Assignment:
Qa) 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.
Qb) 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.
Qc) What is the error term for Composite Simpson's Rule?
Qd) 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)
Provide complete and step by step solution for the question and show calculations and use formulas.