Relation With Derivatives
The derivative of a function f at a point x is defined by the limit
If h has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written
Hence, the forward difference divided by h approximates the derivative when h is small. The error in this approximation can be derived from Taylor's theorem. Assuming that f is continuously differentiable, the error is
The same formula holds for the backward difference:
However, the central difference yields a more accurate approximation. Its error is proportional to square of the spacing (if f is twice continuously differentiable):
The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If for uneven, and for even, then if it is calculated with the central difference scheme. This is particularly troublesome if the domain of is discrete.
Read more about this topic: Finite Difference
Famous quotes containing the word relation:
“The instincts of the ant are very unimportant, considered as the ant’s; but the moment a ray of relation is seen to extend from it to man, and the little drudge is seen to be a monitor, a little body with a mighty heart, then all its habits, even that said to be recently observed, that it never sleeps, become sublime.”
—Ralph Waldo Emerson (1803–1882)