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| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Sun, 01 Aug 2010 20:22:17 -0700 |
| parents | 322f43e0e170 |
| children | b8b08b1ac21f |
| rev | line source |
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| 7018 | 1 @c Copyright (C) 1996, 1997, 1999, 2000, 2002, 2003, 2004, 2006, |
| 9245 | 2 @c 2007, 2008, 2009 John W. Eaton |
| 7018 | 3 @c |
| 4 @c This file is part of Octave. | |
| 5 @c | |
| 6 @c Octave is free software; you can redistribute it and/or modify it | |
| 7 @c under the terms of the GNU General Public License as published by the | |
| 8 @c Free Software Foundation; either version 3 of the License, or (at | |
| 9 @c your option) any later version. | |
| 10 @c | |
| 11 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 12 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 13 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 14 @c for more details. | |
| 15 @c | |
| 16 @c You should have received a copy of the GNU General Public License | |
| 17 @c along with Octave; see the file COPYING. If not, see | |
| 18 @c <http://www.gnu.org/licenses/>. | |
| 3294 | 19 |
| 4167 | 20 @node Expressions |
| 3294 | 21 @chapter Expressions |
| 22 @cindex expressions | |
| 23 | |
| 24 Expressions are the basic building block of statements in Octave. An | |
| 25 expression evaluates to a value, which you can print, test, store in a | |
| 26 variable, pass to a function, or assign a new value to a variable with | |
| 27 an assignment operator. | |
| 28 | |
| 29 An expression can serve as a statement on its own. Most other kinds of | |
| 30 statements contain one or more expressions which specify data to be | |
| 31 operated on. As in other languages, expressions in Octave include | |
| 32 variables, array references, constants, and function calls, as well as | |
| 33 combinations of these with various operators. | |
| 34 | |
| 35 @menu | |
| 36 * Index Expressions:: | |
| 37 * Calling Functions:: | |
| 38 * Arithmetic Ops:: | |
| 39 * Comparison Ops:: | |
| 40 * Boolean Expressions:: | |
| 41 * Assignment Ops:: | |
| 42 * Increment Ops:: | |
| 43 * Operator Precedence:: | |
| 44 @end menu | |
| 45 | |
| 4167 | 46 @node Index Expressions |
| 3294 | 47 @section Index Expressions |
| 48 | |
| 49 @opindex ( | |
| 50 @opindex ) | |
| 51 | |
| 52 An @dfn{index expression} allows you to reference or extract selected | |
| 53 elements of a matrix or vector. | |
| 54 | |
| 55 Indices may be scalars, vectors, ranges, or the special operator | |
| 56 @samp{:}, which may be used to select entire rows or columns. | |
| 57 | |
| 5679 | 58 Vectors are indexed using a single index expression. Matrices may be |
| 59 indexed using one or two indices. When using a single index | |
| 60 expression, the elements of the matrix are taken in column-first order; | |
| 61 the dimensions of the output match those of the index expression. For | |
| 62 example, | |
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63 |
| 5679 | 64 @example |
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65 @group |
| 5679 | 66 a (2) # a scalar |
| 67 a (1:2) # a row vector | |
| 68 a ([1; 2]) # a column vector | |
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69 @end group |
| 5679 | 70 @end example |
| 71 | |
| 72 As a special case, when a colon is used as a single index, the output | |
| 73 is a column vector containing all the elements of the vector or matrix. | |
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74 For example: |
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75 |
| 5679 | 76 @example |
| 77 a (:) # a column vector | |
| 78 @end example | |
| 79 | |
| 3294 | 80 Given the matrix |
| 81 | |
| 82 @example | |
| 83 a = [1, 2; 3, 4] | |
| 84 @end example | |
| 85 | |
| 86 @noindent | |
| 87 all of the following expressions are equivalent | |
| 88 | |
| 89 @example | |
| 90 @group | |
| 91 a (1, [1, 2]) | |
| 92 a (1, 1:2) | |
| 93 a (1, :) | |
| 94 @end group | |
| 95 @end example | |
| 96 | |
| 97 @noindent | |
| 98 and select the first row of the matrix. | |
| 99 | |
| 9159 | 100 In general, an array with @samp{n} dimensions can be indexed using @samp{m} |
| 101 indices. If @code{n == m}, each index corresponds to its respective dimension. | |
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102 The set of index tuples determining the result is formed by the Cartesian |
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103 product of the index vectors (or ranges or scalars). If @code{n < m}, then the |
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104 array is padded by trailing singleton dimensions. If @code{n > m}, the last |
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105 @code{n-m+1} dimensions are folded into a single dimension with extent equal to |
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106 product of extents of the original dimensions. |
| 9159 | 107 |
| 5016 | 108 @c FIXED -- sections on variable prefer_zero_one_indexing were removed |
| 3294 | 109 |
| 5016 | 110 Indexing a scalar with a vector of ones can be used to create a |
| 3294 | 111 vector the same size as the index vector, with each element equal to |
| 112 the value of the original scalar. For example, the following statements | |
| 113 | |
| 114 @example | |
| 115 @group | |
| 116 a = 13; | |
| 9159 | 117 a (ones (1, 4)) |
| 3294 | 118 @end group |
| 119 @end example | |
| 120 | |
| 121 @noindent | |
| 122 produce a vector whose four elements are all equal to 13. | |
| 123 | |
| 124 Similarly, indexing a scalar with two vectors of ones can be used to | |
| 125 create a matrix. For example the following statements | |
| 126 | |
| 127 @example | |
| 128 @group | |
| 129 a = 13; | |
| 9159 | 130 a (ones (1, 2), ones (1, 3)) |
| 3294 | 131 @end group |
| 132 @end example | |
| 133 | |
| 134 @noindent | |
| 135 create a 2 by 3 matrix with all elements equal to 13. | |
| 136 | |
| 9159 | 137 The last example could also be written as |
| 138 | |
| 139 @example | |
| 140 @group | |
| 141 13 (ones (2, 3)) | |
| 142 @end group | |
| 143 @end example | |
| 144 | |
| 145 It should be, noted that @code{ones (1, n)} (a row vector of ones) results in a | |
| 146 range (with zero increment), and is therefore more efficient when used in index | |
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147 expression than other forms of @dfn{ones}. In particular, when @samp{r} is a |
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148 row vector, the expressions |
| 9159 | 149 |
| 150 @example | |
| 151 r(ones (1, n), :) | |
| 152 @end example | |
| 153 | |
| 154 @example | |
| 155 r(ones (n, 1), :) | |
| 156 @end example | |
| 157 | |
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158 @noindent |
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159 will produce identical results, but the first one will be significantly |
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160 faster, at least for @samp{r} and @samp{n} large enough. The reason is that |
| 9159 | 161 in the first case the index is kept in a compressed form, which allows Octave |
| 162 to choose a more efficient algorithm to handle the expression. | |
| 163 | |
| 164 In general, for an user unaware of these subtleties, it is best to use | |
| 165 the function @dfn{repmat} for spreading arrays into bigger ones. | |
| 3294 | 166 |
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167 It is also possible to create a matrix with different values. The |
| 6939 | 168 following example creates a 10 dimensional row vector @math{a} containing |
| 6642 | 169 the values |
| 170 @tex | |
| 171 $a_i = \sqrt{i}$. | |
| 172 @end tex | |
| 173 @ifnottex | |
| 174 a(i) = sqrt(i). | |
| 175 @end ifnottex | |
| 176 | |
| 177 @example | |
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178 @group |
| 6642 | 179 for i = 1:10 |
| 180 a(i) = sqrt (i); | |
| 181 endfor | |
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182 @end group |
| 6642 | 183 @end example |
| 184 | |
| 185 @noindent | |
| 3294 | 186 Note that it is quite inefficient to create a vector using a loop like |
| 187 the one shown in the example above. In this particular case, it would | |
| 188 have been much more efficient to use the expression | |
| 189 | |
| 190 @example | |
| 191 a = sqrt (1:10); | |
| 192 @end example | |
| 193 | |
| 194 @noindent | |
| 195 thus avoiding the loop entirely. In cases where a loop is still | |
| 196 required, or a number of values must be combined to form a larger | |
| 197 matrix, it is generally much faster to set the size of the matrix first, | |
| 198 and then insert elements using indexing commands. For example, given a | |
| 199 matrix @code{a}, | |
| 200 | |
| 201 @example | |
| 202 @group | |
| 203 [nr, nc] = size (a); | |
| 204 x = zeros (nr, n * nc); | |
| 205 for i = 1:n | |
| 3602 | 206 x(:,(i-1)*nc+1:i*nc) = a; |
| 3294 | 207 endfor |
| 208 @end group | |
| 209 @end example | |
| 210 | |
| 211 @noindent | |
| 212 is considerably faster than | |
| 213 | |
| 214 @example | |
| 215 @group | |
| 216 x = a; | |
| 217 for i = 1:n-1 | |
| 218 x = [x, a]; | |
| 219 endfor | |
| 220 @end group | |
| 221 @end example | |
| 222 | |
| 223 @noindent | |
| 224 particularly for large matrices because Octave does not have to | |
| 225 repeatedly resize the result. | |
| 226 | |
| 6549 | 227 @DOCSTRING(sub2ind) |
| 228 | |
| 229 @DOCSTRING(ind2sub) | |
| 230 | |
| 4167 | 231 @node Calling Functions |
| 3294 | 232 @section Calling Functions |
| 233 | |
| 234 A @dfn{function} is a name for a particular calculation. Because it has | |
| 235 a name, you can ask for it by name at any point in the program. For | |
| 236 example, the function @code{sqrt} computes the square root of a number. | |
| 237 | |
| 238 A fixed set of functions are @dfn{built-in}, which means they are | |
| 239 available in every Octave program. The @code{sqrt} function is one of | |
| 240 these. In addition, you can define your own functions. | |
| 241 @xref{Functions and Scripts}, for information about how to do this. | |
| 242 | |
| 243 @cindex arguments in function call | |
| 244 The way to use a function is with a @dfn{function call} expression, | |
| 245 which consists of the function name followed by a list of | |
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246 @dfn{arguments} in parentheses. The arguments are expressions which give |
| 3294 | 247 the raw materials for the calculation that the function will do. When |
| 248 there is more than one argument, they are separated by commas. If there | |
| 249 are no arguments, you can omit the parentheses, but it is a good idea to | |
| 250 include them anyway, to clearly indicate that a function call was | |
| 251 intended. Here are some examples: | |
| 252 | |
| 253 @example | |
| 254 @group | |
| 255 sqrt (x^2 + y^2) # @r{One argument} | |
| 256 ones (n, m) # @r{Two arguments} | |
| 257 rand () # @r{No arguments} | |
| 258 @end group | |
| 259 @end example | |
| 260 | |
| 261 Each function expects a particular number of arguments. For example, the | |
| 262 @code{sqrt} function must be called with a single argument, the number | |
| 263 to take the square root of: | |
| 264 | |
| 265 @example | |
| 266 sqrt (@var{argument}) | |
| 267 @end example | |
| 268 | |
| 269 Some of the built-in functions take a variable number of arguments, | |
| 270 depending on the particular usage, and their behavior is different | |
| 271 depending on the number of arguments supplied. | |
| 272 | |
| 273 Like every other expression, the function call has a value, which is | |
| 274 computed by the function based on the arguments you give it. In this | |
| 275 example, the value of @code{sqrt (@var{argument})} is the square root of | |
| 276 the argument. A function can also have side effects, such as assigning | |
| 277 the values of certain variables or doing input or output operations. | |
| 278 | |
| 279 Unlike most languages, functions in Octave may return multiple values. | |
| 280 For example, the following statement | |
| 281 | |
| 282 @example | |
| 283 [u, s, v] = svd (a) | |
| 284 @end example | |
| 285 | |
| 286 @noindent | |
| 287 computes the singular value decomposition of the matrix @code{a} and | |
| 288 assigns the three result matrices to @code{u}, @code{s}, and @code{v}. | |
| 289 | |
| 290 The left side of a multiple assignment expression is itself a list of | |
| 291 expressions, and is allowed to be a list of variable names or index | |
| 292 expressions. See also @ref{Index Expressions}, and @ref{Assignment Ops}. | |
| 293 | |
| 294 @menu | |
| 295 * Call by Value:: | |
| 296 * Recursion:: | |
| 297 @end menu | |
| 298 | |
| 4167 | 299 @node Call by Value |
| 3294 | 300 @subsection Call by Value |
| 301 | |
| 302 In Octave, unlike Fortran, function arguments are passed by value, which | |
| 303 means that each argument in a function call is evaluated and assigned to | |
| 304 a temporary location in memory before being passed to the function. | |
| 305 There is currently no way to specify that a function parameter should be | |
| 306 passed by reference instead of by value. This means that it is | |
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307 impossible to directly alter the value of a function parameter in the |
| 3294 | 308 calling function. It can only change the local copy within the function |
| 309 body. For example, the function | |
| 310 | |
| 311 @example | |
| 312 @group | |
| 313 function f (x, n) | |
| 314 while (n-- > 0) | |
| 315 disp (x); | |
| 316 endwhile | |
| 317 endfunction | |
| 318 @end group | |
| 319 @end example | |
| 320 | |
| 321 @noindent | |
| 322 displays the value of the first argument @var{n} times. In this | |
| 323 function, the variable @var{n} is used as a temporary variable without | |
| 324 having to worry that its value might also change in the calling | |
| 325 function. Call by value is also useful because it is always possible to | |
| 326 pass constants for any function parameter without first having to | |
| 327 determine that the function will not attempt to modify the parameter. | |
| 328 | |
| 329 The caller may use a variable as the expression for the argument, but | |
| 330 the called function does not know this: it only knows what value the | |
| 331 argument had. For example, given a function called as | |
| 332 | |
| 333 @example | |
| 334 @group | |
| 335 foo = "bar"; | |
| 336 fcn (foo) | |
| 337 @end group | |
| 338 @end example | |
| 339 | |
| 340 @noindent | |
| 341 you should not think of the argument as being ``the variable | |
| 342 @code{foo}.'' Instead, think of the argument as the string value, | |
| 343 @code{"bar"}. | |
| 344 | |
| 345 Even though Octave uses pass-by-value semantics for function arguments, | |
| 346 values are not copied unnecessarily. For example, | |
| 347 | |
| 348 @example | |
| 349 @group | |
| 350 x = rand (1000); | |
| 351 f (x); | |
| 352 @end group | |
| 353 @end example | |
| 354 | |
| 355 @noindent | |
| 356 does not actually force two 1000 by 1000 element matrices to exist | |
| 357 @emph{unless} the function @code{f} modifies the value of its | |
| 358 argument. Then Octave must create a copy to avoid changing the | |
| 359 value outside the scope of the function @code{f}, or attempting (and | |
| 360 probably failing!) to modify the value of a constant or the value of a | |
| 361 temporary result. | |
| 362 | |
| 4167 | 363 @node Recursion |
| 3294 | 364 @subsection Recursion |
| 365 @cindex factorial function | |
| 366 | |
| 6939 | 367 With some restrictions@footnote{Some of Octave's functions are |
| 3294 | 368 implemented in terms of functions that cannot be called recursively. |
| 369 For example, the ODE solver @code{lsode} is ultimately implemented in a | |
| 370 Fortran subroutine that cannot be called recursively, so @code{lsode} | |
| 371 should not be called either directly or indirectly from within the | |
| 372 user-supplied function that @code{lsode} requires. Doing so will result | |
| 6642 | 373 in an error.}, recursive function calls are allowed. A |
| 3294 | 374 @dfn{recursive function} is one which calls itself, either directly or |
| 375 indirectly. For example, here is an inefficient@footnote{It would be | |
| 376 much better to use @code{prod (1:n)}, or @code{gamma (n+1)} instead, | |
| 377 after first checking to ensure that the value @code{n} is actually a | |
| 378 positive integer.} way to compute the factorial of a given integer: | |
| 379 | |
| 380 @example | |
| 381 @group | |
| 382 function retval = fact (n) | |
| 383 if (n > 0) | |
| 384 retval = n * fact (n-1); | |
| 385 else | |
| 386 retval = 1; | |
| 387 endif | |
| 388 endfunction | |
| 389 @end group | |
| 390 @end example | |
| 391 | |
| 392 This function is recursive because it calls itself directly. It | |
| 393 eventually terminates because each time it calls itself, it uses an | |
| 394 argument that is one less than was used for the previous call. Once the | |
| 395 argument is no longer greater than zero, it does not call itself, and | |
| 396 the recursion ends. | |
| 397 | |
| 398 The built-in variable @code{max_recursion_depth} specifies a limit to | |
| 399 the recursion depth and prevents Octave from recursing infinitely. | |
| 400 | |
| 3371 | 401 @DOCSTRING(max_recursion_depth) |
| 3294 | 402 |
| 4167 | 403 @node Arithmetic Ops |
| 3294 | 404 @section Arithmetic Operators |
| 405 @cindex arithmetic operators | |
| 406 @cindex operators, arithmetic | |
| 407 @cindex addition | |
| 408 @cindex subtraction | |
| 409 @cindex multiplication | |
| 410 @cindex matrix multiplication | |
| 411 @cindex division | |
| 412 @cindex quotient | |
| 413 @cindex negation | |
| 414 @cindex unary minus | |
| 415 @cindex exponentiation | |
| 416 @cindex transpose | |
| 417 @cindex Hermitian operator | |
| 418 @cindex transpose, complex-conjugate | |
| 419 @cindex complex-conjugate transpose | |
| 420 | |
| 421 The following arithmetic operators are available, and work on scalars | |
| 422 and matrices. | |
| 423 | |
| 424 @table @code | |
| 425 @item @var{x} + @var{y} | |
| 426 @opindex + | |
| 427 Addition. If both operands are matrices, the number of rows and columns | |
| 428 must both agree. If one operand is a scalar, its value is added to | |
| 429 all the elements of the other operand. | |
| 430 | |
| 431 @item @var{x} .+ @var{y} | |
| 432 @opindex .+ | |
| 433 Element by element addition. This operator is equivalent to @code{+}. | |
| 434 | |
| 435 @item @var{x} - @var{y} | |
| 436 @opindex - | |
| 437 Subtraction. If both operands are matrices, the number of rows and | |
| 438 columns of both must agree. | |
| 439 | |
| 440 @item @var{x} .- @var{y} | |
| 441 Element by element subtraction. This operator is equivalent to @code{-}. | |
| 442 | |
| 443 @item @var{x} * @var{y} | |
| 444 @opindex * | |
| 445 Matrix multiplication. The number of columns of @var{x} must agree | |
| 446 with the number of rows of @var{y}. | |
| 447 | |
| 448 @item @var{x} .* @var{y} | |
| 449 @opindex .* | |
| 450 Element by element multiplication. If both operands are matrices, the | |
| 451 number of rows and columns must both agree. | |
| 452 | |
| 453 @item @var{x} / @var{y} | |
| 454 @opindex / | |
| 455 Right division. This is conceptually equivalent to the expression | |
| 456 | |
| 457 @example | |
| 458 (inverse (y') * x')' | |
| 459 @end example | |
| 460 | |
| 461 @noindent | |
| 462 but it is computed without forming the inverse of @var{y'}. | |
| 463 | |
| 464 If the system is not square, or if the coefficient matrix is singular, | |
| 465 a minimum norm solution is computed. | |
| 466 | |
| 467 @item @var{x} ./ @var{y} | |
| 468 @opindex ./ | |
| 469 Element by element right division. | |
| 470 | |
| 471 @item @var{x} \ @var{y} | |
| 472 @opindex \ | |
| 473 Left division. This is conceptually equivalent to the expression | |
| 474 | |
| 475 @example | |
| 476 inverse (x) * y | |
| 477 @end example | |
| 478 | |
| 479 @noindent | |
| 480 but it is computed without forming the inverse of @var{x}. | |
| 481 | |
| 482 If the system is not square, or if the coefficient matrix is singular, | |
| 483 a minimum norm solution is computed. | |
| 484 | |
| 485 @item @var{x} .\ @var{y} | |
| 486 @opindex .\ | |
| 487 Element by element left division. Each element of @var{y} is divided | |
| 488 by each corresponding element of @var{x}. | |
| 489 | |
| 490 @item @var{x} ^ @var{y} | |
| 491 @itemx @var{x} ** @var{y} | |
| 492 @opindex ** | |
| 493 @opindex ^ | |
| 494 Power operator. If @var{x} and @var{y} are both scalars, this operator | |
| 495 returns @var{x} raised to the power @var{y}. If @var{x} is a scalar and | |
| 496 @var{y} is a square matrix, the result is computed using an eigenvalue | |
| 7001 | 497 expansion. If @var{x} is a square matrix, the result is computed by |
| 3294 | 498 repeated multiplication if @var{y} is an integer, and by an eigenvalue |
| 499 expansion if @var{y} is not an integer. An error results if both | |
| 500 @var{x} and @var{y} are matrices. | |
| 501 | |
| 502 The implementation of this operator needs to be improved. | |
| 503 | |
| 504 @item @var{x} .^ @var{y} | |
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505 @itemx @var{x} .** @var{y} |
| 3294 | 506 @opindex .** |
| 507 @opindex .^ | |
| 508 Element by element power operator. If both operands are matrices, the | |
| 509 number of rows and columns must both agree. | |
| 510 | |
| 511 @item -@var{x} | |
| 512 @opindex - | |
| 513 Negation. | |
| 514 | |
| 515 @item +@var{x} | |
| 516 @opindex + | |
| 517 Unary plus. This operator has no effect on the operand. | |
| 518 | |
| 519 @item @var{x}' | |
| 520 @opindex ' | |
| 521 Complex conjugate transpose. For real arguments, this operator is the | |
| 522 same as the transpose operator. For complex arguments, this operator is | |
| 523 equivalent to the expression | |
| 524 | |
| 525 @example | |
| 526 conj (x.') | |
| 527 @end example | |
| 528 | |
| 529 @item @var{x}.' | |
| 530 @opindex .' | |
| 531 Transpose. | |
| 532 @end table | |
| 533 | |
| 534 Note that because Octave's element by element operators begin with a | |
| 535 @samp{.}, there is a possible ambiguity for statements like | |
| 536 | |
| 537 @example | |
| 538 1./m | |
| 539 @end example | |
| 540 | |
| 541 @noindent | |
| 542 because the period could be interpreted either as part of the constant | |
| 543 or as part of the operator. To resolve this conflict, Octave treats the | |
| 544 expression as if you had typed | |
| 545 | |
| 546 @example | |
| 547 (1) ./ m | |
| 548 @end example | |
| 549 | |
| 550 @noindent | |
| 551 and not | |
| 552 | |
| 553 @example | |
| 554 (1.) / m | |
| 555 @end example | |
| 556 | |
| 557 @noindent | |
| 558 Although this is inconsistent with the normal behavior of Octave's | |
| 559 lexer, which usually prefers to break the input into tokens by | |
| 560 preferring the longest possible match at any given point, it is more | |
| 561 useful in this case. | |
| 562 | |
| 4167 | 563 @node Comparison Ops |
| 3294 | 564 @section Comparison Operators |
| 565 @cindex comparison expressions | |
| 566 @cindex expressions, comparison | |
| 567 @cindex relational operators | |
| 568 @cindex operators, relational | |
| 569 @cindex less than operator | |
| 570 @cindex greater than operator | |
| 571 @cindex equality operator | |
| 572 @cindex tests for equality | |
| 573 @cindex equality, tests for | |
| 574 | |
| 575 @dfn{Comparison operators} compare numeric values for relationships | |
| 576 such as equality. They are written using | |
| 577 @emph{relational operators}. | |
| 578 | |
| 579 All of Octave's comparison operators return a value of 1 if the | |
| 580 comparison is true, or 0 if it is false. For matrix values, they all | |
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581 work on an element-by-element basis. For example: |
| 3294 | 582 |
| 583 @example | |
| 584 @group | |
| 585 [1, 2; 3, 4] == [1, 3; 2, 4] | |
| 586 @result{} 1 0 | |
| 587 0 1 | |
| 588 @end group | |
| 589 @end example | |
| 590 | |
| 591 If one operand is a scalar and the other is a matrix, the scalar is | |
| 592 compared to each element of the matrix in turn, and the result is the | |
| 593 same size as the matrix. | |
| 594 | |
| 595 @table @code | |
| 596 @item @var{x} < @var{y} | |
| 597 @opindex < | |
| 598 True if @var{x} is less than @var{y}. | |
| 599 | |
| 600 @item @var{x} <= @var{y} | |
| 601 @opindex <= | |
| 602 True if @var{x} is less than or equal to @var{y}. | |
| 603 | |
| 604 @item @var{x} == @var{y} | |
| 605 @opindex == | |
| 606 True if @var{x} is equal to @var{y}. | |
| 607 | |
| 608 @item @var{x} >= @var{y} | |
| 609 @opindex >= | |
| 610 True if @var{x} is greater than or equal to @var{y}. | |
| 611 | |
| 612 @item @var{x} > @var{y} | |
| 613 @opindex > | |
| 614 True if @var{x} is greater than @var{y}. | |
| 615 | |
| 616 @item @var{x} != @var{y} | |
| 617 @itemx @var{x} ~= @var{y} | |
| 618 @opindex != | |
| 619 @opindex ~= | |
| 620 True if @var{x} is not equal to @var{y}. | |
| 621 @end table | |
| 622 | |
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623 For complex numbers, the following ordering is defined: |
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624 @var{z1} < @var{z2} |
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625 iff |
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626 |
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627 @example |
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628 @group |
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629 abs(@var{z1}) < abs(@var{z2}) |
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630 || (abs(@var{z1}) == abs(@var{z2}) && arg(@var{z1}) < arg(@var{z2})) |
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631 @end group |
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632 @end example |
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633 |
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634 This is consistent with the ordering used by @dfn{max}, @dfn{min} and |
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635 @dfn{sort}, but is not consistent with @sc{matlab}, which only compares the real |
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636 parts. |
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637 |
| 3294 | 638 String comparisons may also be performed with the @code{strcmp} |
| 639 function, not with the comparison operators listed above. | |
| 640 @xref{Strings}. | |
| 641 | |
| 6550 | 642 @DOCSTRING(isequal) |
| 643 | |
| 644 @DOCSTRING(isequalwithequalnans) | |
| 645 | |
| 4167 | 646 @node Boolean Expressions |
| 3294 | 647 @section Boolean Expressions |
| 648 @cindex expressions, boolean | |
| 649 @cindex boolean expressions | |
| 650 @cindex expressions, logical | |
| 651 @cindex logical expressions | |
| 652 @cindex operators, boolean | |
| 653 @cindex boolean operators | |
| 654 @cindex logical operators | |
| 655 @cindex operators, logical | |
| 656 @cindex and operator | |
| 657 @cindex or operator | |
| 658 @cindex not operator | |
| 659 | |
| 660 @menu | |
| 661 * Element-by-element Boolean Operators:: | |
| 662 * Short-circuit Boolean Operators:: | |
| 663 @end menu | |
| 664 | |
| 4167 | 665 @node Element-by-element Boolean Operators |
| 3294 | 666 @subsection Element-by-element Boolean Operators |
| 667 @cindex element-by-element evaluation | |
| 668 | |
| 669 An @dfn{element-by-element boolean expression} is a combination of | |
| 670 comparison expressions using the boolean | |
| 671 operators ``or'' (@samp{|}), ``and'' (@samp{&}), and ``not'' (@samp{!}), | |
| 672 along with parentheses to control nesting. The truth of the boolean | |
| 673 expression is computed by combining the truth values of the | |
| 674 corresponding elements of the component expressions. A value is | |
| 675 considered to be false if it is zero, and true otherwise. | |
| 676 | |
| 677 Element-by-element boolean expressions can be used wherever comparison | |
| 678 expressions can be used. They can be used in @code{if} and @code{while} | |
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679 statements. However, a matrix value used as the condition in an |
| 3294 | 680 @code{if} or @code{while} statement is only true if @emph{all} of its |
| 681 elements are nonzero. | |
| 682 | |
| 683 Like comparison operations, each element of an element-by-element | |
| 684 boolean expression also has a numeric value (1 if true, 0 if false) that | |
| 685 comes into play if the result of the boolean expression is stored in a | |
| 686 variable, or used in arithmetic. | |
| 687 | |
| 688 Here are descriptions of the three element-by-element boolean operators. | |
| 689 | |
| 690 @table @code | |
| 691 @item @var{boolean1} & @var{boolean2} | |
| 692 @opindex & | |
| 693 Elements of the result are true if both corresponding elements of | |
| 694 @var{boolean1} and @var{boolean2} are true. | |
| 695 | |
| 696 @item @var{boolean1} | @var{boolean2} | |
| 697 @opindex | | |
| 698 Elements of the result are true if either of the corresponding elements | |
| 699 of @var{boolean1} or @var{boolean2} is true. | |
| 700 | |
| 701 @item ! @var{boolean} | |
| 702 @itemx ~ @var{boolean} | |
| 703 @opindex ~ | |
| 704 @opindex ! | |
| 705 Each element of the result is true if the corresponding element of | |
| 706 @var{boolean} is false. | |
| 707 @end table | |
| 708 | |
| 709 For matrix operands, these operators work on an element-by-element | |
| 710 basis. For example, the expression | |
| 711 | |
| 712 @example | |
| 713 [1, 0; 0, 1] & [1, 0; 2, 3] | |
| 714 @end example | |
| 715 | |
| 716 @noindent | |
| 717 returns a two by two identity matrix. | |
| 718 | |
| 719 For the binary operators, the dimensions of the operands must conform if | |
| 720 both are matrices. If one of the operands is a scalar and the other a | |
| 721 matrix, the operator is applied to the scalar and each element of the | |
| 722 matrix. | |
| 723 | |
| 724 For the binary element-by-element boolean operators, both subexpressions | |
| 725 @var{boolean1} and @var{boolean2} are evaluated before computing the | |
| 726 result. This can make a difference when the expressions have side | |
| 727 effects. For example, in the expression | |
| 728 | |
| 729 @example | |
| 730 a & b++ | |
| 731 @end example | |
| 732 | |
| 733 @noindent | |
| 734 the value of the variable @var{b} is incremented even if the variable | |
| 735 @var{a} is zero. | |
| 736 | |
| 737 This behavior is necessary for the boolean operators to work as | |
| 738 described for matrix-valued operands. | |
| 739 | |
| 4167 | 740 @node Short-circuit Boolean Operators |
| 3294 | 741 @subsection Short-circuit Boolean Operators |
| 742 @cindex short-circuit evaluation | |
| 743 | |
| 744 Combined with the implicit conversion to scalar values in @code{if} and | |
| 745 @code{while} conditions, Octave's element-by-element boolean operators | |
| 746 are often sufficient for performing most logical operations. However, | |
| 747 it is sometimes desirable to stop evaluating a boolean expression as | |
| 748 soon as the overall truth value can be determined. Octave's | |
| 749 @dfn{short-circuit} boolean operators work this way. | |
| 750 | |
| 751 @table @code | |
| 752 @item @var{boolean1} && @var{boolean2} | |
| 753 @opindex && | |
| 754 The expression @var{boolean1} is evaluated and converted to a scalar | |
| 6632 | 755 using the equivalent of the operation @code{all (@var{boolean1}(:))}. |
| 3294 | 756 If it is false, the result of the overall expression is 0. If it is |
| 757 true, the expression @var{boolean2} is evaluated and converted to a | |
| 6632 | 758 scalar using the equivalent of the operation @code{all |
| 759 (@var{boolean1}(:))}. If it is true, the result of the overall expression | |
| 3294 | 760 is 1. Otherwise, the result of the overall expression is 0. |
| 761 | |
| 6632 | 762 @strong{Warning:} there is one exception to the rule of evaluating |
| 763 @code{all (@var{boolean1}(:))}, which is when @code{boolean1} is the | |
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764 empty matrix. The truth value of an empty matrix is always @code{false} |
| 6632 | 765 so @code{[] && true} evaluates to @code{false} even though |
| 766 @code{all ([])} is @code{true}. | |
| 767 | |
| 3294 | 768 @item @var{boolean1} || @var{boolean2} |
| 769 @opindex || | |
| 770 The expression @var{boolean1} is evaluated and converted to a scalar | |
| 6632 | 771 using the equivalent of the operation @code{all (@var{boolean1}(:))}. |
| 3294 | 772 If it is true, the result of the overall expression is 1. If it is |
| 773 false, the expression @var{boolean2} is evaluated and converted to a | |
| 6632 | 774 scalar using the equivalent of the operation @code{all |
| 775 (@var{boolean1}(:))}. If it is true, the result of the overall expression | |
| 3294 | 776 is 1. Otherwise, the result of the overall expression is 0. |
| 6632 | 777 |
| 778 @strong{Warning:} the truth value of an empty matrix is always @code{false}, | |
| 779 see the previous list item for details. | |
| 3294 | 780 @end table |
| 781 | |
| 782 The fact that both operands may not be evaluated before determining the | |
| 783 overall truth value of the expression can be important. For example, in | |
| 784 the expression | |
| 785 | |
| 786 @example | |
| 787 a && b++ | |
| 788 @end example | |
| 789 | |
| 790 @noindent | |
| 791 the value of the variable @var{b} is only incremented if the variable | |
| 792 @var{a} is nonzero. | |
| 793 | |
| 794 This can be used to write somewhat more concise code. For example, it | |
| 795 is possible write | |
| 796 | |
| 797 @example | |
| 798 @group | |
| 799 function f (a, b, c) | |
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800 if (nargin > 2 && ischar (c)) |
| 3294 | 801 @dots{} |
| 802 @end group | |
| 803 @end example | |
| 804 | |
| 805 @noindent | |
| 806 instead of having to use two @code{if} statements to avoid attempting to | |
| 807 evaluate an argument that doesn't exist. For example, without the | |
| 808 short-circuit feature, it would be necessary to write | |
| 809 | |
| 810 @example | |
| 811 @group | |
| 812 function f (a, b, c) | |
| 813 if (nargin > 2) | |
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814 if (ischar (c)) |
| 3294 | 815 @dots{} |
| 816 @end group | |
| 817 @end example | |
| 818 | |
| 6632 | 819 @noindent |
| 3294 | 820 Writing |
| 821 | |
| 822 @example | |
| 823 @group | |
| 824 function f (a, b, c) | |
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825 if (nargin > 2 & ischar (c)) |
| 3294 | 826 @dots{} |
| 827 @end group | |
| 828 @end example | |
| 829 | |
| 830 @noindent | |
| 831 would result in an error if @code{f} were called with one or two | |
| 832 arguments because Octave would be forced to try to evaluate both of the | |
| 833 operands for the operator @samp{&}. | |
| 834 | |
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835 The ternary operator (?:) is not supported in Octave. If short-circuiting is |
| 10308 | 836 not important, it can be replaced by the @code{ifelse} function. |
| 837 | |
| 838 @DOCSTRING(ifelse) | |
| 839 | |
| 4167 | 840 @node Assignment Ops |
| 3294 | 841 @section Assignment Expressions |
| 842 @cindex assignment expressions | |
| 843 @cindex assignment operators | |
| 844 @cindex operators, assignment | |
| 845 @cindex expressions, assignment | |
| 846 | |
| 847 @opindex = | |
| 848 | |
| 849 An @dfn{assignment} is an expression that stores a new value into a | |
| 850 variable. For example, the following expression assigns the value 1 to | |
| 851 the variable @code{z}: | |
| 852 | |
| 853 @example | |
| 854 z = 1 | |
| 855 @end example | |
| 856 | |
| 6632 | 857 @noindent |
| 3294 | 858 After this expression is executed, the variable @code{z} has the value 1. |
| 859 Whatever old value @code{z} had before the assignment is forgotten. | |
| 860 The @samp{=} sign is called an @dfn{assignment operator}. | |
| 861 | |
| 862 Assignments can store string values also. For example, the following | |
| 863 expression would store the value @code{"this food is good"} in the | |
| 864 variable @code{message}: | |
| 865 | |
| 866 @example | |
| 867 @group | |
| 868 thing = "food" | |
| 869 predicate = "good" | |
| 870 message = [ "this " , thing , " is " , predicate ] | |
| 871 @end group | |
| 872 @end example | |
| 873 | |
| 874 @noindent | |
| 875 (This also illustrates concatenation of strings.) | |
| 876 | |
| 877 @cindex side effect | |
| 878 Most operators (addition, concatenation, and so on) have no effect | |
| 879 except to compute a value. If you ignore the value, you might as well | |
| 880 not use the operator. An assignment operator is different. It does | |
| 881 produce a value, but even if you ignore the value, the assignment still | |
| 882 makes itself felt through the alteration of the variable. We call this | |
| 883 a @dfn{side effect}. | |
| 884 | |
| 885 @cindex lvalue | |
| 886 The left-hand operand of an assignment need not be a variable | |
| 887 (@pxref{Variables}). It can also be an element of a matrix | |
| 888 (@pxref{Index Expressions}) or a list of return values | |
| 889 (@pxref{Calling Functions}). These are all called @dfn{lvalues}, which | |
| 890 means they can appear on the left-hand side of an assignment operator. | |
| 891 The right-hand operand may be any expression. It produces the new value | |
| 892 which the assignment stores in the specified variable, matrix element, | |
| 893 or list of return values. | |
| 894 | |
| 895 It is important to note that variables do @emph{not} have permanent types. | |
| 896 The type of a variable is simply the type of whatever value it happens | |
| 897 to hold at the moment. In the following program fragment, the variable | |
| 898 @code{foo} has a numeric value at first, and a string value later on: | |
| 899 | |
| 900 @example | |
| 901 @group | |
| 902 octave:13> foo = 1 | |
| 903 foo = 1 | |
| 904 octave:13> foo = "bar" | |
| 905 foo = bar | |
| 906 @end group | |
| 907 @end example | |
| 908 | |
| 909 @noindent | |
| 910 When the second assignment gives @code{foo} a string value, the fact that | |
| 911 it previously had a numeric value is forgotten. | |
| 912 | |
| 913 Assignment of a scalar to an indexed matrix sets all of the elements | |
| 914 that are referenced by the indices to the scalar value. For example, if | |
| 915 @code{a} is a matrix with at least two columns, | |
| 916 | |
| 917 @example | |
| 918 @group | |
| 919 a(:, 2) = 5 | |
| 920 @end group | |
| 921 @end example | |
| 922 | |
| 923 @noindent | |
| 924 sets all the elements in the second column of @code{a} to 5. | |
| 925 | |
| 926 Assigning an empty matrix @samp{[]} works in most cases to allow you to | |
| 927 delete rows or columns of matrices and vectors. @xref{Empty Matrices}. | |
| 928 For example, given a 4 by 5 matrix @var{A}, the assignment | |
| 929 | |
| 930 @example | |
| 931 A (3, :) = [] | |
| 932 @end example | |
| 933 | |
| 934 @noindent | |
| 935 deletes the third row of @var{A}, and the assignment | |
| 936 | |
| 937 @example | |
| 938 A (:, 1:2:5) = [] | |
| 939 @end example | |
| 940 | |
| 941 @noindent | |
| 6672 | 942 deletes the first, third, and fifth columns. |
| 3294 | 943 |
| 944 An assignment is an expression, so it has a value. Thus, @code{z = 1} | |
| 945 as an expression has the value 1. One consequence of this is that you | |
| 946 can write multiple assignments together: | |
| 947 | |
| 948 @example | |
| 949 x = y = z = 0 | |
| 950 @end example | |
| 951 | |
| 952 @noindent | |
| 953 stores the value 0 in all three variables. It does this because the | |
| 954 value of @code{z = 0}, which is 0, is stored into @code{y}, and then | |
| 955 the value of @code{y = z = 0}, which is 0, is stored into @code{x}. | |
| 956 | |
| 957 This is also true of assignments to lists of values, so the following is | |
| 958 a valid expression | |
| 959 | |
| 960 @example | |
| 961 [a, b, c] = [u, s, v] = svd (a) | |
| 962 @end example | |
| 963 | |
| 964 @noindent | |
| 965 that is exactly equivalent to | |
| 966 | |
| 967 @example | |
| 968 @group | |
| 969 [u, s, v] = svd (a) | |
| 970 a = u | |
| 971 b = s | |
| 972 c = v | |
| 973 @end group | |
| 974 @end example | |
| 975 | |
| 976 In expressions like this, the number of values in each part of the | |
| 977 expression need not match. For example, the expression | |
| 978 | |
| 979 @example | |
| 980 [a, b] = [u, s, v] = svd (a) | |
| 981 @end example | |
| 982 | |
| 983 @noindent | |
| 984 is equivalent to | |
| 985 | |
| 986 @example | |
| 987 @group | |
| 988 [u, s, v] = svd (a) | |
| 989 a = u | |
| 990 b = s | |
| 991 @end group | |
| 992 @end example | |
| 993 | |
| 6632 | 994 @noindent |
| 995 The number of values on the left side of the expression can, however, | |
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996 not exceed the number of values on the right side. For example, the |
| 6632 | 997 following will produce an error. |
| 998 | |
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999 @example |
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1000 @group |
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1001 [a, b, c, d] = [u, s, v] = svd (a); |
| 7031 | 1002 @print{} error: element number 4 undefined in return list |
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1003 @end group |
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1004 @end example |
| 6632 | 1005 |
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1006 The symbol @code{~} may be used as a placeholder in the list of lvalues, |
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1007 indicating that the corresponding return value should be ignored and not stored |
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1008 anywhere: |
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1009 |
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1010 @example |
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1011 @group |
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1012 [~, s, v] = svd (a); |
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1013 @end group |
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1014 @end example |
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1015 |
| 10228 | 1016 This is cleaner and more memory efficient than using a dummy variable. |
| 1017 The @code{nargout} value for the right-hand side expression is not affected. | |
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1018 If the assignment is used as an expression, the return value is a |
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1019 comma-separated list with the ignored values dropped. |
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1020 |
| 6642 | 1021 @opindex += |
| 1022 A very common programming pattern is to increment an existing variable | |
| 1023 with a given value, like this | |
| 1024 | |
| 1025 @example | |
| 1026 a = a + 2; | |
| 1027 @end example | |
| 1028 | |
| 1029 @noindent | |
| 1030 This can be written in a clearer and more condensed form using the | |
| 1031 @code{+=} operator | |
| 1032 | |
| 1033 @example | |
| 1034 a += 2; | |
| 1035 @end example | |
| 1036 | |
| 1037 @noindent | |
| 1038 @opindex -= | |
| 1039 @opindex *= | |
| 1040 @opindex /= | |
| 1041 Similar operators also exist for subtraction (@code{-=}), | |
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1042 multiplication (@code{*=}), and division (@code{/=}). An expression |
| 6642 | 1043 of the form |
| 1044 | |
| 1045 @example | |
| 1046 @var{expr1} @var{op}= @var{expr2} | |
| 1047 @end example | |
| 1048 | |
| 1049 @noindent | |
| 1050 is evaluated as | |
| 1051 | |
| 1052 @example | |
| 1053 @var{expr1} = (@var{expr1}) @var{op} (@var{expr2}) | |
| 1054 @end example | |
| 1055 | |
| 1056 @noindent | |
| 1057 where @var{op} can be either @code{+}, @code{-}, @code{*}, or @code{/}. | |
| 1058 So, the expression | |
| 1059 | |
| 1060 @example | |
| 1061 a *= b+1 | |
| 1062 @end example | |
| 1063 | |
| 1064 @noindent | |
| 1065 is evaluated as | |
| 1066 | |
| 1067 @example | |
| 1068 a = a * (b+1) | |
| 1069 @end example | |
| 1070 | |
| 1071 @noindent | |
| 1072 and @emph{not} | |
| 1073 | |
| 1074 @example | |
| 1075 a = a * b + 1 | |
| 1076 @end example | |
| 1077 | |
| 3294 | 1078 You can use an assignment anywhere an expression is called for. For |
| 1079 example, it is valid to write @code{x != (y = 1)} to set @code{y} to 1 | |
| 1080 and then test whether @code{x} equals 1. But this style tends to make | |
| 1081 programs hard to read. Except in a one-shot program, you should rewrite | |
| 1082 it to get rid of such nesting of assignments. This is never very hard. | |
| 1083 | |
| 1084 @cindex increment operator | |
| 1085 @cindex decrement operator | |
| 1086 @cindex operators, increment | |
| 1087 @cindex operators, decrement | |
| 1088 | |
| 4167 | 1089 @node Increment Ops |
| 3294 | 1090 @section Increment Operators |
| 1091 | |
| 1092 @emph{Increment operators} increase or decrease the value of a variable | |
| 1093 by 1. The operator to increment a variable is written as @samp{++}. It | |
| 1094 may be used to increment a variable either before or after taking its | |
| 1095 value. | |
| 1096 | |
| 1097 For example, to pre-increment the variable @var{x}, you would write | |
| 1098 @code{++@var{x}}. This would add one to @var{x} and then return the new | |
| 1099 value of @var{x} as the result of the expression. It is exactly the | |
| 1100 same as the expression @code{@var{x} = @var{x} + 1}. | |
| 1101 | |
| 1102 To post-increment a variable @var{x}, you would write @code{@var{x}++}. | |
| 1103 This adds one to the variable @var{x}, but returns the value that | |
| 1104 @var{x} had prior to incrementing it. For example, if @var{x} is equal | |
| 1105 to 2, the result of the expression @code{@var{x}++} is 2, and the new | |
| 1106 value of @var{x} is 3. | |
| 1107 | |
| 1108 For matrix and vector arguments, the increment and decrement operators | |
| 1109 work on each element of the operand. | |
| 1110 | |
| 1111 Here is a list of all the increment and decrement expressions. | |
| 1112 | |
| 1113 @table @code | |
| 1114 @item ++@var{x} | |
| 1115 @opindex ++ | |
| 1116 This expression increments the variable @var{x}. The value of the | |
| 1117 expression is the @emph{new} value of @var{x}. It is equivalent to the | |
| 1118 expression @code{@var{x} = @var{x} + 1}. | |
| 1119 | |
| 1120 @item --@var{x} | |
| 1121 @opindex @code{--} | |
| 1122 This expression decrements the variable @var{x}. The value of the | |
| 1123 expression is the @emph{new} value of @var{x}. It is equivalent to the | |
| 1124 expression @code{@var{x} = @var{x} - 1}. | |
| 1125 | |
| 1126 @item @var{x}++ | |
| 1127 @opindex ++ | |
| 1128 This expression causes the variable @var{x} to be incremented. The | |
| 1129 value of the expression is the @emph{old} value of @var{x}. | |
| 1130 | |
| 1131 @item @var{x}-- | |
| 1132 @opindex @code{--} | |
| 1133 This expression causes the variable @var{x} to be decremented. The | |
| 1134 value of the expression is the @emph{old} value of @var{x}. | |
| 1135 @end table | |
| 1136 | |
| 4167 | 1137 @node Operator Precedence |
| 3294 | 1138 @section Operator Precedence |
| 1139 @cindex operator precedence | |
| 1140 | |
| 1141 @dfn{Operator precedence} determines how operators are grouped, when | |
| 1142 different operators appear close by in one expression. For example, | |
| 1143 @samp{*} has higher precedence than @samp{+}. Thus, the expression | |
| 1144 @code{a + b * c} means to multiply @code{b} and @code{c}, and then add | |
| 1145 @code{a} to the product (i.e., @code{a + (b * c)}). | |
| 1146 | |
| 1147 You can overrule the precedence of the operators by using parentheses. | |
| 1148 You can think of the precedence rules as saying where the parentheses | |
| 1149 are assumed if you do not write parentheses yourself. In fact, it is | |
| 1150 wise to use parentheses whenever you have an unusual combination of | |
| 1151 operators, because other people who read the program may not remember | |
| 1152 what the precedence is in this case. You might forget as well, and then | |
| 1153 you too could make a mistake. Explicit parentheses will help prevent | |
| 1154 any such mistake. | |
| 1155 | |
| 1156 When operators of equal precedence are used together, the leftmost | |
| 1157 operator groups first, except for the assignment and exponentiation | |
| 1158 operators, which group in the opposite order. Thus, the expression | |
| 1159 @code{a - b + c} groups as @code{(a - b) + c}, but the expression | |
| 1160 @code{a = b = c} groups as @code{a = (b = c)}. | |
| 1161 | |
| 1162 The precedence of prefix unary operators is important when another | |
| 1163 operator follows the operand. For example, @code{-x^2} means | |
| 1164 @code{-(x^2)}, because @samp{-} has lower precedence than @samp{^}. | |
| 1165 | |
| 1166 Here is a table of the operators in Octave, in order of increasing | |
| 1167 precedence. | |
| 1168 | |
| 1169 @table @code | |
| 1170 @item statement separators | |
| 1171 @samp{;}, @samp{,}. | |
| 1172 | |
| 1173 @item assignment | |
| 6642 | 1174 @samp{=}, @samp{+=}, @samp{-=}, @samp{*=},@samp{/=}. This operator |
| 1175 groups right to left. | |
| 3294 | 1176 |
| 1177 @item logical "or" and "and" | |
| 1178 @samp{||}, @samp{&&}. | |
| 1179 | |
| 1180 @item element-wise "or" and "and" | |
| 1181 @samp{|}, @samp{&}. | |
| 1182 | |
| 1183 @item relational | |
| 1184 @samp{<}, @samp{<=}, @samp{==}, @samp{>=}, @samp{>}, @samp{!=}, | |
| 7594 | 1185 @samp{~=}. |
| 3294 | 1186 |
| 1187 @item colon | |
| 1188 @samp{:}. | |
| 1189 | |
| 1190 @item add, subtract | |
| 1191 @samp{+}, @samp{-}. | |
| 1192 | |
| 1193 @item multiply, divide | |
| 1194 @samp{*}, @samp{/}, @samp{\}, @samp{.\}, @samp{.*}, @samp{./}. | |
| 1195 | |
| 1196 @item transpose | |
| 1197 @samp{'}, @samp{.'} | |
| 1198 | |
| 1199 @item unary plus, minus, increment, decrement, and ``not'' | |
| 1200 @samp{+}, @samp{-}, @samp{++}, @samp{--}, @samp{!}, @samp{~}. | |
| 1201 | |
| 1202 @item exponentiation | |
| 1203 @samp{^}, @samp{**}, @samp{.^}, @samp{.**}. | |
| 1204 @end table |