Using Matlab algorithm Find an approximating of integral0pi

Using Matlab (algorithm) Find an approximating of integral_0^pi sin x dx with an absolute error less than 0.00002 using composite midpoint rule?

Solution

Fortunately, expressions vulnerable to cancellation can often be recast in a mathematically equivalent form that is no longer affected by cancellation. There we studied several examples, and this problem gives some more.

We consider the function f1(x₀, h) = sin(x₀ + h) sin(x₀).

It can the transformed into another form, f₂(x₀, h), using the trigonometric identity sin() sin() = 2 cos ( + 2 ) sin ( 2 ) .

Thus, f1 and f2 give the same values, in exact arithmetic, for any given argument values x₀ and h. 1.

Derive f₂(x₀, h), which does no longer involve the difference of return values of trigonometric functions.

2. Suggest a formula that avoids cancellation errors for computing the approximation (f(x₀ + h) f(x₀))/h) of the derivative of

f(x) = sin(x) at x = x₀. Integrate in between 0 to pi

a MATLAB program that implements your formula and computes an approximation of f (1.2), for h = 1 10²⁰ , 1 10¹⁹ ,, 1.

  Plot the error (in doubly logarithmic scale using MATLAB’s log plotting function) of the derivative computed with the suggested formula and with the naive implementation using f₁.