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## Simpson

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### Transforming Numerical Methods Education ... Basis of Simpson's 1/3rd Rule ... Just like in multiple segment Trapezoidal Rule, one can subdivide the interval ... – PowerPoint PPT presentation

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Title: Simpson

1
Simpsons 1/3rd Rule of Integration
• Electrical Engineering Majors
• Authors Autar Kaw, Charlie Barker
• http//numericalmethods.eng.usf.edu
• Transforming Numerical Methods Education for STEM

2
Simpsons 1/3rd Rule of Integration
http//numericalmethods.eng.usf.edu
3
What is Integration?
• Integration

The process of measuring the area under a curve.
Where f(x) is the integrand a lower limit of
integration b upper limit of integration
4
• Simpsons 1/3rd Rule

5
Basis of Simpsons 1/3rd Rule
• Trapezoidal rule was based on approximating the
integrand by a first
• order polynomial, and then integrating the
polynomial in the interval of
• integration. Simpsons 1/3rd rule is an
extension of Trapezoidal rule
• where the integrand is approximated by a second
order polynomial.

Hence
6
Basis of Simpsons 1/3rd Rule
Choose
and
as the three points of the function to evaluate
a0, a1 and a2.
7
Basis of Simpsons 1/3rd Rule
Solving the previous equations for a0, a1 and a2
give
8
Basis of Simpsons 1/3rd Rule
Then
9
Basis of Simpsons 1/3rd Rule
Substituting values of a0, a1, a 2 give
Since for Simpsons 1/3rd Rule, the interval a,
b is broken
into 2 segments, the segment width
10
Basis of Simpsons 1/3rd Rule
11
Example 1
The probability for an oscillator to have its
frequency within 5 of the target of 1kHz is
determined by finding total area under the normal
distribution function for the range in question
• Use Simpsons 1/3rd rule to find the frequency
• Find the true error, for part (a).
• Find the absolute relative true error, for
part (a).

12
Solution

a)

13
Solution (cont)

b)

True Error
14
Solution (cont)
c) Absolute relative true error,

15
• Multiple Segment Simpsons 1/3rd Rule

16
Multiple Segment Simpsons 1/3rd Rule
Just like in multiple segment Trapezoidal Rule,
one can subdivide the interval
a, b into n segments and apply Simpsons 1/3rd
Rule repeatedly over
every two segments. Note that n needs to be
even. Divide interval
a, b into equal segments, hence the segment
width
where
17
Multiple Segment Simpsons 1/3rd Rule
Apply Simpsons 1/3rd Rule over each interval,
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Multiple Segment Simpsons 1/3rd Rule
Since
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Multiple Segment Simpsons 1/3rd Rule
Then
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Multiple Segment Simpsons 1/3rd Rule
21
Example 2
The probability for an oscillator to have its
frequency within 5 of the target of 1kHz is
determined by finding total area under the normal
distribution function for the range in question
• Use four segment Simpsons 1/3rd Rule to find the
approximate value of x.
• Find the true error, for part (a).
• Find the absolute relative true error, for
part (a).

22
Solution

Using n segment Simpsons 1/3rd Rule,
a)

So

23
Solution (cont.)

24
Solution (cont.)
In this case, the true error is
b)
The absolute relative true error
c)
25
Solution (cont.)
Table Values of Simpsons 1/3rd Rule for Example
2 with multiple segments
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Error in the Multiple Segment Simpsons 1/3rd Rule
The true error in a single application of
Simpsons 1/3rd Rule is given as
In Multiple Segment Simpsons 1/3rd Rule, the
error is the sum of the errors
in each application of Simpsons 1/3rd Rule. The
error in n segment Simpsons
1/3rd Rule is given by
27
Error in the Multiple Segment Simpsons 1/3rd Rule
. . .
28
Error in the Multiple Segment Simpsons 1/3rd Rule
Hence, the total error in Multiple Segment
Simpsons 1/3rd Rule is
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Error in the Multiple Segment Simpsons 1/3rd Rule
The term
is an approximate average value of
Hence
where
30