Iterate

The result of repeated application of a mathematical operation. Thus, if $$ y = f(x) \equiv f_1(x) $$ is a function of $x$, then the functions $$ f_2(x) = f[f_1(x)] \,,\ \ldots\ ,\,f_n(x) = f[f_{n-1}(x)] $$ are called the second, $\ldots$, $n$-th iterates of $f(x)$. E.g., putting $f(x) = x^\alpha$ one obtains $$ f_2(x) = x^{\alpha^2} \,,\ \ldots\ ,\,f_n(x) = x^{\alpha^n} \ . $$

The index $n$ is called the exponent of the iterate, while transition from $f(x)$ to $f_2(x),f_3(x),\ldots$ is called iteration. Iterates with an arbitrary real, or even complex, exponent can be defined for certain classes of functions. Iterates are used in the solution of various kinds of equations and systems of equations by iteration methods. For more information, see Sequential approximation, method of.

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