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Basic examples of stationary iterative methods use a splitting of the matrix such as = + +,:= (()) where is only the diagonal part of , and is the strict lower triangular part of .
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.
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.
The method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems with objective functions of the form of a p-norm: = | |, by an iterative method in which each step involves solving a weighted least squares problem of the form: [1]
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
Iterative design has long been used in engineering fields. One example is the plan–do–check–act cycle implemented in the 1960s. Most New product development or existing product improvement programs have a checking loop which is used for iterative purposes. DMAIC uses the Six Sigma framework and has such a checking function.
Relaxation (iterative method) S. Shanks transformation; Superiorization This page was last edited on 31 December 2018, at 21:42 (UTC). Text is available under the ...