forked from Reference-LAPACK/lapack-www
-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathlapack-3.10.0.html
More file actions
1037 lines (954 loc) · 27.9 KB
/
lapack-3.10.0.html
File metadata and controls
1037 lines (954 loc) · 27.9 KB
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
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN"
"http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<meta http-equiv="Content-Type" content="application/xhtml+xml; charset=UTF-8" />
<meta name="generator" content="AsciiDoc 9.1.0" />
<title>LAPACK 3.10.0</title>
<style type="text/css">
/* Shared CSS for AsciiDoc xhtml11 and html5 backends */
/* Default font. */
body {
font-family: Georgia,serif;
}
/* Title font. */
h1, h2, h3, h4, h5, h6,
div.title, caption.title,
thead, p.table.header,
#toctitle,
#author, #revnumber, #revdate, #revremark,
#footer {
font-family: Arial,Helvetica,sans-serif;
}
body {
margin: 1em 5% 1em 5%;
}
a {
color: blue;
text-decoration: underline;
}
a:visited {
color: fuchsia;
}
em {
font-style: italic;
color: navy;
}
strong {
font-weight: bold;
color: #083194;
}
h1, h2, h3, h4, h5, h6 {
color: #527bbd;
margin-top: 1.2em;
margin-bottom: 0.5em;
line-height: 1.3;
}
h1, h2, h3 {
border-bottom: 2px solid silver;
}
h2 {
padding-top: 0.5em;
}
h3 {
float: left;
}
h3 + * {
clear: left;
}
h5 {
font-size: 1.0em;
}
div.sectionbody {
margin-left: 0;
}
hr {
border: 1px solid silver;
}
p {
margin-top: 0.5em;
margin-bottom: 0.5em;
}
ul, ol, li > p {
margin-top: 0;
}
ul > li { color: #aaa; }
ul > li > * { color: black; }
.monospaced, code, pre {
font-family: "Courier New", Courier, monospace;
font-size: inherit;
color: navy;
padding: 0;
margin: 0;
}
pre {
white-space: pre-wrap;
}
#author {
color: #527bbd;
font-weight: bold;
font-size: 1.1em;
}
#email {
}
#revnumber, #revdate, #revremark {
}
#footer {
font-size: small;
border-top: 2px solid silver;
padding-top: 0.5em;
margin-top: 4.0em;
}
#footer-text {
float: left;
padding-bottom: 0.5em;
}
#footer-badges {
float: right;
padding-bottom: 0.5em;
}
#preamble {
margin-top: 1.5em;
margin-bottom: 1.5em;
}
div.imageblock, div.exampleblock, div.verseblock,
div.quoteblock, div.literalblock, div.listingblock, div.sidebarblock,
div.admonitionblock {
margin-top: 1.0em;
margin-bottom: 1.5em;
}
div.admonitionblock {
margin-top: 2.0em;
margin-bottom: 2.0em;
margin-right: 10%;
color: #606060;
}
div.content { /* Block element content. */
padding: 0;
}
/* Block element titles. */
div.title, caption.title {
color: #527bbd;
font-weight: bold;
text-align: left;
margin-top: 1.0em;
margin-bottom: 0.5em;
}
div.title + * {
margin-top: 0;
}
td div.title:first-child {
margin-top: 0.0em;
}
div.content div.title:first-child {
margin-top: 0.0em;
}
div.content + div.title {
margin-top: 0.0em;
}
div.sidebarblock > div.content {
background: #ffffee;
border: 1px solid #dddddd;
border-left: 4px solid #f0f0f0;
padding: 0.5em;
}
div.listingblock > div.content {
border: 1px solid #dddddd;
border-left: 5px solid #f0f0f0;
background: #f8f8f8;
padding: 0.5em;
}
div.quoteblock, div.verseblock {
padding-left: 1.0em;
margin-left: 1.0em;
margin-right: 10%;
border-left: 5px solid #f0f0f0;
color: #888;
}
div.quoteblock > div.attribution {
padding-top: 0.5em;
text-align: right;
}
div.verseblock > pre.content {
font-family: inherit;
font-size: inherit;
}
div.verseblock > div.attribution {
padding-top: 0.75em;
text-align: left;
}
/* DEPRECATED: Pre version 8.2.7 verse style literal block. */
div.verseblock + div.attribution {
text-align: left;
}
div.admonitionblock .icon {
vertical-align: top;
font-size: 1.1em;
font-weight: bold;
text-decoration: underline;
color: #527bbd;
padding-right: 0.5em;
}
div.admonitionblock td.content {
padding-left: 0.5em;
border-left: 3px solid #dddddd;
}
div.exampleblock > div.content {
border-left: 3px solid #dddddd;
padding-left: 0.5em;
}
div.imageblock div.content { padding-left: 0; }
span.image img { border-style: none; vertical-align: text-bottom; }
a.image:visited { color: white; }
dl {
margin-top: 0.8em;
margin-bottom: 0.8em;
}
dt {
margin-top: 0.5em;
margin-bottom: 0;
font-style: normal;
color: navy;
}
dd > *:first-child {
margin-top: 0.1em;
}
ul, ol {
list-style-position: outside;
}
ol.arabic {
list-style-type: decimal;
}
ol.loweralpha {
list-style-type: lower-alpha;
}
ol.upperalpha {
list-style-type: upper-alpha;
}
ol.lowerroman {
list-style-type: lower-roman;
}
ol.upperroman {
list-style-type: upper-roman;
}
div.compact ul, div.compact ol,
div.compact p, div.compact p,
div.compact div, div.compact div {
margin-top: 0.1em;
margin-bottom: 0.1em;
}
tfoot {
font-weight: bold;
}
td > div.verse {
white-space: pre;
}
div.hdlist {
margin-top: 0.8em;
margin-bottom: 0.8em;
}
div.hdlist tr {
padding-bottom: 15px;
}
dt.hdlist1.strong, td.hdlist1.strong {
font-weight: bold;
}
td.hdlist1 {
vertical-align: top;
font-style: normal;
padding-right: 0.8em;
color: navy;
}
td.hdlist2 {
vertical-align: top;
}
div.hdlist.compact tr {
margin: 0;
padding-bottom: 0;
}
.comment {
background: yellow;
}
.footnote, .footnoteref {
font-size: 0.8em;
}
span.footnote, span.footnoteref {
vertical-align: super;
}
#footnotes {
margin: 20px 0 20px 0;
padding: 7px 0 0 0;
}
#footnotes div.footnote {
margin: 0 0 5px 0;
}
#footnotes hr {
border: none;
border-top: 1px solid silver;
height: 1px;
text-align: left;
margin-left: 0;
width: 20%;
min-width: 100px;
}
div.colist td {
padding-right: 0.5em;
padding-bottom: 0.3em;
vertical-align: top;
}
div.colist td img {
margin-top: 0.3em;
}
@media print {
#footer-badges { display: none; }
}
#toc {
margin-bottom: 2.5em;
}
#toctitle {
color: #527bbd;
font-size: 1.1em;
font-weight: bold;
margin-top: 1.0em;
margin-bottom: 0.1em;
}
div.toclevel0, div.toclevel1, div.toclevel2, div.toclevel3, div.toclevel4 {
margin-top: 0;
margin-bottom: 0;
}
div.toclevel2 {
margin-left: 2em;
font-size: 0.9em;
}
div.toclevel3 {
margin-left: 4em;
font-size: 0.9em;
}
div.toclevel4 {
margin-left: 6em;
font-size: 0.9em;
}
span.aqua { color: aqua; }
span.black { color: black; }
span.blue { color: blue; }
span.fuchsia { color: fuchsia; }
span.gray { color: gray; }
span.green { color: green; }
span.lime { color: lime; }
span.maroon { color: maroon; }
span.navy { color: navy; }
span.olive { color: olive; }
span.purple { color: purple; }
span.red { color: red; }
span.silver { color: silver; }
span.teal { color: teal; }
span.white { color: white; }
span.yellow { color: yellow; }
span.aqua-background { background: aqua; }
span.black-background { background: black; }
span.blue-background { background: blue; }
span.fuchsia-background { background: fuchsia; }
span.gray-background { background: gray; }
span.green-background { background: green; }
span.lime-background { background: lime; }
span.maroon-background { background: maroon; }
span.navy-background { background: navy; }
span.olive-background { background: olive; }
span.purple-background { background: purple; }
span.red-background { background: red; }
span.silver-background { background: silver; }
span.teal-background { background: teal; }
span.white-background { background: white; }
span.yellow-background { background: yellow; }
span.big { font-size: 2em; }
span.small { font-size: 0.6em; }
span.underline { text-decoration: underline; }
span.overline { text-decoration: overline; }
span.line-through { text-decoration: line-through; }
div.unbreakable { page-break-inside: avoid; }
/*
* xhtml11 specific
*
* */
div.tableblock {
margin-top: 1.0em;
margin-bottom: 1.5em;
}
div.tableblock > table {
border: 3px solid #527bbd;
}
thead, p.table.header {
font-weight: bold;
color: #527bbd;
}
p.table {
margin-top: 0;
}
/* Because the table frame attribute is overridden by CSS in most browsers. */
div.tableblock > table[frame="void"] {
border-style: none;
}
div.tableblock > table[frame="hsides"] {
border-left-style: none;
border-right-style: none;
}
div.tableblock > table[frame="vsides"] {
border-top-style: none;
border-bottom-style: none;
}
/*
* html5 specific
*
* */
table.tableblock {
margin-top: 1.0em;
margin-bottom: 1.5em;
}
thead, p.tableblock.header {
font-weight: bold;
color: #527bbd;
}
p.tableblock {
margin-top: 0;
}
table.tableblock {
border-width: 3px;
border-spacing: 0px;
border-style: solid;
border-color: #527bbd;
border-collapse: collapse;
}
th.tableblock, td.tableblock {
border-width: 1px;
padding: 4px;
border-style: solid;
border-color: #527bbd;
}
table.tableblock.frame-topbot {
border-left-style: hidden;
border-right-style: hidden;
}
table.tableblock.frame-sides {
border-top-style: hidden;
border-bottom-style: hidden;
}
table.tableblock.frame-none {
border-style: hidden;
}
th.tableblock.halign-left, td.tableblock.halign-left {
text-align: left;
}
th.tableblock.halign-center, td.tableblock.halign-center {
text-align: center;
}
th.tableblock.halign-right, td.tableblock.halign-right {
text-align: right;
}
th.tableblock.valign-top, td.tableblock.valign-top {
vertical-align: top;
}
th.tableblock.valign-middle, td.tableblock.valign-middle {
vertical-align: middle;
}
th.tableblock.valign-bottom, td.tableblock.valign-bottom {
vertical-align: bottom;
}
/*
* manpage specific
*
* */
body.manpage h1 {
padding-top: 0.5em;
padding-bottom: 0.5em;
border-top: 2px solid silver;
border-bottom: 2px solid silver;
}
body.manpage h2 {
border-style: none;
}
body.manpage div.sectionbody {
margin-left: 3em;
}
@media print {
body.manpage div#toc { display: none; }
}
</style>
<script type="text/javascript">
/*<+'])');
// Function that scans the DOM tree for header elements (the DOM2
// nodeIterator API would be a better technique but not supported by all
// browsers).
var iterate = function (el) {
for (var i = el.firstChild; i != null; i = i.nextSibling) {
if (i.nodeType == 1 /* Node.ELEMENT_NODE */) {
var mo = re.exec(i.tagName);
if (mo && (i.getAttribute("class") || i.getAttribute("className")) != "float") {
result[result.length] = new TocEntry(i, getText(i), mo[1]-1);
}
iterate(i);
}
}
}
iterate(el);
return result;
}
var toc = document.getElementById("toc");
if (!toc) {
return;
}
// Delete existing TOC entries in case we're reloading the TOC.
var tocEntriesToRemove = [];
var i;
for (i = 0; i < toc.childNodes.length; i++) {
var entry = toc.childNodes[i];
if (entry.nodeName.toLowerCase() == 'div'
&& entry.getAttribute("class")
&& entry.getAttribute("class").match(/^toclevel/))
tocEntriesToRemove.push(entry);
}
for (i = 0; i < tocEntriesToRemove.length; i++) {
toc.removeChild(tocEntriesToRemove[i]);
}
// Rebuild TOC entries.
var entries = tocEntries(document.getElementById("content"), toclevels);
for (var i = 0; i < entries.length; ++i) {
var entry = entries[i];
if (entry.element.id == "")
entry.element.id = "_toc_" + i;
var a = document.createElement("a");
a.href = "#" + entry.element.id;
a.appendChild(document.createTextNode(entry.text));
var div = document.createElement("div");
div.appendChild(a);
div.className = "toclevel" + entry.toclevel;
toc.appendChild(div);
}
if (entries.length == 0)
toc.parentNode.removeChild(toc);
},
/////////////////////////////////////////////////////////////////////
// Footnotes generator
/////////////////////////////////////////////////////////////////////
/* Based on footnote generation code from:
* http://www.brandspankingnew.net/archive/2005/07/format_footnote.html
*/
footnotes: function () {
// Delete existing footnote entries in case we're reloading the footnodes.
var i;
var noteholder = document.getElementById("footnotes");
if (!noteholder) {
return;
}
var entriesToRemove = [];
for (i = 0; i < noteholder.childNodes.length; i++) {
var entry = noteholder.childNodes[i];
if (entry.nodeName.toLowerCase() == 'div' && entry.getAttribute("class") == "footnote")
entriesToRemove.push(entry);
}
for (i = 0; i < entriesToRemove.length; i++) {
noteholder.removeChild(entriesToRemove[i]);
}
// Rebuild footnote entries.
var cont = document.getElementById("content");
var spans = cont.getElementsByTagName("span");
var refs = {};
var n = 0;
for (i=0; i<spans.length; i++) {
if (spans[i].className == "footnote") {
n++;
var note = spans[i].getAttribute("data-note");
if (!note) {
// Use [\s\S] in place of . so multi-line matches work.
// Because JavaScript has no s (dotall) regex flag.
note = spans[i].innerHTML.match(/\s*\[([\s\S]*)]\s*/)[1];
spans[i].innerHTML =
"[<a id='_footnoteref_" + n + "' href='#_footnote_" + n +
"' title='View footnote' class='footnote'>" + n + "</a>]";
spans[i].setAttribute("data-note", note);
}
noteholder.innerHTML +=
"<div class='footnote' id='_footnote_" + n + "'>" +
"<a href='#_footnoteref_" + n + "' title='Return to text'>" +
n + "</a>. " + note + "</div>";
var id =spans[i].getAttribute("id");
if (id != null) refs["#"+id] = n;
}
}
if (n == 0)
noteholder.parentNode.removeChild(noteholder);
else {
// Process footnoterefs.
for (i=0; i<spans.length; i++) {
if (spans[i].className == "footnoteref") {
var href = spans[i].getElementsByTagName("a")[0].getAttribute("href");
href = href.match(/#.*/)[0]; // Because IE return full URL.
n = refs[href];
spans[i].innerHTML =
"[<a href='#_footnote_" + n +
"' title='View footnote' class='footnote'>" + n + "</a>]";
}
}
}
},
install: function(toclevels) {
var timerId;
function reinstall() {
asciidoc.footnotes();
if (toclevels) {
asciidoc.toc(toclevels);
}
}
function reinstallAndRemoveTimer() {
clearInterval(timerId);
reinstall();
}
timerId = setInterval(reinstall, 500);
if (document.addEventListener)
document.addEventListener("DOMContentLoaded", reinstallAndRemoveTimer, false);
else
window.onload = reinstallAndRemoveTimer;
}
}
asciidoc.install(2);
/*]]>*/
</script>
</head>
<body class="article">
<div id="header">
<h1>LAPACK 3.10.0</h1>
<div id="toc">
<div id="toctitle">Table of Contents</div>
<noscript><p><b>JavaScript must be enabled in your browser to display the table of contents.</b></p></noscript>
</div>
</div>
<div id="content">
<div id="preamble">
<div class="sectionbody">
<div class="paragraph"><p>Release date: 06/28/21.</p></div>
<div class="paragraph"><p>This material is based upon work supported by the National Science Foundation
and the Department of Energy.</p></div>
<div class="paragraph"><p>LAPACK is a software package provided by Univ. of Tennessee, Univ. of
California, Berkeley, Univ. of Colorado Denver and NAG Ltd..</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_1_support_and_questions">1. Support and questions:</h2>
<div class="sectionbody">
<div class="paragraph"><p><a href="http://icl.cs.utk.edu/lapack-forum/">http://icl.cs.utk.edu/lapack-forum/</a></p></div>
<div class="paragraph"><p><a href="https://github.com/Reference-LAPACK/lapack/">https://github.com/Reference-LAPACK/lapack/</a></p></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_lapack_3_10_0_what_8217_s_new">2. LAPACK 3.10.0: What’s new</h2>
<div class="sectionbody">
<div class="ulist"><ul>
<li>
<p>
Download: <a href="https://github.com/Reference-LAPACK/lapack/archive/refs/tags/v3.10.0.tar.gz">lapack-3.10.0.tar.gz</a>
</p>
</li>
</ul></div>
<div class="paragraph"><p>This is a major release and also addressing multiple bug fixes.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_1_safe_scaling_algorithms">2.1. Safe scaling algorithms</h2>
<div class="sectionbody">
<div class="paragraph"><p>Contribution by Ed Anderson</p></div>
<div class="literalblock">
<div class="content">
<pre><code>Added SRC/la_constants.f90
Added SRC/la_xisnan.F90</code></pre>
</div></div>
<div class="literalblock">
<div class="content">
<pre><code>Deleted BLAS/SRC/dnrm2.f Added BLAS/SRC/dnrm2.f90
Deleted BLAS/SRC/dznrm2.f Added BLAS/SRC/dznrm2.f90
Deleted BLAS/SRC/scnrm2.f Added BLAS/SRC/scnrm2.f90
Deleted BLAS/SRC/snrm2.f Added BLAS/SRC/snrm2.f90
Deleted BLAS/SRC/drotg.f Added BLAS/SRC/drotg.f90
Deleted BLAS/SRC/zrotg.f Added BLAS/SRC/zrotg.f90
Deleted BLAS/SRC/crotg.f Added BLAS/SRC/crotg.f90
Deleted BLAS/SRC/srotg.f Added BLAS/SRC/srotg.f90
Deleted SRC/dlassq.f Added SRC/dlassq.f90
Deleted SRC/zlassq.f Added SRC/zlassq.f90
Deleted SRC/classq.f Added SRC/classq.f90
Deleted SRC/slassq.f Added SRC/slassq.f90
Deleted SRC/dlartg.f Added SRC/dlartg.f90
Deleted SRC/zlartg.f Added SRC/zlartg.f90
Deleted SRC/clartg.f Added SRC/clartg.f90
Deleted SRC/slartg.f Added SRC/slartg.f90</code></pre>
</div></div>
<div class="paragraph"><p>Edward Anderson stated in his paper <a href="https://doi.org/10.1145/3061665">https://doi.org/10.1145/3061665</a>:</p></div>
<div class="paragraph"><p>The square root of a sum of squares is well known to be prone to overflow and underflow. Ad hoc scaling
of intermediate results, as has been done in numerical software such as the BLAS and LAPACK, mostly
avoids the problem, but it can still occur at extreme values in the range of representable numbers. More
careful scaling, as has been implemented in recent versions of the standard algorithms, may come at the
expense of performance or clarity. This work reimplements the vector 2-norm and the generation of Givens
rotations from the Level 1 BLAS to improve their performance and design. In addition, support for negative
increments is extended to the Level 1 BLAS operations on a single vector, and a comprehensive test suite
for all the Level 1 BLAS is included.</p></div>
<div class="paragraph"><p>This contribution replaces the original xNRM2, xROTG, xLARTG, xLASSQ routines with Edward’s safe scaling codes plus some minor modifications.</p></div>
<div class="paragraph"><p>Note: this code follows Fortran90 standard.</p></div>
<div class="ulist"><ul>
<li>
<p>
The new Fortran90 module <code>SRC/la_constants.f90</code> expands the functionality of <code>INSTALL/xLAMCH.f</code> by adding safe scaling constants.
</p>
</li>
<li>
<p>
The new Fortran90 module <code>SRC/la_xisnan.F90</code> expands the functionality of <code>SRC/xISNAN.f</code> and <code>SRC/xLAISNAN.f</code> by adding the option of using the <code>ieee_is_nan</code> routine.
</p>
</li>
</ul></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_2_implementation_of_the_multishift_qz_algorithm_with_aed">2.2. Implementation of the multishift QZ algorithm with AED</h2>
<div class="sectionbody">
<div class="paragraph"><p>Contribution by <a href="https://github.com/thijssteel">thijssteel</a></p></div>
<div class="paragraph"><p>It is loosely based on my implementation of the <strong>rational QZ algorithm</strong> (<a href="https://github.com/thijssteel/multishift-multipole-rqz">https://github.com/thijssteel/multishift-multipole-rqz</a>).</p></div>
<div class="paragraph"><p>It features:</p></div>
<div class="ulist"><ul>
<li>
<p>
Agressive early deflation
</p>
</li>
<li>
<p>
Multishift QZ sweeps using optimal packing of the bulges
</p>
</li>
<li>
<p>
A new heuristic to select the number of positions in the sweep windows
</p>
</li>
</ul></div>
<div class="paragraph"><p>It does not feature:</p></div>
<div class="ulist"><ul>
<li>
<p>
A windowed deflation of infinite eigenvalues (that is only useful if many infinite eigenvalues are to be deflated and in that case you should probably do some preprocessing anyway).
</p>
</li>
</ul></div>
<div class="paragraph"><p>Also <strong>two accuracy improvements to xtgex2</strong>:</p></div>
<div class="ulist"><ul>
<li>
<p>
Daan Camps has proven in his dissertation that replacing the condition in xtgex2 when swapping 1x1 blocks can improve the accuracy. This condition and the respective error tolerances have been changed to reflect it.
</p>
</li>
<li>
<p>
Also by Daan Camps, "SWAPPING 2 × 2 BLOCKS IN THE SCHUR AND GENERALIZED SCHUR FORM", shows that the accuracy of swapping 2x2 blocks can be improved by iterative refinement and by replacing the QR factorisation with an SVD based method. Only the first improvement was implemented as the paper does not mention how to avoid overflow.
</p>
</li>
</ul></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_3_householder_reconstruction">2.3 Householder Reconstruction</h2>
<div class="sectionbody">
<div class="literalblock">
<div class="content">
<pre><code>new file: SRC/sorhr_col.f
new file: SRC/dorhr_col.f
new file: SRC/cunhr_col.f
new file: SRC/zunhr_col.f
new file: SRC/dorgtsqr.f
new file: SRC/sorgtsqr.f</code></pre>
</div></div>
<div class="literalblock">
<div class="content">
<pre><code>new file: SRC/dgetsqrhrt.f
new file: SRC/dlarfb_gett.f
new file: SRC/dorgtsqr_row.f
new file: SRC/sgetsqrhrt.f
new file: SRC/slarfb_gett.f
new file: SRC/sorgtsqr_row.f
new file: SRC/cgetsqrhrt.f
new file: SRC/clarfb_gett.f
new file: SRC/cungtsqr_row.f
new file: SRC/zgetsqrhrt.f
new file: SRC/zlarfb_gett.f
new file: SRC/zungtsqr_row.f</code></pre>
</div></div>
<div class="paragraph"><p>GETSQRHRT is a QR factorization routine for tall-skinny matrices. It is based on LATSQR, but returns the Q and R factors in the same format as GEQRT, i.e. using the compact WY representation of Q.</p></div>
<div class="paragraph"><p>This is the same format used by other LAPACK routines that depend on QR factorizations. GETSQRHRT uses ORGTSQR_ROW (which in turn calls LARFB_GETT) to construct an orthonormal matrix from a TSQR factorization, which is a more efficient version of ORGTSQR.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_4_change_in_the_return_behavior_of_gesdd">2.4 Change in the return behavior of GESDD</h2>
<div class="sectionbody">
<div class="paragraph"><p>GESDD now returns with INFO = -4 if A has a NaN entry.</p></div>
<div class="paragraph"><p>In version 3.9.0 and master, GESDD would call <code>exit(0)</code> instead of returning an error code on some matrices. See <a href="https://github.com/Reference-LAPACK/lapack/issues/469#issue-781977643">Issue #469</a></p></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_5_bug_fixes">2.5 Bug fixes</h2>
<div class="sectionbody">
<div class="paragraph"><p>For details please see our Github repository</p></div>
<div class="ulist"><ul>
<li>
<p>
<a href="https://github.com/Reference-LAPACK/lapack">LAPACK Github Repository</a>
</p>
</li>
<li>
<p>
<a href="https://github.com/Reference-LAPACK/lapack/issues?q=is%3Aclosed+milestone%3A%22LAPACK+3.10.0%22">LAPACK 3.10.0 Milestone</a>
</p>
</li>
</ul></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_6_notes_about_compiler_dependency">2.6 Notes about compiler dependency</h2>
<div class="sectionbody">
<div class="paragraph"><p>Some LAPACK routines rely on trustworthy complex division and ABS routines in the FORTRAN compiler.
This [link](<a href="https://github.com/Reference-LAPACK/lapack/files/6672436/complexDivisionFound.txt">https://github.com/Reference-LAPACK/lapack/files/6672436/complexDivisionFound.txt</a>) lists the LAPACK COMPLEX*16 algorithms that contain compiler dependent complex divisions of the form</p></div>
<div class="literalblock">
<div class="content">
<pre><code>REAL / COMPLEX or COMPLEX / COMPLEX</code></pre>
</div></div>
<div class="paragraph"><p>See <a href="https://github.com/Reference-LAPACK/lapack/issues/575">Issue #575</a> and <a href="https://github.com/Reference-LAPACK/lapack/issues/577">Issue #577</a> for a more complete discussion on this topic.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_3_developer_list">3. Developer list</h2>
<div class="sectionbody">
<div class="ulist"><div class="title">LAPACK developers involved in this release</div><ul>
<li>
<p>
Weslley da Silva Pereira (University of Colorado Denver, USA)
</p>
</li>
<li>
<p>
Julie Langou (University of Tennessee, USA)
</p>
</li>
<li>
<p>
Igor Kozachenko (University of California, Berkeley, USA)
</p>
</li>
</ul></div>
<div class="ulist"><div class="title">Principal Investigators</div><ul>
<li>
<p>
Jim Demmel (University of California, Berkeley, USA)
</p>
</li>
<li>
<p>
Jack Dongarra (University of Tennessee and ORNL, USA)
</p>
</li>
<li>
<p>
Julien Langou (University of Colorado Denver, USA)
</p>
</li>
</ul></div>
</div>
</div>
<div class="sect1">
<h2 id="_4_thanks">4. Thanks</h2>
<div class="sectionbody">
<div class="ulist"><ul>
<li>
<p>
MathWorks: Penny Anderson, Amanda Barry, Mary Ann Freeman, Bobby Cheng,
Pat Quillen, Christine Tobler.
</p>
</li>
<li>
<p>
Ed Anderson
</p>
</li>
<li>
<p>
GitHub Users: <a href="https://github.com/thijssteel">thijssteel</a>, <a href="https://github.com/jip">jip</a>, <a href="https://github.com/eshpc">eshpc</a>, <a href="https://github.com/martin-frbg">martin-frbg</a>,
<a href="https://github.com/Matthew-Badin">Matthew-Badin</a>,
<a href="https://github.com/msk">msk</a>, <a href="https://github.com/ZTaylor39">ZTaylor39</a>,
<a href="https://github.com/sergey-v-kuznetsov">sergey-v-kuznetsov</a>, <a href="https://github.com/5tefan">5tefan</a>
</p>