Let Q_n be the equal spacing composite trapezoidal rule: where x = linspace (a,

Question

Let Q_n be the equal spacing composite trapezoidal rule: where x = linspace (a, b, n) and we assume that n greaterthanorequalto 2. Assume that there is a constant C (independent of n), such that I = integral^b_a f(x) fx = Q_n + Ch^2. (a) Give an expression for |I - Q_2n| in terms of |Q_2n - Q_n|. (b) Write an efficient script that computes Q_2k + 1, where k is the smallest positive integer so that |I - Q_2k + 1 | is smaller than a given positive tolerance to1. You may assume that such a k exists. You may assume that the integrand function is available in f.m and that it accepts vector arguments.

Explanation / Answer

The trapezodial rule also known as the trapezoid rule or trapezium ruls is a technique for approximating the definite integral {display styleint_{a}^{b}f(x),dx.}int_{a}^{b}f(x),dx.

The trapezodial rule works by approximating the region under the graph of the function {display style f(x)} f(x) as a trapezoid and calculating its area. It follows that {display styleint_{a}^{b}f(x),dxpprox (b-a)left[{rac {f(a)+f(b)}{2}} ight].} int_{a}^{b}f(x),dxpprox(b-a)left[{rac{f(a)+f(b)}{2}} ight].

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