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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]
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a common use and a well-known example.
There are several techniques for convergence acceleration of the sequences produced by fixed point iteration. [10] For example, the Aitken method applied to an iterated fixed point is known as Steffensen's method, and produces quadratic convergence.
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel .
Pages in category "Iterative methods" The following 14 pages are in this category, out of 14 total. This list may not reflect recent changes. ...
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges.
Iterative and incremental development is any combination of both iterative design (or iterative method) and incremental build model for development. Usage of the term began in software development , with a long-standing combination of the two terms iterative and incremental [ 1 ] having been widely suggested for large development efforts.