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HW#7
EGM6341
Spring 2009
Due:
3/26/09
Additional problem (on numerical differentiation):
A1.
Using Taylor series expansion for
f
(
x
±
2
h
) &
f
(
x
±
h
)
to derive a forth order accurate
formula for the 1
st
and 2
nd
order derivatives,
)
(
x
f
′
)
(
x
f
′
′
in terms of
f
(
x
+2
h
),
f
(
x
+
h
),
f
(
x
),
f
(
x

h
)
and
f
(
x
2
h
). One way you can approach it is as follows:
*
Expand, for example,
f
(
x
+2
h
) as
f
(
x
+2
h
) =
f
(
x
) +2
h
)
(
x
f
′
+ 4
!
2
2
h
)
(
x
f
′
′
+ 8
3
3!
h
)
(
x
f
′
′
′
+16
4
4!
h
(4)
( )
f
x
+…
*
You then do the same for
f
(
x
+
h
),
f
(
x

h
)
and
f
(
x
2
h
).
*
Now you will have 4 equations and you can solve for 4 unknowns:
)
(
x
f
′
,
)
(
x
f
′
′
,
)
(
x
f
′
′
′
,
and
(4)
( )
f
x
by neglecting the higher order terms. Just get the result for
)
(
x
f
′
&
)
(
x
f
′
′
.
*
Verify your result: consider f(x)=
4
x
so that the exact result should be
3
( )
4
f
x
x
f
=
2
( )
12
f
x
x
f
=
.
Compare your numerical results with the exact results to see if your formulae are
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This note was uploaded on 04/23/2009 for the course EGM 6341 taught by Professor Mei during the Spring '09 term at University of Florida.
 Spring '09
 MEI

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