Find the error estimate by approximating $f(x)$ and derive a numerical integration formula for $\int_0^l f(x) \,dx$ based on approximating $f(x)$ by the straight line joining $(x_0, f(x_0))$ and $(x_1, f(x_1))$, where the two points $x_0$ and $x_1 = h – x_0$ are chosen so that $x_0, x_1 \in (0, l)$, $x_0 < x_1$ and $\int_0^l {(x – x_0) (x – x_1)} dx = 0$.
Derive the error estimate, asymptotic error and the Peano kernel error formula for the composite rule for $\int_a^b f(x) \,dx$.
Use the asymptotic error estimate to improve the integration formula. Find the values of $x_0$, $x_1$.
I know the Peano Kernel formula will take the form $E_n(f)=1/2($$\int_a^b K(t)\ f''(t) \,dx$$)$ with $K(t)$ being the Peano kernel but am having a tough time getting started on the question. Any help will be greatly appreciated. Thanks a lot!
Best Answer
For this I think you can use trapezoidal rule. You can approximate $f(x)$ by the straight line joining $(a,f(a))$ and $(b,f(b))$ Then by integrating the formula for this straight line, we get the approximation $$I_1(f)=\frac{(b-a)}{2}[f(a)+f(b)].$$ To get the error formula we get $$f(x)-\frac{b-x)f(a)+(x-a)f(b)}{b-a}=(x-a)(x-b)f[a,b,x]$$
I am not sure if this is absolutely correct, maybe someone can verify my answer?