Built-in Mathematica Symbol

FixedPoint

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

MORE INFORMATION

FixedPoint[f, expr, n] 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[f, expr] 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

:

Fixed point of an integer-valued function:

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

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 [more info]:

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: