1. Forward/Backward-Difference formula

Hi,
I'm learning some Numerical Analysis but this is a very calculus-like question so I'll put it here.

Forward/Backward - Difference formula,

Assumptions:

$\displaystyle f$ is continuous on $\displaystyle [a,b]$ and $\displaystyle f''(x)$ exists for all $\displaystyle x\in[a,b]$.
$\displaystyle x_0\in[a,b]$ and $\displaystyle x_1=x_0+h$ for $\displaystyle h$ such that $\displaystyle x_1\in[a,b]$.
$\displaystyle \xi$ is also in $\displaystyle [a,b]$.

$\displaystyle f'(x_0)=\frac{f(x_0+h)-f(x_0)}{h}-\frac{h}{2}f''(\xi).$

Since I do not have any information on f$\displaystyle ''(\xi)$, the error in my approximation is bounded by $\displaystyle \frac{M|h|}{2}$ where $\displaystyle M$ is the maximum value of $\displaystyle f''(x)$ on my interval. How can I find the error bound without actually taking the second derivative of $\displaystyle f$?

Thanks.

2. Originally Posted by Mollier
Hi,
I'm learning some Numerical Analysis but this is a very calculus-like question so I'll put it here.

Forward/Backward - Difference formula,

Assumptions:

$\displaystyle f$ is continuous on $\displaystyle [a,b]$ and $\displaystyle f''(x)$ exists for all $\displaystyle x\in[a,b]$.
$\displaystyle x_0\in[a,b]$ and $\displaystyle x_1=x_0+h$ for $\displaystyle h$ such that $\displaystyle x_1\in[a,b]$.
$\displaystyle \xi$ is also in $\displaystyle [a,b]$.

$\displaystyle f'(x_0)=\frac{f(x_0+h)-f(x_0)}{h}-\frac{h}{2}f''(\xi).$

Since I do not have any information on f$\displaystyle ''(\xi)$, the error in my approximation is bounded by $\displaystyle \frac{M|h|}{2}$ where $\displaystyle M$ is the maximum value of $\displaystyle f''(x)$ on my interval. How can I find the error bound without actually taking the second derivative of $\displaystyle f$?

Thanks.
Without knowing the function, and its second derivative I think you have done all that can be expected.

CB

3. Originally Posted by CaptainBlack
Without knowing the function, and its second derivative I think you have done all that can be expected.

CB
What if have a function but am unable to analytically find the second derivative?
Thanks