Iterative method

An iterative method repeatedly applies a rule to produce a sequence of approximations or states. It is used when repeated updates can approach a solution or desired outcome.

Fixed-point iteration: A method that repeatedly evaluates a function at the current estimate to seek a fixed point. It expresses iteration directly as repeated application of one function.
Newton's method: A root-finding algorithm that updates an estimate using the function and its derivative. Its repeated tangent-based updates can rapidly refine an approximate root.
Sequence: An ordered list of mathematical objects indexed by integers or another ordered set. An iterative method produces successive states that can be studied as a sequence.
Direct method: A computational approach that seeks a solution through a finite prescribed sequence of operations. It contrasts with methods that refine an estimate through repeated updates.
Convergence: The tendency of a sequence or process to approach a limit or stable state. It describes whether successive approximations settle toward a meaningful result.
Gradient descent: An optimization algorithm that moves parameters opposite the gradient of an objective function. Each update aims to reduce the objective until a stationary solution is approached.
Recurrence relation: An equation that defines terms of a sequence using earlier terms. It formalizes the rule that generates each new approximation from prior states.
Divergence: The failure of a sequence or process to approach a finite limit or stable state. An iteration may move away from any solution rather than improve its estimates.
Contraction mapping: A function that brings distances between points closer by a uniform factor less than one. Its distance-shrinking property guarantees convergence for fixed-point iteration.
Jacobi method: An iterative algorithm for solving linear systems by updating each variable from the previous iteration's values. It solves large linear systems through repeated, parallelizable estimates.

Linked from 8 pages