## iteration

iteration
1. The repetition of a numerical or nonnumerical process where the results from one or more stages are used to form the input to the next. Generally the recycling of the process continues until some preset bound is achieved, or the process result is constantly repeated. This is one of the key ideas used in the design of numerical methods (see also iterative methods).

An iterative process is m-stage if the new value is derived from m previous values; it is m-stage, sequential if the new value depends upon the last m values, i.e. xk+1 = Gk(xk, xk–1, …, xkm+1)

The iteration is stationary if the function Gk is independent of k, i.e. the new value is calculated from the old values using the same formula. For example, xk+1 = ½(xk + a/xk)

is a stationary, one-stage iteration (used for evaluating the square root of a); this is a particular application of Newton's method. The secant method is a stationary two-stage sequential iteration. False position is an example of a nonsequential iteration.

2. of a formal language. See Kleene star.

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