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Computer Arithmetic and Validity
book

Computer Arithmetic and Validity

by Ulrich Kulisch
April 2013
Intermediate to advanced content levelIntermediate to advanced
456 pages
16h 7m
English
De Gruyter
Content preview from Computer Arithmetic and Validity
Section 5 .3 Floating-point operations 177
We now use the abbreviation
:D
n
1 n
,
and distinguish two index sets:
I
C
:Dfi j x
i
y
i
0g, I
:Dfi j x
i
y
i
< 0g.
Generalizing relation (5.3.5) we obtain
n
P
iD1
x
i
y
i
D
n
P
iD1
x
i
y
i
.1
i1
/ D
n
X
iD1
x
i
y
i
n
Y
kD1
.1
ik
/. (5.3.8)
Here, the formula on the right-hand side has been completed by factors .1
ik
/ to
obtain n factors for every summand where possibly
ik
D 0forsomek.
We are now going to derive upper and lower bounds for equation (5.3.8).
Upper bound:
n
P
iD1
x
i
y
i
D
X
i2I
C
x
i
y
i
n
Y
kD1
.1
ik
/
X
i2I
jx
i
y
i
j
n
Y
kD1
.1
ik
/.
Applying Lemma 5.8 and the abbreviation we now obtain
n
P
iD1
x
i
y
i
X
i2I
C
x
i
y
i
.1 C /
X
i2I
jx
i
y
i
j.1
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Publisher Resources

ISBN: 9783110301731