A finite difference is a mathematical expression of the form f(x + b) − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
Recurrence relations can be written as difference equations by replacing iteration notation with finite differences.
Forward, backward, and central 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 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
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 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
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