Ayle and Victoria are interested in Simpson's rule Qx (f) = fou; f(;) for approximating an integral of the form I(A) = l

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Ayle And Victoria Are Interested In Simpson S Rule Qx F Fou F For Approximating An Integral Of The Form I A L 1 (142.75 KiB) Viewed 23 times

Ayle and Victoria are interested in Simpson's rule Qx (f) = fou; f(;) for approximating an integral of the form I(A) = lo f(T) dr. Let En = \|(1) - Qn. ()| denote the quadature error. (a) [1 mark] What is the general convergence theory for Simpson's rule? O For fe C'([a,b]]), the quadrature error is O(N-1) and depends on F'. For f e C?(a,b]), the quadrature error is O(N 2) and depends on f" For fe cº([a,b]), the quadrature error is O(N-4) and depends on f(4), For f € C2N ([a, b]), the quadrature error depends on f(2%) and converges asymptotically faster than O(NT) for any value of r. O Victoria remindes everyone that when the theory holds, if they double the number of points, they expect the ratio of the errors to be approximately EN 16 E2N (b) [4 marks] Noah suggests that they consider the specific function $(2) = 12 5 sin(3x) over the interval 1, 4]. The exact integral is I(F) 4.9376007342125. Write a MATLAB script to compute Simpson's rule QN(S) and then complete the following entries. E128 = 4.938 E256 = Number Give your answers to at least 4 significant figures. You may use scientific notation e.g. 1.234e-5. Compute the ratio E 128 E256 for Noah's function. Is this observed rate of convergence consistent with the theory? True False (c) (2 marks] Now Kyle replaces Noah's function by its absolute value, that is, f(x) = In() ? 12 5 sin(3:2) over the interval (1,4) The exact integral is now 1) = 10.508001997701 If you were to repeat part (b), do you expect a similar ratiostor Kyle's function? Give reasons for your answer. Refer to the two ratios as Noah's ratio and Kyle's ratio in your answer Easo