Numerical Analysis - Simpson's Rule
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a) Explain how we arrive at the formula for Simpson's rule (standard formula) using the Lagrange Interpolating Polynomial of degree 2. Ignore the error term, and do not compute any integral.
b) We define the Composite Simpson's Rule by splitting the interval [a,b] into smaller sub-intervals, applying Simpson's Rule on those sub-intervals, and then summing up the results. Write down Composite Simpson's Rule applied to the sub-intervals [x0,x2], [x2, x4],....[x2-n, x2n], where xk= a + k(b-a) / 2n.
c) What is the error term for Composite Simpson's Rule?
d) Richard Extrapolation: Explain how we can use multiple instances of Composite Simpson's Rule (with point-spacing h, h/2, h/4....) to generate a scheme with an error term = O(h^6)
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a) Explain how we arrive at the formula for Simpson's rule (standard formula) using the Lagrange Interpolating Polynomial of degree 2. Ignore the error term, and do not compute any integral. The idea is to replace the integral of an unknown function into a polynomial, which can be easily evaluated. This is:
(see attached file for equation)
Pn is the Lagrange Interpolating Polynomial of degree n, and En is the error. These are given by:
(see attached file for equation)
If we ignore the error term and use the definition for the Lagrange Interpolating polynomial, we get:
(see attached file for equation)
If we rename the last integral as ai (because an definite integral leads to a constant), then we get:
(see attached file for equation)
Thus, to develop a method, it's simply a matter of finding the coefficients. For Simpson's Rule, we use n = 2 (a quadratic interpolating polynomial). Recall that,
(see attached file for equation)
And for n = 2,
(see attached file for ...
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