Récursives

Récursives refers to notions in mathematics and computer science that rely on recursion, a process where a concept is defined in terms of itself. A recursive definition typically includes a base case, which stops the recursion, and a recursive clause, which reduces the problem to simpler instances.

In mathematics, recursive definitions describe sequences or sets by initial values followed by recurrence relations. A common example is the Fibonacci sequence: F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2 for n > 1. More generally, many processes can be specified by iterative rules that reuse previously computed terms.

In computer science, recursive functions are named for their use of self-reference: a function calls itself with a smaller input until a base case is reached. This approach can lead to concise and clear solutions for problems such as tree traversal, factorials, and divide-and-conquer algorithms like quicksort or binary search. Recursion can be mutual when two or more functions call each other in a cycle.

Well-foundedness and termination are essential for recursive definitions to be meaningful. If the base case is unreachable or the recursive step does not reduce the problem, the process may not terminate. Practical considerations include memory usage and performance overhead from repeated calls; some languages optimize tail-recursive patterns to behave like iteration, while others do not.

Applications of recursion span formal mathematics, algorithm design, and programming languages. They provide a powerful paradigm for expressing hierarchical structures and self-similar processes.

See also: recursion in programming, primitive recursion, tail recursion, recurrence relation, recursive definition.