Finite-difference approximation

finite-difference approximation

The difference between the values of a function at two discrete points, used to approximate the derivative of the function.

The derivative f′(x) of a function f(x) at an arbitrary point x is usually approximated by finite differences in one of three ways:

where Δf is called a forward difference;

where δf is called a centered difference;

where ∇f is called a backward difference (not to be confused with the gradient). Of these approximations, the centered difference is the most accurate; whether it is the most convenient or accurate for the problem as a whole depends on the character of the equations involved. Higher derivatives are approximated by iteration of these formulas.

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