Using the following equations find the expressions for in terms of and where i=0,1,2
|
(1 p8-3)
|
|
(3 p8-3)
|
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)
|
For the Lagrange Interpolation Error verify the following:
|
(1)
|
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
[edit | edit source]
Given the polynomial where determine the exact integral and the integral using Simpson's Rule
|
(1)
|
We have the Simple Simpson's rule as
|
(2 p7-2)
|
where
we know substituting we get
|
(2)
|