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1 @c Copyright (C) 1996, 1997, 2007 John W. Eaton |
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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 the |
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7 @c Free Software Foundation; either version 3 of the License, or (at |
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8 @c 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 WITHOUT |
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11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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13 @c 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 <http://www.gnu.org/licenses/>. |
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18 |
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19 @node Numeric Data Types |
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20 @chapter Numeric Data Types |
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21 @cindex numeric constant |
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22 @cindex numeric value |
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23 |
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24 A @dfn{numeric constant} may be a scalar, a vector, or a matrix, and it |
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25 may contain complex values. |
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26 |
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27 The simplest form of a numeric constant, a scalar, is a single number |
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28 that can be an integer, a decimal fraction, a number in scientific |
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29 (exponential) notation, or a complex number. Note that by default numeric |
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30 constants are represented within Octave in double-precision floating |
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31 point format (complex constants are stored as pairs of double-precision |
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32 floating point values). It is however possible to represent real |
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33 integers as described in @ref{Integer Data Types}. Here are some examples |
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34 of real-valued numeric constants, which all have the same value: |
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35 |
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36 @example |
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37 @group |
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38 105 |
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39 1.05e+2 |
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40 1050e-1 |
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41 @end group |
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42 @end example |
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43 |
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44 To specify complex constants, you can write an expression of the form |
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45 |
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46 @example |
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47 @group |
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48 3 + 4i |
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49 3.0 + 4.0i |
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50 0.3e1 + 40e-1i |
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51 @end group |
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52 @end example |
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53 |
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54 @noindent |
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55 all of which are equivalent. The letter @samp{i} in the previous example |
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56 stands for the pure imaginary constant, defined as |
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57 @iftex |
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58 @tex |
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59 $\sqrt{-1}$. |
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60 @end tex |
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61 @end iftex |
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62 @ifnottex |
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63 @code{sqrt (-1)}. |
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64 @end ifnottex |
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65 |
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66 For Octave to recognize a value as the imaginary part of a complex |
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67 constant, a space must not appear between the number and the @samp{i}. |
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68 If it does, Octave will print an error message, like this: |
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69 |
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70 @example |
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71 @group |
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72 octave:13> 3 + 4 i |
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73 |
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74 parse error: |
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75 |
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76 3 + 4 i |
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77 ^ |
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78 @end group |
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79 @end example |
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80 |
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81 @noindent |
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82 You may also use @samp{j}, @samp{I}, or @samp{J} in place of the |
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83 @samp{i} above. All four forms are equivalent. |
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84 |
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85 @DOCSTRING(double) |
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86 |
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87 @DOCSTRING(single) |
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88 |
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89 @DOCSTRING(complex) |
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90 |
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91 @menu |
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92 * Matrices:: |
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93 * Ranges:: |
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94 * Integer Data Types:: |
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95 * Bit Manipulations:: |
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96 * Logical Values:: |
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97 * Predicates for Numeric Objects:: |
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98 @end menu |
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99 |
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100 @node Matrices |
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101 @section Matrices |
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102 @cindex matrices |
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103 |
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104 @opindex [ |
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105 @opindex ] |
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106 @opindex ; |
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107 @opindex , |
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108 |
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109 It is easy to define a matrix of values in Octave. The size of the |
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110 matrix is determined automatically, so it is not necessary to explicitly |
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111 state the dimensions. The expression |
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112 |
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113 @example |
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114 a = [1, 2; 3, 4] |
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115 @end example |
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116 |
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117 @noindent |
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118 results in the matrix |
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119 @iftex |
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120 @tex |
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121 $$ a = \left[ \matrix{ 1 & 2 \cr 3 & 4 } \right] $$ |
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122 @end tex |
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123 @end iftex |
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124 @ifnottex |
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125 |
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126 @example |
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127 @group |
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128 |
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129 / \ |
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130 | 1 2 | |
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131 a = | | |
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132 | 3 4 | |
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133 \ / |
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134 |
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135 @end group |
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136 @end example |
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137 @end ifnottex |
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138 |
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139 Elements of a matrix may be arbitrary expressions, provided that the |
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140 dimensions all make sense when combining the various pieces. For |
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141 example, given the above matrix, the expression |
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142 |
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143 @example |
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144 [ a, a ] |
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145 @end example |
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146 |
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147 @noindent |
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148 produces the matrix |
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149 |
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150 @example |
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151 @group |
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152 ans = |
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153 |
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154 1 2 1 2 |
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155 3 4 3 4 |
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156 @end group |
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157 @end example |
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158 |
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159 @noindent |
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160 but the expression |
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161 |
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162 @example |
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163 [ a, 1 ] |
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164 @end example |
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165 |
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166 @noindent |
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167 produces the error |
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168 |
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169 @example |
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170 error: number of rows must match near line 13, column 6 |
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171 @end example |
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172 |
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173 @noindent |
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174 (assuming that this expression was entered as the first thing on line |
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175 13, of course). |
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176 |
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177 Inside the square brackets that delimit a matrix expression, Octave |
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178 looks at the surrounding context to determine whether spaces and newline |
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179 characters should be converted into element and row separators, or |
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180 simply ignored, so an expression like |
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181 |
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182 @example |
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183 @group |
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184 a = [ 1 2 |
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185 3 4 ] |
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186 @end group |
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187 @end example |
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188 |
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189 @noindent |
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190 will work. However, some possible sources of confusion remain. For |
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191 example, in the expression |
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192 |
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193 @example |
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194 [ 1 - 1 ] |
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195 @end example |
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196 |
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197 @noindent |
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198 the @samp{-} is treated as a binary operator and the result is the |
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199 scalar 0, but in the expression |
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200 |
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201 @example |
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202 [ 1 -1 ] |
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203 @end example |
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204 |
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205 @noindent |
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206 the @samp{-} is treated as a unary operator and the result is the |
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207 vector @code{[ 1, -1 ]}. Similarly, the expression |
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208 |
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209 @example |
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210 [ sin (pi) ] |
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211 @end example |
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212 |
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213 @noindent |
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214 will be parsed as |
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215 |
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216 @example |
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217 [ sin, (pi) ] |
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218 @end example |
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219 |
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220 @noindent |
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221 and will result in an error since the @code{sin} function will be |
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222 called with no arguments. To get around this, you must omit the space |
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223 between @code{sin} and the opening parenthesis, or enclose the |
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224 expression in a set of parentheses: |
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225 |
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226 @example |
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227 [ (sin (pi)) ] |
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228 @end example |
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229 |
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230 Whitespace surrounding the single quote character (@samp{'}, used as a |
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231 transpose operator and for delimiting character strings) can also cause |
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232 confusion. Given @code{a = 1}, the expression |
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233 |
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234 @example |
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235 [ 1 a' ] |
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236 @end example |
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237 |
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238 @noindent |
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239 results in the single quote character being treated as a |
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240 transpose operator and the result is the vector @code{[ 1, 1 ]}, but the |
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241 expression |
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242 |
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243 @example |
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244 [ 1 a ' ] |
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245 @end example |
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246 |
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247 @noindent |
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248 produces the error message |
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249 |
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250 @example |
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251 error: unterminated string constant |
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252 @end example |
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253 |
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254 @noindent |
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255 because not doing so would cause trouble when parsing the valid expression |
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256 |
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257 @example |
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258 [ a 'foo' ] |
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259 @end example |
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260 |
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261 For clarity, it is probably best to always use commas and semicolons to |
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262 separate matrix elements and rows. |
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263 |
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264 When you type a matrix or the name of a variable whose value is a |
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265 matrix, Octave responds by printing the matrix in with neatly aligned |
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266 rows and columns. If the rows of the matrix are too large to fit on the |
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267 screen, Octave splits the matrix and displays a header before each |
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268 section to indicate which columns are being displayed. You can use the |
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269 following variables to control the format of the output. |
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270 |
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271 @DOCSTRING(output_max_field_width) |
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272 |
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273 @DOCSTRING(output_precision) |
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274 |
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275 It is possible to achieve a wide range of output styles by using |
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276 different values of @code{output_precision} and |
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277 @code{output_max_field_width}. Reasonable combinations can be set using |
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278 the @code{format} function. @xref{Basic Input and Output}. |
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279 |
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280 @DOCSTRING(split_long_rows) |
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281 |
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282 Octave automatically switches to scientific notation when values become |
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283 very large or very small. This guarantees that you will see several |
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284 significant figures for every value in a matrix. If you would prefer to |
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285 see all values in a matrix printed in a fixed point format, you can set |
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286 the built-in variable @code{fixed_point_format} to a nonzero value. But |
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287 doing so is not recommended, because it can produce output that can |
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288 easily be misinterpreted. |
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289 |
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290 @DOCSTRING(fixed_point_format) |
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291 |
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292 @menu |
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293 * Empty Matrices:: |
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294 @end menu |
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295 |
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296 @node Empty Matrices |
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297 @subsection Empty Matrices |
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298 |
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299 A matrix may have one or both dimensions zero, and operations on empty |
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300 matrices are handled as described by Carl de Boor in @cite{An Empty |
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301 Exercise}, SIGNUM, Volume 25, pages 2--6, 1990 and C. N. Nett and W. M. |
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302 Haddad, in @cite{A System-Theoretic Appropriate Realization of the Empty |
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303 Matrix Concept}, IEEE Transactions on Automatic Control, Volume 38, |
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304 Number 5, May 1993. |
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305 @iftex |
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306 @tex |
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307 Briefly, given a scalar $s$, an $m\times n$ matrix $M_{m\times n}$, |
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308 and an $m\times n$ empty matrix $[\,]_{m\times n}$ (with either one or |
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309 both dimensions equal to zero), the following are true: |
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310 $$ |
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311 \eqalign{% |
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312 s \cdot [\,]_{m\times n} = [\,]_{m\times n} \cdot s &= [\,]_{m\times n}\cr |
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313 [\,]_{m\times n} + [\,]_{m\times n} &= [\,]_{m\times n}\cr |
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314 [\,]_{0\times m} \cdot M_{m\times n} &= [\,]_{0\times n}\cr |
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315 M_{m\times n} \cdot [\,]_{n\times 0} &= [\,]_{m\times 0}\cr |
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316 [\,]_{m\times 0} \cdot [\,]_{0\times n} &= 0_{m\times n}} |
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317 $$ |
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318 @end tex |
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319 @end iftex |
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320 @ifnottex |
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321 Briefly, given a scalar @var{s}, an @var{m} by |
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322 @var{n} matrix @code{M(mxn)}, and an @var{m} by @var{n} empty matrix |
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323 @code{[](mxn)} (with either one or both dimensions equal to zero), the |
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324 following are true: |
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325 |
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326 @example |
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327 @group |
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328 s * [](mxn) = [](mxn) * s = [](mxn) |
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329 |
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330 [](mxn) + [](mxn) = [](mxn) |
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331 |
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332 [](0xm) * M(mxn) = [](0xn) |
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333 |
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334 M(mxn) * [](nx0) = [](mx0) |
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335 |
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336 [](mx0) * [](0xn) = 0(mxn) |
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337 @end group |
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338 @end example |
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339 @end ifnottex |
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340 |
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341 By default, dimensions of the empty matrix are printed along with the |
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342 empty matrix symbol, @samp{[]}. The built-in variable |
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343 @code{print_empty_dimensions} controls this behavior. |
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344 |
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345 @DOCSTRING(print_empty_dimensions) |
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346 |
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347 Empty matrices may also be used in assignment statements as a convenient |
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348 way to delete rows or columns of matrices. |
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349 @xref{Assignment Ops, ,Assignment Expressions}. |
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350 |
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351 When Octave parses a matrix expression, it examines the elements of the |
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352 list to determine whether they are all constants. If they are, it |
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353 replaces the list with a single matrix constant. |
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354 |
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355 @node Ranges |
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356 @section Ranges |
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357 @cindex range expressions |
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358 @cindex expression, range |
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359 |
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360 @opindex colon |
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361 |
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362 A @dfn{range} is a convenient way to write a row vector with evenly |
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363 spaced elements. A range expression is defined by the value of the first |
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364 element in the range, an optional value for the increment between |
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365 elements, and a maximum value which the elements of the range will not |
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366 exceed. The base, increment, and limit are separated by colons (the |
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367 @samp{:} character) and may contain any arithmetic expressions and |
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368 function calls. If the increment is omitted, it is assumed to be 1. |
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369 For example, the range |
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370 |
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371 @example |
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372 1 : 5 |
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373 @end example |
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374 |
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375 @noindent |
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376 defines the set of values @samp{[ 1, 2, 3, 4, 5 ]}, and the range |
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377 |
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378 @example |
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379 1 : 3 : 5 |
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380 @end example |
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381 |
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382 @noindent |
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383 defines the set of values @samp{[ 1, 4 ]}. |
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384 |
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385 Although a range constant specifies a row vector, Octave does @emph{not} |
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386 convert range constants to vectors unless it is necessary to do so. |
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387 This allows you to write a constant like @samp{1 : 10000} without using |
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388 80,000 bytes of storage on a typical 32-bit workstation. |
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389 |
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390 Note that the upper (or lower, if the increment is negative) bound on |
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391 the range is not always included in the set of values, and that ranges |
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392 defined by floating point values can produce surprising results because |
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393 Octave uses floating point arithmetic to compute the values in the |
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394 range. If it is important to include the endpoints of a range and the |
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395 number of elements is known, you should use the @code{linspace} function |
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396 instead (@pxref{Special Utility Matrices}). |
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397 |
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398 When Octave parses a range expression, it examines the elements of the |
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399 expression to determine whether they are all constants. If they are, it |
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400 replaces the range expression with a single range constant. |
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401 |
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402 @node Integer Data Types |
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403 @section Integer Data Types |
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404 |
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405 Octave supports integer matrices as an alternative to using double |
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406 precision. It is possible to use both signed and unsigned integers |
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407 represented by 8, 16, 32, or 64 bits. It should be noted that most |
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408 computations require floating point data, meaning that integers will |
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409 often change type when involved in numeric computations. For this |
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410 reason integers are most often used to store data, and not for |
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411 calculations. |
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412 |
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413 In general most integer matrices are created by casting |
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414 existing matrices to integers. The following example shows how to cast |
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415 a matrix into 32 bit integers. |
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416 |
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417 @example |
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418 float = rand (2, 2) |
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419 @result{} float = 0.37569 0.92982 |
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420 0.11962 0.50876 |
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421 integer = int32 (float) |
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422 @result{} integer = 0 1 |
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423 0 1 |
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424 @end example |
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425 |
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426 @noindent |
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427 As can be seen, floating point values are rounded to the nearest integer |
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428 when converted. |
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429 |
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430 @DOCSTRING(isinteger) |
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431 |
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432 @DOCSTRING(int8) |
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433 |
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434 @DOCSTRING(uint8) |
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435 |
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436 @DOCSTRING(int16) |
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437 |
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438 @DOCSTRING(uint16) |
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439 |
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440 @DOCSTRING(int32) |
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441 |
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442 @DOCSTRING(uint32) |
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443 |
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444 @DOCSTRING(int64) |
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445 |
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446 @DOCSTRING(uint64) |
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447 |
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448 @DOCSTRING(intmax) |
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449 |
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450 @DOCSTRING(intmin) |
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451 |
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452 @menu |
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453 * Integer Arithmetic:: |
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454 @end menu |
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455 |
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456 @node Integer Arithmetic |
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457 @subsection Integer Arithmetic |
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458 |
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459 While many numerical computations can't be carried out in integers, |
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460 Octave does support basic operations like addition and multiplication |
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461 on integers. The operators @code{+}, @code{-}, @code{.*}, and @code{./} |
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462 works on integers of the same type. So, it is possible to add two 32 bit |
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463 integers, but not to add a 32 bit integer and a 16 bit integer. |
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464 |
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465 The arithmetic operations on integers are performed by casting the |
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466 integer values to double precision values, performing the operation, and |
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467 then re-casting the values back to the original integer type. As the |
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468 double precision type of Octave is only capable of representing integers |
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469 with up to 53 bits of precision, it is not possible to perform |
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470 arithmetic of the 64 bit integer types. |
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471 |
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472 When doing integer arithmetic one should consider the possibility of |
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473 underflow and overflow. This happens when the result of the computation |
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474 can't be represented using the chosen integer type. As an example it is |
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475 not possible to represent the result of @math{10 - 20} when using |
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476 unsigned integers. Octave makes sure that the result of integer |
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477 computations is the integer that is closest to the true result. So, the |
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478 result of @math{10 - 20} when using unsigned integers is zero. |
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479 |
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480 When doing integer division Octave will round the result to the nearest |
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481 integer. This is different from most programming languages, where the |
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482 result is often floored to the nearest integer. So, the result of |
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483 @code{int32(5)./int32(8)} is @code{1}. |
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484 |
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6781
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485 @node Bit Manipulations |
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486 @section Bit Manipulations |
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487 |
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488 Octave provides a number of functions for the manipulation of numeric |
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489 values on a bit by bit basis. The basic functions to set and obtain the |
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490 values of individual bits are @code{bitset} and @code{bitget}. |
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491 |
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492 @DOCSTRING(bitset) |
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493 |
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494 @DOCSTRING(bitget) |
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495 |
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496 The arguments to all of Octave's bitwise operations can be scalar or |
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497 arrays, except for @code{bitcmp}, whose @var{k} argument must a |
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498 scalar. In the case where more than one argument is an array, then all |
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499 arguments must have the same shape, and the bitwise operator is applied |
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500 to each of the elements of the argument individually. If at least one |
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501 argument is a scalar and one an array, then the scalar argument is |
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|
502 duplicated. Therefore |
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503 |
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504 @example |
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505 bitget (100, 8:-1:1) |
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506 @end example |
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507 |
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508 is the same as |
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509 |
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510 @example |
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511 bitget (100 * ones (1, 8), 8:-1:1) |
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512 @end example |
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513 |
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|
514 It should be noted that all values passed to the bit manipulation |
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515 functions of Octave are treated as integers. Therefore, even though the |
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516 example for @code{bitset} above passes the floating point value |
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517 @code{10}, it is treated as the bits @code{[1, 0, 1, 0]} rather than the |
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518 bits of the native floating point format representation of @code{10}. |
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|
519 |
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520 As the maximum number that can be represented by a number is important |
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521 for bit manipulation, particularly when forming masks, Octave supplies |
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522 the function @code{bitmax}. |
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523 |
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524 @DOCSTRING(bitmax) |
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525 |
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526 This is the double precision version of the functions @code{intmax}, |
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|
527 previously discussed. |
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528 |
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529 Octave also include the basic bitwise 'and', 'or' and 'exclusive or' |
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|
530 operators. |
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531 |
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532 @DOCSTRING(bitand) |
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533 |
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534 @DOCSTRING(bitor) |
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535 |
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536 @DOCSTRING(bitxor) |
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537 |
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|
538 The bitwise 'not' operator is unary operator that performs a logial |
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|
539 negation of each of the bits of the value. For this to make sense, the |
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|
540 mask against which the value is negated must be defined. Octave's |
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|
541 bitwise 'not' operator is @code{bitcmp}. |
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542 |
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543 @DOCSTRING(bitcmp) |
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544 |
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545 Octave also includes the ability to left and right values bitwise. |
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546 |
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|
547 @DOCSTRING(bitshift) |
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548 |
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549 Bits that are shifted out of either end of the value are lost. Octave |
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|
550 also uses arithmetic shifts, where the sign bit of the value is keep |
|
|
551 during a right shift. For example |
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|
552 |
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553 @example |
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|
554 @group |
|
|
555 bitshift (-10, -1) |
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|
556 @result{} -5 |
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557 bitshift (int8 (-1), -1) |
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|
558 @result{} -1 |
|
|
559 @end group |
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560 @end example |
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561 |
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|
562 Note that @code{bitshift (int8 (-1), -1)} is @code{-1} since the bit |
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|
563 representation of @code{-1} in the @code{int8} data type is @code{[1, 1, |
|
|
564 1, 1, 1, 1, 1, 1]}. |
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|
565 |
|
4167
|
566 @node Logical Values |
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3428
|
567 @section Logical Values |
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|
568 |
|
6620
|
569 Octave has built-in support for logical values, i.e. variables that |
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|
570 are either @code{true} or @code{false}. When comparing two variables, |
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|
571 the result will be a logical value whose value depends on whether or |
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|
572 not the comparison is true. |
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|
573 |
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|
574 The basic logical operations are @code{&}, @code{|}, and @code{!}, |
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|
575 that corresponds to ``Logical And'', ``Logical Or'', and ``Logical |
|
|
576 Negation''. These operations all follow the rules of logic. |
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|
577 |
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|
578 It is also possible to use logical values as part of standard numerical |
|
|
579 calculations. In this case @code{true} is converted to @code{1}, and |
|
|
580 @code{false} to 0, both represented using double precision floating |
|
|
581 point numbers. So, the result of @code{true*22 - false/6} is @code{22}. |
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|
582 |
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|
583 Logical values can also be used to index matrices and cell arrays. |
|
|
584 When indexing with a logical array the result will be a vector containing |
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|
585 the values corresponding to @code{true} parts of the logical array. |
|
|
586 The following example illustrates this. |
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|
587 |
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|
588 @example |
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|
589 data = [ 1, 2; 3, 4 ]; |
|
|
590 idx = (data <= 2); |
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|
591 data(idx) |
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|
592 @result{} ans = [ 1; 4 ] |
|
|
593 @end example |
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|
594 |
|
|
595 @noindent |
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|
596 Instead of creating the @code{idx} array it is possible to replace |
|
|
597 @code{data(idx)} with @code{data( data <= 2 )} in the above code. |
|
|
598 |
|
|
599 Besides when doing comparisons, logical values can be constructed by |
|
|
600 casting numeric objects to logical values, or by using the @code{true} |
|
|
601 or @code{false} functions. |
|
|
602 |
|
6549
|
603 @DOCSTRING(logical) |
|
|
604 |
|
3428
|
605 @DOCSTRING(true) |
|
|
606 |
|
|
607 @DOCSTRING(false) |
|
|
608 |
|
4167
|
609 @node Predicates for Numeric Objects |
|
3294
|
610 @section Predicates for Numeric Objects |
|
|
611 |
|
6620
|
612 Since the type of a variable may change during the execution of a |
|
|
613 program, it can be necessary to type checking at run-time. Doing this |
|
|
614 also allows you to change the behaviour of a function depending on the |
|
|
615 type of the input. As an example, this naive implementation of @code{abs} |
|
7001
|
616 returns the absolute value of the input if it is a real number, and the |
|
6620
|
617 length of the input if it is a complex number. |
|
|
618 |
|
|
619 @example |
|
|
620 function a = abs (x) |
|
|
621 if (isreal (x)) |
|
|
622 a = sign (x) .* x; |
|
|
623 elseif (iscomplex (x)) |
|
|
624 a = sqrt (real(x).^2 + imag(x).^2); |
|
|
625 endif |
|
|
626 endfunction |
|
|
627 @end example |
|
|
628 |
|
|
629 The following functions are available for determining the type of a |
|
|
630 variable. |
|
|
631 |
|
3428
|
632 @DOCSTRING(isnumeric) |
|
|
633 |
|
|
634 @DOCSTRING(isreal) |
|
|
635 |
|
4029
|
636 @DOCSTRING(iscomplex) |
|
3428
|
637 |
|
4029
|
638 @DOCSTRING(ismatrix) |
|
3294
|
639 |
|
4029
|
640 @DOCSTRING(isvector) |
|
3294
|
641 |
|
4029
|
642 @DOCSTRING(isscalar) |
|
3294
|
643 |
|
4029
|
644 @DOCSTRING(issquare) |
|
3294
|
645 |
|
4029
|
646 @DOCSTRING(issymmetric) |
|
3428
|
647 |
|
6550
|
648 @DOCSTRING(isdefinite) |
|
|
649 |
|
6688
|
650 @DOCSTRING(islogical) |
|
6550
|
651 |
|
|
652 @DOCSTRING(isprime) |
