Mercurial > octave
annotate doc/interpreter/numbers.txi @ 9158:48ee8c73ff38
info about ranges
| author | Jaroslav Hajek <highegg@gmail.com> |
|---|---|
| date | Wed, 29 Apr 2009 07:44:34 +0200 |
| parents | 537a73861cb7 |
| children | 923c7cb7f13f |
| 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 @iftex | |
| 59 @tex | |
| 60 $\sqrt{-1}$. | |
| 61 @end tex | |
| 62 @end iftex | |
| 6620 | 63 @ifnottex |
| 3294 | 64 @code{sqrt (-1)}. |
| 6620 | 65 @end ifnottex |
| 3294 | 66 |
| 67 For Octave to recognize a value as the imaginary part of a complex | |
| 68 constant, a space must not appear between the number and the @samp{i}. | |
| 69 If it does, Octave will print an error message, like this: | |
| 70 | |
| 71 @example | |
| 72 @group | |
| 73 octave:13> 3 + 4 i | |
| 74 | |
| 75 parse error: | |
| 76 | |
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77 syntax error |
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78 |
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79 >>> 3 + 4 i |
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80 ^ |
| 3294 | 81 @end group |
| 82 @end example | |
| 83 | |
| 6620 | 84 @noindent |
| 3294 | 85 You may also use @samp{j}, @samp{I}, or @samp{J} in place of the |
| 86 @samp{i} above. All four forms are equivalent. | |
| 87 | |
| 6549 | 88 @DOCSTRING(double) |
| 89 | |
| 90 @DOCSTRING(complex) | |
| 91 | |
| 3294 | 92 @menu |
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93 * Matrices:: |
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94 * Ranges:: |
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95 * Single Precision Data Types:: |
| 6549 | 96 * Integer Data Types:: |
| 6781 | 97 * Bit Manipulations:: |
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98 * Logical Values:: |
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99 * Promotion and Demotion of Data Types:: |
| 3294 | 100 * Predicates for Numeric Objects:: |
| 101 @end menu | |
| 102 | |
| 4167 | 103 @node Matrices |
| 3294 | 104 @section Matrices |
| 105 @cindex matrices | |
| 106 | |
| 107 @opindex [ | |
| 108 @opindex ] | |
| 109 @opindex ; | |
| 110 @opindex , | |
| 111 | |
| 112 It is easy to define a matrix of values in Octave. The size of the | |
| 113 matrix is determined automatically, so it is not necessary to explicitly | |
| 114 state the dimensions. The expression | |
| 115 | |
| 116 @example | |
| 117 a = [1, 2; 3, 4] | |
| 118 @end example | |
| 119 | |
| 120 @noindent | |
| 121 results in the matrix | |
| 122 @iftex | |
| 123 @tex | |
| 124 $$ a = \left[ \matrix{ 1 & 2 \cr 3 & 4 } \right] $$ | |
| 125 @end tex | |
| 126 @end iftex | |
| 6620 | 127 @ifnottex |
| 3294 | 128 |
| 129 @example | |
| 130 @group | |
| 131 | |
| 132 / \ | |
| 133 | 1 2 | | |
| 134 a = | | | |
| 135 | 3 4 | | |
| 136 \ / | |
| 137 | |
| 138 @end group | |
| 139 @end example | |
| 6620 | 140 @end ifnottex |
| 3294 | 141 |
| 142 Elements of a matrix may be arbitrary expressions, provided that the | |
| 143 dimensions all make sense when combining the various pieces. For | |
| 144 example, given the above matrix, the expression | |
| 145 | |
| 146 @example | |
| 147 [ a, a ] | |
| 148 @end example | |
| 149 | |
| 150 @noindent | |
| 151 produces the matrix | |
| 152 | |
| 153 @example | |
| 154 @group | |
| 155 ans = | |
| 156 | |
| 157 1 2 1 2 | |
| 158 3 4 3 4 | |
| 159 @end group | |
| 160 @end example | |
| 161 | |
| 162 @noindent | |
| 163 but the expression | |
| 164 | |
| 165 @example | |
| 166 [ a, 1 ] | |
| 167 @end example | |
| 168 | |
| 169 @noindent | |
| 170 produces the error | |
| 171 | |
| 172 @example | |
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173 error: number of rows must match (1 != 2) near line 13, column 6 |
| 3294 | 174 @end example |
| 175 | |
| 176 @noindent | |
| 177 (assuming that this expression was entered as the first thing on line | |
| 178 13, of course). | |
| 179 | |
| 180 Inside the square brackets that delimit a matrix expression, Octave | |
| 181 looks at the surrounding context to determine whether spaces and newline | |
| 182 characters should be converted into element and row separators, or | |
| 4476 | 183 simply ignored, so an expression like |
| 3294 | 184 |
| 185 @example | |
| 186 @group | |
| 187 a = [ 1 2 | |
| 188 3 4 ] | |
| 189 @end group | |
| 190 @end example | |
| 191 | |
| 192 @noindent | |
| 193 will work. However, some possible sources of confusion remain. For | |
| 194 example, in the expression | |
| 195 | |
| 196 @example | |
| 197 [ 1 - 1 ] | |
| 198 @end example | |
| 199 | |
| 200 @noindent | |
| 201 the @samp{-} is treated as a binary operator and the result is the | |
| 202 scalar 0, but in the expression | |
| 203 | |
| 204 @example | |
| 205 [ 1 -1 ] | |
| 206 @end example | |
| 207 | |
| 208 @noindent | |
| 209 the @samp{-} is treated as a unary operator and the result is the | |
| 4476 | 210 vector @code{[ 1, -1 ]}. Similarly, the expression |
| 211 | |
| 212 @example | |
| 213 [ sin (pi) ] | |
| 214 @end example | |
| 215 | |
| 216 @noindent | |
| 217 will be parsed as | |
| 218 | |
| 219 @example | |
| 220 [ sin, (pi) ] | |
| 221 @end example | |
| 3294 | 222 |
| 4476 | 223 @noindent |
| 224 and will result in an error since the @code{sin} function will be | |
| 225 called with no arguments. To get around this, you must omit the space | |
| 226 between @code{sin} and the opening parenthesis, or enclose the | |
| 227 expression in a set of parentheses: | |
| 228 | |
| 229 @example | |
| 230 [ (sin (pi)) ] | |
| 231 @end example | |
| 232 | |
| 233 Whitespace surrounding the single quote character (@samp{'}, used as a | |
| 234 transpose operator and for delimiting character strings) can also cause | |
| 235 confusion. Given @code{a = 1}, the expression | |
| 3294 | 236 |
| 237 @example | |
| 238 [ 1 a' ] | |
| 239 @end example | |
| 240 | |
| 241 @noindent | |
| 4476 | 242 results in the single quote character being treated as a |
| 3294 | 243 transpose operator and the result is the vector @code{[ 1, 1 ]}, but the |
| 244 expression | |
| 245 | |
| 246 @example | |
| 247 [ 1 a ' ] | |
| 248 @end example | |
| 249 | |
| 250 @noindent | |
| 251 produces the error message | |
| 252 | |
| 253 @example | |
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254 parse error: |
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255 |
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256 syntax error |
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257 |
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258 >>> [ 1 a ' ] |
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259 ^ |
| 3294 | 260 @end example |
| 261 | |
| 262 @noindent | |
| 6620 | 263 because not doing so would cause trouble when parsing the valid expression |
| 3294 | 264 |
| 265 @example | |
| 266 [ a 'foo' ] | |
| 267 @end example | |
| 268 | |
| 269 For clarity, it is probably best to always use commas and semicolons to | |
| 4476 | 270 separate matrix elements and rows. |
| 3294 | 271 |
| 272 When you type a matrix or the name of a variable whose value is a | |
| 273 matrix, Octave responds by printing the matrix in with neatly aligned | |
| 274 rows and columns. If the rows of the matrix are too large to fit on the | |
| 275 screen, Octave splits the matrix and displays a header before each | |
| 276 section to indicate which columns are being displayed. You can use the | |
| 277 following variables to control the format of the output. | |
| 278 | |
| 3321 | 279 @DOCSTRING(output_max_field_width) |
| 3294 | 280 |
| 3321 | 281 @DOCSTRING(output_precision) |
| 3294 | 282 |
| 283 It is possible to achieve a wide range of output styles by using | |
| 284 different values of @code{output_precision} and | |
| 285 @code{output_max_field_width}. Reasonable combinations can be set using | |
| 286 the @code{format} function. @xref{Basic Input and Output}. | |
| 287 | |
| 3321 | 288 @DOCSTRING(split_long_rows) |
| 3294 | 289 |
| 290 Octave automatically switches to scientific notation when values become | |
| 291 very large or very small. This guarantees that you will see several | |
| 292 significant figures for every value in a matrix. If you would prefer to | |
| 293 see all values in a matrix printed in a fixed point format, you can set | |
| 294 the built-in variable @code{fixed_point_format} to a nonzero value. But | |
| 295 doing so is not recommended, because it can produce output that can | |
| 296 easily be misinterpreted. | |
| 297 | |
| 3321 | 298 @DOCSTRING(fixed_point_format) |
| 3294 | 299 |
| 300 @menu | |
| 301 * Empty Matrices:: | |
| 302 @end menu | |
| 303 | |
| 4167 | 304 @node Empty Matrices |
| 3294 | 305 @subsection Empty Matrices |
| 306 | |
| 307 A matrix may have one or both dimensions zero, and operations on empty | |
| 308 matrices are handled as described by Carl de Boor in @cite{An Empty | |
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309 Exercise}, SIGNUM, Volume 25, pages 2-6, 1990 and C. N. Nett and W. M. |
| 3294 | 310 Haddad, in @cite{A System-Theoretic Appropriate Realization of the Empty |
| 311 Matrix Concept}, IEEE Transactions on Automatic Control, Volume 38, | |
| 312 Number 5, May 1993. | |
| 313 @iftex | |
| 314 @tex | |
| 315 Briefly, given a scalar $s$, an $m\times n$ matrix $M_{m\times n}$, | |
| 316 and an $m\times n$ empty matrix $[\,]_{m\times n}$ (with either one or | |
| 317 both dimensions equal to zero), the following are true: | |
| 318 $$ | |
| 319 \eqalign{% | |
| 320 s \cdot [\,]_{m\times n} = [\,]_{m\times n} \cdot s &= [\,]_{m\times n}\cr | |
| 321 [\,]_{m\times n} + [\,]_{m\times n} &= [\,]_{m\times n}\cr | |
| 322 [\,]_{0\times m} \cdot M_{m\times n} &= [\,]_{0\times n}\cr | |
| 323 M_{m\times n} \cdot [\,]_{n\times 0} &= [\,]_{m\times 0}\cr | |
| 324 [\,]_{m\times 0} \cdot [\,]_{0\times n} &= 0_{m\times n}} | |
| 325 $$ | |
| 326 @end tex | |
| 327 @end iftex | |
| 6620 | 328 @ifnottex |
| 3294 | 329 Briefly, given a scalar @var{s}, an @var{m} by |
| 330 @var{n} matrix @code{M(mxn)}, and an @var{m} by @var{n} empty matrix | |
| 331 @code{[](mxn)} (with either one or both dimensions equal to zero), the | |
| 332 following are true: | |
| 333 | |
| 334 @example | |
| 335 @group | |
| 336 s * [](mxn) = [](mxn) * s = [](mxn) | |
| 337 | |
| 338 [](mxn) + [](mxn) = [](mxn) | |
| 339 | |
| 340 [](0xm) * M(mxn) = [](0xn) | |
| 341 | |
| 342 M(mxn) * [](nx0) = [](mx0) | |
| 343 | |
| 344 [](mx0) * [](0xn) = 0(mxn) | |
| 345 @end group | |
| 346 @end example | |
| 6620 | 347 @end ifnottex |
| 3294 | 348 |
| 349 By default, dimensions of the empty matrix are printed along with the | |
| 350 empty matrix symbol, @samp{[]}. The built-in variable | |
| 351 @code{print_empty_dimensions} controls this behavior. | |
| 352 | |
| 3321 | 353 @DOCSTRING(print_empty_dimensions) |
| 3294 | 354 |
| 355 Empty matrices may also be used in assignment statements as a convenient | |
| 356 way to delete rows or columns of matrices. | |
| 357 @xref{Assignment Ops, ,Assignment Expressions}. | |
| 358 | |
| 359 When Octave parses a matrix expression, it examines the elements of the | |
| 360 list to determine whether they are all constants. If they are, it | |
| 361 replaces the list with a single matrix constant. | |
| 362 | |
| 4167 | 363 @node Ranges |
| 3294 | 364 @section Ranges |
| 365 @cindex range expressions | |
| 366 @cindex expression, range | |
| 367 | |
| 3920 | 368 @opindex colon |
| 3294 | 369 |
| 370 A @dfn{range} is a convenient way to write a row vector with evenly | |
| 371 spaced elements. A range expression is defined by the value of the first | |
| 372 element in the range, an optional value for the increment between | |
| 373 elements, and a maximum value which the elements of the range will not | |
| 374 exceed. The base, increment, and limit are separated by colons (the | |
| 375 @samp{:} character) and may contain any arithmetic expressions and | |
| 376 function calls. If the increment is omitted, it is assumed to be 1. | |
| 377 For example, the range | |
| 378 | |
| 379 @example | |
| 380 1 : 5 | |
| 381 @end example | |
| 382 | |
| 383 @noindent | |
| 384 defines the set of values @samp{[ 1, 2, 3, 4, 5 ]}, and the range | |
| 385 | |
| 386 @example | |
| 387 1 : 3 : 5 | |
| 388 @end example | |
| 389 | |
| 390 @noindent | |
| 391 defines the set of values @samp{[ 1, 4 ]}. | |
| 392 | |
| 393 Although a range constant specifies a row vector, Octave does @emph{not} | |
| 394 convert range constants to vectors unless it is necessary to do so. | |
| 395 This allows you to write a constant like @samp{1 : 10000} without using | |
| 396 80,000 bytes of storage on a typical 32-bit workstation. | |
| 397 | |
| 398 Note that the upper (or lower, if the increment is negative) bound on | |
| 399 the range is not always included in the set of values, and that ranges | |
| 400 defined by floating point values can produce surprising results because | |
| 401 Octave uses floating point arithmetic to compute the values in the | |
| 402 range. If it is important to include the endpoints of a range and the | |
| 403 number of elements is known, you should use the @code{linspace} function | |
| 404 instead (@pxref{Special Utility Matrices}). | |
| 405 | |
| 9158 | 406 When adding a scalar to a range, subtracting a scalar from it (or subtracting a |
| 407 range from a scalar) and multiplying by scalar, Octave will attempt to avoid | |
| 408 unpacking the range and keep the result as a range, too, if it can determine | |
| 409 that it is safe to do so. For instance, doing | |
| 410 | |
| 411 @example | |
| 412 a = 2*(1:1e7) - 1; | |
| 413 @end example | |
| 414 | |
| 415 will produce the same result as @samp{1:2:2e7-1}, but without ever forming a | |
| 416 vector with ten million elements. | |
| 417 | |
| 418 Using zero as an increment in the colon notation, as @samp{1:0:1} is not | |
| 419 allowed, because a division by zero would occur in determining the number of | |
| 420 range elements. However, ranges with zero increment (i.e. all elements equal) | |
| 421 are useful, especially in indexing, and Octave allows them to be constructed | |
| 422 using the built-in function @dfn{ones}. Note that because a range must be a row | |
| 423 vector, @samp{ones (1, 10)} produces a range, while @samp{ones (10, 1)} does not. | |
| 424 | |
| 3294 | 425 When Octave parses a range expression, it examines the elements of the |
| 426 expression to determine whether they are all constants. If they are, it | |
| 427 replaces the range expression with a single range constant. | |
| 428 | |
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429 @node Single Precision Data Types |
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430 @section Single Precision Data Types |
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431 |
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432 Octave includes support for single precision data types, and most of the |
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433 functions in Octave accept single precision values and return single |
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434 precision answers. A single precision variable is created with the |
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435 @code{single} function. |
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436 |
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437 @DOCSTRING(single) |
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438 |
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439 for example |
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440 |
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441 @example |
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442 sngl = single (rand (2, 2)) |
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443 @result{} sngl = |
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444 0.37569 0.92982 |
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445 0.11962 0.50876 |
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446 class (sngl) |
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447 @result{} single |
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448 @end example |
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449 |
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450 Many functions can also return single precision values directly. For |
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451 example |
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452 |
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453 @example |
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454 ones (2, 2, "single") |
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455 zeros (2, 2, "single") |
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456 eye (2, 2, "single") |
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457 rand (2, 2, "single") |
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458 NaN (2, 2, "single") |
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459 NA (2, 2, "single") |
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460 Inf (2, 2, "single") |
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461 @end example |
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462 |
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463 @noindent |
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464 will all return single precision matrices. |
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465 |
| 6549 | 466 @node Integer Data Types |
| 467 @section Integer Data Types | |
| 468 | |
| 6620 | 469 Octave supports integer matrices as an alternative to using double |
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470 precision. It is possible to use both signed and unsigned integers |
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471 represented by 8, 16, 32, or 64 bits. It should be noted that most |
| 6620 | 472 computations require floating point data, meaning that integers will |
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473 often change type when involved in numeric computations. For this |
| 6620 | 474 reason integers are most often used to store data, and not for |
| 475 calculations. | |
| 476 | |
| 477 In general most integer matrices are created by casting | |
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478 existing matrices to integers. The following example shows how to cast |
| 6620 | 479 a matrix into 32 bit integers. |
| 480 | |
| 481 @example | |
| 482 float = rand (2, 2) | |
| 483 @result{} float = 0.37569 0.92982 | |
| 484 0.11962 0.50876 | |
| 485 integer = int32 (float) | |
| 486 @result{} integer = 0 1 | |
| 487 0 1 | |
| 488 @end example | |
| 489 | |
| 490 @noindent | |
| 491 As can be seen, floating point values are rounded to the nearest integer | |
| 492 when converted. | |
| 493 | |
| 6549 | 494 @DOCSTRING(isinteger) |
| 495 | |
| 496 @DOCSTRING(int8) | |
| 497 | |
| 498 @DOCSTRING(uint8) | |
| 499 | |
| 500 @DOCSTRING(int16) | |
| 501 | |
| 502 @DOCSTRING(uint16) | |
| 503 | |
| 504 @DOCSTRING(int32) | |
| 505 | |
| 506 @DOCSTRING(uint32) | |
| 507 | |
| 508 @DOCSTRING(int64) | |
| 509 | |
| 510 @DOCSTRING(uint64) | |
| 511 | |
| 512 @DOCSTRING(intmax) | |
| 513 | |
| 514 @DOCSTRING(intmin) | |
| 515 | |
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516 @DOCSTRING(intwarning) |
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517 |
| 6624 | 518 @menu |
| 519 * Integer Arithmetic:: | |
| 520 @end menu | |
| 521 | |
| 6620 | 522 @node Integer Arithmetic |
| 523 @subsection Integer Arithmetic | |
| 524 | |
| 525 While many numerical computations can't be carried out in integers, | |
| 526 Octave does support basic operations like addition and multiplication | |
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527 on integers. The operators @code{+}, @code{-}, @code{.*}, and @code{./} |
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528 work on integers of the same type. So, it is possible to add two 32 bit |
| 6620 | 529 integers, but not to add a 32 bit integer and a 16 bit integer. |
| 530 | |
| 6715 | 531 The arithmetic operations on integers are performed by casting the |
| 532 integer values to double precision values, performing the operation, and | |
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533 then re-casting the values back to the original integer type. As the |
| 6715 | 534 double precision type of Octave is only capable of representing integers |
| 535 with up to 53 bits of precision, it is not possible to perform | |
| 8828 | 536 arithmetic with 64 bit integer types. |
| 6715 | 537 |
| 6620 | 538 When doing integer arithmetic one should consider the possibility of |
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539 underflow and overflow. This happens when the result of the computation |
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540 can't be represented using the chosen integer type. As an example it is |
| 6620 | 541 not possible to represent the result of @math{10 - 20} when using |
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542 unsigned integers. Octave makes sure that the result of integer |
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543 computations is the integer that is closest to the true result. So, the |
| 6620 | 544 result of @math{10 - 20} when using unsigned integers is zero. |
| 545 | |
| 546 When doing integer division Octave will round the result to the nearest | |
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547 integer. This is different from most programming languages, where the |
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548 result is often floored to the nearest integer. So, the result of |
| 6620 | 549 @code{int32(5)./int32(8)} is @code{1}. |
| 550 | |
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551 @DOCSTRING(idivide) |
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552 |
| 6781 | 553 @node Bit Manipulations |
| 554 @section Bit Manipulations | |
| 555 | |
| 556 Octave provides a number of functions for the manipulation of numeric | |
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557 values on a bit by bit basis. The basic functions to set and obtain the |
| 6781 | 558 values of individual bits are @code{bitset} and @code{bitget}. |
| 559 | |
| 560 @DOCSTRING(bitset) | |
| 561 | |
| 562 @DOCSTRING(bitget) | |
| 563 | |
| 564 The arguments to all of Octave's bitwise operations can be scalar or | |
| 565 arrays, except for @code{bitcmp}, whose @var{k} argument must a | |
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566 scalar. In the case where more than one argument is an array, then all |
| 6781 | 567 arguments must have the same shape, and the bitwise operator is applied |
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568 to each of the elements of the argument individually. If at least one |
| 6781 | 569 argument is a scalar and one an array, then the scalar argument is |
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570 duplicated. Therefore |
| 6781 | 571 |
| 572 @example | |
| 573 bitget (100, 8:-1:1) | |
| 574 @end example | |
| 575 | |
| 576 is the same as | |
| 577 | |
| 578 @example | |
| 579 bitget (100 * ones (1, 8), 8:-1:1) | |
| 580 @end example | |
| 581 | |
| 582 It should be noted that all values passed to the bit manipulation | |
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583 functions of Octave are treated as integers. Therefore, even though the |
| 6781 | 584 example for @code{bitset} above passes the floating point value |
| 585 @code{10}, it is treated as the bits @code{[1, 0, 1, 0]} rather than the | |
| 586 bits of the native floating point format representation of @code{10}. | |
| 587 | |
| 8828 | 588 As the maximum value that can be represented by a number is important |
| 6781 | 589 for bit manipulation, particularly when forming masks, Octave supplies |
| 590 the function @code{bitmax}. | |
| 591 | |
| 592 @DOCSTRING(bitmax) | |
| 593 | |
| 594 This is the double precision version of the functions @code{intmax}, | |
| 595 previously discussed. | |
| 596 | |
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597 Octave also includes the basic bitwise 'and', 'or' and 'exclusive or' |
| 6781 | 598 operators. |
| 599 | |
| 600 @DOCSTRING(bitand) | |
| 601 | |
| 602 @DOCSTRING(bitor) | |
| 603 | |
| 604 @DOCSTRING(bitxor) | |
| 605 | |
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606 The bitwise 'not' operator is a unary operator that performs a logical |
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607 negation of each of the bits of the value. For this to make sense, the |
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608 mask against which the value is negated must be defined. Octave's |
| 6781 | 609 bitwise 'not' operator is @code{bitcmp}. |
| 610 | |
| 611 @DOCSTRING(bitcmp) | |
| 612 | |
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613 Octave also includes the ability to left-shift and right-shift values bitwise. |
| 6781 | 614 |
| 615 @DOCSTRING(bitshift) | |
| 616 | |
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617 Bits that are shifted out of either end of the value are lost. Octave |
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618 also uses arithmetic shifts, where the sign bit of the value is kept |
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619 during a right shift. For example |
| 6781 | 620 |
| 621 @example | |
| 622 @group | |
| 623 bitshift (-10, -1) | |
| 624 @result{} -5 | |
| 625 bitshift (int8 (-1), -1) | |
| 626 @result{} -1 | |
| 627 @end group | |
| 628 @end example | |
| 629 | |
| 630 Note that @code{bitshift (int8 (-1), -1)} is @code{-1} since the bit | |
| 631 representation of @code{-1} in the @code{int8} data type is @code{[1, 1, | |
| 632 1, 1, 1, 1, 1, 1]}. | |
| 633 | |
| 4167 | 634 @node Logical Values |
| 3428 | 635 @section Logical Values |
| 636 | |
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637 Octave has built-in support for logical values, i.e., variables that |
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638 are either @code{true} or @code{false}. When comparing two variables, |
| 6620 | 639 the result will be a logical value whose value depends on whether or |
| 640 not the comparison is true. | |
| 641 | |
| 642 The basic logical operations are @code{&}, @code{|}, and @code{!}, | |
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643 which correspond to ``Logical And'', ``Logical Or'', and ``Logical |
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644 Negation''. These operations all follow the usual rules of logic. |
| 6620 | 645 |
| 646 It is also possible to use logical values as part of standard numerical | |
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647 calculations. In this case @code{true} is converted to @code{1}, and |
| 6620 | 648 @code{false} to 0, both represented using double precision floating |
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649 point numbers. So, the result of @code{true*22 - false/6} is @code{22}. |
| 6620 | 650 |
| 651 Logical values can also be used to index matrices and cell arrays. | |
| 652 When indexing with a logical array the result will be a vector containing | |
| 653 the values corresponding to @code{true} parts of the logical array. | |
| 654 The following example illustrates this. | |
| 655 | |
| 656 @example | |
| 657 data = [ 1, 2; 3, 4 ]; | |
| 658 idx = (data <= 2); | |
| 659 data(idx) | |
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660 @result{} ans = [ 1; 2 ] |
| 6620 | 661 @end example |
| 662 | |
| 663 @noindent | |
| 664 Instead of creating the @code{idx} array it is possible to replace | |
| 665 @code{data(idx)} with @code{data( data <= 2 )} in the above code. | |
| 666 | |
| 8828 | 667 Logical values can also be constructed by |
| 6620 | 668 casting numeric objects to logical values, or by using the @code{true} |
| 669 or @code{false} functions. | |
| 670 | |
| 6549 | 671 @DOCSTRING(logical) |
| 672 | |
| 3428 | 673 @DOCSTRING(true) |
| 674 | |
| 675 @DOCSTRING(false) | |
| 676 | |
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677 @node Promotion and Demotion of Data Types |
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678 @section Promotion and Demotion of Data Types |
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679 |
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680 Many operators and functions can work with mixed data types. For example |
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681 |
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682 @example |
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683 uint8 (1) + 1 |
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684 @result{} 2 |
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685 @end example |
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686 |
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687 @noindent |
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688 where the above operator works with an 8-bit integer and a double precision |
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689 value and returns an 8-bit integer value. Note that the type is demoted |
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690 to an 8-bit integer, rather than promoted to a double precision value as |
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691 might be expected. The reason is that if Octave promoted values in |
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692 expressions like the above with all numerical constants would need to be |
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693 explicitly cast to the appropriate data type like |
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694 |
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695 @example |
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696 uint8 (1) + uint8 (1) |
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697 @result{} 2 |
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698 @end example |
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699 |
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700 @noindent |
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701 which becomes difficult for the user to apply uniformly and might allow |
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702 hard to find bugs to be introduced. The same applies to single precision |
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703 values where a mixed operation such as |
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704 |
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705 @example |
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706 single (1) + 1 |
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707 @result{} 2 |
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708 @end example |
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709 |
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710 @noindent |
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711 returns a single precision value. The mixed operations that are valid |
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712 and their returned data types are |
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713 |
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714 @multitable @columnfractions .2 .3 .3 .2 |
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715 @item @tab Mixed Operation @tab Result @tab |
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716 @item @tab double OP single @tab single @tab |
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717 @item @tab double OP integer @tab integer @tab |
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718 @item @tab double OP char @tab double @tab |
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719 @item @tab double OP logical @tab double @tab |
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720 @item @tab single OP integer @tab integer @tab |
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721 @item @tab single OP char @tab single @tab |
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722 @item @tab single OP logical @tab single @tab |
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723 @end multitable |
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724 |
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725 The same logic applies to functions with mixed arguments such as |
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726 |
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727 @example |
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728 min (single (1), 0) |
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729 @result{} 0 |
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730 @end example |
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731 |
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732 @noindent |
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733 where the returned value is single precision. |
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734 |
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735 In the case of mixed type indexed assignments, the type is not |
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736 changed. For example |
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737 |
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738 @example |
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739 x = ones (2, 2); |
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740 x (1, 1) = single (2) |
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741 @result{} x = 2 1 |
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742 1 1 |
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743 @end example |
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744 |
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745 @noindent |
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746 where @code{x} remains of the double precision type. |
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747 |
| 4167 | 748 @node Predicates for Numeric Objects |
| 3294 | 749 @section Predicates for Numeric Objects |
| 750 | |
| 6620 | 751 Since the type of a variable may change during the execution of a |
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752 program, it can be necessary to do type checking at run-time. Doing this |
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753 also allows you to change the behavior of a function depending on the |
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754 type of the input. As an example, this naive implementation of @code{abs} |
| 7001 | 755 returns the absolute value of the input if it is a real number, and the |
| 6620 | 756 length of the input if it is a complex number. |
| 757 | |
| 758 @example | |
| 759 function a = abs (x) | |
| 760 if (isreal (x)) | |
| 761 a = sign (x) .* x; | |
| 762 elseif (iscomplex (x)) | |
| 763 a = sqrt (real(x).^2 + imag(x).^2); | |
| 764 endif | |
| 765 endfunction | |
| 766 @end example | |
| 767 | |
| 768 The following functions are available for determining the type of a | |
| 769 variable. | |
| 770 | |
| 3428 | 771 @DOCSTRING(isnumeric) |
| 772 | |
| 773 @DOCSTRING(isreal) | |
| 774 | |
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775 @DOCSTRING(isfloat) |
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776 |
| 4029 | 777 @DOCSTRING(iscomplex) |
| 3428 | 778 |
| 4029 | 779 @DOCSTRING(ismatrix) |
| 3294 | 780 |
| 4029 | 781 @DOCSTRING(isvector) |
| 3294 | 782 |
| 4029 | 783 @DOCSTRING(isscalar) |
| 3294 | 784 |
| 4029 | 785 @DOCSTRING(issquare) |
| 3294 | 786 |
| 4029 | 787 @DOCSTRING(issymmetric) |
| 3428 | 788 |
| 6550 | 789 @DOCSTRING(isdefinite) |
| 790 | |
| 6688 | 791 @DOCSTRING(islogical) |
| 6550 | 792 |
| 793 @DOCSTRING(isprime) |