Mercurial > octave
annotate doc/interpreter/numbers.txi @ 11179:6ead75935ebf
Explain that ranges in vectors do get expanded
| author | Jordi Gutiérrez Hermoso <jordigh@gmail.com> |
|---|---|
| date | Mon, 30 Aug 2010 13:49:07 -0500 |
| parents | a4f482e66b65 |
| children | 567ca09a97aa |
| rev | line source |
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| 8920 | 1 @c Copyright (C) 1996, 1997, 2007, 2008, 2009 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 3294 | 18 |
| 4167 | 19 @node Numeric Data Types |
| 3294 | 20 @chapter Numeric Data Types |
| 21 @cindex numeric constant | |
| 22 @cindex numeric value | |
| 23 | |
| 24 A @dfn{numeric constant} may be a scalar, a vector, or a matrix, and it | |
| 25 may contain complex values. | |
| 26 | |
| 27 The simplest form of a numeric constant, a scalar, is a single number | |
| 28 that can be an integer, a decimal fraction, a number in scientific | |
| 6620 | 29 (exponential) notation, or a complex number. Note that by default numeric |
| 3294 | 30 constants are represented within Octave in double-precision floating |
| 31 point format (complex constants are stored as pairs of double-precision | |
| 6620 | 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 |
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34 examples of real-valued numeric constants, which all have the same |
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35 value: |
| 3294 | 36 |
| 37 @example | |
| 38 @group | |
| 39 105 | |
| 40 1.05e+2 | |
| 41 1050e-1 | |
| 42 @end group | |
| 43 @end example | |
| 44 | |
| 45 To specify complex constants, you can write an expression of the form | |
| 46 | |
| 47 @example | |
| 48 @group | |
| 49 3 + 4i | |
| 50 3.0 + 4.0i | |
| 51 0.3e1 + 40e-1i | |
| 52 @end group | |
| 53 @end example | |
| 54 | |
| 6620 | 55 @noindent |
| 3294 | 56 all of which are equivalent. The letter @samp{i} in the previous example |
| 57 stands for the pure imaginary constant, defined as | |
| 58 @tex | |
| 59 $\sqrt{-1}$. | |
| 60 @end tex | |
| 6620 | 61 @ifnottex |
| 3294 | 62 @code{sqrt (-1)}. |
| 6620 | 63 @end ifnottex |
| 3294 | 64 |
| 65 For Octave to recognize a value as the imaginary part of a complex | |
| 66 constant, a space must not appear between the number and the @samp{i}. | |
| 67 If it does, Octave will print an error message, like this: | |
| 68 | |
| 69 @example | |
| 70 @group | |
| 71 octave:13> 3 + 4 i | |
| 72 | |
| 73 parse error: | |
| 74 | |
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75 syntax error |
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76 |
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77 >>> 3 + 4 i |
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78 ^ |
| 3294 | 79 @end group |
| 80 @end example | |
| 81 | |
| 6620 | 82 @noindent |
| 3294 | 83 You may also use @samp{j}, @samp{I}, or @samp{J} in place of the |
| 84 @samp{i} above. All four forms are equivalent. | |
| 85 | |
| 6549 | 86 @DOCSTRING(double) |
| 87 | |
| 88 @DOCSTRING(complex) | |
| 89 | |
| 3294 | 90 @menu |
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91 * Matrices:: |
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92 * Ranges:: |
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93 * Single Precision Data Types:: |
| 6549 | 94 * Integer Data Types:: |
| 6781 | 95 * Bit Manipulations:: |
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96 * Logical Values:: |
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97 * Promotion and Demotion of Data Types:: |
| 3294 | 98 * Predicates for Numeric Objects:: |
| 99 @end menu | |
| 100 | |
| 4167 | 101 @node Matrices |
| 3294 | 102 @section Matrices |
| 103 @cindex matrices | |
| 104 | |
| 105 @opindex [ | |
| 106 @opindex ] | |
| 107 @opindex ; | |
| 108 @opindex , | |
| 109 | |
| 110 It is easy to define a matrix of values in Octave. The size of the | |
| 111 matrix is determined automatically, so it is not necessary to explicitly | |
| 112 state the dimensions. The expression | |
| 113 | |
| 114 @example | |
| 115 a = [1, 2; 3, 4] | |
| 116 @end example | |
| 117 | |
| 118 @noindent | |
| 119 results in the matrix | |
| 120 @tex | |
| 121 $$ a = \left[ \matrix{ 1 & 2 \cr 3 & 4 } \right] $$ | |
| 122 @end tex | |
| 6620 | 123 @ifnottex |
| 3294 | 124 |
| 125 @example | |
| 126 @group | |
| 127 | |
| 128 / \ | |
| 129 | 1 2 | | |
| 130 a = | | | |
| 131 | 3 4 | | |
| 132 \ / | |
| 133 | |
| 134 @end group | |
| 135 @end example | |
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136 |
| 6620 | 137 @end ifnottex |
| 3294 | 138 |
| 139 Elements of a matrix may be arbitrary expressions, provided that the | |
| 140 dimensions all make sense when combining the various pieces. For | |
| 141 example, given the above matrix, the expression | |
| 142 | |
| 143 @example | |
| 144 [ a, a ] | |
| 145 @end example | |
| 146 | |
| 147 @noindent | |
| 148 produces the matrix | |
| 149 | |
| 150 @example | |
| 151 @group | |
| 152 ans = | |
| 153 | |
| 154 1 2 1 2 | |
| 155 3 4 3 4 | |
| 156 @end group | |
| 157 @end example | |
| 158 | |
| 159 @noindent | |
| 160 but the expression | |
| 161 | |
| 162 @example | |
| 163 [ a, 1 ] | |
| 164 @end example | |
| 165 | |
| 166 @noindent | |
| 167 produces the error | |
| 168 | |
| 169 @example | |
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170 error: number of rows must match (1 != 2) near line 13, column 6 |
| 3294 | 171 @end example |
| 172 | |
| 173 @noindent | |
| 174 (assuming that this expression was entered as the first thing on line | |
| 175 13, of course). | |
| 176 | |
| 177 Inside the square brackets that delimit a matrix expression, Octave | |
| 178 looks at the surrounding context to determine whether spaces and newline | |
| 179 characters should be converted into element and row separators, or | |
| 4476 | 180 simply ignored, so an expression like |
| 3294 | 181 |
| 182 @example | |
| 183 @group | |
| 184 a = [ 1 2 | |
| 185 3 4 ] | |
| 186 @end group | |
| 187 @end example | |
| 188 | |
| 189 @noindent | |
| 190 will work. However, some possible sources of confusion remain. For | |
| 191 example, in the expression | |
| 192 | |
| 193 @example | |
| 194 [ 1 - 1 ] | |
| 195 @end example | |
| 196 | |
| 197 @noindent | |
| 198 the @samp{-} is treated as a binary operator and the result is the | |
| 199 scalar 0, but in the expression | |
| 200 | |
| 201 @example | |
| 202 [ 1 -1 ] | |
| 203 @end example | |
| 204 | |
| 205 @noindent | |
| 206 the @samp{-} is treated as a unary operator and the result is the | |
| 4476 | 207 vector @code{[ 1, -1 ]}. Similarly, the expression |
| 208 | |
| 209 @example | |
| 210 [ sin (pi) ] | |
| 211 @end example | |
| 212 | |
| 213 @noindent | |
| 214 will be parsed as | |
| 215 | |
| 216 @example | |
| 217 [ sin, (pi) ] | |
| 218 @end example | |
| 3294 | 219 |
| 4476 | 220 @noindent |
| 221 and will result in an error since the @code{sin} function will be | |
| 222 called with no arguments. To get around this, you must omit the space | |
| 223 between @code{sin} and the opening parenthesis, or enclose the | |
| 224 expression in a set of parentheses: | |
| 225 | |
| 226 @example | |
| 227 [ (sin (pi)) ] | |
| 228 @end example | |
| 229 | |
| 230 Whitespace surrounding the single quote character (@samp{'}, used as a | |
| 231 transpose operator and for delimiting character strings) can also cause | |
| 232 confusion. Given @code{a = 1}, the expression | |
| 3294 | 233 |
| 234 @example | |
| 235 [ 1 a' ] | |
| 236 @end example | |
| 237 | |
| 238 @noindent | |
| 4476 | 239 results in the single quote character being treated as a |
| 3294 | 240 transpose operator and the result is the vector @code{[ 1, 1 ]}, but the |
| 241 expression | |
| 242 | |
| 243 @example | |
| 244 [ 1 a ' ] | |
| 245 @end example | |
| 246 | |
| 247 @noindent | |
| 248 produces the error message | |
| 249 | |
| 250 @example | |
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251 @group |
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252 parse error: |
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253 |
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254 syntax error |
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255 |
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256 >>> [ 1 a ' ] |
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257 ^ |
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258 @end group |
| 3294 | 259 @end example |
| 260 | |
| 261 @noindent | |
| 6620 | 262 because not doing so would cause trouble when parsing the valid expression |
| 3294 | 263 |
| 264 @example | |
| 265 [ a 'foo' ] | |
| 266 @end example | |
| 267 | |
| 268 For clarity, it is probably best to always use commas and semicolons to | |
| 4476 | 269 separate matrix elements and rows. |
| 3294 | 270 |
| 271 When you type a matrix or the name of a variable whose value is a | |
| 272 matrix, Octave responds by printing the matrix in with neatly aligned | |
| 273 rows and columns. If the rows of the matrix are too large to fit on the | |
| 274 screen, Octave splits the matrix and displays a header before each | |
| 275 section to indicate which columns are being displayed. You can use the | |
| 276 following variables to control the format of the output. | |
| 277 | |
| 3321 | 278 @DOCSTRING(output_max_field_width) |
| 3294 | 279 |
| 3321 | 280 @DOCSTRING(output_precision) |
| 3294 | 281 |
| 282 It is possible to achieve a wide range of output styles by using | |
| 283 different values of @code{output_precision} and | |
| 284 @code{output_max_field_width}. Reasonable combinations can be set using | |
| 285 the @code{format} function. @xref{Basic Input and Output}. | |
| 286 | |
| 3321 | 287 @DOCSTRING(split_long_rows) |
| 3294 | 288 |
| 289 Octave automatically switches to scientific notation when values become | |
| 290 very large or very small. This guarantees that you will see several | |
| 291 significant figures for every value in a matrix. If you would prefer to | |
| 292 see all values in a matrix printed in a fixed point format, you can set | |
| 293 the built-in variable @code{fixed_point_format} to a nonzero value. But | |
| 294 doing so is not recommended, because it can produce output that can | |
| 295 easily be misinterpreted. | |
| 296 | |
| 3321 | 297 @DOCSTRING(fixed_point_format) |
| 3294 | 298 |
| 299 @menu | |
| 300 * Empty Matrices:: | |
| 301 @end menu | |
| 302 | |
| 4167 | 303 @node Empty Matrices |
| 3294 | 304 @subsection Empty Matrices |
| 305 | |
| 306 A matrix may have one or both dimensions zero, and operations on empty | |
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307 matrices are handled as described by Carl @nospell{de} Boor in @cite{An Empty |
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308 Exercise}, SIGNUM, Volume 25, pages 2-6, 1990 and C. N. Nett and W. M. |
| 3294 | 309 Haddad, in @cite{A System-Theoretic Appropriate Realization of the Empty |
| 310 Matrix Concept}, IEEE Transactions on Automatic Control, Volume 38, | |
| 311 Number 5, May 1993. | |
| 312 @tex | |
| 313 Briefly, given a scalar $s$, an $m\times n$ matrix $M_{m\times n}$, | |
| 314 and an $m\times n$ empty matrix $[\,]_{m\times n}$ (with either one or | |
| 315 both dimensions equal to zero), the following are true: | |
| 316 $$ | |
| 317 \eqalign{% | |
| 318 s \cdot [\,]_{m\times n} = [\,]_{m\times n} \cdot s &= [\,]_{m\times n}\cr | |
| 319 [\,]_{m\times n} + [\,]_{m\times n} &= [\,]_{m\times n}\cr | |
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320 [\,]_{0\times m} \cdot M_{m\times n} &= [\,]_{0\times n}\cr |
| 3294 | 321 M_{m\times n} \cdot [\,]_{n\times 0} &= [\,]_{m\times 0}\cr |
| 322 [\,]_{m\times 0} \cdot [\,]_{0\times n} &= 0_{m\times n}} | |
| 323 $$ | |
| 324 @end tex | |
| 6620 | 325 @ifnottex |
| 3294 | 326 Briefly, given a scalar @var{s}, an @var{m} by |
| 327 @var{n} matrix @code{M(mxn)}, and an @var{m} by @var{n} empty matrix | |
| 328 @code{[](mxn)} (with either one or both dimensions equal to zero), the | |
| 329 following are true: | |
| 330 | |
| 331 @example | |
| 332 @group | |
| 333 s * [](mxn) = [](mxn) * s = [](mxn) | |
| 334 | |
| 335 [](mxn) + [](mxn) = [](mxn) | |
| 336 | |
| 337 [](0xm) * M(mxn) = [](0xn) | |
| 338 | |
| 339 M(mxn) * [](nx0) = [](mx0) | |
| 340 | |
| 341 [](mx0) * [](0xn) = 0(mxn) | |
| 342 @end group | |
| 343 @end example | |
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344 |
| 6620 | 345 @end ifnottex |
| 3294 | 346 |
| 347 By default, dimensions of the empty matrix are printed along with the | |
| 348 empty matrix symbol, @samp{[]}. The built-in variable | |
| 349 @code{print_empty_dimensions} controls this behavior. | |
| 350 | |
| 3321 | 351 @DOCSTRING(print_empty_dimensions) |
| 3294 | 352 |
| 353 Empty matrices may also be used in assignment statements as a convenient | |
| 354 way to delete rows or columns of matrices. | |
| 355 @xref{Assignment Ops, ,Assignment Expressions}. | |
| 356 | |
| 357 When Octave parses a matrix expression, it examines the elements of the | |
| 358 list to determine whether they are all constants. If they are, it | |
| 359 replaces the list with a single matrix constant. | |
| 360 | |
| 4167 | 361 @node Ranges |
| 3294 | 362 @section Ranges |
| 363 @cindex range expressions | |
| 364 @cindex expression, range | |
| 365 | |
| 3920 | 366 @opindex colon |
| 3294 | 367 |
| 368 A @dfn{range} is a convenient way to write a row vector with evenly | |
| 369 spaced elements. A range expression is defined by the value of the first | |
| 370 element in the range, an optional value for the increment between | |
| 371 elements, and a maximum value which the elements of the range will not | |
| 372 exceed. The base, increment, and limit are separated by colons (the | |
| 373 @samp{:} character) and may contain any arithmetic expressions and | |
| 374 function calls. If the increment is omitted, it is assumed to be 1. | |
| 375 For example, the range | |
| 376 | |
| 377 @example | |
| 378 1 : 5 | |
| 379 @end example | |
| 380 | |
| 381 @noindent | |
| 382 defines the set of values @samp{[ 1, 2, 3, 4, 5 ]}, and the range | |
| 383 | |
| 384 @example | |
| 385 1 : 3 : 5 | |
| 386 @end example | |
| 387 | |
| 388 @noindent | |
| 389 defines the set of values @samp{[ 1, 4 ]}. | |
| 390 | |
| 391 Although a range constant specifies a row vector, Octave does @emph{not} | |
| 392 convert range constants to vectors unless it is necessary to do so. | |
| 393 This allows you to write a constant like @samp{1 : 10000} without using | |
| 394 80,000 bytes of storage on a typical 32-bit workstation. | |
| 395 | |
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396 A common example of when it does become necessary to convert ranges into |
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397 vectors occurs when they appear within a vector (i.e. inside square |
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398 brackets). For instance, whereas |
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399 |
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400 @example |
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401 x = 0 : 0.1 : 1; |
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402 @end example |
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403 |
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404 defines @var{x} to be a variable of type @code{range} and occupies 24 |
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405 bytes of memory, the expression |
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406 |
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407 @example |
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408 y = [ 0 : 0.1 : 1]; |
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409 @end example |
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410 |
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411 defines @var{y} to be of type @code{matrix} and occupies 88 bytes of |
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412 memory. |
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413 |
| 3294 | 414 Note that the upper (or lower, if the increment is negative) bound on |
| 415 the range is not always included in the set of values, and that ranges | |
| 416 defined by floating point values can produce surprising results because | |
| 417 Octave uses floating point arithmetic to compute the values in the | |
| 418 range. If it is important to include the endpoints of a range and the | |
| 419 number of elements is known, you should use the @code{linspace} function | |
| 420 instead (@pxref{Special Utility Matrices}). | |
| 421 | |
| 9158 | 422 When adding a scalar to a range, subtracting a scalar from it (or subtracting a |
| 423 range from a scalar) and multiplying by scalar, Octave will attempt to avoid | |
| 424 unpacking the range and keep the result as a range, too, if it can determine | |
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425 that it is safe to do so. For instance, doing |
| 9158 | 426 |
| 427 @example | |
| 428 a = 2*(1:1e7) - 1; | |
| 429 @end example | |
| 430 | |
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431 @noindent |
| 9158 | 432 will produce the same result as @samp{1:2:2e7-1}, but without ever forming a |
| 433 vector with ten million elements. | |
| 434 | |
| 435 Using zero as an increment in the colon notation, as @samp{1:0:1} is not | |
| 436 allowed, because a division by zero would occur in determining the number of | |
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437 range elements. However, ranges with zero increment (i.e., all elements equal) |
| 9158 | 438 are useful, especially in indexing, and Octave allows them to be constructed |
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439 using the built-in function @dfn{ones}. Note that because a range must be a row |
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440 vector, @samp{ones (1, 10)} produces a range, while @samp{ones (10, 1)} does |
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441 not. |
| 9158 | 442 |
| 3294 | 443 When Octave parses a range expression, it examines the elements of the |
| 444 expression to determine whether they are all constants. If they are, it | |
| 445 replaces the range expression with a single range constant. | |
| 446 | |
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447 @node Single Precision Data Types |
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448 @section Single Precision Data Types |
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449 |
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450 Octave includes support for single precision data types, and most of the |
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451 functions in Octave accept single precision values and return single |
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452 precision answers. A single precision variable is created with the |
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453 @code{single} function. |
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454 |
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455 @DOCSTRING(single) |
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456 |
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457 for example: |
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458 |
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459 @example |
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460 @group |
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461 sngl = single (rand (2, 2)) |
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462 @result{} sngl = |
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463 0.37569 0.92982 |
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464 0.11962 0.50876 |
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465 class (sngl) |
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466 @result{} single |
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467 @end group |
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468 @end example |
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469 |
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470 Many functions can also return single precision values directly. For |
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471 example |
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472 |
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473 @example |
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474 @group |
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475 ones (2, 2, "single") |
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476 zeros (2, 2, "single") |
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477 eye (2, 2, "single") |
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478 rand (2, 2, "single") |
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479 NaN (2, 2, "single") |
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480 NA (2, 2, "single") |
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481 Inf (2, 2, "single") |
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482 @end group |
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483 @end example |
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484 |
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485 @noindent |
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486 will all return single precision matrices. |
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487 |
| 6549 | 488 @node Integer Data Types |
| 489 @section Integer Data Types | |
| 490 | |
| 6620 | 491 Octave supports integer matrices as an alternative to using double |
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492 precision. It is possible to use both signed and unsigned integers |
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493 represented by 8, 16, 32, or 64 bits. It should be noted that most |
| 6620 | 494 computations require floating point data, meaning that integers will |
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495 often change type when involved in numeric computations. For this |
| 6620 | 496 reason integers are most often used to store data, and not for |
| 497 calculations. | |
| 498 | |
| 499 In general most integer matrices are created by casting | |
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500 existing matrices to integers. The following example shows how to cast |
| 6620 | 501 a matrix into 32 bit integers. |
| 502 | |
| 503 @example | |
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504 @group |
| 6620 | 505 float = rand (2, 2) |
| 506 @result{} float = 0.37569 0.92982 | |
| 507 0.11962 0.50876 | |
| 508 integer = int32 (float) | |
| 509 @result{} integer = 0 1 | |
| 510 0 1 | |
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511 @end group |
| 6620 | 512 @end example |
| 513 | |
| 514 @noindent | |
| 515 As can be seen, floating point values are rounded to the nearest integer | |
| 516 when converted. | |
| 517 | |
| 6549 | 518 @DOCSTRING(isinteger) |
| 519 | |
| 520 @DOCSTRING(int8) | |
| 521 | |
| 522 @DOCSTRING(uint8) | |
| 523 | |
| 524 @DOCSTRING(int16) | |
| 525 | |
| 526 @DOCSTRING(uint16) | |
| 527 | |
| 528 @DOCSTRING(int32) | |
| 529 | |
| 530 @DOCSTRING(uint32) | |
| 531 | |
| 532 @DOCSTRING(int64) | |
| 533 | |
| 534 @DOCSTRING(uint64) | |
| 535 | |
| 536 @DOCSTRING(intmax) | |
| 537 | |
| 538 @DOCSTRING(intmin) | |
| 539 | |
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540 @DOCSTRING(intwarning) |
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541 |
| 6624 | 542 @menu |
| 543 * Integer Arithmetic:: | |
| 544 @end menu | |
| 545 | |
| 6620 | 546 @node Integer Arithmetic |
| 547 @subsection Integer Arithmetic | |
| 548 | |
| 549 While many numerical computations can't be carried out in integers, | |
| 550 Octave does support basic operations like addition and multiplication | |
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551 on integers. The operators @code{+}, @code{-}, @code{.*}, and @code{./} |
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552 work on integers of the same type. So, it is possible to add two 32 bit |
| 6620 | 553 integers, but not to add a 32 bit integer and a 16 bit integer. |
| 554 | |
| 6715 | 555 The arithmetic operations on integers are performed by casting the |
| 556 integer values to double precision values, performing the operation, and | |
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557 then re-casting the values back to the original integer type. As the |
| 6715 | 558 double precision type of Octave is only capable of representing integers |
| 559 with up to 53 bits of precision, it is not possible to perform | |
| 8828 | 560 arithmetic with 64 bit integer types. |
| 6715 | 561 |
| 6620 | 562 When doing integer arithmetic one should consider the possibility of |
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563 underflow and overflow. This happens when the result of the computation |
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564 can't be represented using the chosen integer type. As an example it is |
| 6620 | 565 not possible to represent the result of @math{10 - 20} when using |
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566 unsigned integers. Octave makes sure that the result of integer |
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567 computations is the integer that is closest to the true result. So, the |
| 6620 | 568 result of @math{10 - 20} when using unsigned integers is zero. |
| 569 | |
| 570 When doing integer division Octave will round the result to the nearest | |
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571 integer. This is different from most programming languages, where the |
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572 result is often floored to the nearest integer. So, the result of |
| 6620 | 573 @code{int32(5)./int32(8)} is @code{1}. |
| 574 | |
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575 @DOCSTRING(idivide) |
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576 |
| 6781 | 577 @node Bit Manipulations |
| 578 @section Bit Manipulations | |
| 579 | |
| 580 Octave provides a number of functions for the manipulation of numeric | |
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581 values on a bit by bit basis. The basic functions to set and obtain the |
| 6781 | 582 values of individual bits are @code{bitset} and @code{bitget}. |
| 583 | |
| 584 @DOCSTRING(bitset) | |
| 585 | |
| 586 @DOCSTRING(bitget) | |
| 587 | |
| 588 The arguments to all of Octave's bitwise operations can be scalar or | |
| 589 arrays, except for @code{bitcmp}, whose @var{k} argument must a | |
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590 scalar. In the case where more than one argument is an array, then all |
| 6781 | 591 arguments must have the same shape, and the bitwise operator is applied |
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592 to each of the elements of the argument individually. If at least one |
| 6781 | 593 argument is a scalar and one an array, then the scalar argument is |
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594 duplicated. Therefore |
| 6781 | 595 |
| 596 @example | |
| 597 bitget (100, 8:-1:1) | |
| 598 @end example | |
| 599 | |
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600 @noindent |
| 6781 | 601 is the same as |
| 602 | |
| 603 @example | |
| 604 bitget (100 * ones (1, 8), 8:-1:1) | |
| 605 @end example | |
| 606 | |
| 607 It should be noted that all values passed to the bit manipulation | |
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608 functions of Octave are treated as integers. Therefore, even though the |
| 6781 | 609 example for @code{bitset} above passes the floating point value |
| 610 @code{10}, it is treated as the bits @code{[1, 0, 1, 0]} rather than the | |
| 611 bits of the native floating point format representation of @code{10}. | |
| 612 | |
| 8828 | 613 As the maximum value that can be represented by a number is important |
| 6781 | 614 for bit manipulation, particularly when forming masks, Octave supplies |
| 615 the function @code{bitmax}. | |
| 616 | |
| 617 @DOCSTRING(bitmax) | |
| 618 | |
| 619 This is the double precision version of the functions @code{intmax}, | |
| 620 previously discussed. | |
| 621 | |
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622 Octave also includes the basic bitwise 'and', 'or' and 'exclusive or' |
| 6781 | 623 operators. |
| 624 | |
| 625 @DOCSTRING(bitand) | |
| 626 | |
| 627 @DOCSTRING(bitor) | |
| 628 | |
| 629 @DOCSTRING(bitxor) | |
| 630 | |
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631 The bitwise 'not' operator is a unary operator that performs a logical |
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632 negation of each of the bits of the value. For this to make sense, the |
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633 mask against which the value is negated must be defined. Octave's |
| 6781 | 634 bitwise 'not' operator is @code{bitcmp}. |
| 635 | |
| 636 @DOCSTRING(bitcmp) | |
| 637 | |
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638 Octave also includes the ability to left-shift and right-shift values bitwise. |
| 6781 | 639 |
| 640 @DOCSTRING(bitshift) | |
| 641 | |
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642 Bits that are shifted out of either end of the value are lost. Octave |
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643 also uses arithmetic shifts, where the sign bit of the value is kept |
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644 during a right shift. For example: |
| 6781 | 645 |
| 646 @example | |
| 647 @group | |
| 648 bitshift (-10, -1) | |
| 649 @result{} -5 | |
| 650 bitshift (int8 (-1), -1) | |
| 651 @result{} -1 | |
| 652 @end group | |
| 653 @end example | |
| 654 | |
| 655 Note that @code{bitshift (int8 (-1), -1)} is @code{-1} since the bit | |
| 656 representation of @code{-1} in the @code{int8} data type is @code{[1, 1, | |
| 657 1, 1, 1, 1, 1, 1]}. | |
| 658 | |
| 4167 | 659 @node Logical Values |
| 3428 | 660 @section Logical Values |
| 661 | |
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662 Octave has built-in support for logical values, i.e., variables that |
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663 are either @code{true} or @code{false}. When comparing two variables, |
| 6620 | 664 the result will be a logical value whose value depends on whether or |
| 665 not the comparison is true. | |
| 666 | |
| 667 The basic logical operations are @code{&}, @code{|}, and @code{!}, | |
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668 which correspond to ``Logical And'', ``Logical Or'', and ``Logical |
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669 Negation''. These operations all follow the usual rules of logic. |
| 6620 | 670 |
| 671 It is also possible to use logical values as part of standard numerical | |
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672 calculations. In this case @code{true} is converted to @code{1}, and |
| 6620 | 673 @code{false} to 0, both represented using double precision floating |
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674 point numbers. So, the result of @code{true*22 - false/6} is @code{22}. |
| 6620 | 675 |
| 676 Logical values can also be used to index matrices and cell arrays. | |
| 677 When indexing with a logical array the result will be a vector containing | |
| 678 the values corresponding to @code{true} parts of the logical array. | |
| 679 The following example illustrates this. | |
| 680 | |
| 681 @example | |
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682 @group |
| 6620 | 683 data = [ 1, 2; 3, 4 ]; |
| 684 idx = (data <= 2); | |
| 685 data(idx) | |
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686 @result{} ans = [ 1; 2 ] |
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687 @end group |
| 6620 | 688 @end example |
| 689 | |
| 690 @noindent | |
| 691 Instead of creating the @code{idx} array it is possible to replace | |
| 692 @code{data(idx)} with @code{data( data <= 2 )} in the above code. | |
| 693 | |
| 8828 | 694 Logical values can also be constructed by |
| 6620 | 695 casting numeric objects to logical values, or by using the @code{true} |
| 696 or @code{false} functions. | |
| 697 | |
| 6549 | 698 @DOCSTRING(logical) |
| 699 | |
| 3428 | 700 @DOCSTRING(true) |
| 701 | |
| 702 @DOCSTRING(false) | |
| 703 | |
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704 @node Promotion and Demotion of Data Types |
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705 @section Promotion and Demotion of Data Types |
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706 |
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707 Many operators and functions can work with mixed data types. For example, |
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708 |
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709 @example |
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710 @group |
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711 uint8 (1) + 1 |
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712 @result{} 2 |
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713 @end group |
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714 @end example |
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715 |
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716 @noindent |
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717 where the above operator works with an 8-bit integer and a double precision |
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718 value and returns an 8-bit integer value. Note that the type is demoted |
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719 to an 8-bit integer, rather than promoted to a double precision value as |
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720 might be expected. The reason is that if Octave promoted values in |
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721 expressions like the above with all numerical constants would need to be |
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722 explicitly cast to the appropriate data type like |
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723 |
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724 @example |
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725 @group |
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726 uint8 (1) + uint8 (1) |
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727 @result{} 2 |
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728 @end group |
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729 @end example |
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730 |
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731 @noindent |
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732 which becomes difficult for the user to apply uniformly and might allow |
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733 hard to find bugs to be introduced. The same applies to single precision |
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734 values where a mixed operation such as |
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735 |
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736 @example |
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737 @group |
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738 single (1) + 1 |
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739 @result{} 2 |
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740 @end group |
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741 @end example |
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742 |
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743 @noindent |
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744 returns a single precision value. The mixed operations that are valid |
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745 and their returned data types are |
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746 |
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747 @multitable @columnfractions .2 .3 .3 .2 |
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748 @item @tab Mixed Operation @tab Result @tab |
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749 @item @tab double OP single @tab single @tab |
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750 @item @tab double OP integer @tab integer @tab |
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751 @item @tab double OP char @tab double @tab |
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752 @item @tab double OP logical @tab double @tab |
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753 @item @tab single OP integer @tab integer @tab |
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754 @item @tab single OP char @tab single @tab |
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755 @item @tab single OP logical @tab single @tab |
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756 @end multitable |
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757 |
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758 The same logic applies to functions with mixed arguments such as |
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759 |
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760 @example |
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761 @group |
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762 min (single (1), 0) |
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763 @result{} 0 |
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764 @end group |
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765 @end example |
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766 |
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767 @noindent |
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768 where the returned value is single precision. |
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769 |
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770 In the case of mixed type indexed assignments, the type is not |
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771 changed. For example, |
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772 |
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773 @example |
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774 @group |
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775 x = ones (2, 2); |
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776 x (1, 1) = single (2) |
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777 @result{} x = 2 1 |
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778 1 1 |
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779 @end group |
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780 @end example |
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781 |
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782 @noindent |
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783 where @code{x} remains of the double precision type. |
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784 |
| 4167 | 785 @node Predicates for Numeric Objects |
| 3294 | 786 @section Predicates for Numeric Objects |
| 787 | |
| 6620 | 788 Since the type of a variable may change during the execution of a |
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789 program, it can be necessary to do type checking at run-time. Doing this |
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790 also allows you to change the behavior of a function depending on the |
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791 type of the input. As an example, this naive implementation of @code{abs} |
| 7001 | 792 returns the absolute value of the input if it is a real number, and the |
| 6620 | 793 length of the input if it is a complex number. |
| 794 | |
| 795 @example | |
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796 @group |
| 6620 | 797 function a = abs (x) |
| 798 if (isreal (x)) | |
| 799 a = sign (x) .* x; | |
| 800 elseif (iscomplex (x)) | |
| 801 a = sqrt (real(x).^2 + imag(x).^2); | |
| 802 endif | |
| 803 endfunction | |
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804 @end group |
| 6620 | 805 @end example |
| 806 | |
| 807 The following functions are available for determining the type of a | |
| 808 variable. | |
| 809 | |
| 3428 | 810 @DOCSTRING(isnumeric) |
| 811 | |
| 812 @DOCSTRING(isreal) | |
| 813 | |
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814 @DOCSTRING(isfloat) |
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815 |
| 4029 | 816 @DOCSTRING(iscomplex) |
| 3428 | 817 |
| 4029 | 818 @DOCSTRING(ismatrix) |
| 3294 | 819 |
| 4029 | 820 @DOCSTRING(isvector) |
| 3294 | 821 |
| 4029 | 822 @DOCSTRING(isscalar) |
| 3294 | 823 |
| 4029 | 824 @DOCSTRING(issquare) |
| 3294 | 825 |
| 4029 | 826 @DOCSTRING(issymmetric) |
| 3428 | 827 |
| 6550 | 828 @DOCSTRING(isdefinite) |
| 829 | |
| 6688 | 830 @DOCSTRING(islogical) |
| 6550 | 831 |
| 832 @DOCSTRING(isprime) |