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1 @c Copyright (C) 1996, 1997 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 all 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). Here are some examples of real-valued numeric |
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19 constants, which all have the same value: |
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20 |
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21 @example |
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22 @group |
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23 105 |
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24 1.05e+2 |
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25 1050e-1 |
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26 @end group |
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27 @end example |
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28 |
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29 To specify complex constants, you can write an expression of the form |
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30 |
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31 @example |
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32 @group |
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33 3 + 4i |
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34 3.0 + 4.0i |
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35 0.3e1 + 40e-1i |
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36 @end group |
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37 @end example |
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38 |
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39 all of which are equivalent. The letter @samp{i} in the previous example |
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40 stands for the pure imaginary constant, defined as |
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41 @iftex |
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42 @tex |
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43 $\sqrt{-1}$. |
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44 @end tex |
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45 @end iftex |
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46 @ifinfo |
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47 @code{sqrt (-1)}. |
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48 @end ifinfo |
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49 |
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50 For Octave to recognize a value as the imaginary part of a complex |
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51 constant, a space must not appear between the number and the @samp{i}. |
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52 If it does, Octave will print an error message, like this: |
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53 |
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54 @example |
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55 @group |
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56 octave:13> 3 + 4 i |
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57 |
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58 parse error: |
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59 |
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60 3 + 4 i |
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61 ^ |
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62 @end group |
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63 @end example |
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64 |
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65 You may also use @samp{j}, @samp{I}, or @samp{J} in place of the |
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66 @samp{i} above. All four forms are equivalent. |
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67 |
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68 @menu |
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69 * Matrices:: |
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70 * Ranges:: |
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71 * Logical Values:: |
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72 * Predicates for Numeric Objects:: |
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73 @end menu |
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74 |
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75 @node Matrices |
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76 @section Matrices |
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77 @cindex matrices |
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78 |
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79 @opindex [ |
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80 @opindex ] |
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81 @opindex ; |
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82 @opindex , |
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83 |
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84 It is easy to define a matrix of values in Octave. The size of the |
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85 matrix is determined automatically, so it is not necessary to explicitly |
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86 state the dimensions. The expression |
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87 |
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88 @example |
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89 a = [1, 2; 3, 4] |
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90 @end example |
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91 |
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92 @noindent |
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93 results in the matrix |
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94 @iftex |
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95 @tex |
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96 $$ a = \left[ \matrix{ 1 & 2 \cr 3 & 4 } \right] $$ |
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97 @end tex |
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98 @end iftex |
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99 @ifinfo |
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100 |
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101 @example |
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102 @group |
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103 |
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104 / \ |
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105 | 1 2 | |
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106 a = | | |
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107 | 3 4 | |
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108 \ / |
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109 |
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110 @end group |
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111 @end example |
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112 @end ifinfo |
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113 |
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114 Elements of a matrix may be arbitrary expressions, provided that the |
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115 dimensions all make sense when combining the various pieces. For |
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116 example, given the above matrix, the expression |
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117 |
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118 @example |
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119 [ a, a ] |
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120 @end example |
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121 |
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122 @noindent |
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123 produces the matrix |
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124 |
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125 @example |
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126 @group |
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127 ans = |
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128 |
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129 1 2 1 2 |
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130 3 4 3 4 |
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131 @end group |
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132 @end example |
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133 |
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134 @noindent |
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135 but the expression |
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136 |
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137 @example |
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138 [ a, 1 ] |
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139 @end example |
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140 |
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141 @noindent |
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142 produces the error |
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143 |
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144 @example |
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145 error: number of rows must match near line 13, column 6 |
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146 @end example |
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147 |
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148 @noindent |
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149 (assuming that this expression was entered as the first thing on line |
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150 13, of course). |
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151 |
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152 Inside the square brackets that delimit a matrix expression, Octave |
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153 looks at the surrounding context to determine whether spaces and newline |
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154 characters should be converted into element and row separators, or |
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155 simply ignored, so an expression like |
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156 |
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157 @example |
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158 @group |
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159 a = [ 1 2 |
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160 3 4 ] |
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161 @end group |
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162 @end example |
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163 |
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164 @noindent |
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165 will work. However, some possible sources of confusion remain. For |
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166 example, in the expression |
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167 |
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168 @example |
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169 [ 1 - 1 ] |
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170 @end example |
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171 |
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172 @noindent |
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173 the @samp{-} is treated as a binary operator and the result is the |
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174 scalar 0, but in the expression |
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175 |
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176 @example |
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177 [ 1 -1 ] |
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178 @end example |
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179 |
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180 @noindent |
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181 the @samp{-} is treated as a unary operator and the result is the |
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182 vector @code{[ 1, -1 ]}. Similarly, the expression |
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183 |
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184 @example |
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185 [ sin (pi) ] |
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186 @end example |
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187 |
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188 @noindent |
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189 will be parsed as |
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190 |
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191 @example |
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192 [ sin, (pi) ] |
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193 @end example |
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194 |
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195 @noindent |
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196 and will result in an error since the @code{sin} function will be |
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197 called with no arguments. To get around this, you must omit the space |
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198 between @code{sin} and the opening parenthesis, or enclose the |
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199 expression in a set of parentheses: |
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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 Whitespace surrounding the single quote character (@samp{'}, used as a |
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206 transpose operator and for delimiting character strings) can also cause |
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207 confusion. Given @code{a = 1}, the expression |
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208 |
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209 @example |
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210 [ 1 a' ] |
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211 @end example |
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212 |
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213 @noindent |
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214 results in the single quote character being treated as a |
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215 transpose operator and the result is the vector @code{[ 1, 1 ]}, but the |
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216 expression |
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217 |
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218 @example |
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219 [ 1 a ' ] |
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220 @end example |
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221 |
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222 @noindent |
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223 produces the error message |
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224 |
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225 @example |
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226 error: unterminated string constant |
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227 @end example |
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228 |
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229 @noindent |
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230 because to not do so would cause trouble when parsing the valid expression |
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231 |
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232 @example |
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233 [ a 'foo' ] |
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234 @end example |
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235 |
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236 For clarity, it is probably best to always use commas and semicolons to |
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237 separate matrix elements and rows. |
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238 |
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239 @DOCSTRING(warn_separator_insert) |
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240 |
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241 When you type a matrix or the name of a variable whose value is a |
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242 matrix, Octave responds by printing the matrix in with neatly aligned |
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243 rows and columns. If the rows of the matrix are too large to fit on the |
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244 screen, Octave splits the matrix and displays a header before each |
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245 section to indicate which columns are being displayed. You can use the |
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246 following variables to control the format of the output. |
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247 |
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248 @DOCSTRING(output_max_field_width) |
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249 |
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250 @DOCSTRING(output_precision) |
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251 |
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252 It is possible to achieve a wide range of output styles by using |
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253 different values of @code{output_precision} and |
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254 @code{output_max_field_width}. Reasonable combinations can be set using |
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255 the @code{format} function. @xref{Basic Input and Output}. |
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256 |
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257 @DOCSTRING(split_long_rows) |
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258 |
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259 Octave automatically switches to scientific notation when values become |
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260 very large or very small. This guarantees that you will see several |
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261 significant figures for every value in a matrix. If you would prefer to |
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262 see all values in a matrix printed in a fixed point format, you can set |
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263 the built-in variable @code{fixed_point_format} to a nonzero value. But |
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264 doing so is not recommended, because it can produce output that can |
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265 easily be misinterpreted. |
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266 |
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267 @DOCSTRING(fixed_point_format) |
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268 |
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269 @menu |
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270 * Empty Matrices:: |
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271 @end menu |
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272 |
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273 @node Empty Matrices |
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274 @subsection Empty Matrices |
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275 |
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276 A matrix may have one or both dimensions zero, and operations on empty |
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277 matrices are handled as described by Carl de Boor in @cite{An Empty |
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278 Exercise}, SIGNUM, Volume 25, pages 2--6, 1990 and C. N. Nett and W. M. |
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279 Haddad, in @cite{A System-Theoretic Appropriate Realization of the Empty |
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280 Matrix Concept}, IEEE Transactions on Automatic Control, Volume 38, |
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281 Number 5, May 1993. |
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282 @iftex |
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283 @tex |
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284 Briefly, given a scalar $s$, an $m\times n$ matrix $M_{m\times n}$, |
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285 and an $m\times n$ empty matrix $[\,]_{m\times n}$ (with either one or |
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286 both dimensions equal to zero), the following are true: |
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287 $$ |
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288 \eqalign{% |
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289 s \cdot [\,]_{m\times n} = [\,]_{m\times n} \cdot s &= [\,]_{m\times n}\cr |
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290 [\,]_{m\times n} + [\,]_{m\times n} &= [\,]_{m\times n}\cr |
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291 [\,]_{0\times m} \cdot M_{m\times n} &= [\,]_{0\times n}\cr |
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292 M_{m\times n} \cdot [\,]_{n\times 0} &= [\,]_{m\times 0}\cr |
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293 [\,]_{m\times 0} \cdot [\,]_{0\times n} &= 0_{m\times n}} |
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294 $$ |
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295 @end tex |
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296 @end iftex |
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297 @ifinfo |
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298 Briefly, given a scalar @var{s}, an @var{m} by |
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299 @var{n} matrix @code{M(mxn)}, and an @var{m} by @var{n} empty matrix |
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300 @code{[](mxn)} (with either one or both dimensions equal to zero), the |
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301 following are true: |
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302 |
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303 @example |
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304 @group |
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305 s * [](mxn) = [](mxn) * s = [](mxn) |
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306 |
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307 [](mxn) + [](mxn) = [](mxn) |
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308 |
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309 [](0xm) * M(mxn) = [](0xn) |
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310 |
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311 M(mxn) * [](nx0) = [](mx0) |
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312 |
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313 [](mx0) * [](0xn) = 0(mxn) |
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314 @end group |
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315 @end example |
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316 @end ifinfo |
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317 |
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318 By default, dimensions of the empty matrix are printed along with the |
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319 empty matrix symbol, @samp{[]}. The built-in variable |
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320 @code{print_empty_dimensions} controls this behavior. |
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321 |
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322 @DOCSTRING(print_empty_dimensions) |
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323 |
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324 Empty matrices may also be used in assignment statements as a convenient |
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325 way to delete rows or columns of matrices. |
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326 @xref{Assignment Ops, ,Assignment Expressions}. |
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327 |
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328 @DOCSTRING(warn_empty_list_elements) |
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329 |
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330 When Octave parses a matrix expression, it examines the elements of the |
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331 list to determine whether they are all constants. If they are, it |
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332 replaces the list with a single matrix constant. |
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333 |
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334 @node Ranges |
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335 @section Ranges |
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336 @cindex range expressions |
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337 @cindex expression, range |
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338 |
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339 @opindex colon |
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340 |
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341 A @dfn{range} is a convenient way to write a row vector with evenly |
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342 spaced elements. A range expression is defined by the value of the first |
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343 element in the range, an optional value for the increment between |
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344 elements, and a maximum value which the elements of the range will not |
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345 exceed. The base, increment, and limit are separated by colons (the |
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346 @samp{:} character) and may contain any arithmetic expressions and |
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347 function calls. If the increment is omitted, it is assumed to be 1. |
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348 For example, the range |
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349 |
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350 @example |
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351 1 : 5 |
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352 @end example |
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353 |
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354 @noindent |
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355 defines the set of values @samp{[ 1, 2, 3, 4, 5 ]}, and the range |
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356 |
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357 @example |
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358 1 : 3 : 5 |
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359 @end example |
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360 |
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361 @noindent |
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362 defines the set of values @samp{[ 1, 4 ]}. |
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363 |
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364 Although a range constant specifies a row vector, Octave does @emph{not} |
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365 convert range constants to vectors unless it is necessary to do so. |
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366 This allows you to write a constant like @samp{1 : 10000} without using |
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367 80,000 bytes of storage on a typical 32-bit workstation. |
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368 |
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369 Note that the upper (or lower, if the increment is negative) bound on |
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370 the range is not always included in the set of values, and that ranges |
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371 defined by floating point values can produce surprising results because |
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372 Octave uses floating point arithmetic to compute the values in the |
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373 range. If it is important to include the endpoints of a range and the |
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374 number of elements is known, you should use the @code{linspace} function |
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375 instead (@pxref{Special Utility Matrices}). |
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376 |
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377 When Octave parses a range expression, it examines the elements of the |
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378 expression to determine whether they are all constants. If they are, it |
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379 replaces the range expression with a single range constant. |
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380 |
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381 @node Logical Values |
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382 @section Logical Values |
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383 |
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384 @DOCSTRING(true) |
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385 |
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386 @DOCSTRING(false) |
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387 |
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388 @node Predicates for Numeric Objects |
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389 @section Predicates for Numeric Objects |
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390 |
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391 @DOCSTRING(isnumeric) |
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392 |
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393 @DOCSTRING(isreal) |
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394 |
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395 @DOCSTRING(iscomplex) |
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396 |
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397 @DOCSTRING(ismatrix) |
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398 |
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399 @DOCSTRING(isvector) |
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400 |
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401 @DOCSTRING(isscalar) |
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402 |
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403 @DOCSTRING(issquare) |
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404 |
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405 @DOCSTRING(issymmetric) |
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406 |
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407 @DOCSTRING(isbool) |
