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
|
<!DOCTYPE html>
<html lang="en" data-content_root="../../" >
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Exercise 8: Quadratic Eigenvalue Problem — SLEPc 3.24.1 documentation</title>
<script data-cfasync="false">
document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
document.documentElement.dataset.theme = localStorage.getItem("theme") || "";
</script>
<!--
this give us a css class that will be invisible only if js is disabled
-->
<noscript>
<style>
.pst-js-only { display: none !important; }
</style>
</noscript>
<!-- Loaded before other Sphinx assets -->
<link href="../../_static/styles/theme.css?digest=8878045cc6db502f8baf" rel="stylesheet" />
<link href="../../_static/styles/pydata-sphinx-theme.css?digest=8878045cc6db502f8baf" rel="stylesheet" />
<link rel="stylesheet" type="text/css" href="../../_static/pygments.css?v=8f2a1f02" />
<link rel="stylesheet" type="text/css" href="../../_static/copybutton.css?v=76b2166b" />
<link rel="stylesheet" type="text/css" href="../../_static/togglebutton.css?v=13237357" />
<link rel="stylesheet" type="text/css" href="../../_static/sphinx-design.min.css?v=95c83b7e" />
<link rel="stylesheet" type="text/css" href="../../_static/css/slepc.css?v=d285b177" />
<!-- So that users can add custom icons -->
<script src="../../_static/scripts/fontawesome.js?digest=8878045cc6db502f8baf"></script>
<!-- Pre-loaded scripts that we'll load fully later -->
<link rel="preload" as="script" href="../../_static/scripts/bootstrap.js?digest=8878045cc6db502f8baf" />
<link rel="preload" as="script" href="../../_static/scripts/pydata-sphinx-theme.js?digest=8878045cc6db502f8baf" />
<script src="../../_static/documentation_options.js?v=d1c46438"></script>
<script src="../../_static/doctools.js?v=9a2dae69"></script>
<script src="../../_static/sphinx_highlight.js?v=dc90522c"></script>
<script src="../../_static/clipboard.min.js?v=a7894cd8"></script>
<script src="../../_static/copybutton.js?v=a56c686a"></script>
<script>let toggleHintShow = 'Click to show';</script>
<script>let toggleHintHide = 'Click to hide';</script>
<script>let toggleOpenOnPrint = 'true';</script>
<script src="../../_static/togglebutton.js?v=4a39c7ea"></script>
<script src="../../_static/design-tabs.js?v=f930bc37"></script>
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown';</script>
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown';</script>
<script>window.MathJax = {"options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
<script defer="defer" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<script>DOCUMENTATION_OPTIONS.pagename = 'documentation/hands-on/hands-on8';</script>
<link rel="icon" href="../../_static/favicon-slepc.ico"/>
<link rel="index" title="Index" href="../../genindex.html" />
<link rel="search" title="Search" href="../../search.html" />
<link rel="next" title="Frequently Asked Questions" href="../faq.html" />
<link rel="prev" title="Exercise 7: Use of Deflation Subspaces" href="hands-on7.html" />
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<meta name="docsearch:language" content="en"/>
<meta name="docsearch:version" content="3.24" />
</head>
<body data-bs-spy="scroll" data-bs-target=".bd-toc-nav" data-offset="180" data-bs-root-margin="0px 0px -60%" data-default-mode="">
<div id="pst-skip-link" class="skip-link d-print-none"><a href="#main-content">Skip to main content</a></div>
<div id="pst-scroll-pixel-helper"></div>
<button type="button" class="btn rounded-pill" id="pst-back-to-top">
<i class="fa-solid fa-arrow-up"></i>Back to top</button>
<dialog id="pst-search-dialog">
<form class="bd-search d-flex align-items-center"
action="../../search.html"
method="get">
<i class="fa-solid fa-magnifying-glass"></i>
<input type="search"
class="form-control"
name="q"
placeholder="Search the docs ..."
aria-label="Search the docs ..."
autocomplete="off"
autocorrect="off"
autocapitalize="off"
spellcheck="false"/>
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd>K</kbd></span>
</form>
</dialog>
<div class="pst-async-banner-revealer d-none">
<aside id="bd-header-version-warning" class="d-none d-print-none" aria-label="Version warning"></aside>
</div>
<header class="bd-header navbar navbar-expand-lg bd-navbar d-print-none">
<div class="bd-header__inner bd-page-width">
<button class="pst-navbar-icon sidebar-toggle primary-toggle" aria-label="Site navigation">
<span class="fa-solid fa-bars"></span>
</button>
<div class="col-lg-3 navbar-header-items__start">
<div class="navbar-item">
<a class="navbar-brand logo" href="../../index.html">
<img src="../../_static/logo-slepc.gif" class="logo__image only-light" alt="SLEPc Home"/>
<img src="../../_static/logo-slepc.gif" class="logo__image only-dark pst-js-only" alt="SLEPc Home"/>
</a></div>
</div>
<div class="col-lg-9 navbar-header-items">
<div class="me-auto navbar-header-items__center">
<div class="navbar-item">
<nav>
<ul class="bd-navbar-elements navbar-nav">
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../about/index.html">
About
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../installation/index.html">
Installation
</a>
</li>
<li class="nav-item current active">
<a class="nav-link nav-internal" href="../index.html">
Documentation
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../manualpages/index.html">
C/Fortran API
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../slepc4py/index.html">
slepc4py API
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../material/index.html">
Material
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../contact/index.html">
Contact
</a>
</li>
</ul>
</nav></div>
</div>
<div class="navbar-header-items__end">
<div class="navbar-item navbar-persistent--container">
<button class="btn search-button-field search-button__button pst-js-only" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass"></i>
<span class="search-button__default-text">Search</span>
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd class="kbd-shortcut__modifier">K</kbd></span>
</button>
</div>
<div class="navbar-item">
<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button pst-js-only" aria-label="Color mode" data-bs-title="Color mode" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light" title="Light"></i>
<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark" title="Dark"></i>
<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto" title="System Settings"></i>
</button></div>
<div class="navbar-item"><ul class="navbar-icon-links"
aria-label="Icon Links">
<li class="nav-item">
<a href="https://gitlab.com/slepc/slepc" title="GitLab" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><i class="fa-brands fa-square-gitlab fa-lg" aria-hidden="true"></i>
<span class="sr-only">GitLab</span></a>
</li>
<li class="nav-item">
<a href="https://www.upv.es" title="UPV" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><img src="https://www.upv.es/favicon.ico" class="icon-link-image" alt="UPV"/></a>
</li>
<li class="nav-item">
<a href="https://slepc.upv.es/release/_static/rss/slepc-news.xml" title="Feed" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><i class="fa-solid fa-square-rss fa-lg" aria-hidden="true"></i>
<span class="sr-only">Feed</span></a>
</li>
</ul></div>
</div>
</div>
<div class="navbar-persistent--mobile">
<button class="btn search-button-field search-button__button pst-js-only" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass"></i>
<span class="search-button__default-text">Search</span>
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd class="kbd-shortcut__modifier">K</kbd></span>
</button>
</div>
<button class="pst-navbar-icon sidebar-toggle secondary-toggle" aria-label="On this page">
<span class="fa-solid fa-outdent"></span>
</button>
</div>
</header>
<div class="bd-container">
<div class="bd-container__inner bd-page-width">
<dialog id="pst-primary-sidebar-modal"></dialog>
<div id="pst-primary-sidebar" class="bd-sidebar-primary bd-sidebar">
<div class="sidebar-header-items sidebar-primary__section">
<div class="sidebar-header-items__center">
<div class="navbar-item">
<nav>
<ul class="bd-navbar-elements navbar-nav">
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../about/index.html">
About
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../installation/index.html">
Installation
</a>
</li>
<li class="nav-item current active">
<a class="nav-link nav-internal" href="../index.html">
Documentation
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../manualpages/index.html">
C/Fortran API
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../slepc4py/index.html">
slepc4py API
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../material/index.html">
Material
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../../contact/index.html">
Contact
</a>
</li>
</ul>
</nav></div>
</div>
<div class="sidebar-header-items__end">
<div class="navbar-item">
<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button pst-js-only" aria-label="Color mode" data-bs-title="Color mode" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light" title="Light"></i>
<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark" title="Dark"></i>
<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto" title="System Settings"></i>
</button></div>
<div class="navbar-item"><ul class="navbar-icon-links"
aria-label="Icon Links">
<li class="nav-item">
<a href="https://gitlab.com/slepc/slepc" title="GitLab" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><i class="fa-brands fa-square-gitlab fa-lg" aria-hidden="true"></i>
<span class="sr-only">GitLab</span></a>
</li>
<li class="nav-item">
<a href="https://www.upv.es" title="UPV" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><img src="https://www.upv.es/favicon.ico" class="icon-link-image" alt="UPV"/></a>
</li>
<li class="nav-item">
<a href="https://slepc.upv.es/release/_static/rss/slepc-news.xml" title="Feed" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><i class="fa-solid fa-square-rss fa-lg" aria-hidden="true"></i>
<span class="sr-only">Feed</span></a>
</li>
</ul></div>
</div>
</div>
<div class="sidebar-primary-items__start sidebar-primary__section">
<div class="sidebar-primary-item">
<nav class="bd-docs-nav bd-links"
aria-label="Section Navigation">
<p class="bd-links__title" role="heading" aria-level="1">Section Navigation</p>
<div class="bd-toc-item navbar-nav"><ul class="current nav bd-sidenav">
<li class="toctree-l1 has-children"><a class="reference internal" href="../manual/index.html">SLEPc Users Manual</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../manual/intro.html">Getting Started</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/eps.html">EPS: Eigenvalue Problem Solver</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/st.html">ST: Spectral Transformation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/svd.html">SVD: Singular Value Decomposition</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/pep.html">PEP: Polynomial Eigenvalue Problems</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/nep.html">NEP: Nonlinear Eigenvalue Problems</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/mfn.html">MFN: Matrix Function</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/lme.html">LME: Linear Matrix Equation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/aux.html">Auxiliary Classes</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/extra.html">Additional Information</a></li>
</ul>
</details></li>
<li class="toctree-l1 current active has-children"><a class="reference internal" href="index.html">Hands-on exercises</a><details open="open"><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="hands-on0.html">Exercise 0: Hello World</a></li>
<li class="toctree-l2"><a class="reference internal" href="hands-on1.html">Exercise 1: Standard Symmetric Eigenvalue Problem</a></li>
<li class="toctree-l2"><a class="reference internal" href="hands-on2.html">Exercise 2: Standard Non-Symmetric Eigenvalue Problem</a></li>
<li class="toctree-l2"><a class="reference internal" href="hands-on3.html">Exercise 3: Generalized Eigenvalue Problem Stored in a File</a></li>
<li class="toctree-l2"><a class="reference internal" href="hands-on4.html">Exercise 4: Singular Value Decomposition</a></li>
<li class="toctree-l2"><a class="reference internal" href="hands-on5.html">Exercise 5: Problem without Explicit Matrix Storage</a></li>
<li class="toctree-l2"><a class="reference internal" href="hands-on6.html">Exercise 6: Parallel Execution</a></li>
<li class="toctree-l2"><a class="reference internal" href="hands-on7.html">Exercise 7: Use of Deflation Subspaces</a></li>
<li class="toctree-l2 current active"><a class="current reference internal" href="#">Exercise 8: Quadratic Eigenvalue Problem</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../faq.html">Frequently Asked Questions</a></li>
<li class="toctree-l1"><a class="reference internal" href="../presentations.html">Presentations</a></li>
<li class="toctree-l1"><a class="reference internal" href="../license.html">License</a></li>
</ul>
</div>
</nav></div>
</div>
<div class="sidebar-primary-items__end sidebar-primary__section">
<div class="sidebar-primary-item">
<div id="ethical-ad-placement"
class="flat"
data-ea-publisher="readthedocs"
data-ea-type="readthedocs-sidebar"
data-ea-manual="true">
</div></div>
</div>
</div>
<main id="main-content" class="bd-main" role="main">
<div class="bd-content">
<div class="bd-article-container">
<div class="bd-header-article d-print-none">
<div class="header-article-items header-article__inner">
<div class="header-article-items__start">
<div class="header-article-item">
<nav aria-label="Breadcrumb" class="d-print-none">
<ul class="bd-breadcrumbs">
<li class="breadcrumb-item breadcrumb-home">
<a href="../../index.html" class="nav-link" aria-label="Home">
<i class="fa-solid fa-home"></i>
</a>
</li>
<li class="breadcrumb-item"><a href="../index.html" class="nav-link">Documentation</a></li>
<li class="breadcrumb-item"><a href="index.html" class="nav-link">Hands-on exercises</a></li>
<li class="breadcrumb-item active" aria-current="page"><span class="ellipsis">Exercise 8: Quadratic Eigenvalue Problem</span></li>
</ul>
</nav>
</div>
</div>
</div>
</div>
<div id="searchbox"></div>
<article class="bd-article">
<section class="tex2jax_ignore mathjax_ignore" id="exercise-8-quadratic-eigenvalue-problem">
<h1>Exercise 8: Quadratic Eigenvalue Problem<a class="headerlink" href="#exercise-8-quadratic-eigenvalue-problem" title="Link to this heading">#</a></h1>
<p>Now we are going to focus on the solution of quadratic eigenvalue problems with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code> solvers. In this case, the problem to be solved is formulated as <span class="math notranslate nohighlight">\((\lambda^{2}M+ \lambda C+K)x=0\)</span>. In our simple example, <span class="math notranslate nohighlight">\(M\)</span> is a diagonal matrix, <span class="math notranslate nohighlight">\(C\)</span> is tridiagonal, and <span class="math notranslate nohighlight">\(K\)</span> is the 2-D Laplacian.</p>
<section id="compiling">
<h2>Compiling<a class="headerlink" href="#compiling" title="Link to this heading">#</a></h2>
<p>Copy the file <a class="reference external" href="https://slepc.upv.es/release/src/pep/tutorials/ex16.c.html" target="_blank">ex16.c</a> to your directory and add these lines to the makefile</p>
<div class="highlight-make notranslate"><div class="highlight"><pre><span></span><span class="nf">ex16</span><span class="o">:</span><span class="w"> </span><span class="n">ex</span>16.<span class="n">o</span>
<span class="w"> </span>-<span class="si">${</span><span class="nv">CLINKER</span><span class="si">}</span><span class="w"> </span>-o<span class="w"> </span>ex16<span class="w"> </span>ex16.o<span class="w"> </span><span class="si">${</span><span class="nv">SLEPC_PEP_LIB</span><span class="si">}</span>
<span class="w"> </span><span class="si">${</span><span class="nv">RM</span><span class="si">}</span><span class="w"> </span>ex16.o
</pre></div>
</div>
<p>Build the executable with the command</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>make<span class="w"> </span>ex16
</pre></div>
</div>
</section>
<section id="running-the-program">
<h2>Running the Program<a class="headerlink" href="#running-the-program" title="Link to this heading">#</a></h2>
<p>Run the program without arguments and see the output:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">Quadratic</span> <span class="n">Eigenproblem</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="mi">100</span> <span class="p">(</span><span class="mi">10</span><span class="n">x10</span> <span class="n">grid</span><span class="p">)</span>
<span class="n">Number</span> <span class="n">of</span> <span class="n">requested</span> <span class="n">eigenvalues</span><span class="p">:</span> <span class="mi">1</span>
<span class="n">k</span> <span class="o">||</span><span class="n">P</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="n">x</span><span class="o">||/||</span><span class="n">kx</span><span class="o">||</span>
<span class="o">-----------------</span> <span class="o">------------------</span>
<span class="o">-</span><span class="mf">1.164037</span><span class="o">+</span><span class="mf">1.653625</span><span class="n">i</span> <span class="mf">2.02021e-12</span>
<span class="o">-</span><span class="mf">1.164037</span><span class="o">-</span><span class="mf">1.653625</span><span class="n">i</span> <span class="mf">2.02021e-12</span>
</pre></div>
</div>
</section>
<section id="source-code-details">
<h2>Source Code Details<a class="headerlink" href="#source-code-details" title="Link to this heading">#</a></h2>
<p>The <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code> object is used very much like <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> or <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/SVD/SVD.html">SVD</a></span></code>, as can be seen in the source code. Here is a summary of the main function calls:</p>
<ul class="simple">
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPCreate.html">PEPCreate</a>(<a href="https://petsc.org/release/manualpages/Sys/MPI_Comm/">MPI_Comm</a></span> <span class="pre">comm,<a href="../../manualpages/PEP/PEP.html">PEP</a></span> <span class="pre">*pep);</span></code></p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPSetOperators.html">PEPSetOperators</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code> pep,<a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscInt/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscInt</span></a> <code class="docutils notranslate"><span class="pre">nmat</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Mat/Mat/" title="(in PETSc v3.24)"><span class="xref std std-doc">Mat</span></a> <code class="docutils notranslate"><span class="pre">A[]</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPSetProblemType.html">PEPSetProblemType</a>(<a href="../../manualpages/PEP/PEP.html">PEP</a></span> <span class="pre">pep,<a href="../../manualpages/PEP/PEPProblemType.html">PEPProblemType</a></span> <span class="pre">type);</span></code></p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPSetFromOptions.html">PEPSetFromOptions</a>(<a href="../../manualpages/PEP/PEP.html">PEP</a></span> <span class="pre">pep);</span></code></p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPSolve.html">PEPSolve</a>(<a href="../../manualpages/PEP/PEP.html">PEP</a></span> <span class="pre">pep);</span></code></p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPGetConverged.html">PEPGetConverged</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code> <code class="docutils notranslate"><span class="pre">pep</span></code>, <a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscInt/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscInt</span></a> <code class="docutils notranslate"><span class="pre">*nconv</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPGetEigenpair.html">PEPGetEigenpair</a></span></code>(<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code> <code class="docutils notranslate"><span class="pre">pep</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscInt/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscInt</span></a> <code class="docutils notranslate"><span class="pre">i</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscScalar/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscScalar</span></a> <code class="docutils notranslate"><span class="pre">*kr</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Sys/PetscScalar/" title="(in PETSc v3.24)"><span class="xref std std-doc">PetscScalar</span></a> <code class="docutils notranslate"><span class="pre">*ki</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Vec/Vec/" title="(in PETSc v3.24)"><span class="xref std std-doc">Vec</span></a> <code class="docutils notranslate"><span class="pre">xr</span></code>,<a class="reference external" href="https://petsc.org/release/manualpages/Vec/Vec/" title="(in PETSc v3.24)"><span class="xref std std-doc">Vec</span></a> <code class="docutils notranslate"><span class="pre">xi</span></code>);</p></li>
<li><p><code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPDestroy.html">PEPDestroy</a>(<a href="../../manualpages/PEP/PEP.html">PEP</a></span> <span class="pre">pep)</span></code>;</p></li>
</ul>
<p>First, the solver context (<code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code>) is created and the three problem matrices are specified. Then various options are set for customized solution. After that, the program solves the problem, retrieves the solution, and finally destroys the context.</p>
</section>
<section id="pep-options">
<h2>PEP Options<a class="headerlink" href="#pep-options" title="Link to this heading">#</a></h2>
<p>Most of the options available in the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> object have their equivalent in <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code>. A full list of command-line options can be obtained by running the example
with the option <code class="docutils notranslate"><span class="pre">-help</span></code>.</p>
<p>To show information about the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code> solver, add the <code class="docutils notranslate"><span class="pre">-pep_view</span></code> option:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n"><a href="../../manualpages/PEP/PEP.html">PEP</a></span> <span class="n">Object</span><span class="p">:</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">toar</span>
<span class="mi">50</span><span class="o">%</span> <span class="n">of</span> <span class="n">basis</span> <span class="n">vectors</span> <span class="n">kept</span> <span class="n">after</span> <span class="n">restart</span>
<span class="n">using</span> <span class="n">the</span> <span class="n">locking</span> <span class="n">variant</span>
<span class="n">problem</span> <span class="nb">type</span><span class="p">:</span> <span class="n">symmetric</span> <span class="n">polynomial</span> <span class="n">eigenvalue</span> <span class="n">problem</span>
<span class="n">polynomial</span> <span class="n">represented</span> <span class="ow">in</span> <span class="n">MONOMIAL</span> <span class="n">basis</span>
<span class="n">selected</span> <span class="n">portion</span> <span class="n">of</span> <span class="n">the</span> <span class="n">spectrum</span><span class="p">:</span> <span class="n">largest</span> <span class="n">eigenvalues</span> <span class="ow">in</span> <span class="n">magnitude</span>
<span class="n">number</span> <span class="n">of</span> <span class="n">eigenvalues</span> <span class="p">(</span><span class="n">nev</span><span class="p">):</span> <span class="mi">1</span>
<span class="n">number</span> <span class="n">of</span> <span class="n">column</span> <span class="n">vectors</span> <span class="p">(</span><span class="n">ncv</span><span class="p">):</span> <span class="mi">16</span>
<span class="n">maximum</span> <span class="n">dimension</span> <span class="n">of</span> <span class="n">projected</span> <span class="n">problem</span> <span class="p">(</span><span class="n">mpd</span><span class="p">):</span> <span class="mi">16</span>
<span class="n">maximum</span> <span class="n">number</span> <span class="n">of</span> <span class="n">iterations</span><span class="p">:</span> <span class="mi">100</span>
<span class="n">tolerance</span><span class="p">:</span> <span class="mf">1e-08</span>
<span class="n">convergence</span> <span class="n">test</span><span class="p">:</span> <span class="n">relative</span> <span class="n">to</span> <span class="n">the</span> <span class="n">eigenvalue</span>
<span class="n">extraction</span> <span class="nb">type</span><span class="p">:</span> <span class="n">NORM</span>
<span class="n"><a href="../../manualpages/BV/BV.html">BV</a></span> <span class="n">Object</span><span class="p">:</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">svec</span>
<span class="mi">18</span> <span class="n">columns</span> <span class="n">of</span> <span class="k">global</span> <span class="n">length</span> <span class="mi">100</span>
<span class="n">vector</span> <span class="n">orthogonalization</span> <span class="n">method</span><span class="p">:</span> <span class="n">classical</span> <span class="n">Gram</span><span class="o">-</span><span class="n">Schmidt</span>
<span class="n">orthogonalization</span> <span class="n">refinement</span><span class="p">:</span> <span class="k">if</span> <span class="n">needed</span> <span class="p">(</span><span class="n">eta</span><span class="p">:</span> <span class="mf">0.7071</span><span class="p">)</span>
<span class="n">block</span> <span class="n">orthogonalization</span> <span class="n">method</span><span class="p">:</span> <span class="n">GS</span>
<span class="n">doing</span> <span class="n">matmult</span> <span class="k">as</span> <span class="n">a</span> <span class="n">single</span> <span class="n">matrix</span><span class="o">-</span><span class="n">matrix</span> <span class="n">product</span>
<span class="n"><a href="../../manualpages/DS/DS.html">DS</a></span> <span class="n">Object</span><span class="p">:</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">nhep</span>
<span class="n"><a href="../../manualpages/ST/ST.html">ST</a></span> <span class="n">Object</span><span class="p">:</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">shift</span>
<span class="n">shift</span><span class="p">:</span> <span class="mf">0.</span>
<span class="n">number</span> <span class="n">of</span> <span class="n">matrices</span><span class="p">:</span> <span class="mi">3</span>
<span class="nb">all</span> <span class="n">matrices</span> <span class="n">have</span> <span class="n">different</span> <span class="n">nonzero</span> <span class="n">pattern</span>
<span class="n"><a href="https://petsc.org/release/manualpages/KSP/KSP/">KSP</a></span> <span class="n">Object</span><span class="p">:</span> <span class="p">(</span><span class="n">st_</span><span class="p">)</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">preonly</span>
<span class="n">maximum</span> <span class="n">iterations</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span> <span class="n">initial</span> <span class="n">guess</span> <span class="ow">is</span> <span class="n">zero</span>
<span class="n">tolerances</span><span class="p">:</span> <span class="n">relative</span><span class="o">=</span><span class="mf">1e-08</span><span class="p">,</span> <span class="n">absolute</span><span class="o">=</span><span class="mf">1e-50</span><span class="p">,</span> <span class="n">divergence</span><span class="o">=</span><span class="mf">10000.</span>
<span class="n">left</span> <span class="n">preconditioning</span>
<span class="n">using</span> <span class="n">NONE</span> <span class="n">norm</span> <span class="nb">type</span> <span class="k">for</span> <span class="n">convergence</span> <span class="n">test</span>
<span class="n"><a href="https://petsc.org/release/manualpages/PC/PC/">PC</a></span> <span class="n">Object</span><span class="p">:</span> <span class="p">(</span><span class="n">st_</span><span class="p">)</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">lu</span>
<span class="n">out</span><span class="o">-</span><span class="n">of</span><span class="o">-</span><span class="n">place</span> <span class="n">factorization</span>
<span class="n">tolerance</span> <span class="k">for</span> <span class="n">zero</span> <span class="n">pivot</span> <span class="mf">2.22045e-14</span>
<span class="n">matrix</span> <span class="n">ordering</span><span class="p">:</span> <span class="n">nd</span>
<span class="n">factor</span> <span class="n">fill</span> <span class="n">ratio</span> <span class="n">given</span> <span class="mf">5.</span><span class="p">,</span> <span class="n">needed</span> <span class="mf">1.</span>
<span class="n">Factored</span> <span class="n">matrix</span> <span class="n">follows</span><span class="p">:</span>
<span class="n"><a href="https://petsc.org/release/manualpages/Mat/Mat/">Mat</a></span> <span class="n">Object</span><span class="p">:</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">seqaij</span>
<span class="n">rows</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">cols</span><span class="o">=</span><span class="mi">100</span>
<span class="n">package</span> <span class="n">used</span> <span class="n">to</span> <span class="n">perform</span> <span class="n">factorization</span><span class="p">:</span> <span class="n">petsc</span>
<span class="n">total</span><span class="p">:</span> <span class="n">nonzeros</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">allocated</span> <span class="n">nonzeros</span><span class="o">=</span><span class="mi">100</span>
<span class="n">total</span> <span class="n">number</span> <span class="n">of</span> <span class="n">mallocs</span> <span class="n">used</span> <span class="n">during</span> <span class="n"><a href="https://petsc.org/release/manualpages/Mat/MatSetValues/">MatSetValues</a></span> <span class="n">calls</span> <span class="o">=</span><span class="mi">0</span>
<span class="ow">not</span> <span class="n">using</span> <span class="n">I</span><span class="o">-</span><span class="n">node</span> <span class="n">routines</span>
<span class="n">linear</span> <span class="n">system</span> <span class="n">matrix</span> <span class="o">=</span> <span class="n">precond</span> <span class="n">matrix</span><span class="p">:</span>
<span class="n"><a href="https://petsc.org/release/manualpages/Mat/Mat/">Mat</a></span> <span class="n">Object</span><span class="p">:</span> <span class="mi">1</span> <span class="n">MPI</span> <span class="n">processes</span>
<span class="nb">type</span><span class="p">:</span> <span class="n">seqaij</span>
<span class="n">rows</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">cols</span><span class="o">=</span><span class="mi">100</span>
<span class="n">total</span><span class="p">:</span> <span class="n">nonzeros</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">allocated</span> <span class="n">nonzeros</span><span class="o">=</span><span class="mi">500</span>
<span class="n">total</span> <span class="n">number</span> <span class="n">of</span> <span class="n">mallocs</span> <span class="n">used</span> <span class="n">during</span> <span class="n"><a href="https://petsc.org/release/manualpages/Mat/MatSetValues/">MatSetValues</a></span> <span class="n">calls</span> <span class="o">=</span><span class="mi">0</span>
<span class="ow">not</span> <span class="n">using</span> <span class="n">I</span><span class="o">-</span><span class="n">node</span> <span class="n">routines</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>All the command-line options related to the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP.html">PEP</a></span></code> object have the <code class="docutils notranslate"><span class="pre">-pep_</span></code> prefix.</p>
</div>
<p>Try changing some of the values, for example:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex16<span class="w"> </span>-pep_nev<span class="w"> </span><span class="m">4</span><span class="w"> </span>-pep_ncv<span class="w"> </span><span class="m">24</span><span class="w"> </span>-pep_smallest_magnitude<span class="w"> </span>-pep_tol<span class="w"> </span>1e-5
</pre></div>
</div>
</section>
<section id="choosing-the-solver-method">
<h2>Choosing the Solver Method<a class="headerlink" href="#choosing-the-solver-method" title="Link to this heading">#</a></h2>
<p>Several polynomial eigensolvers are available, which can be selected in the source code with the function <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPSetType.html">PEPSetType</a>()</span></code>, or at run time:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex16<span class="w"> </span>-pep_type<span class="w"> </span>qarnoldi
</pre></div>
</div>
<p>Check the table of <a class="reference internal" href="../manual/pep.html#tab-solversp"><span class="std std-ref">Polynomial eigenvalue solvers available in the PEP module</span></a>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The default solver is <code class="docutils notranslate"><span class="pre">toar</span></code>.</p>
</div>
<p>The <code class="docutils notranslate"><span class="pre">linear</span></code> solver performs an explicit linearization of the quadratic eigenproblem, resulting in a generalized eigenproblem. This linearization can be customized, see section <a class="reference internal" href="../manual/pep.html#sec-linearization"><span class="std std-ref">Linearization</span></a> in the Users Manual for details. For instance:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex16<span class="w"> </span>-pep_type<span class="w"> </span>linear<span class="w"> </span>-pep_linear_linearization<span class="w"> </span><span class="m">1</span>,0<span class="w"> </span>-pep_linear_explicitmatrix
</pre></div>
</div>
<p>Since in this problem all matrices are symmetric, the problem type is set to <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEP_HERMITIAN.html">PEP_HERMITIAN</a></span></code> in the source code with <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPSetProblemType.html">PEPSetProblemType</a>()</span></code>, and this obliges us to set the explicit matrix flag, see <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/PEP/PEPLinearSetExplicitMatrix.html">PEPLinearSetExplicitMatrix</a>()</span></code>. It is also possible to use a non-symmetric linearization by choosing the corresponding problem type:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex16<span class="w"> </span>-pep_type<span class="w"> </span>linear<span class="w"> </span>-pep_general
</pre></div>
</div>
<p>In the <code class="docutils notranslate"><span class="pre">linear</span></code> solver it is also possible to tune any of the <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/EPS/EPS.html">EPS</a></span></code> options, including those corresponding to <code class="docutils notranslate"><span class="pre"><a href="../../manualpages/ST/ST.html">ST</a></span></code> and the linear solvers. For instance:</p>
<div class="highlight-console notranslate"><div class="highlight"><pre><span></span><span class="gp">$ </span>./ex16<span class="w"> </span>-pep_type<span class="w"> </span>linear<span class="w"> </span>-pep_general<span class="w"> </span>-pep_linear_st_ksp_type<span class="w"> </span>bcgs<span class="w"> </span>-pep_linear_st_pc_type<span class="w"> </span>jacobi
</pre></div>
</div>
</section>
</section>
</article>
<footer class="prev-next-footer d-print-none">
<div class="prev-next-area">
<a class="left-prev"
href="hands-on7.html"
title="previous page">
<i class="fa-solid fa-angle-left"></i>
<div class="prev-next-info">
<p class="prev-next-subtitle">previous</p>
<p class="prev-next-title">Exercise 7: Use of Deflation Subspaces</p>
</div>
</a>
<a class="right-next"
href="../faq.html"
title="next page">
<div class="prev-next-info">
<p class="prev-next-subtitle">next</p>
<p class="prev-next-title">Frequently Asked Questions</p>
</div>
<i class="fa-solid fa-angle-right"></i>
</a>
</div>
</footer>
</div>
<dialog id="pst-secondary-sidebar-modal"></dialog>
<div id="pst-secondary-sidebar" class="bd-sidebar-secondary bd-toc"><div class="sidebar-secondary-items sidebar-secondary__inner">
<div class="sidebar-secondary-item">
<div
id="pst-page-navigation-heading-2"
class="page-toc tocsection onthispage">
<i class="fa-solid fa-list"></i> On this page
</div>
<nav class="bd-toc-nav page-toc" aria-labelledby="pst-page-navigation-heading-2">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#compiling">Compiling</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#running-the-program">Running the Program</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#source-code-details">Source Code Details</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pep-options">PEP Options</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#choosing-the-solver-method">Choosing the Solver Method</a></li>
</ul>
</nav></div>
<div class="sidebar-secondary-item">
<div class="tocsection editthispage">
<a href="https://gitlab.com/slepc/slepc/-/edit/release/doc/source/documentation/hands-on/hands-on8.md">
<i class="fa-solid fa-pencil"></i>
Edit on GitLab
</a>
</div>
</div>
<div class="sidebar-secondary-item">
<div role="note" aria-label="source link">
<h3>This Page</h3>
<ul class="this-page-menu">
<li><a href="../../_sources/documentation/hands-on/hands-on8.md.txt"
rel="nofollow">Show Source</a></li>
</ul>
</div></div>
</div></div>
</div>
<footer class="bd-footer-content">
</footer>
</main>
</div>
</div>
<!-- Scripts loaded after <body> so the DOM is not blocked -->
<script defer src="../../_static/scripts/bootstrap.js?digest=8878045cc6db502f8baf"></script>
<script defer src="../../_static/scripts/pydata-sphinx-theme.js?digest=8878045cc6db502f8baf"></script>
<footer class="bd-footer">
<div class="bd-footer__inner bd-page-width">
<div class="footer-items__start">
<div class="footer-item">
<p class="copyright">
© Copyright 2002-2025, Universitat Politecnica de Valencia, Spain.
<br/>
</p>
</div>
<div class="footer-item">
<p class="sphinx-version">
Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 7.3.7.
<br/>
</p>
</div>
</div>
<div class="footer-items__end">
<div class="footer-item">
<p class="theme-version">
<!-- # L10n: Setting the PST URL as an argument as this does not need to be localized -->
Built with the <a href="https://pydata-sphinx-theme.readthedocs.io/en/stable/index.html">PyData Sphinx Theme</a> 0.16.1.
</p></div>
</div>
</div>
</footer>
</body>
</html>
|
