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1 @c Copyright (C) 1996, 1997, 2007 John W. Eaton |
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2 @c This is part of the Octave manual. |
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3 @c For copying conditions, see the file gpl.texi. |
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4 |
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5 @node Numeric Data Types |
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6 @chapter Numeric Data Types |
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7 @cindex numeric constant |
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8 @cindex numeric value |
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9 |
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10 A @dfn{numeric constant} may be a scalar, a vector, or a matrix, and it |
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11 may contain complex values. |
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12 |
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13 The simplest form of a numeric constant, a scalar, is a single number |
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14 that can be an integer, a decimal fraction, a number in scientific |
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15 (exponential) notation, or a complex number. Note that by default numeric |
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16 constants are represented within Octave in double-precision floating |
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17 point format (complex constants are stored as pairs of double-precision |
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18 floating point values). It is however possible to represent real |
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19 integers as described in @ref{Integer Data Types}. Here are some examples |
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20 of real-valued numeric constants, which all have the same value: |
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21 |
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22 @example |
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23 @group |
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24 105 |
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25 1.05e+2 |
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26 1050e-1 |
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27 @end group |
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28 @end example |
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29 |
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30 To specify complex constants, you can write an expression of the form |
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31 |
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32 @example |
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33 @group |
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34 3 + 4i |
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35 3.0 + 4.0i |
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36 0.3e1 + 40e-1i |
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37 @end group |
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38 @end example |
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39 |
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40 @noindent |
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41 all of which are equivalent. The letter @samp{i} in the previous example |
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42 stands for the pure imaginary constant, defined as |
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43 @iftex |
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44 @tex |
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45 $\sqrt{-1}$. |
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46 @end tex |
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47 @end iftex |
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48 @ifnottex |
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49 @code{sqrt (-1)}. |
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50 @end ifnottex |
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51 |
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52 For Octave to recognize a value as the imaginary part of a complex |
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53 constant, a space must not appear between the number and the @samp{i}. |
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54 If it does, Octave will print an error message, like this: |
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55 |
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56 @example |
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57 @group |
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58 octave:13> 3 + 4 i |
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59 |
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60 parse error: |
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61 |
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62 3 + 4 i |
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63 ^ |
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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 You may also use @samp{j}, @samp{I}, or @samp{J} in place of the |
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69 @samp{i} above. All four forms are equivalent. |
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70 |
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71 @DOCSTRING(double) |
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72 |
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73 @DOCSTRING(single) |
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74 |
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75 @DOCSTRING(complex) |
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76 |
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77 @menu |
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78 * Matrices:: |
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79 * Ranges:: |
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80 * Integer Data Types:: |
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81 * Logical Values:: |
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82 * Predicates for Numeric Objects:: |
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83 @end menu |
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84 |
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85 @node Matrices |
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86 @section Matrices |
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87 @cindex matrices |
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88 |
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89 @opindex [ |
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90 @opindex ] |
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91 @opindex ; |
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92 @opindex , |
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93 |
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94 It is easy to define a matrix of values in Octave. The size of the |
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95 matrix is determined automatically, so it is not necessary to explicitly |
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96 state the dimensions. The expression |
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97 |
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98 @example |
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99 a = [1, 2; 3, 4] |
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100 @end example |
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101 |
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102 @noindent |
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103 results in the matrix |
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104 @iftex |
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105 @tex |
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106 $$ a = \left[ \matrix{ 1 & 2 \cr 3 & 4 } \right] $$ |
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107 @end tex |
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108 @end iftex |
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109 @ifnottex |
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110 |
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111 @example |
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112 @group |
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113 |
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114 / \ |
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115 | 1 2 | |
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116 a = | | |
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117 | 3 4 | |
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118 \ / |
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119 |
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120 @end group |
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121 @end example |
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122 @end ifnottex |
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123 |
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124 Elements of a matrix may be arbitrary expressions, provided that the |
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125 dimensions all make sense when combining the various pieces. For |
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126 example, given the above matrix, the expression |
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127 |
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128 @example |
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129 [ a, a ] |
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130 @end example |
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131 |
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132 @noindent |
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133 produces the matrix |
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134 |
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135 @example |
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136 @group |
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137 ans = |
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138 |
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139 1 2 1 2 |
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140 3 4 3 4 |
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141 @end group |
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142 @end example |
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143 |
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144 @noindent |
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145 but the expression |
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146 |
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147 @example |
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148 [ a, 1 ] |
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149 @end example |
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150 |
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151 @noindent |
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152 produces the error |
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153 |
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154 @example |
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155 error: number of rows must match near line 13, column 6 |
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156 @end example |
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157 |
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158 @noindent |
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159 (assuming that this expression was entered as the first thing on line |
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160 13, of course). |
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161 |
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162 Inside the square brackets that delimit a matrix expression, Octave |
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163 looks at the surrounding context to determine whether spaces and newline |
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164 characters should be converted into element and row separators, or |
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165 simply ignored, so an expression like |
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166 |
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167 @example |
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168 @group |
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169 a = [ 1 2 |
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170 3 4 ] |
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171 @end group |
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172 @end example |
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173 |
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174 @noindent |
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175 will work. However, some possible sources of confusion remain. For |
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176 example, in the expression |
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177 |
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178 @example |
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179 [ 1 - 1 ] |
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180 @end example |
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181 |
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182 @noindent |
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183 the @samp{-} is treated as a binary operator and the result is the |
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184 scalar 0, but in the expression |
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185 |
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186 @example |
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187 [ 1 -1 ] |
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188 @end example |
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189 |
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190 @noindent |
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191 the @samp{-} is treated as a unary operator and the result is the |
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192 vector @code{[ 1, -1 ]}. Similarly, the expression |
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193 |
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194 @example |
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195 [ sin (pi) ] |
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196 @end example |
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197 |
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198 @noindent |
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199 will be parsed as |
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200 |
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201 @example |
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202 [ sin, (pi) ] |
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203 @end example |
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204 |
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205 @noindent |
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206 and will result in an error since the @code{sin} function will be |
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207 called with no arguments. To get around this, you must omit the space |
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208 between @code{sin} and the opening parenthesis, or enclose the |
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209 expression in a set of parentheses: |
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210 |
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211 @example |
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212 [ (sin (pi)) ] |
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213 @end example |
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214 |
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215 Whitespace surrounding the single quote character (@samp{'}, used as a |
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216 transpose operator and for delimiting character strings) can also cause |
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217 confusion. Given @code{a = 1}, the expression |
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218 |
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219 @example |
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220 [ 1 a' ] |
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221 @end example |
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222 |
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223 @noindent |
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224 results in the single quote character being treated as a |
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225 transpose operator and the result is the vector @code{[ 1, 1 ]}, but the |
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226 expression |
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227 |
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228 @example |
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229 [ 1 a ' ] |
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230 @end example |
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231 |
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232 @noindent |
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233 produces the error message |
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234 |
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235 @example |
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236 error: unterminated string constant |
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237 @end example |
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238 |
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239 @noindent |
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240 because not doing so would cause trouble when parsing the valid expression |
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241 |
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242 @example |
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243 [ a 'foo' ] |
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244 @end example |
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245 |
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246 For clarity, it is probably best to always use commas and semicolons to |
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247 separate matrix elements and rows. |
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248 |
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249 When you type a matrix or the name of a variable whose value is a |
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250 matrix, Octave responds by printing the matrix in with neatly aligned |
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251 rows and columns. If the rows of the matrix are too large to fit on the |
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252 screen, Octave splits the matrix and displays a header before each |
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253 section to indicate which columns are being displayed. You can use the |
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254 following variables to control the format of the output. |
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255 |
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256 @DOCSTRING(output_max_field_width) |
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257 |
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258 @DOCSTRING(output_precision) |
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259 |
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260 It is possible to achieve a wide range of output styles by using |
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261 different values of @code{output_precision} and |
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262 @code{output_max_field_width}. Reasonable combinations can be set using |
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263 the @code{format} function. @xref{Basic Input and Output}. |
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264 |
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265 @DOCSTRING(split_long_rows) |
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266 |
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267 Octave automatically switches to scientific notation when values become |
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268 very large or very small. This guarantees that you will see several |
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269 significant figures for every value in a matrix. If you would prefer to |
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270 see all values in a matrix printed in a fixed point format, you can set |
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271 the built-in variable @code{fixed_point_format} to a nonzero value. But |
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272 doing so is not recommended, because it can produce output that can |
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273 easily be misinterpreted. |
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274 |
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275 @DOCSTRING(fixed_point_format) |
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276 |
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277 @menu |
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278 * Empty Matrices:: |
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279 @end menu |
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280 |
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281 @node Empty Matrices |
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282 @subsection Empty Matrices |
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283 |
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284 A matrix may have one or both dimensions zero, and operations on empty |
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285 matrices are handled as described by Carl de Boor in @cite{An Empty |
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286 Exercise}, SIGNUM, Volume 25, pages 2--6, 1990 and C. N. Nett and W. M. |
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287 Haddad, in @cite{A System-Theoretic Appropriate Realization of the Empty |
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288 Matrix Concept}, IEEE Transactions on Automatic Control, Volume 38, |
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289 Number 5, May 1993. |
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290 @iftex |
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291 @tex |
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292 Briefly, given a scalar $s$, an $m\times n$ matrix $M_{m\times n}$, |
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293 and an $m\times n$ empty matrix $[\,]_{m\times n}$ (with either one or |
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294 both dimensions equal to zero), the following are true: |
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295 $$ |
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296 \eqalign{% |
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297 s \cdot [\,]_{m\times n} = [\,]_{m\times n} \cdot s &= [\,]_{m\times n}\cr |
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298 [\,]_{m\times n} + [\,]_{m\times n} &= [\,]_{m\times n}\cr |
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299 [\,]_{0\times m} \cdot M_{m\times n} &= [\,]_{0\times n}\cr |
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300 M_{m\times n} \cdot [\,]_{n\times 0} &= [\,]_{m\times 0}\cr |
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301 [\,]_{m\times 0} \cdot [\,]_{0\times n} &= 0_{m\times n}} |
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302 $$ |
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303 @end tex |
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304 @end iftex |
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305 @ifnottex |
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306 Briefly, given a scalar @var{s}, an @var{m} by |
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307 @var{n} matrix @code{M(mxn)}, and an @var{m} by @var{n} empty matrix |
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308 @code{[](mxn)} (with either one or both dimensions equal to zero), the |
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309 following are true: |
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310 |
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311 @example |
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312 @group |
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313 s * [](mxn) = [](mxn) * s = [](mxn) |
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314 |
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315 [](mxn) + [](mxn) = [](mxn) |
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316 |
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317 [](0xm) * M(mxn) = [](0xn) |
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318 |
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319 M(mxn) * [](nx0) = [](mx0) |
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320 |
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321 [](mx0) * [](0xn) = 0(mxn) |
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322 @end group |
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323 @end example |
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324 @end ifnottex |
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325 |
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326 By default, dimensions of the empty matrix are printed along with the |
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327 empty matrix symbol, @samp{[]}. The built-in variable |
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328 @code{print_empty_dimensions} controls this behavior. |
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329 |
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330 @DOCSTRING(print_empty_dimensions) |
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331 |
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332 Empty matrices may also be used in assignment statements as a convenient |
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333 way to delete rows or columns of matrices. |
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334 @xref{Assignment Ops, ,Assignment Expressions}. |
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335 |
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336 When Octave parses a matrix expression, it examines the elements of the |
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337 list to determine whether they are all constants. If they are, it |
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338 replaces the list with a single matrix constant. |
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339 |
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340 @node Ranges |
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341 @section Ranges |
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342 @cindex range expressions |
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343 @cindex expression, range |
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344 |
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345 @opindex colon |
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346 |
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347 A @dfn{range} is a convenient way to write a row vector with evenly |
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348 spaced elements. A range expression is defined by the value of the first |
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349 element in the range, an optional value for the increment between |
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350 elements, and a maximum value which the elements of the range will not |
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351 exceed. The base, increment, and limit are separated by colons (the |
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352 @samp{:} character) and may contain any arithmetic expressions and |
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353 function calls. If the increment is omitted, it is assumed to be 1. |
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354 For example, the range |
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355 |
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356 @example |
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357 1 : 5 |
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358 @end example |
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359 |
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360 @noindent |
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361 defines the set of values @samp{[ 1, 2, 3, 4, 5 ]}, and the range |
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362 |
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363 @example |
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364 1 : 3 : 5 |
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365 @end example |
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366 |
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367 @noindent |
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368 defines the set of values @samp{[ 1, 4 ]}. |
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369 |
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370 Although a range constant specifies a row vector, Octave does @emph{not} |
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371 convert range constants to vectors unless it is necessary to do so. |
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372 This allows you to write a constant like @samp{1 : 10000} without using |
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373 80,000 bytes of storage on a typical 32-bit workstation. |
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374 |
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375 Note that the upper (or lower, if the increment is negative) bound on |
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376 the range is not always included in the set of values, and that ranges |
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377 defined by floating point values can produce surprising results because |
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378 Octave uses floating point arithmetic to compute the values in the |
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379 range. If it is important to include the endpoints of a range and the |
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380 number of elements is known, you should use the @code{linspace} function |
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381 instead (@pxref{Special Utility Matrices}). |
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382 |
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383 When Octave parses a range expression, it examines the elements of the |
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384 expression to determine whether they are all constants. If they are, it |
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385 replaces the range expression with a single range constant. |
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386 |
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387 @node Integer Data Types |
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388 @section Integer Data Types |
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389 |
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390 Octave supports integer matrices as an alternative to using double |
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391 precision. It is possible to use both signed and unsigned integers |
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392 represented by 8, 16, 32, or 64 bits. It should be noted that most |
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393 computations require floating point data, meaning that integers will |
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394 often change type when involved in numeric computations. For this |
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395 reason integers are most often used to store data, and not for |
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396 calculations. |
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397 |
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398 In general most integer matrices are created by casting |
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399 existing matrices to integers. The following example shows how to cast |
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400 a matrix into 32 bit integers. |
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401 |
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402 @example |
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403 float = rand (2, 2) |
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404 @result{} float = 0.37569 0.92982 |
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405 0.11962 0.50876 |
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406 integer = int32 (float) |
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407 @result{} integer = 0 1 |
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408 0 1 |
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409 @end example |
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410 |
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411 @noindent |
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412 As can be seen, floating point values are rounded to the nearest integer |
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413 when converted. |
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414 |
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415 @DOCSTRING(isinteger) |
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416 |
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417 @DOCSTRING(int8) |
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418 |
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419 @DOCSTRING(uint8) |
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420 |
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421 @DOCSTRING(int16) |
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422 |
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423 @DOCSTRING(uint16) |
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424 |
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425 @DOCSTRING(int32) |
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426 |
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427 @DOCSTRING(uint32) |
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428 |
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429 @DOCSTRING(int64) |
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430 |
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431 @DOCSTRING(uint64) |
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432 |
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433 @DOCSTRING(intmax) |
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434 |
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435 @DOCSTRING(intmin) |
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436 |
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437 @menu |
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438 * Integer Arithmetic:: |
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439 @end menu |
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440 |
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441 @node Integer Arithmetic |
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442 @subsection Integer Arithmetic |
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443 |
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444 While many numerical computations can't be carried out in integers, |
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445 Octave does support basic operations like addition and multiplication |
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446 on integers. The operators @code{+}, @code{-}, @code{.*}, and @code{./} |
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447 works on integers of the same type. So, it is possible to add two 32 bit |
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448 integers, but not to add a 32 bit integer and a 16 bit integer. |
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449 |
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450 The arithmetic operations on integers are performed by casting the |
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451 integer values to double precision values, performing the operation, and |
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452 then re-casting the values back to the original integer type. As the |
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453 double precision type of Octave is only capable of representing integers |
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454 with up to 53 bits of precision, it is not possible to perform |
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455 arithmetic of the 64 bit integer types. |
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456 |
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457 When doing integer arithmetic one should consider the possibility of |
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458 underflow and overflow. This happens when the result of the computation |
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459 can't be represented using the chosen integer type. As an example it is |
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460 not possible to represent the result of @math{10 - 20} when using |
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461 unsigned integers. Octave makes sure that the result of integer |
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462 computations is the integer that is closest to the true result. So, the |
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463 result of @math{10 - 20} when using unsigned integers is zero. |
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464 |
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465 When doing integer division Octave will round the result to the nearest |
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466 integer. This is different from most programming languages, where the |
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467 result is often floored to the nearest integer. So, the result of |
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468 @code{int32(5)./int32(8)} is @code{1}. |
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469 |
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470 @node Logical Values |
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471 @section Logical Values |
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472 |
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473 Octave has built-in support for logical values, i.e. variables that |
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474 are either @code{true} or @code{false}. When comparing two variables, |
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475 the result will be a logical value whose value depends on whether or |
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476 not the comparison is true. |
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477 |
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478 The basic logical operations are @code{&}, @code{|}, and @code{!}, |
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479 that corresponds to ``Logical And'', ``Logical Or'', and ``Logical |
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480 Negation''. These operations all follow the rules of logic. |
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481 |
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482 It is also possible to use logical values as part of standard numerical |
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483 calculations. In this case @code{true} is converted to @code{1}, and |
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484 @code{false} to 0, both represented using double precision floating |
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485 point numbers. So, the result of @code{true*22 - false/6} is @code{22}. |
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486 |
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487 Logical values can also be used to index matrices and cell arrays. |
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488 When indexing with a logical array the result will be a vector containing |
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489 the values corresponding to @code{true} parts of the logical array. |
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490 The following example illustrates this. |
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491 |
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492 @example |
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493 data = [ 1, 2; 3, 4 ]; |
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494 idx = (data <= 2); |
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495 data(idx) |
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496 @result{} ans = [ 1; 4 ] |
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497 @end example |
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498 |
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499 @noindent |
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500 Instead of creating the @code{idx} array it is possible to replace |
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501 @code{data(idx)} with @code{data( data <= 2 )} in the above code. |
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502 |
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503 Besides when doing comparisons, logical values can be constructed by |
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504 casting numeric objects to logical values, or by using the @code{true} |
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505 or @code{false} functions. |
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506 |
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6549
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507 @DOCSTRING(logical) |
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508 |
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3428
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509 @DOCSTRING(true) |
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510 |
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511 @DOCSTRING(false) |
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512 |
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4167
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513 @node Predicates for Numeric Objects |
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3294
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514 @section Predicates for Numeric Objects |
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515 |
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6620
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516 Since the type of a variable may change during the execution of a |
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517 program, it can be necessary to type checking at run-time. Doing this |
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518 also allows you to change the behaviour of a function depending on the |
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519 type of the input. As an example, this naive implementation of @code{abs} |
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520 return the absolute value of the input if it is a real number, and the |
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521 length of the input if it is a complex number. |
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522 |
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523 @example |
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524 function a = abs (x) |
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525 if (isreal (x)) |
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526 a = sign (x) .* x; |
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527 elseif (iscomplex (x)) |
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528 a = sqrt (real(x).^2 + imag(x).^2); |
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529 endif |
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530 endfunction |
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531 @end example |
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532 |
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533 The following functions are available for determining the type of a |
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534 variable. |
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535 |
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3428
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536 @DOCSTRING(isnumeric) |
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537 |
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538 @DOCSTRING(isreal) |
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539 |
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4029
|
540 @DOCSTRING(iscomplex) |
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3428
|
541 |
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4029
|
542 @DOCSTRING(ismatrix) |
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3294
|
543 |
|
4029
|
544 @DOCSTRING(isvector) |
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3294
|
545 |
|
4029
|
546 @DOCSTRING(isscalar) |
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3294
|
547 |
|
4029
|
548 @DOCSTRING(issquare) |
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3294
|
549 |
|
4029
|
550 @DOCSTRING(issymmetric) |
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3428
|
551 |
|
6550
|
552 @DOCSTRING(isdefinite) |
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553 |
|
6688
|
554 @DOCSTRING(islogical) |
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6550
|
555 |
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|
556 @DOCSTRING(isprime) |
