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Thus:
dfx
1
dx
()
±
1,
dfx
2
dx
()
±–
x
1
2
,
B
Fyy
(
B
y
1
1
,
2
)
±
yy
2
1
y
2
–
1
and
B
Fyy
(
B
y
1
2
,
2
)
±
ln
()
yy
1
1
y
2
andconsequently:
dy
dx
±
1
x
x
1
x
–
1
+
ln
()
xx
1
x
()
–
x
1
2
±
x
1
x
–
2
(
1ln
–
()
x
)
Tx
()
±
xdy
ydx
±
xx
f
L
1
x
–
21ln
(
x
–
1
x
()
x
)
1
J
±
1ln
–
x
()
x
Solution#2:ThesameresultmaybeobtainedinamoreefficientwaybydifferentiationoftheRHSloga-
rithm:
dx
d
ln
()
y
±
dx
d
ln
()
x
1
x
±
dxx
d
f
|
L
1
ln
()
x
1
|
J
±
dxx
d
(N
||
k)
1
ln
()
x
+
1
xdx
d
f
L
ln
()
x
1±–
J
x
1
2
ln
()
x
+
x
1
2
±
1ln
–
x
2
()
x
Since:
dx
d
ln
()
y
±
1
ydx
dy
thesought-forfunctionmaybegiventheform:
Tx
()
±
xdy
ydx
±
x
1ln
–
x
2
()
x
±
1ln
–
x
()
x
TheaboveresultmaybeconfirmedbymeansofthefollowingMATLABscript:
%COEFFICIENTOFDATAERRORPROPAGATIONT(x)
%COMPUTEDFORTHEFORMULAy=x^(1/x)
symsf(x);
f(x)=x^(1/x);
T(x)=x*(diff(f(x))/f(x));
fplot(T(x),[02]);gridon
axis([0,2,0,20]);gridon
xlabel('x');ylabel('T(x)');
title('y=x^1^/^x');
fprintf('T(x)=%s\n',simplify(T(x)))
T(x)=-(log(x)-1)/x
Problem1.5:Assesstherelativeerrorofcomputing:
y
~
±
dxxb
dxa
(
|
k
+
+
~
~
N
|
)
for
xE
[]
0,1
causedbytherelativeerrorsinthedata:
a
~
±+
1
O
and
b
~
±
21
(
+
B
)
,
where
O
Ś
1%
and
B
Ś
1%.
Solution#1:Sincethepropagationofthedataerrorsdoesnotdependonthenumericalalgorithm,thefor-
muladefiningy
~maybesimplifiedbyperformingthedifferentiation:
y
~
±
dxxb
d
(
|
k
xa
+
+
~
~
N
|
)
±
(
xb
ba
~
+
–
~
~
)
2
±
21
(
(
x
+
+
B
21
(
)(
–
+
1
B
+
)
)
O
2
)
±
(
(
x
12
+
+
2
BO
)
+
–
2
B
)
2
±
(
x
+
12
2
)
+
2
(
|
k
1
BO
+
–
x
2
+
B
2
N
|
)
2
Hencethefirst-orderformulaforerrorpropagation:
δ
[]
y
~
±
2
BO
–
–
x
4
+
B
2
±–+
O
(
|
k
2
–
x
+
4
2
N
|
)
B
±–+
O
x
2
+
x
2
B
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