Iterate a function
This functionality does not run in MATLAB.
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.
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.
For a nonnegative integer n, f@@n is equivalent to an _fconcat call:
@@ 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
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)
@@ 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: