In mathematical analysis, operators involving finite differences are studied. A difference operator is an operator which maps a function f to a function whose values are the corresponding finite differences.
Only three forms are commonly considered: forward, backward, and central differences.
A forward difference is an expression of the form
Depending on the application, the spacing h may be variable or held constant.
A backward difference uses the function values at x and x − h, instead of the values at x + h and x:
Finally, the central difference is given by
If h has a fixed (non-zero) value, instead of approaching zero, then the right-hand side is
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):
In an analogous way one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for and and applying a central difference formula for the derivative of at x, we obtain the central difference approximation of the second derivative of f:
More generally, the nth-order forward, backward, and central differences are respectively given by:
Note that the central difference will, for odd , have multiplied by non-integers. If this is a problem (usually it is), it may be remedied taking the average of and .
The relationship of these higher-order differences with the respective derivatives is very straightforward:
Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order h. However, the combination
If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.
An important application of finite differences is in numerical analysis, especially in numerical differential equations, which aim at the numerical solution of ordinary and partial differential equations respectively. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called finite difference methods.
The forward difference can be considered as a difference operator, which maps the function f to Δh[f]. This operator satisfies
Finite difference of higher orders can be defined in recursive manner as or, in operators notation, Another possible (and equivalent) definition is
Taylor's theorem can now be expressed by the formula
where D denotes the derivative operator, mapping f to its derivative f'. Formally inverting the exponential suggests that
This formula holds in the sense that both operators give the same result when applied to a polynomial. Even for analytic functions, the series on the right is not guaranteed to converge; it may be an asymptotic series. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to mentioned at the end of the section Higher-order differences.
The analogous formulas for the backward and central difference operators are
The calculus of finite differences is related to the umbral calculus in combinatorics.
A generalized finite difference is usually defined as
Finite differences can be considered in more than one variable. They are analogous to partial derivatives in several variables.
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