Question

1. Simpson's rule. Simpson's rule is a different formula for numerical integration of lºf (d.x which is based on approximating f(2) with a piecewise quadratic function. We will now derive Simpson's rule and relate it to Romberg integration: a. Suppose that (2) is a quadratic polynomial so that q(-h) = f(-h), q0) = f(0) and q(h) = f(h). Prove that 92 f(-h) + 4f(0) + f(h)). -h b. Suppose that the interval [a, b] is divided by a = 20, 21, ..., 22n = b into 2n intervals of equal width h. Suppose that Q(x) is the piecewise quadratic function defined by Q(x) = f(xi) for all i E 0,..., 2n and Q(x) is quadratic on 22k, L2k+2) for all k € 0,..., n. Prove that + 4f(21) +2f (22) + ... +2f (x2n-2) + 4f (22n-1) + f(22n)). The approximation SQ(x) dx ~ S f() dx is called “Simpson's rule”. c. Prove that the entry R(n,1) in the Romberg table is equal to the result of Simpson's rule with 2" intervals.

Answer 1

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hence it is proved