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1
Truncation Errors and the Taylor
Series
Lecture 4
•
Nonelementary functions such as trigonometric,
exponential, and others are expressed in an
approximate fashion using Taylor series when their
values, derivatives, and integrals are computed.
•
Any smooth function can be approximated as a
polynomial. Taylor series provides a means to predict
the value of a function at one point in terms of the
function value and its derivatives at another point.
x
F(x)
The approximation to F(x)=0.1x
4
0.15x
3
0.5x
2
0.25x+1
at x=1, by 0
th
, 1
st
and 2
nd
order Taylor expansion
F(x
i+1
) = F(x
i
)
F(x
i+1
)= F(x
i
)+F’(x
i
)h
F(x
i+1
)= F(x
i
)+F’(x
i
)h+1/2F’(x
i
)h
2
True value
x
i+1
=1
x
i
=0
h = x
i+1
 x
i
Example:
To get the
cos
(x) for small x:
If x=0.5
cos
(0.5)
=10.125+0.00260410.0000127+ …
=0.877582
From the supporting theory, for this series, the error
is no greater than the first omitted term.
L
+
−
+
−
=
!
6
!
4
!
2
1
cos
6
4
2
x
x
x
x
0000001
.
0
5
.
0
!
8
8
=
=
∴
x
for
x
•
Any smooth function can be approximated as a
polynomial.
f(x
i+1
)
≈
f(x
i
)
zero order
approximation, only
true if x
i+1
and x
i
are very close
to each other.
f(x
i+1
)
≈
f(x
i
) + f
′
(x
i
) (x
i+1
x
i
)
first order
approximation, in form of a
straight line
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n
n
i
i
n
i
i
i
i
i
i
i
R
x
x
n
f
x
x
f
x
x
x
f
x
f
x
f
+
−
+
+
−
′
′
+
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 Spring '07
 seminario
 Numerical Analysis, Derivative, Taylor Series, Analytic function

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