@@, _fnest

Iterate a function

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f @@ n
_fnest(f, n)


f@@n represents the n-fold iterate x -> f(f(...(f(x))...)) of the function f.

The statement f@@n is equivalent to the call _fnest(f, n).

For positive n, f@@n is also equivalent to _fconcat(f $ n).

f@@0 returns the identity map id.

If f is a function environment with the slot "inverse" set, n can also be negative. Cf. Example 2.

Iteration is only reasonable for functions that accept their own return values as input. Note that fp::fixargs is a handy tool for converting functions with parameters to univariate functions which may be suitable for iteration. Cf. Example 3.


Example 1

For a nonnegative integer n, f@@n is equivalent to an _fconcat call:

f@@4, (f@@4)(x)

@@ simplifies the composition of symbolic iterates:


The iterate may be called like any other MuPAD® function. If f evaluates to a procedure and n to an integer, a corresponding value is computed:

f := x -> x^2:  (f@@n)(x) $ n = 0..10

delete f:

Example 2

For functions with a known inverse function, n may be negative. The function f must have been declared as a function environment with the "inverse" slot. Examples of such functions include the trigonometric functions which are implemented as function environments in MuPAD:

sin::"inverse", sin@@-3, (sin@@(-3))(x)

Example 3

@@ can only be used for functions that accept their own output domain as an input, i.e., for some set M. If you want to use @@ with a function which needs additional parameters, fp::fixargs is a handy tool to generate a corresponding univariate function. In the following call, the function f: x -> g(x, p) is iterated:

g := (x, y) -> x^2 + y: f := fp::fixargs(g, 1, p): (f@@4)(x)

delete g, f:



A function


An integer or a symbolic name

Return Values


See Also

MuPAD Functions