University of Florida/Egm6341/s10.team2.niki/HW2

Problem 1

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Statement

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Using the following equations find the expressions for   in terms of   and   where i=0,1,2

 

(1 p8-3)

 

(3 p8-3)

Solution

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We have the general formula for the Lagrange basis function   as

 

(2 p7-3)

for the case of Simple Simpson's Rule, n =2 i.e i=0,1,2. For the given interval  


 

 

 

Expanding equation 3 p8-3 we get:

  where,


 ;

 ;

 

Thus we have the polynomial as  

Grouping coefficients of  ,  

Comparing this equation with eqn 1p8-3 we see that

 

(1)

 

(2)


 

(3)

Problem 6

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Statement

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For the Lagrange Interpolation Error verify the following:

 

(1)

Solution

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We can write the Lagrange Interpolation error as


 

differentiating the above expression once we get

 

differentiating the expression (n+1) times we get

 

But since   is a polynomial of degree n the (n+1)th derivative is zero

 

(1)

Problem 11:To show that Simpson's rule can be used to integrate a cubic polynomial exactly

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Statement

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Given the polynomial   where   determine the exact integral   and the integral using Simpson's Rule  

Solution

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Case A: Determination of Exact Integral

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(1)

Case B: Using Simple Simpson's rule

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We have the Simple Simpson's rule as

 

(2 p7-2)

where  

  we know   substituting we get

 

 

(2)