# FixedPointList

FixedPointList[f,expr]

generates a list giving the results of applying f repeatedly, starting with expr, until the results no longer change.

# Details and Options

• FixedPointList[f,expr] gives expr as the first element of the list it produces.
• The last two elements in the list produced by FixedPointList are always the same.
• FixedPointList[f,expr,n] stops after at most n steps.
• FixedPointList[f,expr] applies SameQ to successive pairs of results to determine whether a fixed point has been reached.
• FixedPointList[f,expr,,SameTest->s] applies s to successive pairs of results.

# Examples

open allclose all

## Basic Examples(2)

Find the sequence of values leading to a fixed point:

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

## Generalizations & Extensions(1)

Stop after at most 5 steps:

## Options(1)

### SameTest(1)

Stop as soon as successive iterations differ by less than :

## Applications(6)

Show convergence to in Newtons 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:

## Properties & Relations(4)

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:

## Possible Issues(3)

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:

## Neat Examples(1)

Wolfram Research (1991), FixedPointList, Wolfram Language function, https://reference.wolfram.com/language/ref/FixedPointList.html (updated 1996).

#### Text

Wolfram Research (1991), FixedPointList, Wolfram Language function, https://reference.wolfram.com/language/ref/FixedPointList.html (updated 1996).

#### CMS

Wolfram Language. 1991. "FixedPointList." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/FixedPointList.html.

#### APA

Wolfram Language. (1991). FixedPointList. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FixedPointList.html

#### BibTeX

@misc{reference.wolfram_2021_fixedpointlist, author="Wolfram Research", title="{FixedPointList}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/FixedPointList.html}", note=[Accessed: 23-May-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2021_fixedpointlist, organization={Wolfram Research}, title={FixedPointList}, year={1996}, url={https://reference.wolfram.com/language/ref/FixedPointList.html}, note=[Accessed: 23-May-2022 ]}