ceph/src/boost/libs/numeric/ublas/doc/expression_concept.html

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  10 <title>Expression Concepts</title>
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  13 <h1><img src="../../../../boost.png" align="middle" />Expression Concepts</h1>
  14 <div class="toc" id="toc"></div>
  15 <h2><a name="scalar_expression"></a>Scalar Expression</h2>
  16 <h4>Description</h4>
  17 <p>A Scalar Expression is an expression convertible to a scalar
  18 type.</p>
  19 <h4>Refinement of</h4>
  20 <p>Default Constructible.</p>
  21 <h4>Associated types</h4>
  22 <table border="1" summary="associated types">
  23 <tbody>
  24 <tr>
  25 <td>Public base</td>
  26 <td>scaler_expression&lt;S&gt;</td>
  27 <td>S must be derived from this public base type.</td>
  28 </tr>
  29 <tr>
  30 <td>Value type</td>
  31 <td><code>value_type</code></td>
  32 <td>The type of the scalar expression.</td>
  33 </tr>
  34 </tbody>
  35 </table>
  36 <h4>Notation</h4>
  37 <table border="0" summary="notation">
  38 <tbody>
  39 <tr>
  40 <td><code>S</code></td>
  41 <td>A type that is a model of Scalar Expression</td>
  42 </tr>
  43 </tbody>
  44 </table>
  45 <h4>Definitions</h4>
  46 <h4>Valid expressions</h4>
  47 <p>In addition to the expressions defined in Default Constructible
  48 the following expressions must be valid.</p>
  49 <table border="1" summary="expressions">
  50 <tbody>
  51 <tr>
  52 <th>Name</th>
  53 <th>Expression</th>
  54 <th>Type requirements</th>
  55 <th>Return type</th>
  56 </tr>
  57 <tr>
  58 <td>Evaluation</td>
  59 <td><code>operator value_type () const</code></td>
  60 <td>&nbsp;</td>
  61 <td><code>value_type</code></td>
  62 </tr>
  63 </tbody>
  64 </table>
  65 <h4>Expression semantics</h4>
  66 <p>Semantics of an expression is defined only where it differs
  67 from, or is not defined in Default Constructible.</p>
  68 <table border="1" summary="semantics">
  69 <tbody>
  70 <tr>
  71 <th>Name</th>
  72 <th>Expression</th>
  73 <th>Precondition</th>
  74 <th>Semantics</th>
  75 <th>Postcondition</th>
  76 </tr>
  77 <tr>
  78 <td>Evaluation</td>
  79 <td><code>operator value_type () const</code></td>
  80 <td>&nbsp;</td>
  81 <td>&nbsp; Evaluates the scalar expression.</td>
  82 <td>&nbsp;</td>
  83 </tr>
  84 </tbody>
  85 </table>
  86 <h4>Complexity guarantees</h4>
  87 <p>The run-time complexity of the evaluation is specific for the
  88 evaluated scalar expression.</p>
  89 <h4>Invariants</h4>
  90 <h4>Models</h4>
  91 <ul>
  92 <li><code>vector_scalar_unary</code></li>
  93 <li><code>vector_scalar_binary</code></li>
  94 </ul>
  95 <h2><a name="vector_expression"></a>Vector Expression</h2>
  96 <h4>Description</h4>
  97 <p>A Vector Expression is an expression evaluatable to a vector.
  98 Vector Expression provides an <a href=
  99 "iterator_concept.html#indexed_bidirectional_iterator">Indexed Bidirectional
 100 Iterator</a> or an <a href=
 101 "iterator_concept.html#indexed_random_access_iterator">Indexed Random Access
 102 Iterator</a> .</p>
 103 <h4>Refinement of</h4>
 104 <p>Default Constructible.</p>
 105 <h4>Associated types</h4>
 106 <table border="1" summary="associated types">
 107 <tbody>
 108 <tr>
 109 <td>Public base</td>
 110 <td>vector_expression&lt;V&gt;</td>
 111 <td>V must be derived from this public base type.</td>
 112 </tr>
 113 <tr>
 114 <td>Value type</td>
 115 <td><code>value_type</code></td>
 116 <td>
 117 The element type of the vector expression.
 118 </td>
 119 </tr>
 120 <tr>
 121 <td>Reference type</td>
 122 <td><code>reference</code></td>
 123 <td>
 124 The return type when accessing an element of a vector expression.
 125 <br />
 126 Convertable to a<code>value_type</code>.
 127 </td>
 128 </tr>
 129 <tr>
 130 <td>Const reference type</td>
 131 <td><code>const_reference</code></td>
 132 <td>
 133 The return type when accessing an element of a constant vector expression.
 134 <br />
 135 Convertable to a<code>value_type</code>.
 136 </td>
 137 </tr>
 138 <tr>
 139 <td>Size type</td>
 140 <td><code>size_type</code></td>
 141 <td>
 142 The index type of the vector expression. Am unsigned integral type used to represent size and index values.
 143 <br />
 144 Can represent any nonnegative value of <code>difference_type</code>.
 145 </td>
 146 </tr>
 147 <tr>
 148 <td>Distance type</td>
 149 <td><code>difference_type</code></td>
 150 <td>
 151 A signed integral type used to represent the distance between two of the vector expression&#039;s iterators.
 152 </td>
 153 </tr>
 154 <tr>
 155 <td>Const iterator type</td>
 156 <td><code>const_iterator</code></td>
 157 <td>A type of iterator that may be used to examine a vector
 158 expression's elements.</td>
 159 </tr>
 160 <tr>
 161 <td>Iterator type</td>
 162 <td><code>iterator</code></td>
 163 <td>A type of iterator that may be used to modify a vector
 164 expression's elements.</td>
 165 </tr>
 166 <tr>
 167 <td>Const reverse iterator type</td>
 168 <td><code>const_reverse_iterator</code></td>
 169 <td>A Reverse Iterator adaptor whose base iterator type is the
 170 vector expression's const iterator type.</td>
 171 </tr>
 172 <tr>
 173 <td>Reverse iterator type</td>
 174 <td><code>reverse_iterator</code></td>
 175 <td>A Reverse Iterator adaptor whose base iterator type is the
 176 vector expression's iterator type.</td>
 177 </tr>
 178 </tbody>
 179 </table>
 180 <h4>Notation</h4>
 181 <table border="0" summary="notation">
 182 <tbody>
 183 <tr>
 184 <td><code>V</code></td>
 185 <td>A type that is a model of Vector Expression</td>
 186 </tr>
 187 <tr>
 188 <td><code>v, v1, v2</code></td>
 189 <td>Object of type <code>V</code></td>
 190 </tr>
 191 <tr>
 192 <td><code>i</code></td>
 193 <td>Object of a type convertible to <code>size_type</code></td>
 194 </tr>
 195 <tr>
 196 <td><code>t</code></td>
 197 <td>Object of a type convertible to <code>value_type</code></td>
 198 </tr>
 199 </tbody>
 200 </table>
 201 <h4>Definitions</h4>
 202 <h4>Valid expressions</h4>
 203 <p>In addition to the expressions defined in Default Constructible
 204 the following expressions must be valid.</p>
 205 <table border="1" summary="expressions">
 206 <tbody>
 207 <tr>
 208 <th>Name</th>
 209 <th>Expression</th>
 210 <th>Type requirements</th>
 211 <th>Return type</th>
 212 </tr>
 213 <tr>
 214 <td rowspan="2">Beginning of range</td>
 215 <td><code>v.begin ()</code></td>
 216 <td>&nbsp;</td>
 217 <td><code>const_iterator</code></td>
 218 </tr>
 219 <tr>
 220 <td><code>v.begin ()</code></td>
 221 <td><code>v</code> is mutable.</td>
 222 <td><code>iterator</code></td>
 223 </tr>
 224 <tr>
 225 <td rowspan="2">End of range</td>
 226 <td><code>v.end ()</code></td>
 227 <td>&nbsp;</td>
 228 <td><code>const_iterator</code></td>
 229 </tr>
 230 <tr>
 231 <td><code>v.end ()</code></td>
 232 <td><code>v</code> is mutable.</td>
 233 <td><code>iterator</code></td>
 234 </tr>
 235 <tr>
 236 <td>Size</td>
 237 <td><code>v.size ()</code></td>
 238 <td>&nbsp;</td>
 239 <td><code>size_type</code></td>
 240 </tr>
 241 <tr>
 242 <td>Swap</td>
 243 <td><code>v1.swap (v2)</code></td>
 244 <td><code>v1</code> and <code>v2</code> are mutable.</td>
 245 <td><code>void</code></td>
 246 </tr>
 247 <tr>
 248 <td rowspan="2">Beginning of reverse range</td>
 249 <td><code>v.rbegin ()</code></td>
 250 <td>&nbsp;</td>
 251 <td><code>const_reverse_iterator</code></td>
 252 </tr>
 253 <tr>
 254 <td><code>v.rbegin ()</code></td>
 255 <td><code>v</code> is mutable.</td>
 256 <td><code>reverse_iterator</code></td>
 257 </tr>
 258 <tr>
 259 <td rowspan="2">End of reverse range</td>
 260 <td><code>v.rend ()</code></td>
 261 <td>&nbsp;</td>
 262 <td><code>const_reverse_iterator</code></td>
 263 </tr>
 264 <tr>
 265 <td><code>v.rend ()</code></td>
 266 <td><code>v</code> is mutable.</td>
 267 <td><code>reverse_iterator</code></td>
 268 </tr>
 269 <tr>
 270 <td>Element access</td>
 271 <td><code>v (i)</code></td>
 272 <td><code>i</code> is convertible to <code>size_type</code>.</td>
 273 <td>Convertible to <code>value_type</code>.</td>
 274 </tr>
 275 <tr>
 276 <td rowspan="2">Assignment</td>
 277 <td><code>v2 = v1</code></td>
 278 <td><code>v2</code> is mutable and <code>v1</code> is convertible
 279 to <code>V</code>.</td>
 280 <td><code>V &amp;</code></td>
 281 </tr>
 282 <tr>
 283 <td><code>v2.assign (v1)</code></td>
 284 <td><code>v2</code> is mutable and <code>v1</code> is convertible
 285 to <code>V</code>.</td>
 286 <td><code>V &amp;</code></td>
 287 </tr>
 288 <tr>
 289 <td rowspan="5">Computed assignment</td>
 290 <td><code>v2 += v1</code></td>
 291 <td><code>v2</code> is mutable and <code>v1</code> is convertible
 292 to <code>V</code>.</td>
 293 <td><code>V &amp;</code></td>
 294 </tr>
 295 <tr>
 296 <td><code>v2.plus_assign (v1)</code></td>
 297 <td><code>v2</code> is mutable and <code>v1</code> is convertible
 298 to <code>V</code>.</td>
 299 <td><code>V &amp;</code></td>
 300 </tr>
 301 <tr>
 302 <td><code>v2 -= v1</code></td>
 303 <td><code>v2</code> is mutable and <code>v1</code> is convertible
 304 to <code>V</code>.</td>
 305 <td><code>V &amp;</code></td>
 306 </tr>
 307 <tr>
 308 <td><code>v2.minus_assign (v1)</code></td>
 309 <td><code>v2</code> is mutable and <code>v1</code> is convertible
 310 to <code>V</code>.</td>
 311 <td><code>V &amp;</code></td>
 312 </tr>
 313 <tr>
 314 <td><code>v *= t</code></td>
 315 <td><code>v</code> is mutable and <code>t</code> is convertible to
 316 <code>value_type</code>.</td>
 317 <td><code>V &amp;</code></td>
 318 </tr>
 319 </tbody>
 320 </table>
 321 <h4>Expression semantics</h4>
 322 <p>Semantics of an expression is defined only where it differs
 323 from, or is not defined in Default Constructible.</p>
 324 <table border="1" summary="semantics">
 325 <tbody>
 326 <tr>
 327 <th>Name</th>
 328 <th>Expression</th>
 329 <th>Precondition</th>
 330 <th>Semantics</th>
 331 <th>Postcondition</th>
 332 </tr>
 333 <tr>
 334 <td>Beginning of range</td>
 335 <td><code>v.begin ()</code></td>
 336 <td>&nbsp;</td>
 337 <td>Returns an iterator pointing to the first element in the vector
 338 expression.</td>
 339 <td><code>v.begin ()</code> is either dereferenceable or
 340 past-the-end. It is past-the-end if and only if <code>v.size () ==
 341 0</code>.</td>
 342 </tr>
 343 <tr>
 344 <td>End of range</td>
 345 <td><code>v.end ()</code></td>
 346 <td>&nbsp;</td>
 347 <td>Returns an iterator pointing one past the last element in the
 348 vector expression.</td>
 349 <td><code>v.end ()</code> is past-the-end.</td>
 350 </tr>
 351 <tr>
 352 <td>Size</td>
 353 <td><code>v.size ()</code></td>
 354 <td>&nbsp;</td>
 355 <td>Returns the size of the vector expression, that is, its number
 356 of elements.</td>
 357 <td><code>v.size () &gt;= 0</code></td>
 358 </tr>
 359 <tr>
 360 <td>Swap</td>
 361 <td><code>v1.swap (v2)</code></td>
 362 <td>&nbsp;</td>
 363 <td>Equivalent to <code>swap (v1, v2)</code>.</td>
 364 <td>&nbsp;</td>
 365 </tr>
 366 <tr>
 367 <td>Beginning of reverse range</td>
 368 <td><code>v.rbegin ()</code></td>
 369 <td>&nbsp;</td>
 370 <td>Equivalent to <code>reverse_iterator (v.end ())</code>.</td>
 371 <td><code>v.rbegin ()</code> is either dereferenceable or
 372 past-the-end. It is past-the-end if and only if <code>v.size () ==
 373 0</code>.</td>
 374 </tr>
 375 <tr>
 376 <td>End of reverse range</td>
 377 <td><code>v.rend ()</code></td>
 378 <td>&nbsp;</td>
 379 <td>Equivalent to <code>reverse_iterator (v.begin ())</code>.</td>
 380 <td><code>v.rend ()</code> is past-the-end.</td>
 381 </tr>
 382 <tr>
 383 <td>Element access</td>
 384 <td><code>v (i)</code></td>
 385 <td><code>0 &lt;= i &lt; v.size ()</code></td>
 386 <td>Returns the <code>i</code>-th element of the vector
 387 expression.</td>
 388 <td>&nbsp;</td>
 389 </tr>
 390 <tr>
 391 <td rowspan="2">Assignment</td>
 392 <td><code>v2 = v1</code></td>
 393 <td><code>v1.size () == v2.size ()</code></td>
 394 <td>Assigns every element of the evaluated vector expression
 395 <code>v1</code> to the corresponding element of <code>v2</code>
 396 .</td>
 397 <td>&nbsp;</td>
 398 </tr>
 399 <tr>
 400 <td><code>v2.assign (v1)</code></td>
 401 <td><code>v1.size () == v2.size ()</code></td>
 402 <td>Assigns every element of <code>v1</code> to the corresponding
 403 element of <code>v2</code>.</td>
 404 <td>&nbsp;</td>
 405 </tr>
 406 <tr>
 407 <td rowspan="5">Computed assignment</td>
 408 <td><code>v2 += v1</code></td>
 409 <td><code>v1.size () == v2.size ()</code></td>
 410 <td>Adds every element of the evaluated vector expression
 411 <code>v1</code> to the corresponding element of
 412 <code>v2</code>.</td>
 413 <td>&nbsp;</td>
 414 </tr>
 415 <tr>
 416 <td><code>v2.plus_assign (v1)</code></td>
 417 <td><code>v1.size () == v2.size ()</code></td>
 418 <td>Adds every element of <code>v1</code> to the corresponding
 419 element of <code>v2</code>.</td>
 420 <td>&nbsp;</td>
 421 </tr>
 422 <tr>
 423 <td><code>v2 -= v1</code></td>
 424 <td><code>v1.size () == v2.size ()</code></td>
 425 <td>Subtracts every element of the evaluated vector expression
 426 <code>v1</code> from the corresponding element of <code>v2</code>
 427 .</td>
 428 <td>&nbsp;</td>
 429 </tr>
 430 <tr>
 431 <td><code>v2.minus_assign (v1)</code></td>
 432 <td><code>v1.size () == v2.size ()</code></td>
 433 <td>Subtracts every element of <code>v1</code> from the
 434 corresponding element of <code>v2</code>.</td>
 435 <td>&nbsp;</td>
 436 </tr>
 437 <tr>
 438 <td><code>v *= t</code></td>
 439 <td>&nbsp;</td>
 440 <td>Multiplies every element of <code>v</code> with <code>t</code>
 441 .</td>
 442 <td>&nbsp;</td>
 443 </tr>
 444 </tbody>
 445 </table>
 446 <h4>Complexity guarantees</h4>
 447 <p>The run-time complexity of <code>begin ()</code> and <code>end
 448 ()</code> is specific for the evaluated vector expression,
 449 typically amortized constant time.</p>
 450 <p>The run-time complexity of <code>size ()</code> is constant
 451 time.</p>
 452 <p>The run-time complexity of <code>swap ()</code> is specific for
 453 the evaluated vector expression, typically constant time.</p>
 454 <p>The run-time complexity of <code>rbegin ()</code> and <code>rend
 455 ()</code> is specific for the evaluated vector expression,
 456 typically amortized constant time.</p>
 457 <p>The run-time complexity of the element access is specific for
 458 the evaluated vector expression, typically amortized constant time
 459 for the dense and logarithmic for the sparse case.</p>
 460 <p>The run-time complexity of the arithmetic operations is specific
 461 for the evaluated vector expressions, typically linear in the size
 462 of the expressions.</p>
 463 <h4>Invariants</h4>
 464 <table border="1" summary="invariants">
 465 <tbody>
 466 <tr>
 467 <td>Valid range</td>
 468 <td>For any vector expression <code>v</code>, <code>[v.begin (),
 469 v.end ())</code> is a valid range.</td>
 470 </tr>
 471 <tr>
 472 <td>Completeness</td>
 473 <td>An algorithm that iterates through the range <code>[v.begin (),
 474 v.end ())</code> will pass through every element of <code>v</code>
 475 .</td>
 476 </tr>
 477 <tr>
 478 <td>Valid reverse range</td>
 479 <td><code>[v.rbegin (), v.rend ())</code> is a valid range.</td>
 480 </tr>
 481 <tr>
 482 <td>Equivalence of ranges</td>
 483 <td>The distance from <code>v.begin ()</code> to <code>v.end
 484 ()</code> is the same as the distance from <code>v.rbegin ()</code>
 485 to <code>v.rend ()</code>.</td>
 486 </tr>
 487 </tbody>
 488 </table>
 489 <h4>Models</h4>
 490 <ul>
 491 <li><code>vector_range;</code></li>
 492 <li><code>vector_slice</code></li>
 493 <li><code>matrix_row</code></li>
 494 <li><code>matrix_column</code></li>
 495 <li><code>matrix_vector_range</code></li>
 496 <li><code>matrix_vector_slice</code></li>
 497 <li><code>vector_unary</code></li>
 498 <li><code>vector_binary</code></li>
 499 <li><code>vector_binary_scalar1</code></li>
 500 <li><code>vector_binary_scalar2</code></li>
 501 <li><code>matrix_vector_unary1</code></li>
 502 <li><code>matrix_vector_unary2</code></li>
 503 <li><code>matrix_vector_binary1</code></li>
 504 <li><code>matrix_vector_binary2</code></li>
 505 </ul>
 506
 507 <h2><a name="matrix_expression"></a>Matrix Expression</h2>
 508 <h4>Description</h4>
 509 <p>A Matrix Expression is an expression evaluatable to a matrix.
 510 Matrix Expression provides an <a href=
 511 "iterator_concept.html#indexed_bidirectional_cr_iterator">Indexed
 512 Bidirectional Column/Row Iterator</a> or an <a href=
 513 "iterator_concept.html#indexed_random_access_cr_iterator">Indexed Random
 514 Access Column/Row Iterator</a> .</p>
 515 <h4>Refinement of</h4>
 516 <p>Default Constructible.</p>
 517
 518 <h4>Associated types</h4>
 519 <h5>immutable types</h5>
 520 <table border="1" summary="associated immutable types" title="">
 521 <tbody>
 522 <tr>
 523 <td>Public base</td>
 524 <td><code>matrix_expression&lt;M&gt;</code></td>
 525 <td>M must be derived from this public base type.</td>
 526 </tr>
 527 <tr>
 528 <td>Value type</td>
 529 <td><code>value_type</code></td>
 530 <td>
 531 The element type of the matrix expression.
 532 </td>
 533 </tr>
 534 <tr>
 535 <td>Const reference type</td>
 536 <td><code>const_reference</code></td>
 537 <td>
 538 The return type when accessing an element of a constant matrix expression.
 539 <br />
 540 Convertable to a <code>value_type</code>.
 541 </td>
 542 </tr>
 543 <tr>
 544 <td>Size type</td>
 545 <td><code>size_type</code></td>
 546 <td>
 547 The index type of the matrix expression. Am unsigned integral type used to represent size and index values.
 548 <br />
 549 Can represent any nonnegative value of <code>difference_type</code>.
 550 </td>
 551 </tr>
 552 <tr>
 553 <td>Distance type</td>
 554 <td><code>difference_type</code></td>
 555 <td>
 556 A signed integral type used to represent the distance between two of the matrix expression&#039;s iterators.
 557 </td>
 558 </tr>
 559 <tr>
 560 <td rowspan="2">Const iterator types</td>
 561 <td><code>const_iterator1</code></td>
 562 <td>A type of column iterator that may be used to examine a matrix
 563 expression's elements.</td>
 564 </tr>
 565 <tr>
 566 <td><code>const_iterator2</code></td>
 567 <td>A type of row iterator that may be used to examine a matrix
 568 expression's elements.</td>
 569 </tr>
 570 <tr>
 571 <td rowspan="2">Const reverse iterator types</td>
 572 <td><code>const_reverse_iterator1</code></td>
 573 <td>A Reverse Iterator adaptor whose base iterator type is the
 574 matrix expression's const column iterator type.</td>
 575 </tr>
 576 <tr>
 577 <td><code>const_reverse_iterator2</code></td>
 578 <td>A Reverse Iterator adaptor whose base iterator type is the
 579 matrix expression's const row iterator type.</td>
 580 </tr>
 581 </tbody>
 582 </table>
 583
 584 <h5>mutable types</h5>
 585 <table border="1" summary="associated mutable types">
 586 <tbody>
 587 <tr>
 588 <td>Reference type</td>
 589 <td><code>reference</code></td>
 590 <td>
 591 The return type when accessing an element of a matrix expression.
 592 <br />
 593 Convertable to a <code>value_type</code>.
 594 </td>
 595 </tr>
 596 <tr>
 597 <td rowspan="2">Iterator types</td>
 598 <td><code>iterator1</code></td>
 599 <td>A type of column iterator that may be used to modify a matrix
 600 expression's elements.</td>
 601 </tr>
 602 <tr>
 603 <td><code>iterator2</code></td>
 604 <td>A type of row iterator that may be used to modify a matrix
 605 expression's elements.</td>
 606 </tr>
 607 <tr>
 608 <td rowspan="2">Reverse iterator types</td>
 609 <td><code>reverse_iterator1</code></td>
 610 <td>A Reverse Iterator adaptor whose base iterator type is the
 611 matrix expression's column iterator type.</td>
 612 </tr>
 613 <tr>
 614 <td><code>reverse_iterator2</code></td>
 615 <td>A Reverse Iterator adaptor whose base iterator type is the
 616 matrix expression's row iterator type.</td>
 617 </tr>
 618 </tbody>
 619 </table>
 620
 621
 622 <h4>Notation</h4>
 623 <table border="0" summary="notation">
 624 <tbody>
 625 <tr>
 626 <td><code>M</code></td>
 627 <td>A type that is a model of Matrix Expression</td>
 628 </tr>
 629 <tr>
 630 <td><code>m, m1, m2</code></td>
 631 <td>Object of type <code>M</code></td>
 632 </tr>
 633 <tr>
 634 <td><code>i, j</code></td>
 635 <td>Objects of a type convertible to <code>size_type</code></td>
 636 </tr>
 637 <tr>
 638 <td><code>t</code></td>
 639 <td>Object of a type convertible to <code>value_type</code></td>
 640 </tr>
 641 </tbody>
 642 </table>
 643 <h4>Definitions</h4>
 644 <h4>Valid expressions</h4>
 645 <p>In addition to the expressions defined in Default Constructible
 646 the following expressions must be valid.</p>
 647
 648 <h5>immutable expressions</h5>
 649 <table border="1" summary="expressions">
 650 <thead>
 651 <tr>
 652 <th>Name</th>
 653 <th>Expression</th>
 654 <th>Type requirements</th>
 655 <th>Return type</th>
 656 </tr>
 657 </thead>
 658 <tbody>
 659 <tr>
 660 <td rowspan="2">Size</td>
 661 <td><code>m.size1 ()</code></td>
 662 <td>&nbsp;</td>
 663 <td><code>size_type</code></td>
 664 </tr>
 665 <tr>
 666 <td><code>m.size2 ()</code></td>
 667 <td>&nbsp;</td>
 668 <td><code>size_type</code></td>
 669 </tr>
 670 </tbody>
 671 </table>
 672
 673 <h5>possibly mutable expressions</h5>
 674 <table border="1" summary="expressions">
 675 <tbody>
 676 <tr>
 677 <th>Name</th>
 678 <th>Expression</th>
 679 <th>Type requirements</th>
 680 <th>Return type</th>
 681 </tr>
 682 <tr>
 683 <td rowspan="4">Beginning of range</td>
 684 <td><code>m.begin1 ()</code></td>
 685 <td>&nbsp;</td>
 686 <td><code>const_iterator1</code></td>
 687 </tr>
 688 <tr>
 689 <td><code>m.begin2 ()</code></td>
 690 <td>&nbsp;</td>
 691 <td><code>const_iterator2</code></td>
 692 </tr>
 693 <tr>
 694 <td><code>m.begin1 ()</code></td>
 695 <td><code>m</code> is mutable.&nbsp;</td>
 696 <td><code>iterator1</code></td>
 697 </tr>
 698 <tr>
 699 <td><code>m.begin2 ()</code></td>
 700 <td><code>m</code> is mutable.</td>
 701 <td><code>iterator2</code></td>
 702 </tr>
 703 <tr>
 704 <td rowspan="4">End of range</td>
 705 <td><code>m.end1 ()</code></td>
 706 <td>&nbsp;</td>
 707 <td><code>const_iterator1</code></td>
 708 </tr>
 709 <tr>
 710 <td><code>m.end2 ()</code></td>
 711 <td>&nbsp;</td>
 712 <td><code>const_iterator2</code></td>
 713 </tr>
 714 <tr>
 715 <td><code>m.end1 ()</code></td>
 716 <td><code>m</code> is mutable.&nbsp;</td>
 717 <td><code>iterator1</code></td>
 718 </tr>
 719 <tr>
 720 <td><code>m.end2 ()</code></td>
 721 <td><code>m</code> is mutable.</td>
 722 <td><code>iterator2</code></td>
 723 </tr>
 724 <tr>
 725 <td>Swap</td>
 726 <td><code>m1.swap (m2)</code></td>
 727 <td><code>m1</code> and <code>m2</code> are mutable.&nbsp;</td>
 728 <td><code>void</code></td>
 729 </tr>
 730 <tr>
 731 <td rowspan="4">Beginning of reverse range</td>
 732 <td><code>m.rbegin1 ()</code></td>
 733 <td>&nbsp;</td>
 734 <td><code>const_reverse_iterator1</code></td>
 735 </tr>
 736 <tr>
 737 <td><code>m.rbegin2 ()</code></td>
 738 <td>&nbsp;</td>
 739 <td><code>const_reverse_iterator2</code></td>
 740 </tr>
 741 <tr>
 742 <td><code>m.rbegin1 ()</code></td>
 743 <td><code>m</code> is mutable.&nbsp;</td>
 744 <td><code>reverse_iterator1</code></td>
 745 </tr>
 746 <tr>
 747 <td><code>m.rbegin2 ()</code></td>
 748 <td><code>m</code> is mutable.</td>
 749 <td><code>reverse_iterator2</code></td>
 750 </tr>
 751 <tr>
 752 <td rowspan="4">End of reverse range</td>
 753 <td><code>m.rend1 ()</code></td>
 754 <td>&nbsp;</td>
 755 <td><code>const_reverse_iterator1</code></td>
 756 </tr>
 757 <tr>
 758 <td><code>m.rend2 ()</code></td>
 759 <td>&nbsp;</td>
 760 <td><code>const_reverse_iterator2</code></td>
 761 </tr>
 762 <tr>
 763 <td><code>m.rend1 ()</code></td>
 764 <td><code>m</code> is mutable.</td>
 765 <td><code>reverse_iterator1</code></td>
 766 </tr>
 767 <tr>
 768 <td><code>m.rend2 ()</code></td>
 769 <td><code>m</code> is mutable.</td>
 770 <td><code>reverse_iterator2</code></td>
 771 </tr>
 772 <tr>
 773 <td>Element access</td>
 774 <td><code>m (i, j)</code></td>
 775 <td><code>i</code> and <code>j</code> are convertible to
 776 <code>size_type</code> .</td>
 777 <td>Convertible to <code>value_type</code>.</td>
 778 </tr>
 779 <tr>
 780 <td rowspan="2">Assignment</td>
 781 <td><code>m2 = m1</code></td>
 782 <td><code>m2</code> is mutable and <code>m1</code> is convertible
 783 to <code>M</code>.</td>
 784 <td><code>M &amp;</code></td>
 785 </tr>
 786 <tr>
 787 <td><code>m2.assign (m1)</code></td>
 788 <td><code>m2</code> is mutable and <code>m1</code> is convertible
 789 to <code>M</code>.</td>
 790 <td><code>M &amp;</code></td>
 791 </tr>
 792 <tr>
 793 <td rowspan="5">Computed assignment</td>
 794 <td><code>m2 += m1</code></td>
 795 <td><code>m2</code> is mutable and <code>m1</code> is convertible
 796 to <code>M</code>.</td>
 797 <td><code>M &amp;</code></td>
 798 </tr>
 799 <tr>
 800 <td><code>m2.plus_assign (m1)</code></td>
 801 <td><code>m2</code> is mutable and <code>m1</code> is convertible
 802 to <code>M</code>.</td>
 803 <td><code>M &amp;</code></td>
 804 </tr>
 805 <tr>
 806 <td><code>m2 -= m1</code></td>
 807 <td><code>m2</code> is mutable and <code>m1</code> is convertible
 808 to <code>M</code>.</td>
 809 <td><code>M &amp;</code></td>
 810 </tr>
 811 <tr>
 812 <td><code>m2.minus_assign (m1)</code></td>
 813 <td><code>m2</code> is mutable and <code>m1</code> is convertible
 814 to <code>M</code>.</td>
 815 <td><code>M &amp;</code></td>
 816 </tr>
 817 <tr>
 818 <td><code>m *= t</code></td>
 819 <td><code>m</code> is mutable and <code>t</code> is convertible to
 820 <code>value_type</code>.</td>
 821 <td><code>M &amp;</code></td>
 822 </tr>
 823 </tbody>
 824 </table>
 825 <h4>Expression semantics</h4>
 826 <p>Semantics of an expression is defined only where it differs
 827 from, or is not defined in Default Constructible.</p>
 828 <table border="1" summary="semantics">
 829 <tbody>
 830 <tr>
 831 <th>Name</th>
 832 <th>Expression</th>
 833 <th>Precondition</th>
 834 <th>Semantics</th>
 835 <th>Postcondition</th>
 836 </tr>
 837 <tr>
 838 <td rowspan="2">Beginning of range</td>
 839 <td><code>m.begin1 ()</code></td>
 840 <td>&nbsp;</td>
 841 <td>Returns an iterator pointing to the first element in the first
 842 column of a matrix expression.</td>
 843 <td><code>m.begin1 ()</code> is either dereferenceable or
 844 past-the-end. It is past-the-end if and only if <code>m.size1 () ==
 845 0</code>.</td>
 846 </tr>
 847 <tr>
 848 <td><code>m.begin2 ()</code></td>
 849 <td>&nbsp;</td>
 850 <td>Returns an iterator pointing to the first element in the first
 851 row of a matrix expression.</td>
 852 <td><code>m.begin2 ()</code> is either dereferenceable or
 853 past-the-end. It is past-the-end if and only if <code>m.size2 () ==
 854 0</code>.</td>
 855 </tr>
 856 <tr>
 857 <td rowspan="2">End of range</td>
 858 <td><code>m.end1 ()</code></td>
 859 <td>&nbsp;</td>
 860 <td>Returns an iterator pointing one past the last element in the
 861 matrix expression.</td>
 862 <td><code>m.end1 ()</code> is past-the-end.</td>
 863 </tr>
 864 <tr>
 865 <td><code>m.end2 ()</code></td>
 866 <td>&nbsp;</td>
 867 <td>Returns an iterator pointing one past the last element in the
 868 matrix expression.</td>
 869 <td><code>m.end2 ()</code> is past-the-end.</td>
 870 </tr>
 871 <tr>
 872 <td rowspan="2">Size</td>
 873 <td><code>m.size1 ()</code></td>
 874 <td>&nbsp;</td>
 875 <td>Returns the number of rows of the matrix expression.</td>
 876 <td><code>m.size1 () &gt;= 0</code></td>
 877 </tr>
 878 <tr>
 879 <td><code>m.size2 ()</code></td>
 880 <td>&nbsp;</td>
 881 <td>Returns the number of columns of the matrix expression.</td>
 882 <td><code>m.size2 () &gt;= 0</code></td>
 883 </tr>
 884 <tr>
 885 <td>Swap</td>
 886 <td><code>m1.swap (m2)</code></td>
 887 <td>&nbsp;</td>
 888 <td>Equivalent to <code>swap (m1, m2)</code>.</td>
 889 <td>&nbsp;</td>
 890 </tr>
 891 <tr>
 892 <td rowspan="2">Beginning of reverse range</td>
 893 <td><code>m.rbegin1 ()</code></td>
 894 <td>&nbsp;</td>
 895 <td>Equivalent to <code>reverse_iterator1 (m.end1 ())</code>.</td>
 896 <td><code>m.rbegin1 ()</code> is either dereferenceable or
 897 past-the-end. It is past-the-end if and only if <code>m.size1 () ==
 898 0</code>.</td>
 899 </tr>
 900 <tr>
 901 <td><code>m.rbegin2 ()</code></td>
 902 <td>&nbsp;</td>
 903 <td>Equivalent to <code>reverse_iterator2 (m.end2 ())</code>.</td>
 904 <td><code>m.rbegin2 ()</code> is either dereferenceable or
 905 past-the-end. It is past-the-end if and only if <code>m.size2 () ==
 906 0</code>.</td>
 907 </tr>
 908 <tr>
 909 <td rowspan="2">End of reverse range</td>
 910 <td><code>m.rend1 ()</code></td>
 911 <td>&nbsp;</td>
 912 <td>Equivalent to <code>reverse_iterator1 (m.begin1
 913 ())</code>.</td>
 914 <td><code>m.rend1 ()</code> is past-the-end.</td>
 915 </tr>
 916 <tr>
 917 <td><code>m.rend2 ()</code></td>
 918 <td>&nbsp;</td>
 919 <td>Equivalent to <code>reverse_iterator2 (m.begin2
 920 ())</code>.</td>
 921 <td><code>m.rend2 ()</code> is past-the-end.</td>
 922 </tr>
 923 <tr>
 924 <td>Element access</td>
 925 <td><code>m (i, j)</code></td>
 926 <td><code>0 &lt;= i &lt; m.size1 ()</code> and <code>0 &lt;= j &lt;
 927 m.size2 ()</code></td>
 928 <td>Returns the <code>j</code>-th element of the <code>i</code>-th
 929 row of the matrix expression.</td>
 930 <td>&nbsp;</td>
 931 </tr>
 932 <tr>
 933 <td rowspan="2">Assignment</td>
 934 <td><code>m2 = m1</code></td>
 935 <td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
 936 m1.size2 () == m2.size2 ()</code></td>
 937 <td>Assigns every element of the evaluated matrix expression
 938 <code>m1</code> to the corresponding element of <code>m2</code>
 939 .</td>
 940 <td>&nbsp;</td>
 941 </tr>
 942 <tr>
 943 <td><code>m2.assign (m1)</code></td>
 944 <td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
 945 m1.size2 () == m2.size2 ()</code></td>
 946 <td>Assigns every element of <code>m1</code> to the corresponding
 947 element of <code>m2</code>.</td>
 948 <td>&nbsp;</td>
 949 </tr>
 950 <tr>
 951 <td rowspan="5">Computed assignment</td>
 952 <td><code>m2 += m1</code></td>
 953 <td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
 954 m1.size2 () == m2.size2 ()</code></td>
 955 <td>Adds every element of the evaluated matrix expression
 956 <code>m1</code> to the corresponding element of
 957 <code>m2</code>.</td>
 958 <td>&nbsp;</td>
 959 </tr>
 960 <tr>
 961 <td><code>m2.plus_assign (m1)</code></td>
 962 <td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
 963 m1.size2 () == m2.size2 ()</code></td>
 964 <td>Adds every element of <code>m1</code> to the corresponding
 965 element of <code>m2</code>.</td>
 966 <td>&nbsp;</td>
 967 </tr>
 968 <tr>
 969 <td><code>m2 -= m1</code></td>
 970 <td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
 971 m1.size2 () == m2.size2 ()</code></td>
 972 <td>Subtracts every element of the evaluated matrix expression
 973 <code>m1</code> from the corresponding element of <code>m2</code>
 974 .</td>
 975 <td>&nbsp;</td>
 976 </tr>
 977 <tr>
 978 <td><code>m2.minus_assign (m1)</code></td>
 979 <td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
 980 m1.size2 () == m2.size2 ()</code></td>
 981 <td>Subtracts every element of <code>m1</code> from the
 982 corresponding element of <code>m2</code>.</td>
 983 <td>&nbsp;</td>
 984 </tr>
 985 <tr>
 986 <td><code>m *= t</code></td>
 987 <td>&nbsp;</td>
 988 <td>Multiplies every element of <code>m</code> with <code>t</code>
 989 .</td>
 990 <td>&nbsp;</td>
 991 </tr>
 992 </tbody>
 993 </table>
 994 <h4>Complexity guarantees</h4>
 995 <p>The run-time complexity of <code>begin1 ()</code>, <code>begin2
 996 ()</code> , <code>end1 ()</code> and <code>end2 ()</code> is
 997 specific for the evaluated matrix expression.</p>
 998 <p>The run-time complexity of <code>size1 ()</code> and <code>size2
 999 ()</code> is constant time.</p>
1000 <p>The run-time complexity of <code>swap ()</code> is specific for
1001 the evaluated matrix expression, typically constant time.</p>
1002 <p>The run-time complexity of <code>rbegin1 ()</code>,
1003 <code>rbegin2 ()</code> , <code>rend1 ()</code> and <code>rend2
1004 ()</code> is specific for the evaluated matrix expression.</p>
1005 <p>The run-time complexity of the element access is specific for
1006 the evaluated matrix expression, typically amortized constant time
1007 for the dense and logarithmic for the sparse case.</p>
1008 <p>The run-time complexity of the arithmetic operations is specific
1009 for the evaluated matrix expressions, typically quadratic in the
1010 size of the proxies.</p>
1011 <h4>Invariants</h4>
1012 <table border="1" summary="invariants">
1013 <tbody>
1014 <tr>
1015 <td>Valid range</td>
1016 <td>For any matrix expression <code>m</code>, <code>[m.begin1 (),
1017 m.end1 ())</code> and <code>[m.begin2 (), m.end2 ())</code> are
1018 valid ranges.</td>
1019 </tr>
1020 <tr>
1021 <td>Completeness</td>
1022 <td>An algorithm that iterates through the range <code>[m.begin1
1023 (), m.end1 ())</code> will pass through every row of <code>m</code>
1024 , an algorithm that iterates through the range <code>[m.begin2 (),
1025 m.end2 ())</code> will pass through every column of <code>m</code>
1026 .</td>
1027 </tr>
1028 <tr>
1029 <td>Valid reverse range</td>
1030 <td><code>[m.rbegin1 (), m.rend1 ())</code> and <code>[m.rbegin2
1031 (), m.rend2 ())</code> are valid ranges.</td>
1032 </tr>
1033 <tr>
1034 <td>Equivalence of ranges</td>
1035 <td>The distance from <code>m.begin1 ()</code> to <code>m.end1
1036 ()</code> is the same as the distance from <code>m.rbegin1
1037 ()</code> to <code>m.rend1 ()</code> and the distance from
1038 <code>m.begin2 ()</code> to <code>m.end2 ()</code> is the same as
1039 the distance from <code>m.rbegin2 ()</code> to <code>m.rend2
1040 ()</code>.</td>
1041 </tr>
1042 </tbody>
1043 </table>
1044 <h4>Models</h4>
1045 <ul>
1046 <li><code>matrix_range</code></li>
1047 <li><code>matrix_slice;</code></li>
1048 <li><code>triangular_adaptor</code></li>
1049 <li><code>symmetric_adaptor</code></li>
1050 <li><code>banded_adaptor</code></li>
1051 <li><code>vector_matrix_binary</code></li>
1052 <li><code>matrix_unary1</code></li>
1053 <li><code>matrix_unary2</code></li>
1054 <li><code>matrix_binary</code></li>
1055 <li><code>matrix_binary_scalar1</code></li>
1056 <li><code>matrix_binary_scalar2</code></li>
1057 <li><code>matrix_matrix_binary</code></li>
1058 </ul>
1059 <hr />
1060 <p>Copyright (&copy;) 2000-2002 Joerg Walter, Mathias Koch<br />
1061    Use, modification and distribution are subject to the
1062    Boost Software License, Version 1.0.
1063    (See accompanying file LICENSE_1_0.txt
1064    or copy at <a href="http://www.boost.org/LICENSE_1_0.txt">
1065       http://www.boost.org/LICENSE_1_0.txt
1066    </a>).
1067 </p>
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