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Simpson's Rule - Alternating Pattern


Date: 09/20/97 at 14:49:53
From: Keith Holm
Subject: Simpson's rule

I'm a computer science student who is trying to apply Simpson's rule 
to a program, but I do not understand it well enough to proceed.  Can 
somebody give a 10 minute "this is how it works"?  The book I have 
shows a series from low to high bounds, but does not explain the 
alternating 4F 2F pattern.


Date: 09/20/97 at 16:13:23
From: Doctor Anthony
Subject: Re: Simpson's rule

I will give you the formula to apply without proof.  If you want to 
see the derivation of the formula, write back and I will go through it 
in detail.  

The formula is exact for polynomials up to cubics, so if for some 
strange reason you wanted to use Simpson's rule to integrate  
3x^3 - x^2 -5x -3 then just two equal intervals as wide as you please 
could be used. In the general case we divide the the total interval 
into an EVEN number of equal sub-intervals, and the more of these, 
the more accurate will be the final answer. If you have just two 
sub-intervals, there will be three ordinates y0, y1 and y2 to be 
calculated. If you have 4 sub-intervals, then you would have 5 
ordinates y0, y1,...., y4  to calculate.  

It is important to use a standard system for naming the ordinates, 
e.g. y2 is an even-numbered ordinate, y3 is an odd-numbered ordinate, 
and the formula requires that you treat these in different ways.  
So always start with y0; then the formula will work correctly in the 
form I give it to you.

Suppose you divide the total range into n intervals, where n MUST be 
an even number. This means, of course, that there will be an odd 
number of ordinates. The first ordinate will be y0; the last ordinate 
will be yn. The width of the sub-intervals is h.

The area under the curve is given by:

 A = (h/3)[y0 + yn + 4(y1+y3+...+y(n-1)) + 2(y2+y4+...+y(n-2))]

In words this formula can be remembered as:

 (h/3)[First + Last + 4(odd numbered) + 2(even numbered]

The First and Last must not appear more than once. Do not include them 
in the brackets of even-numbered ordinates.

Example.  Evaluate   INT(0 to 2)[sqrt(x^2+1)dx]

Divide the range into 4 parts each of width 0.5

Calculate the ordinates in a table as follows:

  x      sqrt(x^2+1) = y
------------------------
  0           1    =  y0
 0.5        1.118  =  y1
  1         1.4142 =  y2
 1.5        1.8027 =  y3
  2         2.236  =  y4
------------------------- 

  A = (.5/3)[1 + 2.236 + 4(1.118 + 1.8027) + 2(1.4142)]

    = (1/6)[17.7472]

    = 2.95786
 
-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   


Date: 09/20/97 at 16:57:07
From: keith holm
Subject: Re: Simpson's rule

Thank you very much, Doctor Anthony and the Math Forum; you were a 
great help! And I appreciate the fast response time too!  Thanks 
again.
    
Associated Topics:
High School Calculus

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