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

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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)