Friday, June 25, 2010

Numerical Differentiation Formulas with error estimates.

Let $$f(x)$$ be the function whose first or second derivatives at $$x = a$$ are desired.

Forward difference $$f'(a)=\frac{f(a+h)-f(a}{h}$$$$\frac{-1}{2}h f^{(2)}(\varphi)$$
Central difference $$f'(a)= \frac{f(a+h)-f(a -h)}{2h}$$$$ \frac {-h^2}{6} f^{(3)}(\varphi)$$
Four point $$f'(a)=\frac{-3f(a+4f(a+h)-f(a+2h)}{2h}$$ $$$$ \frac{h^2}{3} f^2(\varphi)
Five point $$f'(a)=\frac{[f(a -2h)-8f(a-h)+8f(a+h)-f(a + 2h)]}{12h}$$ $$f''(a)= \frac{f(a) - 2 f(a +h) + f(a +2h}{h^2} $$$$ \frac{h^2}{6}f^{iv}(\varphi) - hf''' (\nu) \\ $$$$ f''(a)=\frac{f(a-h)-2f(a)+f(a +h)}{h^2} $$$$ \frac{-h^2}{12}f^{iv}(\varphi), |\varphi-a|<|a|$$

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