Finite difference

A finite difference measures how a function’s values change between distinct input points. Difference operators use these changes to analyze data and approximate derivatives.

Forward difference: A difference between a function value and its value at a fixed step to the right. It measures change from the starting input to the next sampled input.
Function: A rule that assigns each input in a domain exactly one output. Finite differences compare outputs assigned to different inputs.
Numerical differentiation: The estimation of derivatives from discrete or approximate function values. Finite differences provide standard formulas for estimating derivatives from samples.
Derivative: The instantaneous rate at which a function changes with respect to its input. A derivative is a limiting rate; a finite difference measures change across a nonzero interval.
Backward difference: A difference between a function value and its value at a fixed step to the left. It estimates change using the current input and a preceding sample.
Sequence: An ordered list of values indexed by integers or another set of positions. Sampling a function at equally spaced inputs produces a sequence to difference.
Finite difference method for ordinary differential equations: A grid-based method that replaces derivatives in an ordinary differential equation with finite difference formulas. It converts a continuous equation into algebraic relations among sampled values.
Differential calculus: The branch of mathematics that studies derivatives and rates of change. Finite differences provide discrete counterparts to ideas developed in differential calculus.
Central difference: A difference-based estimate of change using function values on both sides of an input. Its symmetric samples often yield a more accurate derivative estimate than one-sided differences.
Step size: The spacing between successive input points in a numerical grid or sampling scheme. The chosen spacing determines which function values a finite difference compares.

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