Assignment #7: Numerical Integration (part 2)
Due date: Wednesday, November 3, 2010 (10:15am)
For full credit you must show all of your work.
1.
Estimate
e
cos2
x
dx
0
"
#
using Basic Simpson’s rule.
If the actual solution, to within 5 decimal
places of accuracy, is 3.97746, what is the absolute error in the approximate answer you
obtain?
What is the error term for this approximation?
If –92 <
d
4
dx
4
e
cos2
x
(
)
< 174 for 0
≤
x
≤
π
,
what is the upper bound on the error that you get by using this formula?
What can you
conclude from this about the practicality of using the upper bound on the error in the Basic
Simpson’s rule to estimate the actual amount of error that will occur when applying it?
2.
Using Matlab, write a program that implements the
composite Simpson’s rule
.
Structure
your code so that it can be easily reused to estimate a variety of different definite integrals.
a.
Test your program by using it to estimate
sin(
x
)
dx
0
"
#
.
If the true solution is 2, at how
many points do you need to evaluate the function to obtain an estimate that is accurate to
within 5 decimal places (
ε
≤
½
×
10
–5
), and what is the distance between neighboring
points?
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 Fall '10
 intern
 Numerical Analysis, Romberg's method

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