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Math Help - Forward/Backward-Difference formula

  1. #1
    Member Mollier's Avatar
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    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:

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

    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 ''(\xi), the error in my approximation is bounded by \frac{M|h|}{2} where M is the maximum value of f''(x) on my interval. How can I find the error bound without actually taking the second derivative of f?

    Thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Mollier View Post
    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:

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

    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 ''(\xi), the error in my approximation is bounded by \frac{M|h|}{2} where M is the maximum value of f''(x) on my interval. How can I find the error bound without actually taking the second derivative of f?

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

    CB
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  3. #3
    Member Mollier's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    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
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