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
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.2 beta6 (August 14th, 1998)
original version by: Nikos Drakos, CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Accuracy and Stability</TITLE>
<META NAME="description" CONTENT="Accuracy and Stability">
<META NAME="keywords" CONTENT="lug_l2h">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="lug_l2h.css">
<LINK REL="next" HREF="node109.html">
<LINK REL="previous" HREF="node60.html">
<LINK REL="up" HREF="node5.html">
<LINK REL="next" HREF="node73.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html5167"
HREF="node73.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="next_motif.png"></A>
<A NAME="tex2html5161"
HREF="node5.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="up_motif.png"></A>
<A NAME="tex2html5155"
HREF="node71.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="previous_motif.png"></A>
<A NAME="tex2html5163"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="contents_motif.png"></A>
<A NAME="tex2html5165"
HREF="node152.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
SRC="index_motif.png"></A>
<BR>
<B> Next:</B> <A NAME="tex2html5168"
HREF="node73.html">Sources of Error in</A>
<B> Up:</B> <A NAME="tex2html5162"
HREF="node5.html">Guide</A>
<B> Previous:</B> <A NAME="tex2html5156"
HREF="node71.html">LAPACK Benchmark</A>
  <B> <A NAME="tex2html5164"
HREF="node1.html">Contents</A></B>
  <B> <A NAME="tex2html5166"
HREF="node152.html">Index</A></B>
<BR>
<BR>
<!--End of Navigation Panel-->
<H1><A NAME="SECTION03400000000000000000"></A><A NAME="chapaccstab"></A>
<BR>
Accuracy and Stability
</H1>
<P>
<P>
<P>
<P>
In addition to providing faster routines than previously available,
LAPACK provides more comprehensive and better<A NAME="9688"></A>
error<A NAME="9689"></A> bounds<A NAME="9690"></A>.
Our goal is to provide error bounds for most quantities computed by LAPACK.
<P>
In this chapter we explain our overall approach to obtaining error bounds,
and provide enough information to use the software.
The comments at the beginning of the individual routines should be consulted
for more details.
It is beyond the scope of this chapter to justify all the bounds
we present. Instead, we give references to the literature.
For example, standard material on error analysis can be found
in [<A
HREF="node151.html#demmelMA221">25</A>,<A
HREF="node151.html#GVL2">55</A>,<A
HREF="node151.html#higham96">67</A>,<A
HREF="node151.html#stewartsun90">95</A>].
<P>
In order to make this chapter easy to read, we have labeled sections
not essential for a first reading as <B>Further Details</B>. The sections
not labeled as <B>Further Details</B> should provide all the information
needed to understand and use the main error bounds computed by
LAPACK. The <B>Further Details</B> sections provide mathematical background,
references, and tighter but more expensive error bounds, and may be read later.
<P>
In section <A HREF="node73.html#secroundoff">4.1</A>
we discuss the sources of numerical error, in particular roundoff
error.
Section <A HREF="node75.html#secnormnotation">4.2</A> discusses how to measure errors,
as well as some standard notation.
Section <A HREF="node77.html#secwhereerrorboundscomefrom">4.3</A> discusses further details
of how error bounds are derived.
Sections <A HREF="node80.html#secAx_b">4.4</A> through <A HREF="node106.html#secGSVDbound">4.12</A> present error bounds
for linear equations, linear least squares problems,
generalized linear least squares problems,
the symmetric eigenproblem,
the nonsymmetric eigenproblem,
the singular value decomposition,
the generalized symmetric definite eigenproblem,
the generalized nonsymmetric eigenproblem
and the
generalized (or quotient) singular value decomposition respectively.
Section <A HREF="node108.html#secfastblas">4.13</A> discusses the impact of fast
Level 3 BLAS<A NAME="9701"></A> on
the accuracy<A NAME="9702"></A> of LAPACK routines.
<P>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"></A>
<UL>
<LI><A NAME="tex2html5169"
HREF="node73.html">Sources of Error in Numerical Calculations</A>
<UL>
<LI><A NAME="tex2html5170"
HREF="node74.html">Further Details: Floating Point Arithmetic</A>
</UL>
<LI><A NAME="tex2html5171"
HREF="node75.html">How to Measure Errors</A>
<UL>
<LI><A NAME="tex2html5172"
HREF="node76.html">Further Details: How to Measure Errors</A>
</UL>
<LI><A NAME="tex2html5173"
HREF="node77.html">Further Details: How Error Bounds Are Derived</A>
<UL>
<LI><A NAME="tex2html5174"
HREF="node78.html">Standard Error Analysis</A>
<LI><A NAME="tex2html5175"
HREF="node79.html">Improved Error Bounds</A>
</UL>
<LI><A NAME="tex2html5176"
HREF="node80.html">Error Bounds for Linear Equation Solving</A>
<UL>
<LI><A NAME="tex2html5177"
HREF="node81.html">Further Details: Error Bounds for Linear Equation Solving</A>
</UL>
<LI><A NAME="tex2html5178"
HREF="node82.html">Error Bounds for Linear Least Squares Problems</A>
<UL>
<LI><A NAME="tex2html5179"
HREF="node83.html">Further Details: Error Bounds for Linear Least Squares
Problems</A>
</UL>
<LI><A NAME="tex2html5180"
HREF="node84.html">Error Bounds for Generalized Least Squares Problems</A>
<UL>
<LI><A NAME="tex2html5181"
HREF="node85.html">Linear Equality Constrained Least Squares Problem</A>
<UL>
<LI><A NAME="tex2html5182"
HREF="node86.html">Further Details:
Error Bounds for Linear Equality Constrained Least Squares Problems</A>
</UL>
<LI><A NAME="tex2html5183"
HREF="node87.html">General Linear Model Problem</A>
<UL>
<LI><A NAME="tex2html5184"
HREF="node88.html">Further Details: Error Bounds for General Linear Model Problems</A>
</UL>
</UL>
<LI><A NAME="tex2html5185"
HREF="node89.html">Error Bounds for the Symmetric Eigenproblem </A>
<UL>
<LI><A NAME="tex2html5186"
HREF="node90.html">Further Details: Error Bounds for the Symmetric Eigenproblem </A>
</UL>
<LI><A NAME="tex2html5187"
HREF="node91.html">Error Bounds for the Nonsymmetric Eigenproblem </A>
<UL>
<LI><A NAME="tex2html5188"
HREF="node92.html">Further Details: Error Bounds for the Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5189"
HREF="node93.html">Overview</A>
<LI><A NAME="tex2html5190"
HREF="node94.html">Balancing and Conditioning</A>
<LI><A NAME="tex2html5191"
HREF="node95.html">Computing <B><I>s</I></B> and <IMG
WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img2.png"
ALT="${\rm sep}$"></A>
</UL>
</UL>
<LI><A NAME="tex2html5192"
HREF="node96.html">Error Bounds for the Singular Value Decomposition</A>
<UL>
<LI><A NAME="tex2html5193"
HREF="node97.html">Further Details: Error Bounds for the Singular Value Decomposition</A>
</UL>
<LI><A NAME="tex2html5194"
HREF="node98.html">Error Bounds for the Generalized Symmetric Definite
Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5195"
HREF="node99.html">Further Details: Error Bounds for the Generalized Symmetric Definite
Eigenproblem</A>
</UL>
<LI><A NAME="tex2html5196"
HREF="node100.html">Error Bounds for the Generalized Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5197"
HREF="node101.html">Further Details: Error Bounds for the Generalized
Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5198"
HREF="node102.html">Overview</A>
<LI><A NAME="tex2html5199"
HREF="node103.html">Balancing and Conditioning</A>
<LI><A NAME="tex2html5200"
HREF="node104.html">Computing <B><I>s</I><SUB><I>i</I></SUB></B>, <IMG
WIDTH="19" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img4.png"
ALT="$l_{\cal I}$">,
<IMG
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$r_{\cal I}$">
and
<IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
ALT="${\rm Dif}_u$">,
<IMG
WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img7.png"
ALT="${\rm Dif}_l$"></A>
<LI><A NAME="tex2html5201"
HREF="node105.html">Singular Eigenproblems</A>
</UL>
</UL>
<LI><A NAME="tex2html5202"
HREF="node106.html">Error Bounds for the Generalized Singular Value Decomposition</A>
<UL>
<LI><A NAME="tex2html5203"
HREF="node107.html">Further Details: Error Bounds for the Generalized Singular Value Decomposition</A>
</UL>
<LI><A NAME="tex2html5204"
HREF="node108.html">Error Bounds for Fast Level 3 BLAS</A>
</UL>
<!--End of Table of Child-Links-->
<HR>
<!--Navigation Panel-->
<A NAME="tex2html5167"
HREF="node73.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="next_motif.png"></A>
<A NAME="tex2html5161"
HREF="node5.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="up_motif.png"></A>
<A NAME="tex2html5155"
HREF="node71.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="previous_motif.png"></A>
<A NAME="tex2html5163"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="contents_motif.png"></A>
<A NAME="tex2html5165"
HREF="node152.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
SRC="index_motif.png"></A>
<BR>
<B> Next:</B> <A NAME="tex2html5168"
HREF="node73.html">Sources of Error in</A>
<B> Up:</B> <A NAME="tex2html5162"
HREF="node5.html">Guide</A>
<B> Previous:</B> <A NAME="tex2html5156"
HREF="node71.html">LAPACK Benchmark</A>
  <B> <A NAME="tex2html5164"
HREF="node1.html">Contents</A></B>
  <B> <A NAME="tex2html5166"
HREF="node152.html">Index</A></B>
<!--End of Navigation Panel-->
<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
</BODY>
</HTML>
|
