Algorithm in which each approximation of the solution is derived from prior approximations
It has been suggested that Methods of successive approximation be merged into this article. (Discuss) Proposed since December 2023.
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.
A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common.
In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. However, iterative methods are often useful even for linear problems involving many variables (sometimes on the order of millions), where direct methods would be prohibitively expensive (and in some cases impossible) even with the best available computing power.[1]
^Amritkar, Amit; de Sturler, Eric; Świrydowicz, Katarzyna; Tafti, Danesh; Ahuja, Kapil (2015). "Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver". Journal of Computational Physics. 303: 222. arXiv:1501.03358. Bibcode:2015JCoPh.303..222A. doi:10.1016/j.jcp.2015.09.040.
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