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FixedPoint
starts with expr, then applies f repeatedly until the result no longer changes.
Details and Options
- 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
open allclose allBasic Examples (3)Summary of the most common use cases
https://wolfram.com/xid/0i1l57sdu-x9nmen
https://wolfram.com/xid/0i1l57sdu-5ndal9
Fixed point of an integer-valued function:
https://wolfram.com/xid/0i1l57sdu-i8m
https://wolfram.com/xid/0i1l57sdu-r2s
Repeated application of a rule until the result no longer changes:
https://wolfram.com/xid/0i1l57sdu-umoip1
https://wolfram.com/xid/0i1l57sdu-k65kz1
Scope (2)Survey of the scope of standard use cases
Generalizations & Extensions (1)Generalized and extended use cases
Options (2)Common values & functionality for each option
SameTest (2)
Stop as soon as successive iterations differ by less than :
https://wolfram.com/xid/0i1l57sdu-q3avny
https://wolfram.com/xid/0i1l57sdu-ykgd5u
Perform exact arithmetic, but use a numerical comparison function:
https://wolfram.com/xid/0i1l57sdu-uee27z
Applications (8)Sample problems that can be solved with this function
https://wolfram.com/xid/0i1l57sdu-it9
Fixed point of a complex iteration:
https://wolfram.com/xid/0i1l57sdu-nwk
Matrix-multiplication convergence:
https://wolfram.com/xid/0i1l57sdu-sp1
Root of the current directory tree (the result will depend on computer system):
https://wolfram.com/xid/0i1l57sdu-rzs
https://wolfram.com/xid/0i1l57sdu-s2l
Find the minimum of with the steepest-descent method (vector notation):
https://wolfram.com/xid/0i1l57sdu-4up6f6
https://wolfram.com/xid/0i1l57sdu-bdvi01
Evaluate combinators [more info]:
https://wolfram.com/xid/0i1l57sdu-pu
Connected components in a graph:
https://wolfram.com/xid/0i1l57sdu-z57fwr
https://wolfram.com/xid/0i1l57sdu-9hetmt
Properties & Relations (3)Properties of the function, and connections to other functions
FixedPoint gives the last element of FixedPointList:
https://wolfram.com/xid/0i1l57sdu-o30avt
https://wolfram.com/xid/0i1l57sdu-hy3x62
Apply rules repeatedly until the result no longer changes using ReplaceRepeated (//.):
https://wolfram.com/xid/0i1l57sdu-c52
https://wolfram.com/xid/0i1l57sdu-gfl
https://wolfram.com/xid/0i1l57sdu-gke
FixedPoint is equivalent to a particular choice of arguments of NestWhile:
https://wolfram.com/xid/0i1l57sdu-wu7qgb
Possible Issues (2)Common pitfalls and unexpected behavior
Calculations may not converge in a finite number of steps:
https://wolfram.com/xid/0i1l57sdu-j18
Providing a maximum number of iterations will guarantee termination:
https://wolfram.com/xid/0i1l57sdu-u1jam2
Using a numerical test for convergence works in this case as well:
https://wolfram.com/xid/0i1l57sdu-bpccda
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:
https://wolfram.com/xid/0i1l57sdu-fcdxdz
Wolfram Research (1988), FixedPoint, Wolfram Language function, https://reference.wolfram.com/language/ref/FixedPoint.html (updated 1996).
Text
Wolfram Research (1988), FixedPoint, Wolfram Language function, https://reference.wolfram.com/language/ref/FixedPoint.html (updated 1996).
Wolfram Research (1988), FixedPoint, Wolfram Language function, https://reference.wolfram.com/language/ref/FixedPoint.html (updated 1996).
CMS
Wolfram Language. 1988. "FixedPoint." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/FixedPoint.html.
Wolfram Language. 1988. "FixedPoint." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/FixedPoint.html.
APA
Wolfram Language. (1988). FixedPoint. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FixedPoint.html
Wolfram Language. (1988). FixedPoint. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FixedPoint.html
BibTeX
@misc{reference.wolfram_2024_fixedpoint, author="Wolfram Research", title="{FixedPoint}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/FixedPoint.html}", note=[Accessed: 07-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_fixedpoint, organization={Wolfram Research}, title={FixedPoint}, year={1996}, url={https://reference.wolfram.com/language/ref/FixedPoint.html}, note=[Accessed: 07-January-2025
]}