Finite Difference Approximation


Definition of a derivative

This is not an article, but rather a friendly post to remind the interested reader of the finite difference method. Remember back in school when you learned about derivatives thru the following formula

    \[f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}\]

Approximations kick in !

Well, the finite difference approximation says that we don’t need “to take things too extreme” in a sense that we can drop the limit above and agree on the approximation learned about derivatives thru the following formula

    \[f'(a) \simeq \frac{f(a+h) - f(a)}{h}\]

but keeping in mind that h is very small, i.e. small enough (that “small enough” phrase controls how good of an approximation we have).

Applications: Differential Equations and beyond !

The finite difference is all over the place to deal with differential equations, one of which is the heat equation. Other applications of interest is the Secant method, where one bypasses the knowledge of the derivative of f(x) in Newton’s method, thanks to this approximation.

Finite Difference Approximation and the Secant Method

How does the finite difference approximation generate the Secant method ? Well at each iteration x_n in Newton’s method, we could struggle with the computation f'(x_n). Well, using the above finite difference approximation, we have that where h is “small enough”. Let’s define our “small enough” factor as the signed interval length h=  x_{n+1} - x_n so that we getwhich reads.For more on Secant method, visit my article here.

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