University of Florida/Egm6341/s10.team3.aks/HW3

(4)Use Error estimate for Taylor series, composite trapezoidal and composite Simpsons rule

Problem Statement

edit

Use Error estimate for Taylor series, composite trapezoidal and composite Simpson's rule to find n such that

   

and compare to numerical results.

Solution

edit
 

Taylor Series

 
 
  with   and  
 
 
  for  
 
 ,  
 ,  
 
 
 
 

Below are the values from Numerical Analysis of Taylor series from HW_1

 
 
 
 

We are getting same values from both analysis for Taylor series

Trapazoidal Rule

Error for Composite Trapazoidal rule is given by

 


where   for  

For the given function  , we have


 


For the given interval [0,1] the maximum value of function   is achieved at  


 


 

 

 

 

 

 

 

 

 

 

Below are the results from Numerical analysis from HW 1

 

 

 

 

 

 

 

 


We can see that we are getting order   at n=128 from both the analysis.

Composite Simpsons Rule

The error estimate of the Composite Simpson's rule is given as

 

where    for  

For the given function  , we have


 

The function   has maximum value at  

 


 

 

 

Below are the results from Numerical analysis from HW 1

 

 

We can see that we are getting order   at n=4 from both the analysis.