numerical differentiation with the second derivative

Question

derive the following differentiation formula

$$f''(0)=f(-1)-2f(0)+f(1) $$

a) using interpolation

b) using method of undetermined coefficients

c) find an upper bound for the error $\lvert e''(0)\rvert$

Answer 1

hint

$$f (1)\approx f (0)+f'(0)+\frac{f''(0)}{2}$$

$$f (-1)\approx f (0)-f (0)+\frac {f''(0)}{2} $$

by sum, we find

$$f (1)+f (-1)=2f (0)+f''(0) $$

Answer 2

Taylor series expansion

$$ f(x) = f\left(x_{0} \right) + \sum_{k=1}^{\infty} \frac{\left(x-x_{0}\right)^{k}}{k!}f^{k}(x_{0}) $$

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Look right: $$ f'(x_{0} \color{blue}{+} h) = f\left(x_{0} \right) \color{blue}{+} f'\left(x_{0} \right) h + f''\left(x_{0} \right) \frac{h^{2}}{2!} \color{blue}{+} f'''\left(x_{0} \right) \frac{h^{3}}{3!} + \dots \tag{1} $$

Look left: $$ f'(x_{0} \color{red}{-} h) = f\left(x_{0} \right) \color{red}{-} f'\left(x_{0} \right) h + f''\left(x_{0} \right) \frac{h^{2}}{2!}\color{red}{-} f'''\left(x_{0} \right) \frac{h^{3}}{3!} + \dots \tag{2} $$

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Centered difference approximations

$(1) \color{red}{-} (2)$ $$ f'(x_{0}) = \frac{f(x_{0} + h) - f(x_{0} - h)}{2h} + \mathcal{O}\left( h^{2} \right) \tag{3} $$

$(1) \color{blue}{+} (2)$ $$ f''(x_{0}) = \frac{f(x_{0} - h) - 2f(x_{0}) + f(x_{0} + h)}{h^{2}} + \mathcal{O}\left( h^{2} \right) \tag{4} $$

The stated question is in the form $(4)$ with the assignments $$ x_{0} = 0, \quad h = 1 $$

Forward and backward differences

Rearrange $(1)$ for the forward difference approximation $$ f'_{R}(x_{0}) = \frac{f(x_{0}+h) - f(x_{0})}{h} + \mathcal{O}\left( h \right) \tag{5} $$ Rearrange $(2)$ for the backward difference approximation $$ f'_{L}(x_{0}) = \frac{f(x_{0}) - f(x_{0}-h)}{h} + \mathcal{O}\left( h \right) \tag{6} $$ Derive $(4)$ from $(5)$ and $(6)$ $$ \begin{align} % f''(x_{0}) &= \left( f'(x_{0}) \right)' \\ &= \left( \frac{f(x_{0} + h) - f(x_{0} - h)}{h} \right)' \\ &= % \frac{f'_{R}(x_{0}) - f'_{L}(x_{0})}{h} \\ % &= \frac{ \frac{f(x_{0}+h) - f(x_{0})}{h} - \frac{f(x_{0}) - f(x_{0}-h)}{h}} {h} \\ % & = \frac{f(x_{0} - h) - 2f(x_{0}) + f(x_{0} + h)}{h^{2}} % \end{align} $$

Error bound

The truncation error in $(1)$ and $(2)$ is majorized by $$ M \frac{h^{2}}{3!} $$