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Math Help - Numerical differentiation

  1. #1
    Senior Member yeKciM's Avatar
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    Red face Numerical differentiation

    Hello,
    I'm having "big" issue recalling formula for numerical differentiation... using front differences (y[n+1]-y[n])... and can't find any of my old textbooks... so if anyone could guide me a little bit, i would be very grateful

    let's say i have function given like :

    xi -2 -1 0 1 2
    f(xi) -3 -1 0 -1 0
    and i need to get formula for numerical differentiation using front differences (or back)... and get approximate value of derivate at point x=-2....

    well...
    i assume this...

    f'(x) = \frac{f(x)}{x[i+1]- x[i]}

    it's probably wrong, but even if it's correct i don't know what to do next ... because I think need only two points from this table to determine the value of the derivate?

    P.S. i don't see why do i get latex error


    edit:
    or i have to use table of front differences ?

    y_1
    \Delta y_1
    y_2 \Delta^2 y_1
    \Delta y_2 \Delta^3 y_1
    y_3 \Delta^2 y_2 \Delta^4 y_1
    \Delta  y_3 \Delta^3 y_2
    y_4 \Delta^2 y_3
    \Delta y_4
    y_5
    Last edited by yeKciM; August 17th 2013 at 03:14 PM.
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  2. #2
    MHF Contributor

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    Re: Numerical differentiation

    First, no, the formula is NOT \frac{f(x)}{x[i+1]- x[i]} because "f(x)" is not a numerical value. The simplest formula is \frac{f(x[i+1])- f(x[i])}{x[i]- x[i]} but that is not very accurate and there is, of course, no way to make it more accurate. My personal preference would be to do as you suggest- use a difference table. By Newton's divided difference formula, that function can be approximated by
    y1+ \Delta y1 (x- x_1)+ \Delta^2 y_1(x- x_1)(x- x_2)/2+ \Delta^3 y_1(x- x_1)(x- x_2)(x- x_3)/6+ \Delta^4 y_1(x- x_1)(x- x_2)(x- x_3)(x- x_4)/24.

    You can then differentiate that polynomial and evaluate the derivative at x= x_1.
    Last edited by HallsofIvy; August 17th 2013 at 03:31 PM.
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  3. #3
    Senior Member yeKciM's Avatar
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    Re: Numerical differentiation

    thanks
    That interpolated polynomial you wrote there .... would be the same thing if u use back difference, it would be something like this.... (with different table) ?
      \Delta y_5 (x-x_1) + \Delta^2 y_5 (x-x_1)(x-x_2)/2 ....
    and so on and those  \Delta y_i are just values that i calculate based on those tables ?

    (i know it's silly question but... )
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