1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
|
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta name="generator" content=
"HTML Tidy for Linux/x86 (vers 1st March 2004), see www.w3.org" />
<meta http-equiv="Content-Type" content=
"text/html; charset=us-ascii" />
<link href="ublas.css" type="text/css" />
<title>Triangular Matrix</title>
</head>
<body>
<h1><img src="../../../../boost.png" align="middle" />
Triangular Matrix</h1>
<h2><a name="triangular_matrix" id="triangular_matrix"></a>Triangular Matrix</h2>
<h4>Description</h4>
<p>The templated class <code>triangular_matrix<T, F1, F2,
A></code> is the base container adaptor for triangular matrices.
For a <em>(n x n</em> )-dimensional lower triangular matrix and
<em>0 <= i < n</em>,<em>0 <= j < n</em> holds
<em>t</em><sub><em>i, j</em></sub> <em>= 0</em> , if <em>i >
j</em>. If furthermore holds t<sub><em>i, i</em></sub><em>= 1</em>
the matrix is called unit lower triangular. For a <em>(n x n</em>
)-dimensional upper triangular matrix and <em>0 <= i <
n</em>,<em>0 <= j < n</em> holds <em>t</em><sub><em>i,
j</em></sub> <em>= 0</em> , if <em>i < j</em>. If furthermore
holds t<sub><em>i, i</em></sub><em>= 1</em> the matrix is called
unit lower triangular. The storage of triangular matrices is
packed.</p>
<h4>Example</h4>
<pre>
#include <boost/numeric/ublas/triangular.hpp>
#include <boost/numeric/ublas/io.hpp>
int main () {
using namespace boost::numeric::ublas;
triangular_matrix<double, lower> ml (3, 3);
for (unsigned i = 0; i < ml.size1 (); ++ i)
for (unsigned j = 0; j <= i; ++ j)
ml (i, j) = 3 * i + j;
std::cout << ml << std::endl;
triangular_matrix<double, upper> mu (3, 3);
for (unsigned i = 0; i < mu.size1 (); ++ i)
for (unsigned j = i; j < mu.size2 (); ++ j)
mu (i, j) = 3 * i + j;
std::cout << mu << std::endl;
}
</pre>
<h4>Definition</h4>
<p>Defined in the header triangular.hpp.</p>
<h4>Template parameters</h4>
<table border="1" summary="parameters">
<tbody>
<tr>
<th>Parameter</th>
<th>Description</th>
<th>Default</th>
</tr>
<tr>
<td><code>T</code></td>
<td>The type of object stored in the matrix.</td>
<td></td>
</tr>
<tr>
<td><code>F1</code></td>
<td>Functor describing the type of the triangular matrix. <a href=
"#triangular_matrix_1">[1]</a></td>
<td><code>lower</code></td>
</tr>
<tr>
<td><code>F2</code></td>
<td>Functor describing the storage organization. <a href=
"#triangular_matrix_2">[2]</a></td>
<td><code>row_major</code></td>
</tr>
<tr>
<td><code>A</code></td>
<td>The type of the adapted array. <a href=
"#triangular_matrix_3">[3]</a></td>
<td><code>unbounded_array<T></code></td>
</tr>
</tbody>
</table>
<h4>Model of</h4>
<p><a href="container_concept.htm#matrix">Matrix</a> .</p>
<h4>Type requirements</h4>
<p>None, except for those imposed by the requirements of <a href=
"container_concept.htm#matrix">Matrix</a> .</p>
<h4>Public base classes</h4>
<p><code>matrix_container<triangular_matrix<T, F1, F2, A>
></code></p>
<h4>Members</h4>
<table border="1" summary="members">
<tbody>
<tr>
<th>Member</th>
<th>Description</th>
</tr>
<tr>
<td><code>triangular_matrix ()</code></td>
<td>Allocates an uninitialized <code>triangular_matrix</code> that
holds zero rows of zero elements.</td>
</tr>
<tr>
<td><code>triangular_matrix (size_type size1, size_type
size2)</code></td>
<td>Allocates an uninitialized <code>triangular_matrix</code> that
holds <code>size1</code> rows of <code>size2</code> elements.</td>
</tr>
<tr>
<td><code>triangular_matrix (const triangular_matrix
&m)</code></td>
<td>The copy constructor.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_matrix (const matrix_expression<AE>
&ae)</code></td>
<td>The extended copy constructor.</td>
</tr>
<tr>
<td><code>void resize (size_type size1, size_type size2, bool
preserve = true)</code></td>
<td>Reallocates a <code>triangular_matrix</code> to hold
<code>size1</code> rows of <code>size2</code> elements. The
existing elements of the <code>triangular_matrix</code> are
preseved when specified.</td>
</tr>
<tr>
<td><code>size_type size1 () const</code></td>
<td>Returns the number of rows.</td>
</tr>
<tr>
<td><code>size_type size2 () const</code></td>
<td>Returns the number of columns.</td>
</tr>
<tr>
<td><code>const_reference operator () (size_type i, size_type j)
const</code></td>
<td>Returns a <code>const</code> reference of the <code>j</code>
-th element in the <code>i</code>-th row.</td>
</tr>
<tr>
<td><code>reference operator () (size_type i, size_type
j)</code></td>
<td>Returns a reference of the <code>j</code>-th element in the
<code>i</code>-th row.</td>
</tr>
<tr>
<td><code>triangular_matrix &operator = (const
triangular_matrix &m)</code></td>
<td>The assignment operator.</td>
</tr>
<tr>
<td><code>triangular_matrix &assign_temporary
(triangular_matrix &m)</code></td>
<td>Assigns a temporary. May change the triangular matrix
<code>m</code>.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_matrix &operator = (const
matrix_expression<AE> &ae)</code></td>
<td>The extended assignment operator.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_matrix &assign (const matrix_expression<AE>
&ae)</code></td>
<td>Assigns a matrix expression to the triangular matrix. Left and
right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_matrix &operator += (const
matrix_expression<AE> &ae)</code></td>
<td>A computed assignment operator. Adds the matrix expression to
the triangular matrix.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_matrix &plus_assign (const
matrix_expression<AE> &ae)</code></td>
<td>Adds a matrix expression to the triangular matrix. Left and
right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_matrix &operator -= (const
matrix_expression<AE> &ae)</code></td>
<td>A computed assignment operator. Subtracts the matrix expression
from the triangular matrix.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_matrix &minus_assign (const
matrix_expression<AE> &ae)</code></td>
<td>Subtracts a matrix expression from the triangular matrix. Left
and right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template<class AT><br />
triangular_matrix &operator *= (const AT &at)</code></td>
<td>A computed assignment operator. Multiplies the triangular
matrix with a scalar.</td>
</tr>
<tr>
<td><code>template<class AT><br />
triangular_matrix &operator /= (const AT &at)</code></td>
<td>A computed assignment operator. Divides the triangular matrix
through a scalar.</td>
</tr>
<tr>
<td><code>void swap (triangular_matrix &m)</code></td>
<td>Swaps the contents of the triangular matrices.</td>
</tr>
<tr>
<td><code>void insert (size_type i, size_type j, const_reference
t)</code></td>
<td>Inserts the value <code>t</code> at the <code>j</code>-th
element of the <code>i</code>-th row.</td>
</tr>
<tr>
<td><code>void erase (size_type i, size_type j)</code></td>
<td>Erases the value at the <code>j</code>-th elemenst of the
<code>i</code>-th row.</td>
</tr>
<tr>
<td><code>void clear ()</code></td>
<td>Clears the matrix.</td>
</tr>
<tr>
<td><code>const_iterator1 begin1 () const</code></td>
<td>Returns a <code>const_iterator1</code> pointing to the
beginning of the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_iterator1 end1 () const</code></td>
<td>Returns a <code>const_iterator1</code> pointing to the end of
the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>iterator1 begin1 ()</code></td>
<td>Returns a <code>iterator1</code> pointing to the beginning of
the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>iterator1 end1 ()</code></td>
<td>Returns a <code>iterator1</code> pointing to the end of the
<code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_iterator2 begin2 () const</code></td>
<td>Returns a <code>const_iterator2</code> pointing to the
beginning of the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_iterator2 end2 () const</code></td>
<td>Returns a <code>const_iterator2</code> pointing to the end of
the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>iterator2 begin2 ()</code></td>
<td>Returns a <code>iterator2</code> pointing to the beginning of
the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>iterator2 end2 ()</code></td>
<td>Returns a <code>iterator2</code> pointing to the end of the
<code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator1 rbegin1 () const</code></td>
<td>Returns a <code>const_reverse_iterator1</code> pointing to the
beginning of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator1 rend1 () const</code></td>
<td>Returns a <code>const_reverse_iterator1</code> pointing to the
end of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator1 rbegin1 ()</code></td>
<td>Returns a <code>reverse_iterator1</code> pointing to the
beginning of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator1 rend1 ()</code></td>
<td>Returns a <code>reverse_iterator1</code> pointing to the end of
the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator2 rbegin2 () const</code></td>
<td>Returns a <code>const_reverse_iterator2</code> pointing to the
beginning of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator2 rend2 () const</code></td>
<td>Returns a <code>const_reverse_iterator2</code> pointing to the
end of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator2 rbegin2 ()</code></td>
<td>Returns a <code>reverse_iterator2</code> pointing to the
beginning of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator2 rend2 ()</code></td>
<td>Returns a <code>reverse_iterator2</code> pointing to the end of
the reversed <code>triangular_matrix</code>.</td>
</tr>
</tbody>
</table>
<h4>Notes</h4>
<p><a name="triangular_matrix_1" id="triangular_matrix_1">[1]</a>
Supported parameters for the type of the triangular matrix are
<code>lower</code> , <code>unit_lower</code>, <code>upper</code>
and <code>unit_upper</code> .</p>
<p><a name="triangular_matrix_2" id="triangular_matrix_2">[2]</a>
Supported parameters for the storage organization are
<code>row_major</code> and <code>column_major</code>.</p>
<p><a name="triangular_matrix_3" id="triangular_matrix_3">[3]</a>
Supported parameters for the adapted array are
<code>unbounded_array<T></code> ,
<code>bounded_array<T></code> and
<code>std::vector<T></code> .</p>
<h2><a name="triangular_adaptor" id="triangular_adaptor"></a>Triangular Adaptor</h2>
<h4>Description</h4>
<p>The templated class <code>triangular_adaptor<M, F></code>
is a triangular matrix adaptor for other matrices.</p>
<h4>Example</h4>
<pre>
#include <boost/numeric/ublas/triangular.hpp>
#include <boost/numeric/ublas/io.hpp>
int main () {
using namespace boost::numeric::ublas;
matrix<double> m (3, 3);
triangular_adaptor<matrix<double>, lower> tal (m);
for (unsigned i = 0; i < tal.size1 (); ++ i)
for (unsigned j = 0; j <= i; ++ j)
tal (i, j) = 3 * i + j;
std::cout << tal << std::endl;
triangular_adaptor<matrix<double>, upper> tau (m);
for (unsigned i = 0; i < tau.size1 (); ++ i)
for (unsigned j = i; j < tau.size2 (); ++ j)
tau (i, j) = 3 * i + j;
std::cout << tau << std::endl;
}
</pre>
<h4>Definition</h4>
<p>Defined in the header triangular.hpp.</p>
<h4>Template parameters</h4>
<table border="1" summary="parameters">
<tbody>
<tr>
<th>Parameter</th>
<th>Description</th>
<th>Default</th>
</tr>
<tr>
<td><code>M</code></td>
<td>The type of the adapted matrix.</td>
<td></td>
</tr>
<tr>
<td><code>F</code></td>
<td>Functor describing the type of the triangular adaptor. <a href=
"#triangular_adaptor_1">[1]</a></td>
<td><code>lower</code></td>
</tr>
</tbody>
</table>
<h4>Model of</h4>
<p><a href="expression_concept.htm#matrix_expression">Matrix Expression</a>
.</p>
<h4>Type requirements</h4>
<p>None, except for those imposed by the requirements of <a href=
"expression_concept.htm#matrix_expression">Matrix Expression</a> .</p>
<h4>Public base classes</h4>
<p><code>matrix_expression<triangular_adaptor<M, F>
></code></p>
<h4>Members</h4>
<table border="1" summary="members">
<tbody>
<tr>
<th>Member</th>
<th>Description</th>
</tr>
<tr>
<td><code>triangular_adaptor (matrix_type &data)</code></td>
<td>Constructs a <code>triangular_adaptor</code> of a matrix.</td>
</tr>
<tr>
<td><code>triangular_adaptor (const triangular_adaptor
&m)</code></td>
<td>The copy constructor.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_adaptor (const matrix_expression<AE>
&ae)</code></td>
<td>The extended copy constructor.</td>
</tr>
<tr>
<td><code>size_type size1 () const</code></td>
<td>Returns the number of rows.</td>
</tr>
<tr>
<td><code>size_type size2 () const</code></td>
<td>Returns the number of columns.</td>
</tr>
<tr>
<td><code>const_reference operator () (size_type i, size_type j)
const</code></td>
<td>Returns a <code>const</code> reference of the <code>j</code>
-th element in the <code>i</code>-th row.</td>
</tr>
<tr>
<td><code>reference operator () (size_type i, size_type
j)</code></td>
<td>Returns a reference of the <code>j</code>-th element in the
<code>i</code>-th row.</td>
</tr>
<tr>
<td><code>triangular_adaptor &operator = (const
triangular_adaptor &m)</code></td>
<td>The assignment operator.</td>
</tr>
<tr>
<td><code>triangular_adaptor &assign_temporary
(triangular_adaptor &m)</code></td>
<td>Assigns a temporary. May change the triangular adaptor
<code>m</code>.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_adaptor &operator = (const
matrix_expression<AE> &ae)</code></td>
<td>The extended assignment operator.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_adaptor &assign (const matrix_expression<AE>
&ae)</code></td>
<td>Assigns a matrix expression to the triangular adaptor. Left and
right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_adaptor &operator += (const
matrix_expression<AE> &ae)</code></td>
<td>A computed assignment operator. Adds the matrix expression to
the triangular adaptor.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_adaptor &plus_assign (const
matrix_expression<AE> &ae)</code></td>
<td>Adds a matrix expression to the triangular adaptor. Left and
right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_adaptor &operator -= (const
matrix_expression<AE> &ae)</code></td>
<td>A computed assignment operator. Subtracts the matrix expression
from the triangular adaptor.</td>
</tr>
<tr>
<td><code>template<class AE><br />
triangular_adaptor &minus_assign (const
matrix_expression<AE> &ae)</code></td>
<td>Subtracts a matrix expression from the triangular adaptor. Left
and right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template<class AT><br />
triangular_adaptor &operator *= (const AT &at)</code></td>
<td>A computed assignment operator. Multiplies the triangular
adaptor with a scalar.</td>
</tr>
<tr>
<td><code>template<class AT><br />
triangular_adaptor &operator /= (const AT &at)</code></td>
<td>A computed assignment operator. Divides the triangular adaptor
through a scalar.</td>
</tr>
<tr>
<td><code>void swap (triangular_adaptor &m)</code></td>
<td>Swaps the contents of the triangular adaptors.</td>
</tr>
<tr>
<td><code>const_iterator1 begin1 () const</code></td>
<td>Returns a <code>const_iterator1</code> pointing to the
beginning of the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_iterator1 end1 () const</code></td>
<td>Returns a <code>const_iterator1</code> pointing to the end of
the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>iterator1 begin1 ()</code></td>
<td>Returns a <code>iterator1</code> pointing to the beginning of
the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>iterator1 end1 ()</code></td>
<td>Returns a <code>iterator1</code> pointing to the end of the
<code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_iterator2 begin2 () const</code></td>
<td>Returns a <code>const_iterator2</code> pointing to the
beginning of the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_iterator2 end2 () const</code></td>
<td>Returns a <code>const_iterator2</code> pointing to the end of
the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>iterator2 begin2 ()</code></td>
<td>Returns a <code>iterator2</code> pointing to the beginning of
the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>iterator2 end2 ()</code></td>
<td>Returns a <code>iterator2</code> pointing to the end of the
<code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator1 rbegin1 () const</code></td>
<td>Returns a <code>const_reverse_iterator1</code> pointing to the
beginning of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator1 rend1 () const</code></td>
<td>Returns a <code>const_reverse_iterator1</code> pointing to the
end of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator1 rbegin1 ()</code></td>
<td>Returns a <code>reverse_iterator1</code> pointing to the
beginning of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator1 rend1 ()</code></td>
<td>Returns a <code>reverse_iterator1</code> pointing to the end of
the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator2 rbegin2 () const</code></td>
<td>Returns a <code>const_reverse_iterator2</code> pointing to the
beginning of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator2 rend2 () const</code></td>
<td>Returns a <code>const_reverse_iterator2</code> pointing to the
end of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator2 rbegin2 ()</code></td>
<td>Returns a <code>reverse_iterator2</code> pointing to the
beginning of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator2 rend2 ()</code></td>
<td>Returns a <code>reverse_iterator2</code> pointing to the end of
the reversed <code>triangular_adaptor</code>.</td>
</tr>
</tbody>
</table>
<h4>Notes</h4>
<p><a name="triangular_adaptor_1" id="triangular_adaptor_1">[1]</a>
Supported parameters for the type of the triangular adaptor are
<code>lower</code> , <code>unit_lower</code>, <code>upper</code>
and <code>unit_upper</code> .</p>
<hr />
<p>Copyright (©) 2000-2002 Joerg Walter, Mathias Koch<br />
Permission to copy, use, modify, sell and distribute this document
is granted provided this copyright notice appears in all copies.
This document is provided ``as is'' without express or implied
warranty, and with no claim as to its suitability for any
purpose.</p>
</body>
</html>
|
