itératives

Iteratives, also known as iterative methods or iterative algorithms, are a class of computational procedures that solve a problem by repeatedly applying a specific set of rules to a sequence of approximations. Instead of calculating a direct solution, iterative methods start with an initial guess and then refine it through a series of steps, progressively getting closer to the true solution. Each step in the process is called an iteration.

The core idea behind iterative methods is to improve upon the previous result. In each iteration, an update rule uses the current approximation to generate a new, hopefully better, approximation. This process continues until a predefined stopping criterion is met. Common stopping criteria include reaching a maximum number of iterations or when the difference between successive approximations falls below a small tolerance, indicating convergence.

Iterative methods are widely used in various fields of science and engineering, particularly in numerical analysis and computer science. They are essential for solving problems that lack direct analytical solutions, such as finding roots of complex equations, solving systems of linear equations, performing optimization, and approximating solutions to differential equations. Examples include the Newton-Raphson method for finding roots and the Gauss-Seidel method for solving linear systems. The efficiency and convergence rate of an iterative method depend heavily on the problem's characteristics and the chosen update rule.