BUILT-IN MATHEMATICA SYMBOL

FixedPoint

starts with expr, then applies f repeatedly until the result no longer changes.

MORE INFORMATION

FixedPoint stops after at most n steps.

FixedPoint always returns the last result it gets.

You can use Throw to exit from FixedPoint before it is finished.

FixedPoint applies SameQ to successive pairs of results to determine whether a fixed point has been reached.

FixedPoint[f, expr, ..., SameTest->s] applies s to successive pairs of results.

EXAMPLES

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Basic Examples (3)

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

:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

Fixed point of an integer-valued function:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

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

In[1]:=

Out[1]=

In[2]:=

Out[2]=

Scope (2)

Numerical fixed point of a function:

Fixed point of a repeated transformation:

Generalizations & Extensions (1)

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:

Properties & Relations (3)

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:

Possible Issues (1)

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: