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From: TideMan <mulgor@gmail.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Problem
Date: Thu, 6 May 2010 01:24:21 -0700 (PDT)
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On May 6, 8:15 pm, "Roger Stafford"
<ellieandrogerxy...@mindspring.com.invalid> wrote:
> "Roger Stafford" <ellieandrogerxy...@mindspring.com.invalid> wrote in message <hrt7e0$dr...@fred.mathworks.com>...
> >   Prepare yourself for a long-winded explanation. .......
>
> - - - - - - - - -
>   Please add this onto the previous discussion I gave.
>
>   If y = f(x) is some continuous function with its first three derivatives also continuous over the interval xa < xb < xc, then by an extension of the famous Rolle's theorem of elementary calculus, there is a value xi (Greek letter) such that the difference between the expression above for an approximation of the derivative at xb and the actual derivative, f'(xb) is equal to
>
>   1/6*f'''(xi)*(xc-xb)*(xb-xa)
>
> where xi lies in xa < xi < xc and f'''(xi) is the third derivative value of f(x) there.  A remarkable fact!
>
>   What this signifies is that if the function whose derivative you are attempting to approximate with the formula I gave you has a third derivative which remains very small - in other words is reasonably smooth - then you are guaranteed a close approximation.
>
> Roger Stafford

But Roger, doesn't your fancy algorithm reduce to central finite
differences if the x are equispaced:
dydx(2:end-1)=(y(3:end) - y(1:end-2))/(2*dx);
without the pretty stuff at the ends of course.