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
|
<!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>Sources of Error in Numerical Calculations</TITLE>
<META NAME="description" CONTENT="Sources of Error in Numerical Calculations">
<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="node75.html">
<LINK REL="previous" HREF="node72.html">
<LINK REL="up" HREF="node72.html">
<LINK REL="next" HREF="node74.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html5217"
HREF="node74.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="next_motif.png"></A>
<A NAME="tex2html5211"
HREF="node72.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="up_motif.png"></A>
<A NAME="tex2html5205"
HREF="node72.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="previous_motif.png"></A>
<A NAME="tex2html5213"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="contents_motif.png"></A>
<A NAME="tex2html5215"
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="tex2html5218"
HREF="node74.html">Further Details: Floating Point</A>
<B> Up:</B> <A NAME="tex2html5212"
HREF="node72.html">Accuracy and Stability</A>
<B> Previous:</B> <A NAME="tex2html5206"
HREF="node72.html">Accuracy and Stability</A>
  <B> <A NAME="tex2html5214"
HREF="node1.html">Contents</A></B>
  <B> <A NAME="tex2html5216"
HREF="node152.html">Index</A></B>
<BR>
<BR>
<!--End of Navigation Panel-->
<H1><A NAME="SECTION03410000000000000000"></A><A NAME="secroundoff"></A>
<BR>
Sources of Error in Numerical Calculations
</H1>
<P>
<A NAME="9705"></A>
<A NAME="9706"></A>
<A NAME="9707"></A>
<A NAME="9708"></A>
<A NAME="9709"></A>
There are two sources of error whose effects can be measured by the bounds in
this chapter: <EM>roundoff error</EM> and <EM>input error</EM>. Roundoff error
arises from rounding results of floating-point operations during
the algorithm.
Input error is error in the input to the algorithm from prior calculations or
measurements. We describe roundoff error first, and then input error.
<P>
Almost all the error bounds LAPACK provides are multiples of
<EM>machine epsilon</EM>,<A NAME="9713"></A><A NAME="9714"></A><A NAME="9715"></A>
which we abbreviate
by <IMG
WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img262.png"
ALT="$\epsilon$">.
Machine epsilon bounds the roundoff in individual
floating-point operations.
It may be loosely defined as the largest relative error
<A NAME="9716"></A><A NAME="9717"></A>
in any floating-point operation that neither overflows nor underflows.
(Overflow means the result is too
large to represent accurately, and underflow means the result is too
small to represent accurately.) Machine epsilon is available either by
the function call
<A NAME="9718"></A><A NAME="9719"></A>SLAMCH('Epsilon') (or simply SLAMCH('E')) in single precision, or
by the function call DLAMCH('Epsilon') (or DLAMCH('E')) in double precision.
See section <A HREF="node74.html#secbackgroundfloatingpoint">4.1.1</A> and
Table <A HREF="node74.html#tabIEEEvalues">4.1</A> for a discussion of common values of machine
epsilon.
<A NAME="9722"></A><A NAME="9723"></A>
<A NAME="9724"></A><A NAME="9725"></A>
<P>
Since underflow is almost always less significant than roundoff,
we will not consider it further.
Overflow usually means the computation is invalid, but there
are some LAPACK routines that routinely generate and handle overflows
using the rules of IEEE arithmetic
(see section <A HREF="node74.html#secbackgroundfloatingpoint">4.1.1</A>).
<P>
Bounds on <EM>input errors</EM>, or errors in the input parameters inherited
from prior computations or measurements, may be easily incorporated
into most LAPACK error bounds.
Suppose the input data is accurate to, say, 5 decimal digits
(we discuss exactly what
this means in section <A HREF="node75.html#secnormnotation">4.2</A>). Then one simply replaces
<IMG
WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img262.png"
ALT="$\epsilon$">
by
<!-- MATH
$\max(\epsilon, 10^{-5})$
-->
<IMG
WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
SRC="img263.png"
ALT="$\max(\epsilon, 10^{-5})$">
in the error bounds.
<P>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"></A>
<UL>
<LI><A NAME="tex2html5219"
HREF="node74.html">Further Details: Floating Point Arithmetic</A>
</UL>
<!--End of Table of Child-Links-->
<HR>
<!--Navigation Panel-->
<A NAME="tex2html5217"
HREF="node74.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="next_motif.png"></A>
<A NAME="tex2html5211"
HREF="node72.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="up_motif.png"></A>
<A NAME="tex2html5205"
HREF="node72.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="previous_motif.png"></A>
<A NAME="tex2html5213"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="contents_motif.png"></A>
<A NAME="tex2html5215"
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="tex2html5218"
HREF="node74.html">Further Details: Floating Point</A>
<B> Up:</B> <A NAME="tex2html5212"
HREF="node72.html">Accuracy and Stability</A>
<B> Previous:</B> <A NAME="tex2html5206"
HREF="node72.html">Accuracy and Stability</A>
  <B> <A NAME="tex2html5214"
HREF="node1.html">Contents</A></B>
  <B> <A NAME="tex2html5216"
HREF="node152.html">Index</A></B>
<!--End of Navigation Panel-->
<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
</BODY>
</HTML>
|
