Second Order Approximation Question

Question

I really don't have any idea how to do this question ? I am already try my best to think but i am still get nothing

Let $f \in C^{3}([a,b])$. Find $\alpha$,$\beta$ and $\gamma$ so that

$$\dfrac{\alpha f(x+3h) + \beta f(x+h) + \gamma f(x)}{h}$$

is second order approximation to $f'(x)$. I.e for any $x \in (a,b)$, There are constants $\delta$ and $C$ so that when $|h|\le \delta $,

$$\bigg | \dfrac{\alpha f(x+3h) + \beta f(x+h) + \gamma f(x)}{h} - f'(x) \bigg | \le Ch^2$$

Can Someone help me to figure this out?

Answer 1

To find a representation accurate to the second order, simply expand the representation in Taylor series to give: $$ f'(x)=\dfrac{\alpha}{h}\left[f(x)+3hf'(x)+\dfrac{(3h)^2}{2!}f''(x)\right] +\dfrac{\beta}{h}\left[f(x)+hf'(x)+\dfrac{h^2}{2!}f''(x)\right] +\dfrac{\gamma}{h}f(x)+\cdots .$$ Equating coefficients gives $$\alpha+\beta+\gamma=0,\quad\text{which eliminates }f(x);$$ $$3\,\alpha +\beta =1,\quad\text{which equates the coefficients of }f'(x);$$ $$\alpha\dfrac{9\,h}{2}+\beta\dfrac{h}{2}=0,\quad\text{which eliminates } f''(x).$$ Solving for $\alpha,\,\beta,\,\gamma,\,$ gives $$\alpha=-\dfrac{1}{6},\quad\beta=\dfrac{3}{2},\quad\gamma=-\dfrac{4}{3}.$$ These values eliminate the terms in $f''(x)$ and the result is therefore accurate to terms of the second order in $h$, as required.

Answer 2

More generally, since $f(x+h) =f(x)+hf'(x)+\frac12h^2f''(x)+O(h^3) $,

$\begin{array}\\ af(x+uh)+bf(x+vh)+cf(x+wh) &=a(f(x)+uhf'(x)+\frac12u^2h^2f''(x))\\ &+b(f(x)+vhf'(x)+\frac12vh^2f''(x))\\ &+c(f(x)+whf'(x)+\frac12wh^2f''(x))+O(h^3))\\ &=(a+b+c)f(x)+(ua+vb+wc)hf'(x)+\frac12(au^2+bv^2+cw^2)h^2f''(x) +O(h^3)\\ \end{array} $

If you want to get an approximation for $f'(x)$ as accurate as possible, then you want to solve $a+b+c=0, au+bv+cw=1, au^2+bv^2+cw^2 =0 $ for $a, b, c$ in terms of $u, v, w$ and you will get

$\begin{array}\\ af(x+uh)+bf(x+vh)+cf(x+wh) &=hf'(x)+O(h^3)\\ \text{so}\\ \dfrac{af(x+uh)+bf(x+vh)+cf(x+wh)}{h} &=f'(x)+O(h^2)\\ \end{array} $

Yours is the case $u=3, v=1, w=0$.