A recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms.
A recurrence relation is a set of equations that recursively define a sequence, with one or more initial terms given, and each subsequent term defined as a function of the preceding terms.
An example is the Fibonacci sequence, which can be defined as:
F(n) = F(n-1) + F(n-2), with F(0) = 1 and F(1) = 1.
Some simply defined recurrence relations can have very complex (chaotic) behaviours and they are a part of the field of mathematics known as nonlinear analysis.
Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function expressed analytically in terms of a finite number of certain "well-known" functions.
Recurrence relations are of fundamental importance in analysis of algorithms. In computer science, recurrence usually emerges when we analyze the complexity of divide-and-conquer algorithms. If an algorithm is designed so that it will break a problem into smaller subproblems (divide and conquer), its running time is described by a recurrence relation.
For some typical recurrence relations, the asymptotic growth can be analyzed via the Master Theorem.
