This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# FixedPoint

 FixedPointstarts with expr, then applies f repeatedly until the result no longer changes.
• FixedPoint always returns the last result it gets.
• FixedPoint applies SameQ to successive pairs of results to determine whether a fixed point has been reached.
Find a value such that :
Fixed point of an integer-valued function:
Repeated application of a rule until the result no longer changes:
Find a value such that :
 Out[1]=
 Out[2]=

Fixed point of an integer-valued function:
 Out[1]=
 Out[2]=

Repeated application of a rule until the result no longer changes:
 Out[1]=
 Out[2]=
 Scope   (2)
Numerical fixed point of a function:
Fixed point of a repeated transformation:
Stop after at most 10 steps:
 Options   (2)
Stop as soon as successive iterations differ by less than :
Perform exact arithmetic, but use a numerical comparison function:
 Applications   (8)
Find using Newton's method:
Fixed point of a complex iteration:
Matrix-multiplication convergence:
Root of the current directory tree (the result will depend on computer system):
Repeated differentiation:
Find the minimum of with the steepest-descent method (vector notation):
Component notation:
Evaluate combinators []:
Connected components in a graph:
FixedPoint gives the last element of FixedPointList:
Apply rules repeatedly until the result no longer changes using ReplaceRepeated (//.):
FixedPoint is equivalent to a particular choice of arguments of NestWhile:
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:
Sometimes convergence may fail on certain platforms due to insufficient accuracy of a machine arithmetic library function. This can be addressed by using a comparison function with a larger tolerance: