 @example
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Carnë Draug <carandraug@octave.org>
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diff changeset
504 @group
6ef177ddc1ee doc: remove section about broadcasting warnings which no longer exists.
Carnë Draug <carandraug@octave.org>
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505 if (isequal (size (a), size (b)))
6ef177ddc1ee doc: remove section about broadcasting warnings which no longer exists.
Carnë Draug <carandraug@octave.org>
parents: 19836
diff changeset
506 c = a .* b;
6ef177ddc1ee doc: remove section about broadcasting warnings which no longer exists.
Carnë Draug <carandraug@octave.org>
parents: 19836
diff changeset
507 else
6ef177ddc1ee doc: remove section about broadcasting warnings which no longer exists.
Carnë Draug <carandraug@octave.org>
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diff changeset
508 c = a .* a;
6ef177ddc1ee doc: remove section about broadcasting warnings which no longer exists.
Carnë Draug <carandraug@octave.org>
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diff changeset
509 endif
6ef177ddc1ee doc: remove section about broadcasting warnings which no longer exists.
Carnë Draug <carandraug@octave.org>
parents: 19836
diff changeset
510 @end group
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511 @end example
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
512
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
513
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
514 @node Function Application
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
515 @section Function Application
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
516 @cindex map
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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diff changeset
517 @cindex apply
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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diff changeset
518 @cindex function application
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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diff changeset
519
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
520 As a general rule, functions should already be written with matrix
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
521 arguments in mind and should consider whole matrix operations in a
14119
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Rik <octave@nomad.inbox5.com>
parents: 14118
diff changeset
522 vectorized manner. Sometimes, writing functions in this way appears
94e2a76f1e5a doc: Final grammarcheck and spellcheck before 3.6.0 release.
Rik <octave@nomad.inbox5.com>
parents: 14118
diff changeset
523 difficult or impossible for various reasons. For those situations,
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
524 Octave provides facilities for applying a function to each element of an
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
525 array, cell, or struct.
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
526
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
527 @DOCSTRING(arrayfun)
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Rik <octave@nomad.inbox5.com>
parents: 14118
diff changeset
528
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diff changeset
529 @DOCSTRING(spfun)
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Rik <octave@nomad.inbox5.com>
parents: 14118
diff changeset
530
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
531 @DOCSTRING(cellfun)
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Rik <octave@nomad.inbox5.com>
parents: 14118
diff changeset
532
14116
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533 @DOCSTRING(structfun)
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
534
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Rik <rik@octave.org>
parents: 18815
diff changeset
535 Consistent with earlier advice, seek to use Octave built-in functions whenever
51a1d1164449 Add extra docs about the performance benefits of operator functions (bug #36439).
Rik <rik@octave.org>
parents: 18815
diff changeset
536 possible for the best performance. This advice applies especially to the four
51a1d1164449 Add extra docs about the performance benefits of operator functions (bug #36439).
Rik <rik@octave.org>
parents: 18815
diff changeset
537 functions above. For example, when adding two arrays together
51a1d1164449 Add extra docs about the performance benefits of operator functions (bug #36439).
Rik <rik@octave.org>
parents: 18815
diff changeset
538 element-by-element one could use a handle to the built-in addition function
51a1d1164449 Add extra docs about the performance benefits of operator functions (bug #36439).
Rik <rik@octave.org>
parents: 18815
diff changeset
539 @code{@@plus} or define an anonymous function @code{@@(x,y) x + y}. But, the
51a1d1164449 Add extra docs about the performance benefits of operator functions (bug #36439).
Rik <rik@octave.org>
parents: 18815
diff changeset
540 anonymous function is 60% slower than the first method.
51a1d1164449 Add extra docs about the performance benefits of operator functions (bug #36439).
Rik <rik@octave.org>
parents: 18815
diff changeset
541 @xref{Operator Overloading}, for a list of basic functions which might be used
51a1d1164449 Add extra docs about the performance benefits of operator functions (bug #36439).
Rik <rik@octave.org>
parents: 18815
diff changeset
542 in place of anonymous ones.
51a1d1164449 Add extra docs about the performance benefits of operator functions (bug #36439).
Rik <rik@octave.org>
parents: 18815
diff changeset
543
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diff changeset
544 @node Accumulation
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
545 @section Accumulation
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
546
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
547 Whenever it's possible to categorize according to indices the elements
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
548 of an array when performing a computation, accumulation functions can be
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
549 useful.
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
550
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
551 @DOCSTRING(accumarray)
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
552
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
553 @DOCSTRING(accumdim)
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
554
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555 @node Memoization
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556 @section Memoization Techniques
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557
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558 Memoization is a technique to cache the results of slow function calls and
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559 return the cached value when the function is called with the same inputs again,
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560 instead of reevaluating it. It is very common to replace function calls with
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561 lookup tables if the same inputs are happening over and over again in a known,
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562 predictable way. Memoization is, at its core, an extension of this practice
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563 where the lookup table is extended even during runtime for new arguments not
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564 seen previously. A basic theoretical background can be found on Wikipedia or
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565 any undergraduate-level computer science textbook.
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566
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567 Octave's @code{memoize} function provides drop-in memoization functionality for
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568 any user function or Octave function, including compiled functions.
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569
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570 @DOCSTRING(memoize)
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571
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572 To memoize a function @code{z = foo(x, y)}, use this general pattern:
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573
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574 @example
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575 @group
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576 foo2 = memoize (@@(@var{x, y}) @var{foo(x, y)});
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577 z = foo2 (x, y);
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578 @end group
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579 @end example
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580
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581 In the above example, the first line creates a memoized version @code{foo2} of
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582 the function @code{foo}. For simple functions with only trivial wrapping, this
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583 line can also be shortened to
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584 @example
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585 @group
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586 foo2 = memoize (@@foo);
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587 @end group
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588 @end example
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589
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590 The second line @code{z = foo2 (x, y);} calls that memoized version @code{foo2}
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591 instead of the original function, allowing @code{memoize} to intercept the call
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592 and replace it with a looked-up value from a table if the inputs have occurred
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593 before, instead of evaluating the original function again. Note that this will
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594 not accelerate the @emph{first} call to the function but only subsequent calls.
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595
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596 Note that due to the overhead incurred by @code{memoize} to create and manage
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597 the lookup tables for each function the user seeks to memoize, this technique
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598 is useful only for functions that take a significant time to execute, at least
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599 a few seconds. Such functions can be replaced by table lookups taking only a
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600 millisecond or less, but if the original function itself was taking only
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601 milliseconds or microseconds, memoizing it will not speed it up.
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602
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603 Octave's memoization also allows the user to clear the cache of lookup values
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604 when it is no longer needed, using the function @code{clearAllMemoizedCaches}.
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605
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606 @DOCSTRING(clearAllMemoizedCaches)
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607
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608 @node Miscellaneous Techniques
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609 @section Miscellaneous Techniques
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610 @cindex execution speed
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611 @cindex speedups
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612 @cindex optimization
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613
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614 Here are some other ways of improving the execution speed of Octave
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615 programs.
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616
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617 @itemize @bullet
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618
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619 @item Avoid computing costly intermediate results multiple times.
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620 Octave currently does not eliminate common subexpressions. Also, certain
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621 internal computation results are cached for variables. For instance, if
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622 a matrix variable is used multiple times as an index, checking the
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623 indices (and internal conversion to integers) is only done once.
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624
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625 @item Be aware of lazy copies (copy-on-write).
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626 @cindex copy-on-write
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627 @cindex COW
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628 @cindex memory management
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629 When a copy of an object is created, the data is not immediately copied, but
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630 rather shared. The actual copying is postponed until the copied data needs to
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631 be modified. For example:
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632
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633 @example
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634 @group
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635 a = zeros (1000); # create a 1000x1000 matrix
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636 b = a; # no copying done here
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637 b(1) = 1; # copying done here
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638 @end group
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639 @end example
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640
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641 Lazy copying applies to whole Octave objects such as matrices, cells,
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642 struct, and also individual cell or struct elements (not array
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643 elements).
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644
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645 Additionally, index expressions also use lazy copying when Octave can
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646 determine that the indexed portion is contiguous in memory. For example:
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647
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648 @example
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649 @group
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650 a = zeros (1000); # create a 1000x1000 matrix
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651 b = a(:,10:100); # no copying done here
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652 b = a(10:100,:); # copying done here
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653 @end group
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654 @end example
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655
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656 This applies to arrays (matrices), cell arrays, and structs indexed
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657 using @samp{()}. Index expressions generating comma-separated lists can also
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658 benefit from shallow copying in some cases. In particular, when @var{a} is a
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659 struct array, expressions like @code{@{a.x@}, @{a(:,2).x@}} will use lazy
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660 copying, so that data can be shared between a struct array and a cell array.
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661
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662 Most indexing expressions do not live longer than their parent
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663 objects. In rare cases, however, a lazily copied slice outlasts its
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664 parent, in which case it becomes orphaned, still occupying unnecessarily
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665 more memory than needed. To provide a remedy working in most real cases,
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666 Octave checks for orphaned lazy slices at certain situations, when a
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667 value is stored into a "permanent" location, such as a named variable or
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668 cell or struct element, and possibly economizes them. For example:
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669
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670 @example
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671 @group
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672 a = zeros (1000); # create a 1000x1000 matrix
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673 b = a(:,10:100); # lazy slice
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674 a = []; # the original "a" array is still allocated
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675 c@{1@} = b; # b is reallocated at this point
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676 @end group
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677 @end example
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678
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679 @item Avoid deep recursion.
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680 Function calls to m-file functions carry a relatively significant overhead, so
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681 rewriting a recursion as a loop often helps. Also, note that the maximum level
94e2a76f1e5a doc: Final grammarcheck and spellcheck before 3.6.0 release.
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682 of recursion is limited.
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683
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684 @item Avoid resizing matrices unnecessarily.
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685 When building a single result matrix from a series of calculations, set the
94e2a76f1e5a doc: Final grammarcheck and spellcheck before 3.6.0 release.
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diff changeset
686 size of the result matrix first, then insert values into it. Write
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687
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
688 @example
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689 @group
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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690 result = zeros (big_n, big_m)
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691 for i = over:and_over
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692 ridx = @dots{}
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693 cidx = @dots{}
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694 result(ridx, cidx) = new_value ();
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695 endfor
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696 @end group
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697 @end example
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diff changeset
698
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diff changeset
699 @noindent
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700 instead of
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
701
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
parents:
diff changeset
702 @example
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diff changeset
703 @group
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diff changeset
704 result = [];
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diff changeset
705 for i = ever:and_ever
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diff changeset
706 result = [ result, new_value() ];
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707 endfor
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
708 @end group
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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709 @end example
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
710
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711 Sometimes the number of items can not be computed in advance, and
94e2a76f1e5a doc: Final grammarcheck and spellcheck before 3.6.0 release.
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712 stack-like operations are needed. When elements are being repeatedly
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713 inserted or removed from the end of an array, Octave detects it as stack
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714 usage and attempts to use a smarter memory management strategy by
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715 pre-allocating the array in bigger chunks. This strategy is also applied
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diff changeset
716 to cell and struct arrays.
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diff changeset
717
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diff changeset
718 @example
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diff changeset
719 @group
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720 a = [];
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diff changeset
721 while (condition)
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
722 @dots{}
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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diff changeset
723 a(end+1) = value; # "push" operation
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Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
724 @dots{}
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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diff changeset
725 a(end) = []; # "pop" operation
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
Jordi Gutiérrez Hermoso <jordigh@octave.org>
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diff changeset
726 @dots{}
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727 endwhile
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728 @end group
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729 @end example
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diff changeset
730
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731 @item Avoid calling @code{eval} or @code{feval} excessively.
94e2a76f1e5a doc: Final grammarcheck and spellcheck before 3.6.0 release.
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732 Parsing input or looking up the name of a function in the symbol table are
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733 relatively expensive operations.
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734
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735 If you are using @code{eval} merely as an exception handling mechanism, and not
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diff changeset
736 because you need to execute some arbitrary text, use the @code{try}
15684
ddc651eecf7a Fix Info index for language statements (bug #37787)
Rik <rik@octave.org>
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diff changeset
737 statement instead. @xref{The try Statement}.
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738
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739 @item Use @code{ignore_function_time_stamp} when appropriate.
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740 If you are calling lots of functions, and none of them will need to change
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741 during your run, set the variable @code{ignore_function_time_stamp} to
22299
9fc91bb2aec3 doc: grammarcheck documentation for 4.2 release.
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diff changeset
742 @qcode{"all"}. This will stop Octave from checking the time stamp of a
9fc91bb2aec3 doc: grammarcheck documentation for 4.2 release.
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diff changeset
743 function file to see if it has been updated while the program is being run.
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744 @end itemize
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diff changeset
745
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diff changeset
746 @node Examples
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diff changeset
747 @section Examples
951eacaf9381 Initial documentation for broadcasting and general vectorization guidelines
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748
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749 The following are examples of vectorization questions asked by actual
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750 users of Octave and their solutions.
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751
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752 @c FIXME: We need a lot more examples here.
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753
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754 @itemize @bullet
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755 @item
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756 For a vector @code{A}, the following loop
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757
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758 @example
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759 @group
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760 n = length (A) - 1;
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761 B = zeros (n, 2);
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762 for i = 1:n
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763 ## this will be two columns, the first is the difference and
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764 ## the second the mean of the two elements used for the diff.
18679
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diff changeset
765 B(i,:) = [A(i+1)-A(i), (A(i+1) + A(i))/2];
14123
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diff changeset
766 endfor
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767 @end group
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768 @end example
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769
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770 @noindent
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diff changeset
771 can be turned into the following one-liner:
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772
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773 @example
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774 B = [diff(A)(:), 0.5*(A(1:end-1)+A(2:end))(:)]
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775 @end example
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776
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777 Note the usage of colon indexing to flatten an intermediate result into
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778 a column vector. This is a common vectorization trick.
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779
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780 @end itemize