SIMPSON’S 1/3RD RULEINTERACTIVE E-BOOK |
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Title |
An interactive e-book for illustrating Simpson’s 1/3rd Rule for numerical integration. |
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Creator |
Autar K Kaw |
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Subject and Keywords |
Simpson’s 1/3rd Rule, Numerical Integration, Mathcad, Maple, Mathematica, Matlab, Simulations. |
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Description |
An interactive E-book for illustrating Simpson’s 1/3rd Rule. It includes links to simulations in Mathcad, Maple, Mathematica and Matlab for the algorithm, multiple choice quizzes, and a PowerPoint presentation. |
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Publisher |
Holistic Numerical Methods Institute, |
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Contributors |
Autar Kaw, Michael Keteltas |
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Format |
Text/HTML |
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Last Revised |
June 14, 2004 |
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Identifier |
http://numericalmethods.eng.usf.edu/ebooks/simpson13_07int_ebook.pdf |
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Language |
English |
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Rights |
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Single-segment Simpson’s 1/3rd Rule Multiple-segment Simpson’s 1/3rd Rule Error in Multiple-segment Simpson’s 1/3rd Rule Method 2: By Newton’s divided difference polynomial method Method 3: By Lagrange polynomial Method 4: Derived from Method of Coefficients Example 1: Single-segment Simpson’s 1/3rd Rule Example 2: 4-segment Simpson’s 1/3rd Rule Simulations Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB] Convergence [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB] Problem Sets |
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This is an example of how an instructor of a Numerical Methods course can develop an E-book on a single topic by using resources that are only available at the Holistic Numerical Methods Institute (HNMI) Website. Although one of the primary advantages of a web-based resource is that one can use documents outside of its own domain, we are particularly using this example as a way to illustrate the holistic nature of the resources available at HNMI. You are welcome to modify and edit this e-book to suit your purpose. Using the textbook document written in MS Word 2000 as a foundation, it was made interactive within MS Word 2000 by putting bookmarks and hyperlinks to background information, PowerPoint presentations, Mathcad simulations, historical anecdotes, multiple choice tests, problem sets, etc. Then the MS Word 2000 file was saved as a webpage and that is what you are about to read here. Back to TOC |
Integration is the process of measuring the area under a function plotted on a graph. Why would we want to do so? Among the most common examples are finding the velocity of a body from acceleration functions, and displacement of a body from velocity data. Throughout the engineering fields, there are (what sometimes seems like) countless applications for integral calculus. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods have been developed to simplify the integral. Here, we will discuss Simpson’s 1/3rd Rule of integral approximation, which improves upon the accuracy of the Trapezoidal Rule.
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Figure 1: Integration of a function
SIMPSON’S 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. Simpson’s 1/3rd rule is an extension of Trapezoidal rule where the integrand is nonapproximated by a second order polynomial.
Hence
where 
is a second order
polynomial.
Choose 
and
as the three points
of the function to evaluate 
and 
.
Solving the above three equations
for unknowns, and give
Then
Substituting values of and give
Since for Simpson’s 1/3rd
Rule, the interval 
is broken into 2
segments, the segment width
Hence the Simpson’s 1/3rd rule is given by
Since the above form has 1/3 in its formula, it is called Simpson’s 1/3rd Rule.
Method 2:
Simpson’s formula could also be derived by approximating 
by a second order
polynomial using
where
Integrating
Substituting values of 
and 
into this equation
yields the same result as before
Method 3:
One could even use the Lagrange polynomial to derive Simpson’s formula. Notice any method of three-point quadratic interpolation can be used to accomplish this task. In this case, the interpolating function becomes
Integrating this function gets
Believe it or not, simplifying and factoring this nightmare gets you the same result as before
Method 4:
Simpson’s 1/3rd Rule can also be derived by the method of coefficients. Assume
Let the right-hand side be an
exact expression for integrals 
and 
. Doing this will
imply that the right hand side will be exact expressions for integrals of any
linear combination of the three integrals, implying it for a general second
order polynomial. Now
Solving the above 3 equations for
and 
give
This gives
The integral from the first method,
can be viewed as the area under the second order polynomial, while the equation from this past method
can be viewed as the sum of the areas of three rectangles.
Simulations
Method [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]
Convergence [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]
Example 1:
The distance covered by a rocket
from 
to 
is given by
a)
Use Simpson’s 1/3rd Rule to find the approximate value of 
.
b)
Find the true error, 
c)
Find the
absolute relative true error, 
.
Solution:
a)
b) The exact value of the above integral is
So the true error is
c) Absolute Relative true error,
Multiple Segment Simpson’s 1/3rd Rule
Just like in multiple-segment
Trapezoidal Rule, one can subdivide the interval into
segments and apply
Simpson’s 1/3rd Rule repeatedly over every two segments. Note
that needs to be
even. Divide interval into
equal segments,
hence the segment width 
.
where
Apply Simpson’s 1/3rd Rule over each interval,
Since
then
Example 2:
Use 4-segment Simpson’s 1/3rd
Rule to approximate the distance covered by a rocket from to as given by
a) Use four segment Simpson’s 1/3rd Rule to find the probability.
b) Find the true error, Et for part (a).
c) Find the absolute relative true error for part (a).
Solution:
a) Using segment Simpson’s
1/3rd Rule,
So
In this case, the true error is
and the absolute relative true error
Table 1: Values of Simpson’s 1/3rd Rule for Example 2 with multiple segments
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Approximate Value |
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2 4 6 8 10 |
11065.72 11061.64 11061.40 11061.35 11061.34 |
4.38 0.30 0.06 0.01 0.00 |
0.0396% 0.0027% 0.0005% 0.0001% 0.0000% |
Error in Multiple Segment Simpson’s 1/3rd Rule
The true error in a single application of Simpson’s 1/3rd Rule is given[1]
In Multiple Segment Simpson’s 1/3rd
Rule, the error is the sum of the errors in each application of Simpson’s 1/3rd
Rule. The error in segment Simpson’s
1/3rd Rule is given by
:
:
:
:
Hence, the total error in Multiple Segment Simpson’s 1/3rd Rule is
The term 
is an approximate
average value of
. Hence
where
