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
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 6.7, http://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Basic Matrix Functions (GNU Octave (version 6.2.0))</title>
<meta name="description" content="Basic Matrix Functions (GNU Octave (version 6.2.0))">
<meta name="keywords" content="Basic Matrix Functions (GNU Octave (version 6.2.0))">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<link href="index.html" rel="start" title="Top">
<link href="Concept-Index.html" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="Linear-Algebra.html" rel="up" title="Linear Algebra">
<link href="Matrix-Factorizations.html" rel="next" title="Matrix Factorizations">
<link href="Techniques-Used-for-Linear-Algebra.html" rel="prev" title="Techniques Used for Linear Algebra">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.indentedblock {margin-right: 0em}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
kbd {font-style: oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
span.nolinebreak {white-space: nowrap}
span.roman {font-family: initial; font-weight: normal}
span.sansserif {font-family: sans-serif; font-weight: normal}
ul.no-bullet {list-style: none}
-->
</style>
<link rel="stylesheet" type="text/css" href="octave.css">
</head>
<body lang="en">
<span id="Basic-Matrix-Functions"></span><div class="header">
<p>
Next: <a href="Matrix-Factorizations.html" accesskey="n" rel="next">Matrix Factorizations</a>, Previous: <a href="Techniques-Used-for-Linear-Algebra.html" accesskey="p" rel="prev">Techniques Used for Linear Algebra</a>, Up: <a href="Linear-Algebra.html" accesskey="u" rel="up">Linear Algebra</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<span id="Basic-Matrix-Functions-1"></span><h3 class="section">18.2 Basic Matrix Functions</h3>
<span id="index-matrix-functions_002c-basic"></span>
<span id="XREFbalance"></span><dl>
<dt id="index-balance">: <em><var>AA</var> =</em> <strong>balance</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-balance-1">: <em><var>AA</var> =</em> <strong>balance</strong> <em>(<var>A</var>, <var>opt</var>)</em></dt>
<dt id="index-balance-2">: <em>[<var>DD</var>, <var>AA</var>] =</em> <strong>balance</strong> <em>(<var>A</var>, <var>opt</var>)</em></dt>
<dt id="index-balance-3">: <em>[<var>D</var>, <var>P</var>, <var>AA</var>] =</em> <strong>balance</strong> <em>(<var>A</var>, <var>opt</var>)</em></dt>
<dt id="index-balance-4">: <em>[<var>CC</var>, <var>DD</var>, <var>AA</var>, <var>BB</var>] =</em> <strong>balance</strong> <em>(<var>A</var>, <var>B</var>, <var>opt</var>)</em></dt>
<dd>
<p>Balance the matrix <var>A</var> to reduce numerical errors in future
calculations.
</p>
<p>Compute <code><var>AA</var> = <var>DD</var> \ <var>A</var> * <var>DD</var></code> in which <var>AA</var>
is a matrix whose row and column norms are roughly equal in magnitude, and
<code><var>DD</var> = <var>P</var> * <var>D</var></code>, in which <var>P</var> is a permutation
matrix and <var>D</var> is a diagonal matrix of powers of two. This allows the
equilibration to be computed without round-off. Results of eigenvalue
calculation are typically improved by balancing first.
</p>
<p>If two output values are requested, <code>balance</code> returns
the diagonal <var>D</var> and the permutation <var>P</var> separately as vectors.
In this case, <code><var>DD</var> = eye(n)(:,<var>P</var>) * diag (<var>D</var>)</code>, where
<em>n</em> is the matrix size.
</p>
<p>If four output values are requested, compute <code><var>AA</var> =
<var>CC</var>*<var>A</var>*<var>DD</var></code> and <code><var>BB</var> = <var>CC</var>*<var>B</var>*<var>DD</var></code>,
in which <var>AA</var> and <var>BB</var> have nonzero elements of approximately the
same magnitude and <var>CC</var> and <var>DD</var> are permuted diagonal matrices as
in <var>DD</var> for the algebraic eigenvalue problem.
</p>
<p>The eigenvalue balancing option <var>opt</var> may be one of:
</p>
<dl compact="compact">
<dt><code>"noperm"</code>, <code>"S"</code></dt>
<dd><p>Scale only; do not permute.
</p>
</dd>
<dt><code>"noscal"</code>, <code>"P"</code></dt>
<dd><p>Permute only; do not scale.
</p></dd>
</dl>
<p>Algebraic eigenvalue balancing uses standard <small>LAPACK</small> routines.
</p>
<p>Generalized eigenvalue problem balancing uses Ward’s algorithm
(SIAM Journal on Scientific and Statistical Computing, 1981).
</p></dd></dl>
<span id="XREFbandwidth"></span><dl>
<dt id="index-bandwidth">: <em><var>bw</var> =</em> <strong>bandwidth</strong> <em>(<var>A</var>, <var>type</var>)</em></dt>
<dt id="index-bandwidth-1">: <em>[<var>lower</var>, <var>upper</var>] =</em> <strong>bandwidth</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the bandwidth of <var>A</var>.
</p>
<p>The <var>type</var> argument is the string <code>"lower"</code> for the lower
bandwidth and <code>"upper"</code> for the upper bandwidth. If no <var>type</var> is
specified return both the lower and upper bandwidth of <var>A</var>.
</p>
<p>The lower/upper bandwidth of a matrix is the number of
subdiagonals/superdiagonals with nonzero entries.
</p>
<p><strong>See also:</strong> <a href="Predicates-for-Numeric-Objects.html#XREFisbanded">isbanded</a>, <a href="Predicates-for-Numeric-Objects.html#XREFisdiag">isdiag</a>, <a href="Predicates-for-Numeric-Objects.html#XREFistril">istril</a>, <a href="Predicates-for-Numeric-Objects.html#XREFistriu">istriu</a>.
</p></dd></dl>
<span id="XREFcond"></span><dl>
<dt id="index-cond">: <em></em> <strong>cond</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-cond-1">: <em></em> <strong>cond</strong> <em>(<var>A</var>, <var>p</var>)</em></dt>
<dd><p>Compute the <var>p</var>-norm condition number of a matrix with respect to
inversion.
</p>
<p><code>cond (<var>A</var>)</code> is defined as
<code>norm (<var>A</var>, <var>p</var>) * norm (inv (<var>A</var>), <var>p</var>)</code>.
</p>
<p>By default, <code><var>p</var> = 2</code> is used which implies a (relatively slow)
singular value decomposition. Other possible selections are
<code><var>p</var> = 1, Inf, "fro"</code> which are generally faster. See <code>norm</code>
for a full discussion of possible <var>p</var> values.
</p>
<p>The condition number of a matrix quantifies the sensitivity of the matrix
inversion operation when small changes are made to matrix elements. Ideally
the condition number will be close to 1. When the number is large this
indicates small changes (such as underflow or round-off error) will produce
large changes in the resulting output. In such cases the solution results
from numerical computing are not likely to be accurate.
</p>
<p><strong>See also:</strong> <a href="Sparse-Linear-Algebra.html#XREFcondest">condest</a>, <a href="#XREFrcond">rcond</a>, <a href="#XREFcondeig">condeig</a>, <a href="#XREFnorm">norm</a>, <a href="Matrix-Factorizations.html#XREFsvd">svd</a>.
</p></dd></dl>
<span id="XREFcondeig"></span><dl>
<dt id="index-condeig">: <em><var>c</var> =</em> <strong>condeig</strong> <em>(<var>a</var>)</em></dt>
<dt id="index-condeig-1">: <em>[<var>v</var>, <var>lambda</var>, <var>c</var>] =</em> <strong>condeig</strong> <em>(<var>a</var>)</em></dt>
<dd><p>Compute condition numbers of a matrix with respect to eigenvalues.
</p>
<p>The condition numbers are the reciprocals of the cosines of the angles
between the left and right eigenvectors; Large values indicate that the
matrix has multiple distinct eigenvalues.
</p>
<p>The input <var>a</var> must be a square numeric matrix.
</p>
<p>The outputs are:
</p>
<ul>
<li> <var>c</var> is a vector of condition numbers for the eigenvalues of
<var>a</var>.
</li><li> <var>v</var> is the matrix of right eigenvectors of <var>a</var>. The result is
equivalent to calling <code>[<var>v</var>, <var>lambda</var>] = eig (<var>a</var>)</code>.
</li><li> <var>lambda</var> is the diagonal matrix of eigenvalues of <var>a</var>. The
result is equivalent to calling
<code>[<var>v</var>, <var>lambda</var>] = eig (<var>a</var>)</code>.
</li></ul>
<p>Example
</p>
<div class="example">
<pre class="example">a = [1, 2; 3, 4];
c = condeig (a)
⇒ c =
1.0150
1.0150
</pre></div>
<p><strong>See also:</strong> <a href="#XREFeig">eig</a>, <a href="#XREFcond">cond</a>, <a href="#XREFbalance">balance</a>.
</p></dd></dl>
<span id="XREFdet"></span><dl>
<dt id="index-det">: <em></em> <strong>det</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-det-1">: <em>[<var>d</var>, <var>rcond</var>] =</em> <strong>det</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the determinant of <var>A</var>.
</p>
<p>Return an estimate of the reciprocal condition number if requested.
</p>
<p>Programming Notes: Routines from <small>LAPACK</small> are used for full matrices and
code from <small>UMFPACK</small> is used for sparse matrices.
</p>
<p>The determinant should not be used to check a matrix for singularity.
For that, use any of the condition number functions: <code>cond</code>,
<code>condest</code>, <code>rcond</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFcond">cond</a>, <a href="Sparse-Linear-Algebra.html#XREFcondest">condest</a>, <a href="#XREFrcond">rcond</a>.
</p></dd></dl>
<span id="XREFeig"></span><dl>
<dt id="index-eig">: <em><var>lambda</var> =</em> <strong>eig</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-eig-1">: <em><var>lambda</var> =</em> <strong>eig</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt id="index-eig-2">: <em>[<var>V</var>, <var>lambda</var>] =</em> <strong>eig</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-eig-3">: <em>[<var>V</var>, <var>lambda</var>] =</em> <strong>eig</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt id="index-eig-4">: <em>[<var>V</var>, <var>lambda</var>, <var>W</var>] =</em> <strong>eig</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-eig-5">: <em>[<var>V</var>, <var>lambda</var>, <var>W</var>] =</em> <strong>eig</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt id="index-eig-6">: <em>[…] =</em> <strong>eig</strong> <em>(<var>A</var>, <var>balanceOption</var>)</em></dt>
<dt id="index-eig-7">: <em>[…] =</em> <strong>eig</strong> <em>(<var>A</var>, <var>B</var>, <var>algorithm</var>)</em></dt>
<dt id="index-eig-8">: <em>[…] =</em> <strong>eig</strong> <em>(…, <var>eigvalOption</var>)</em></dt>
<dd><p>Compute the eigenvalues (<var>lambda</var>) and optionally the right eigenvectors
(<var>V</var>) and the left eigenvectors (<var>W</var>) of a matrix or pair of matrices.
</p>
<p>The flag <var>balanceOption</var> can be one of:
</p>
<dl compact="compact">
<dt><code>"balance"</code> (default)</dt>
<dd><p>Preliminary balancing is on.
</p>
</dd>
<dt><code>"nobalance"</code></dt>
<dd><p>Disables preliminary balancing.
</p></dd>
</dl>
<p>The flag <var>eigvalOption</var> can be one of:
</p>
<dl compact="compact">
<dt><code>"matrix"</code></dt>
<dd><p>Return the eigenvalues in a diagonal matrix. (default if 2 or 3 outputs
are requested)
</p>
</dd>
<dt><code>"vector"</code></dt>
<dd><p>Return the eigenvalues in a column vector. (default if only 1 output is
requested, e.g., <var>lambda</var> = eig (<var>A</var>))
</p></dd>
</dl>
<p>The flag <var>algorithm</var> can be one of:
</p>
<dl compact="compact">
<dt><code>"chol"</code></dt>
<dd><p>Use the Cholesky factorization of B. (default if <var>A</var> is symmetric
(Hermitian) and <var>B</var> is symmetric (Hermitian) positive definite)
</p>
</dd>
<dt><code>"qz"</code></dt>
<dd><p>Use the QZ algorithm. (used whenever <var>A</var> or <var>B</var> are not symmetric)
</p></dd>
</dl>
<table>
<thead><tr><th width="31%"></th><th width="23%">no flag</th><th width="23%">chol</th><th width="23%">qz</th></tr></thead>
<tr><td width="31%">both are symmetric</td><td width="23%"><code>"chol"</code></td><td width="23%"><code>"chol"</code></td><td width="23%"><code>"qz"</code></td></tr>
<tr><td width="31%">at least one is not symmetric</td><td width="23%"><code>"qz"</code></td><td width="23%"><code>"qz"</code></td><td width="23%"><code>"qz"</code></td></tr>
</table>
<p>The eigenvalues returned by <code>eig</code> are not ordered.
</p>
<p><strong>See also:</strong> <a href="Sparse-Linear-Algebra.html#XREFeigs">eigs</a>, <a href="Matrix-Factorizations.html#XREFsvd">svd</a>.
</p></dd></dl>
<span id="XREFgivens"></span><dl>
<dt id="index-givens">: <em><var>G</var> =</em> <strong>givens</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt id="index-givens-1">: <em>[<var>c</var>, <var>s</var>] =</em> <strong>givens</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><p>Compute the Givens rotation matrix <var>G</var>.
</p>
<p>The Givens matrix is a 2-by-2 orthogonal matrix
</p>
<div class="example">
<pre class="example"><var>G</var> = [ <var>c</var> , <var>s</var>
-<var>s</var>', <var>c</var>]
</pre></div>
<p>such that
</p>
<div class="example">
<pre class="example"><var>G</var> * [<var>x</var>; <var>y</var>] = [*; 0]
</pre></div>
<p>with <var>x</var> and <var>y</var> scalars.
</p>
<p>If two output arguments are requested, return the factors <var>c</var> and <var>s</var>
rather than the Givens rotation matrix.
</p>
<p>For example:
</p>
<div class="example">
<pre class="example">givens (1, 1)
⇒ 0.70711 0.70711
-0.70711 0.70711
</pre></div>
<p>Note: The Givens matrix represents a counterclockwise rotation of a 2-D
plane and can be used to introduce zeros into a matrix prior to complete
factorization.
</p>
<p><strong>See also:</strong> <a href="#XREFplanerot">planerot</a>, <a href="Matrix-Factorizations.html#XREFqr">qr</a>.
</p></dd></dl>
<span id="XREFgsvd"></span><dl>
<dt id="index-gsvd">: <em><var>S</var> =</em> <strong>gsvd</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt id="index-gsvd-1">: <em>[<var>U</var>, <var>V</var>, <var>X</var>, <var>C</var>, <var>S</var>] =</em> <strong>gsvd</strong> <em>(<var>A</var>, <var>B</var>)</em></dt>
<dt id="index-gsvd-2">: <em>[<var>U</var>, <var>V</var>, <var>X</var>, <var>C</var>, <var>S</var>] =</em> <strong>gsvd</strong> <em>(<var>A</var>, <var>B</var>, 0)</em></dt>
<dd><p>Compute the generalized singular value decomposition of (<var>A</var>, <var>B</var>).
</p>
<p>The generalized singular value decomposition is defined by the following
relations:
</p>
<div class="example">
<pre class="example">A = U*C*X'
B = V*S*X'
C'*C + S'*S = eye (columns (A))
</pre></div>
<p>The function <code>gsvd</code> normally returns just the vector of generalized
singular values
<code>sqrt (diag (C'*C) ./ diag (S'*S))</code>.
If asked for five return values, it also computes
U, V, X, and C.
</p>
<p>If the optional third input is present, <code>gsvd</code> constructs the
"economy-sized" decomposition where the number of columns of <var>U</var>, <var>V</var>
and the number of rows of <var>C</var>, <var>S</var> is less than or equal to the number
of columns of <var>A</var>. This option is not yet implemented.
</p>
<p>Programming Note: the code is a wrapper to the corresponding <small>LAPACK</small> dggsvd
and zggsvd routines.
</p>
<p><strong>See also:</strong> <a href="Matrix-Factorizations.html#XREFsvd">svd</a>.
</p></dd></dl>
<span id="XREFplanerot"></span><dl>
<dt id="index-planerot">: <em>[<var>G</var>, <var>y</var>] =</em> <strong>planerot</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Compute the Givens rotation matrix for the two-element column vector
<var>x</var>.
</p>
<p>The Givens matrix is a 2-by-2 orthogonal matrix
</p>
<div class="example">
<pre class="example"><var>G</var> = [ <var>c</var> , <var>s</var>
-<var>s</var>', <var>c</var>]
</pre></div>
<p>such that
</p>
<div class="example">
<pre class="example"><var>y</var> = <var>G</var> * [<var>x</var>(1); <var>x</var>(2)] ≡ [*; 0]
</pre></div>
<p>Note: The Givens matrix represents a counterclockwise rotation of a 2-D
plane and can be used to introduce zeros into a matrix prior to complete
factorization.
</p>
<p><strong>See also:</strong> <a href="#XREFgivens">givens</a>, <a href="Matrix-Factorizations.html#XREFqr">qr</a>.
</p></dd></dl>
<span id="XREFinv"></span><dl>
<dt id="index-inv">: <em><var>x</var> =</em> <strong>inv</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-inv-1">: <em>[<var>x</var>, <var>rcond</var>] =</em> <strong>inv</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-inverse">: <em>[…] =</em> <strong>inverse</strong> <em>(…)</em></dt>
<dd><p>Compute the inverse of the square matrix <var>A</var>.
</p>
<p>Return an estimate of the reciprocal condition number if requested,
otherwise warn of an ill-conditioned matrix if the reciprocal condition
number is small.
</p>
<p>In general it is best to avoid calculating the inverse of a matrix directly.
For example, it is both faster and more accurate to solve systems of
equations (<var>A</var>*<em>x</em> = <em>b</em>) with
<code><var>y</var> = <var>A</var> \ <em>b</em></code>, rather than
<code><var>y</var> = inv (<var>A</var>) * <em>b</em></code>.
</p>
<p>If called with a sparse matrix, then in general <var>x</var> will be a full
matrix requiring significantly more storage. Avoid forming the inverse of a
sparse matrix if possible.
</p>
<p><code>inverse</code> is an alias and may be used identically in place of <code>inv</code>.
</p>
<p><strong>See also:</strong> <a href="Arithmetic-Ops.html#XREFldivide">ldivide</a>, <a href="Arithmetic-Ops.html#XREFrdivide">rdivide</a>, <a href="#XREFpinv">pinv</a>.
</p></dd></dl>
<span id="XREFlinsolve"></span><dl>
<dt id="index-linsolve">: <em><var>x</var> =</em> <strong>linsolve</strong> <em>(<var>A</var>, <var>b</var>)</em></dt>
<dt id="index-linsolve-1">: <em><var>x</var> =</em> <strong>linsolve</strong> <em>(<var>A</var>, <var>b</var>, <var>opts</var>)</em></dt>
<dt id="index-linsolve-2">: <em>[<var>x</var>, <var>R</var>] =</em> <strong>linsolve</strong> <em>(…)</em></dt>
<dd><p>Solve the linear system <code>A*x = b</code>.
</p>
<p>With no options, this function is equivalent to the left division operator
(<code>x = A \ b</code>)<!-- /@w --> or the matrix-left-divide function
(<code>x = mldivide (A, b)</code>)<!-- /@w -->.
</p>
<p>Octave ordinarily examines the properties of the matrix <var>A</var> and chooses
a solver that best matches the matrix. By passing a structure <var>opts</var>
to <code>linsolve</code> you can inform Octave directly about the matrix <var>A</var>.
In this case Octave will skip the matrix examination and proceed directly
to solving the linear system.
</p>
<p><strong>Warning:</strong> If the matrix <var>A</var> does not have the properties listed
in the <var>opts</var> structure then the result will not be accurate AND no
warning will be given. When in doubt, let Octave examine the matrix and
choose the appropriate solver as this step takes little time and the result
is cached so that it is only done once per linear system.
</p>
<p>Possible <var>opts</var> fields (set value to true/false):
</p>
<dl compact="compact">
<dt>LT</dt>
<dd><p><var>A</var> is lower triangular
</p>
</dd>
<dt>UT</dt>
<dd><p><var>A</var> is upper triangular
</p>
</dd>
<dt>UHESS</dt>
<dd><p><var>A</var> is upper Hessenberg (currently makes no difference)
</p>
</dd>
<dt>SYM</dt>
<dd><p><var>A</var> is symmetric or complex Hermitian (currently makes no difference)
</p>
</dd>
<dt>POSDEF</dt>
<dd><p><var>A</var> is positive definite
</p>
</dd>
<dt>RECT</dt>
<dd><p><var>A</var> is general rectangular (currently makes no difference)
</p>
</dd>
<dt>TRANSA</dt>
<dd><p>Solve <code>A'*x = b</code> if true rather than <code>A*x = b</code>
</p></dd>
</dl>
<p>The optional second output <var>R</var> is the inverse condition number of
<var>A</var> (zero if matrix is singular).
</p>
<p><strong>See also:</strong> <a href="Arithmetic-Ops.html#XREFmldivide">mldivide</a>, <a href="#XREFmatrix_005ftype">matrix_type</a>, <a href="#XREFrcond">rcond</a>.
</p></dd></dl>
<span id="XREFmatrix_005ftype"></span><dl>
<dt id="index-matrix_005ftype">: <em><var>type</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-matrix_005ftype-1">: <em><var>type</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, "nocompute")</em></dt>
<dt id="index-matrix_005ftype-2">: <em><var>A</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, <var>type</var>)</em></dt>
<dt id="index-matrix_005ftype-3">: <em><var>A</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, "upper", <var>perm</var>)</em></dt>
<dt id="index-matrix_005ftype-4">: <em><var>A</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, "lower", <var>perm</var>)</em></dt>
<dt id="index-matrix_005ftype-5">: <em><var>A</var> =</em> <strong>matrix_type</strong> <em>(<var>A</var>, "banded", <var>nl</var>, <var>nu</var>)</em></dt>
<dd><p>Identify the matrix type or mark a matrix as a particular type.
</p>
<p>This allows more rapid solutions of linear equations involving <var>A</var> to be
performed.
</p>
<p>Called with a single argument, <code>matrix_type</code> returns the type of the
matrix and caches it for future use.
</p>
<p>Called with more than one argument, <code>matrix_type</code> allows the type of
the matrix to be defined.
</p>
<p>If the option <code>"nocompute"</code> is given, the function will not attempt
to guess the type if it is still unknown. This is useful for debugging
purposes.
</p>
<p>The possible matrix types depend on whether the matrix is full or sparse,
and can be one of the following
</p>
<dl compact="compact">
<dt><code>"unknown"</code></dt>
<dd><p>Remove any previously cached matrix type, and mark type as unknown.
</p>
</dd>
<dt><code>"full"</code></dt>
<dd><p>Mark the matrix as full.
</p>
</dd>
<dt><code>"positive definite"</code></dt>
<dd><p>Probable full positive definite matrix.
</p>
</dd>
<dt><code>"diagonal"</code></dt>
<dd><p>Diagonal matrix. (Sparse matrices only)
</p>
</dd>
<dt><code>"permuted diagonal"</code></dt>
<dd><p>Permuted Diagonal matrix. The permutation does not need to be specifically
indicated, as the structure of the matrix explicitly gives this. (Sparse
matrices only)
</p>
</dd>
<dt><code>"upper"</code></dt>
<dd><p>Upper triangular. If the optional third argument <var>perm</var> is given, the
matrix is assumed to be a permuted upper triangular with the permutations
defined by the vector <var>perm</var>.
</p>
</dd>
<dt><code>"lower"</code></dt>
<dd><p>Lower triangular. If the optional third argument <var>perm</var> is given, the
matrix is assumed to be a permuted lower triangular with the permutations
defined by the vector <var>perm</var>.
</p>
</dd>
<dt><code>"banded"</code></dt>
<dt><code>"banded positive definite"</code></dt>
<dd><p>Banded matrix with the band size of <var>nl</var> below the diagonal and <var>nu</var>
above it. If <var>nl</var> and <var>nu</var> are 1, then the matrix is tridiagonal
and treated with specialized code. In addition the matrix can be marked as
probably a positive definite. (Sparse matrices only)
</p>
</dd>
<dt><code>"singular"</code></dt>
<dd><p>The matrix is assumed to be singular and will be treated with a minimum norm
solution.
</p>
</dd>
</dl>
<p>Note that the matrix type will be discovered automatically on the first
attempt to solve a linear equation involving <var>A</var>. Therefore
<code>matrix_type</code> is only useful to give Octave hints of the matrix type.
Incorrectly defining the matrix type will result in incorrect results from
solutions of linear equations; it is entirely <strong>the responsibility of
the user</strong> to correctly identify the matrix type.
</p>
<p>Also, the test for positive definiteness is a low-cost test for a Hermitian
matrix with a real positive diagonal. This does not guarantee that the
matrix is positive definite, but only that it is a probable candidate. When
such a matrix is factorized, a Cholesky factorization is first
attempted, and if that fails the matrix is then treated with an
LU factorization. Once the matrix has been factorized,
<code>matrix_type</code> will return the correct classification of the matrix.
</p></dd></dl>
<span id="XREFnorm"></span><dl>
<dt id="index-norm">: <em></em> <strong>norm</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-norm-1">: <em></em> <strong>norm</strong> <em>(<var>A</var>, <var>p</var>)</em></dt>
<dt id="index-norm-2">: <em></em> <strong>norm</strong> <em>(<var>A</var>, <var>p</var>, <var>opt</var>)</em></dt>
<dd><p>Compute the p-norm of the matrix <var>A</var>.
</p>
<p>If the second argument is not given, <code>p = 2</code><!-- /@w --> is used.
</p>
<p>If <var>A</var> is a matrix (or sparse matrix):
</p>
<dl compact="compact">
<dt><var>p</var> = <code>1</code></dt>
<dd><p>1-norm, the largest column sum of the absolute values of <var>A</var>.
</p>
</dd>
<dt><var>p</var> = <code>2</code></dt>
<dd><p>Largest singular value of <var>A</var>.
</p>
</dd>
<dt><var>p</var> = <code>Inf</code> or <code>"inf"</code></dt>
<dd><span id="index-infinity-norm"></span>
<p>Infinity norm, the largest row sum of the absolute values of <var>A</var>.
</p>
</dd>
<dt><var>p</var> = <code>"fro"</code></dt>
<dd><span id="index-Frobenius-norm"></span>
<p>Frobenius norm of <var>A</var>,
<code>sqrt (sum (diag (<var>A</var>' * <var>A</var>)))</code>.
</p>
</dd>
<dt>other <var>p</var>, <code><var>p</var> > 1</code></dt>
<dd><span id="index-general-p_002dnorm"></span>
<p>maximum <code>norm (A*x, p)</code> such that <code>norm (x, p) == 1</code>
</p></dd>
</dl>
<p>If <var>A</var> is a vector or a scalar:
</p>
<dl compact="compact">
<dt><var>p</var> = <code>Inf</code> or <code>"inf"</code></dt>
<dd><p><code>max (abs (<var>A</var>))</code>.
</p>
</dd>
<dt><var>p</var> = <code>-Inf</code></dt>
<dd><p><code>min (abs (<var>A</var>))</code>.
</p>
</dd>
<dt><var>p</var> = <code>"fro"</code></dt>
<dd><p>Frobenius norm of <var>A</var>, <code>sqrt (sumsq (abs (A)))</code>.
</p>
</dd>
<dt><var>p</var> = 0</dt>
<dd><p>Hamming norm—the number of nonzero elements.
</p>
</dd>
<dt>other <var>p</var>, <code><var>p</var> > 1</code></dt>
<dd><p>p-norm of <var>A</var>, <code>(sum (abs (<var>A</var>) .^ <var>p</var>)) ^ (1/<var>p</var>)</code>.
</p>
</dd>
<dt>other <var>p</var> <code><var>p</var> < 1</code></dt>
<dd><p>the p-pseudonorm defined as above.
</p></dd>
</dl>
<p>If <var>opt</var> is the value <code>"rows"</code>, treat each row as a vector and
compute its norm. The result is returned as a column vector.
Similarly, if <var>opt</var> is <code>"columns"</code> or <code>"cols"</code> then
compute the norms of each column and return a row vector.
</p>
<p><strong>See also:</strong> <a href="Sparse-Linear-Algebra.html#XREFnormest">normest</a>, <a href="Sparse-Linear-Algebra.html#XREFnormest1">normest1</a>, <a href="#XREFvecnorm">vecnorm</a>, <a href="#XREFcond">cond</a>, <a href="Matrix-Factorizations.html#XREFsvd">svd</a>.
</p></dd></dl>
<span id="XREFnull"></span><dl>
<dt id="index-null">: <em></em> <strong>null</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-null-1">: <em></em> <strong>null</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dd><p>Return an orthonormal basis of the null space of <var>A</var>.
</p>
<p>The dimension of the null space is taken as the number of singular values of
<var>A</var> not greater than <var>tol</var>. If the argument <var>tol</var> is missing,
it is computed as
</p>
<div class="example">
<pre class="example">max (size (<var>A</var>)) * max (svd (<var>A</var>)) * eps
</pre></div>
<p><strong>See also:</strong> <a href="#XREForth">orth</a>.
</p></dd></dl>
<span id="XREForth"></span><dl>
<dt id="index-orth">: <em></em> <strong>orth</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-orth-1">: <em></em> <strong>orth</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dd><p>Return an orthonormal basis of the range space of <var>A</var>.
</p>
<p>The dimension of the range space is taken as the number of singular values
of <var>A</var> greater than <var>tol</var>. If the argument <var>tol</var> is missing, it
is computed as
</p>
<div class="example">
<pre class="example">max (size (<var>A</var>)) * max (svd (<var>A</var>)) * eps
</pre></div>
<p><strong>See also:</strong> <a href="#XREFnull">null</a>.
</p></dd></dl>
<span id="XREFmgorth"></span><dl>
<dt id="index-mgorth">: <em>[<var>y</var>, <var>h</var>] =</em> <strong>mgorth</strong> <em>(<var>x</var>, <var>v</var>)</em></dt>
<dd><p>Orthogonalize a given column vector <var>x</var> with respect to a set of
orthonormal vectors comprising the columns of <var>v</var> using the modified
Gram-Schmidt method.
</p>
<p>On exit, <var>y</var> is a unit vector such that:
</p>
<div class="example">
<pre class="example"> norm (<var>y</var>) = 1
<var>v</var>' * <var>y</var> = 0
<var>x</var> = [<var>v</var>, <var>y</var>]*<var>h</var>'
</pre></div>
</dd></dl>
<span id="XREFpinv"></span><dl>
<dt id="index-pinv">: <em></em> <strong>pinv</strong> <em>(<var>x</var>)</em></dt>
<dt id="index-pinv-1">: <em></em> <strong>pinv</strong> <em>(<var>x</var>, <var>tol</var>)</em></dt>
<dd><p>Return the Moore-Penrose pseudoinverse of <var>x</var>.
</p>
<p>Singular values less than <var>tol</var> are ignored.
</p>
<p>If the second argument is omitted, it is taken to be
</p>
<div class="example">
<pre class="example">tol = max ([rows(<var>x</var>), columns(<var>x</var>)]) * norm (<var>x</var>) * eps
</pre></div>
<p><strong>See also:</strong> <a href="#XREFinv">inv</a>, <a href="Arithmetic-Ops.html#XREFldivide">ldivide</a>.
</p></dd></dl>
<span id="index-pseudoinverse"></span>
<span id="XREFrank"></span><dl>
<dt id="index-rank">: <em></em> <strong>rank</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-rank-1">: <em></em> <strong>rank</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dd><p>Compute the rank of matrix <var>A</var>, using the singular value decomposition.
</p>
<p>The rank is taken to be the number of singular values of <var>A</var> that are
greater than the specified tolerance <var>tol</var>. If the second argument is
omitted, it is taken to be
</p>
<div class="example">
<pre class="example">tol = max (size (<var>A</var>)) * sigma(1) * eps;
</pre></div>
<p>where <code>eps</code> is machine precision and <code>sigma(1)</code> is the largest
singular value of <var>A</var>.
</p>
<p>The rank of a matrix is the number of linearly independent rows or columns
and equals the dimension of the row and column space. The function
<code>orth</code> may be used to compute an orthonormal basis of the column space.
</p>
<p>For testing if a system <code><var>A</var>*<var>x</var> = <var>b</var></code> of linear equations
is solvable, one can use
</p>
<div class="example">
<pre class="example">rank (<var>A</var>) == rank ([<var>A</var> <var>b</var>])
</pre></div>
<p>In this case, <code><var>x</var> = <var>A</var> \ <var>b</var></code> finds a particular solution
<var>x</var>. The general solution is <var>x</var> plus the null space of matrix
<var>A</var>. The function <code>null</code> may be used to compute a basis of the
null space.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">A = [1 2 3
4 5 6
7 8 9];
rank (A)
⇒ 2
</pre></div>
<p>In this example, the number of linearly independent rows is only 2 because
the final row is a linear combination of the first two rows:
</p>
<div class="example">
<pre class="example">A(3,:) == -A(1,:) + 2 * A(2,:)
</pre></div>
<p><strong>See also:</strong> <a href="#XREFnull">null</a>, <a href="#XREForth">orth</a>, <a href="Sparse-Linear-Algebra.html#XREFsprank">sprank</a>, <a href="Matrix-Factorizations.html#XREFsvd">svd</a>, <a href="Mathematical-Constants.html#XREFeps">eps</a>.
</p></dd></dl>
<span id="XREFrcond"></span><dl>
<dt id="index-rcond">: <em><var>c</var> =</em> <strong>rcond</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the 1-norm estimate of the reciprocal condition number as returned
by <small>LAPACK</small>.
</p>
<p>If the matrix is well-conditioned then <var>c</var> will be near 1 and if the
matrix is poorly conditioned it will be close to 0.
</p>
<p>The matrix <var>A</var> must not be sparse. If the matrix is sparse then
<code>condest (<var>A</var>)</code> or <code>rcond (full (<var>A</var>))</code> should be used
instead.
</p>
<p><strong>See also:</strong> <a href="#XREFcond">cond</a>, <a href="Sparse-Linear-Algebra.html#XREFcondest">condest</a>.
</p></dd></dl>
<span id="XREFtrace"></span><dl>
<dt id="index-trace">: <em></em> <strong>trace</strong> <em>(<var>A</var>)</em></dt>
<dd><p>Compute the trace of <var>A</var>, the sum of the elements along the main
diagonal.
</p>
<p>The implementation is straightforward: <code>sum (diag (<var>A</var>))</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFeig">eig</a>.
</p></dd></dl>
<span id="XREFrref"></span><dl>
<dt id="index-rref">: <em></em> <strong>rref</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-rref-1">: <em></em> <strong>rref</strong> <em>(<var>A</var>, <var>tol</var>)</em></dt>
<dt id="index-rref-2">: <em>[<var>r</var>, <var>k</var>] =</em> <strong>rref</strong> <em>(…)</em></dt>
<dd><p>Return the reduced row echelon form of <var>A</var>.
</p>
<p><var>tol</var> defaults to
<code>eps * max (size (<var>A</var>)) * norm (<var>A</var>, inf)</code>.
</p>
<p>The optional return argument <var>k</var> contains the vector of
"bound variables", which are those columns on which elimination has been
performed.
</p>
</dd></dl>
<span id="XREFvecnorm"></span><dl>
<dt id="index-vecnorm">: <em><var>n</var> =</em> <strong>vecnorm</strong> <em>(<var>A</var>)</em></dt>
<dt id="index-vecnorm-1">: <em><var>n</var> =</em> <strong>vecnorm</strong> <em>(<var>A</var>, <var>p</var>)</em></dt>
<dt id="index-vecnorm-2">: <em><var>n</var> =</em> <strong>vecnorm</strong> <em>(<var>A</var>, <var>p</var>, <var>dim</var>)</em></dt>
<dd><p>Return the p-norm of the elements of <var>A</var> along dimension <var>dim</var>.
</p>
<p>The p-norm of a vector is defined as
</p>
<div class="example">
<pre class="example"><var>p-norm</var> (<var>A</var>, <var>p</var>) = sum (abs (<var>A</var>) .^ <var>p</var>)) ^ (1/<var>p</var>)
</pre></div>
<p>If <var>p</var> is omitted it defaults to 2 (Euclidean norm). <var>p</var> can be
<code>Inf</code> (absolute value of largest element).
</p>
<p>If <var>dim</var> is omitted the first non-singleton dimension is used.
</p>
<p><strong>See also:</strong> <a href="#XREFnorm">norm</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="Matrix-Factorizations.html" accesskey="n" rel="next">Matrix Factorizations</a>, Previous: <a href="Techniques-Used-for-Linear-Algebra.html" accesskey="p" rel="prev">Techniques Used for Linear Algebra</a>, Up: <a href="Linear-Algebra.html" accesskey="u" rel="up">Linear Algebra</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
