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The version of Steffensen's method implemented in the MATLAB code shown below can be found using the Aitken's delta-squared process for accelerating convergence of a sequence. To compare the following formulae to the formulae in the section above, notice that x n = p − p n . {\displaystyle x_{n}=p\,-\,p_{n}~.}
In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no analytical solution exists. Then one uses root-finding algorithms, such as Newton's method, to find the numerical solution. Crank-Nicolson method. With the Crank-Nicolson method
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 i-th approximation (called an "iterate") is derived from the previous ones.
Most numerical root-finding methods are iterative methods, producing a sequence of numbers that ideally converges towards a root as a limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be ...
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations.It is a popular method for solving the large matrix equations that arise in systems theory and control, [1] and can be formulated to construct solutions in a memory-efficient, factored form.
Numerical analysis is not only the design of numerical methods, but also their analysis. Three central concepts in this analysis are: convergence: whether the method approximates the solution, order: how well it approximates the solution, and; stability: whether errors are damped out.
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 .
In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems. [ 1 ] Relaxation methods were developed for solving large sparse linear systems , which arose as finite-difference discretizations of differential equations .