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Chapter1
ACCURACYOFCOMPUTATION
Inthischapter,themethodologyforevaluationofaccuracyofcomputationisillustratedwiththeproblems
concerning:
−propagationoferrorscorruptingindependentvariablesordata(calledforbrevitydataerrors);
−propagationoferrorscorruptingtheintermediateresultsoffloating-pointcomputation(calledforbrevity
roundingerrors);
−propagationofbothdataerrorsandroundingerrors;
−engineeringapplicationsofthemethodologyforevaluationofaccuracyofcomputation,otherthanas-
sessmentoftheeffectsofdataerrorsandroundingerrors.
1.1.PROPAGATIONOFDATAERRORS
Problem1.1:Thefunction
Tx
()
=
δ
[][]
y
~
δ
x
~
,
characterisingthepropagationoftherelativeerrorfromthe
variablextothevariable
y
±
fx
()
,
maybecomputedaccordingtotheformula:
Tx
()
±
xdy
ydx
.
Demonstrate
that
Tx
()
±
d
d
ln
ln
()
()
y
x
.
Solution#1:Onemaycomputethederivative
d
d
ln
ln
()
()
y
x
bysubstituting
xe
±
z
:
d
d
ln
ln
()
()
x
y
±
d
ln
d
ln
(
fx
()
x
()
)
±
d
ln
(
dz
fe
()
z
)
±
fe
()
1
z
dfxde
dx
()
dz
z
±
1
ydx
dy
xTx
±
()
d
ln
()
y
Solution#2:Alternatively,onemaydeterminethederivative
d
ln
()
x
bycomputingtheratioof:
d
ln
()
y
±
d
ln
dy
()
y
dy
±
1
y
dy
and
d
ln
()
x
±
d
ln
dx
()
x
dx
±
1
x
dx
Hence:
d
d
ln
ln
()
()
x
y
±
1
1
x
y
dx
dy
±
xdy
ydx
±
Tx
()
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