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Lecture 5
●
MEEN 357 Engineering Analysis for Mechanical Engineers
1
Lecture05MEEN3572009.doc
RMB
Lecture 5
Roundoff and Truncation Errors
Chapter 4 of the textbook.
The Taylor’s Theorem
Theorem
:
Given a continuous function
[ ]
:,
fa
b
→
R
that is differentiable of order
1
K
+
on
()
,
ab
and a point
,
ca
b
∈
, then at any point
( )
,
x
∈
( )
1
0
!
k
K
kK
k
fc
fx
xc
O xc
k
+
=
=−
+
−
∑
(5.1)
Examples:
2
0
24
2
0
35
21
1
2
0
1
11
1
!2
1
cos
1
2!
2 4
!
1
sin
!
3
!
5
!
1
cosh
1
!
1
sinh
!
3
!
5
!
xn
n
n
n
n
n
n
n
n
n
n
n
ex
x
x
n
xx
n
x
n
n
x
n
∞
=
∞
=
∞
+
=
∞
=
∞
+
=
==
+
+
+
⋅
⋅
⋅
−
=
= −
+
+⋅⋅⋅
−
−
+
+
⋅
⋅
⋅
+
=
= +
+
=
+
+
∑
∑
∑
∑
∑
(5.2)
Numerical Differentiation
Section 4.3.4 of the textbook.
If you are given a real valued function
( )
N
N
Real
Open interval
numbers
b
→
R
(5.3)
which is differentiable at a point
( )
,
x
∈
, then the derivative is defined by
( ) ( )
lim
h
f
xh fx
h
→∞
+−
′
=
(5.4)
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The formulas we derived during Lecture 4 made use of a partition of the interval
[ ]
,
ab
into equal step sizes
.
We built this partition by first selecting a positive integer
N
which represents the number of partitions.
Given
N
, we defined the step size
h
by the
formula
ba
h
N
−
=
(5.5)
and divided the interval into equal segments by the formulas
0
10
21
1
NN
xa
xh
x
x
x
hx
b
−
=
=+
⋅
⋅
=
+=
(5.6)
The following figure illustrated this construction
Difference Approximation to the First Derivative
Forward Difference
:
()
( ) ( )
1
'
ii
i
fx
f
xO
h
h
+
−
(5.7)
0
ax
=
N
bx
=
( )
f
x
h
i
f
x
1
i
f
x
+
Lecture 5
●
MEEN 357 Engineering Analysis for Mechanical Engineers
3
Lecture05MEEN3572009.doc
RMB
When one adopts (5.7) as the derivative the resulting error is
( )
Oh
.
The following figure
illustrates this error.
Backward Difference
:
()
( ) ( )
1
'
ii
i
fx
f
xO
h
h
−
−
=+
(5.8)
We again reached the conclusion is that when one adopts (5.8) as the derivative the
resulting error is
( )
.
The following figure illustrates this error.
Centered Difference
( ) ( )
11
2
'
2
i
f
h
h
+−
−
(5.9)
i
x
1
i
x
−
f
x
Approximation
True Slope
i
x
1
i
x
+
f
x
Approximation
True Slope
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Equation (5.9) shows that if one adopts the first centered difference as the first derivative,
the resulting error is
()
2
Oh
.
This error is an improvement
of that associated with the
forward and backward differences introduced above.
The following figure illustrates the
error associated with use of the centered difference.
Finite Difference Approximations to Higher Derivatives
Second and higher
derivatives can also be approximated with the use of Taylor’s
theorem.
An application of Taylor’s theorem (5.1) yields
( ) () ()
( )
( )
23
22
2
2
''
'
2!
i
ii
i
i
i
i
i
fx
f
x
f
xf
x
xx
xxO
++
+
+
=+
−
+
−
+
−
(5.10)
Also, (see (5.1))
( )
11
1
1
''
'
2!
i
i
i
i
i
i
i
i
f
x
f
x
O
+
+
−
+
−
+
−
(5.11)
In terms of the step size
h
, these two equations can be written
( )
() ()
2
3
2
''
'2
2
2!
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This note was uploaded on 02/23/2009 for the course MEEN 357 taught by Professor Anamalai during the Fall '07 term at Texas A&M.
 Fall '07
 ANAMALAI

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