Find the sequence of values leading to a fixed point:

Repeated application of a rule until the result no longer changes:

Stop after at most 5 steps:

Show convergence to in Newton’s method:

Find the fixed point of an integer sequence:

Matrix convergence:

Evaluate combinators:

Go up a directory tree to its root (the result will depend on computer system):

Repeated differentiation:

FixedPoint gives the last element of FixedPointList:

NestList goes for a fixed number of steps:

Apply rules repeatedly until the result no longer changes:

FixedPointList is equivalent to a particular choice of arguments of NestWhileList:

Calculations may not converge in a finite number of steps:

Providing a maximum number of iterations will guarantee termination:

Using a numerical test for convergence works in this case as well:

Convergence may fail in machine-precision computations due to oscillations in the final digits. Use a test function with a larger tolerance than SameQ to resolve this:

When finding a fixed point, numerical precision may increase:

The depth of intermediate expressions in a combinator evaluation [more info]: