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A first simple and useful example is the babylonian method for computing the square root of a > 0, which consists in taking , i.e In particular, since it holds that The mean value of x and a/x, to approach the limit (from.

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Matlab supports both external and internal implicit iteration using either native arrays or cell arrays The iteration converges to a solution when The optimization problem was solved in the form using qr decomposition;

Example pseudocode for richardson extrapolation the following pseudocode in matlab style demonstrates richardson extrapolation to help solve the ode , with the trapezoidal method

In this example we halve the step size each iteration and so in the discussion above we'd have that. Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the rayleigh quotient to obtain increasingly accurate eigenvalue estimates. Chebyshev iteration in numerical linear algebra, the chebyshev iteration is an iterative method for determining the solutions of a system of linear equations The method is named after russian mathematician pafnuty chebyshev

Chebyshev iteration avoids the computation of inner products as is necessary for the other nonstationary methods. 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. 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

A randomized version of the method uses non repeated row indices at each iteration