Revision history [back]

In the next version of sage (4.7) you will be able to use the functions self_compose and nest to do this. Use nest when you want to find $f(f(...f(x)...))$ for known $x$, and use self_compose when you want a function that has not been evaluated.

In the meantime you'll need to define them yourself:

def nest(f, n, x):
    """Return `f(f(...f(x)...))`, where the composition occurs n times."""
    for i in xrange(n):
        x = f(x)
    return x

def self_compose(f, n):
    """Return the function `f` composed with itself `n` times."""
    return lambda x: nest(f, n, x)

For usage examples and more documentation, see the patch that adds this functionality.

In the next version of sage (4.7) you will be able to use the functions self_compose and nest to do this. Use nest when you want to find $f(f(...f(x)...))$ for known $x$, and use self_compose when you want a function that has not been evaluated.

In the meantime you'll need to define them yourself:

def nest(f, n, x):
    """Return `f(f(...f(x)...))`, where the composition occurs n times."""
    for i in xrange(n):
        x = f(x)
    return x

def self_compose(f, n):
    """Return the function `f` composed with itself `n` times."""
    return lambda x: nest(f, n, x)

For usage examples and more documentation, see the patch that adds this functionality.functionality. There is one example which is very similar to yours:

    sage: def f(x): return x^2 + 1
    sage: x = var('x')
    sage: nest(f, 3, x)
    ((x^2 + 1)^2 + 1)^2 + 1