FixedPointList

Applies a function repeatedly, collecting every value up to the fixed point.

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FixedPointList(f, x)

\{x, f(x), f(f(x)), \dots\}

, stopping once two consecutive values are structurally the same — the repeated value ends the list once.

Typed signatures

Type	Package	Applies
`(any, any) -> list<any>`	`combinatorics`	always

Domain: Collections

Details

Stops on exact structural equality (Wolfram's $SameQ$ ), not the numeric-precision agreement FixedPoint additionally uses for an inexact sequence — adequate for exact/rational values, which is what every example here uses.
Capped at 4096 applications of $f$ , same as NestWhile.

Examples

Basic #

halve and floor down to 0, ending with the repeated fixed point

\mathrm{FixedPointList}(\operatorname{\_1}↦\lfloor\frac{\operatorname{\_1}}{2}\rfloor, 10)

Out

\bigl\lbrack10, 5, 2, 1, 0, 0\bigr\rbrack

The halving orbit, ending when two consecutive values repeat

\mathrm{FixedPointList}(\operatorname{\_1}↦1+\lfloor\frac{\operatorname{\_1}}{2}\rfloor, 1\,000)

Out

\bigl\lbrack1\,000, 501, 251, 126, 64, 33, 17, 9, 5, 3, 2, 2\bigr\rbrack

Applications #

Orbit of $n \mapsto 11\pi(n) + 1$ from $2$ , growing for many steps before it settles

\mathrm{FixedPointList}(\operatorname{\_1}↦11\mathrm{PrimePi}(\operatorname{\_1})+1, 2)

Out

\bigl\lbrack2, 12, 56, 177, 441, 936, 1\,739, 2\,971, 4\,720, 6\,986, 9\,879, 13\,399, 17\,480, 22\,111, 27\,270, 32\,825, 38\,710, 44\,859, 51\,272, 57\,707, 64\,285, 70\,808, 77\,111, 83\,348, 89\,497, 95\,316, 101\,080, 106\,459, 111\,651, 116\,469, 121\,012, 125\,302, 129\,361, 133\,178, 136\,720, 140\,075, 143\,166, 146\,026, 148\,622, 150\,943, 153\,198, 155\,365, 157\,312, 159\,149, 160\,678, 162\,141, 163\,472, 164\,671, 165\,793, 166\,739, 167\,553, 168\,356, 169\,016, 169\,544, 170\,017, 170\,479, 170\,941, 171\,392, 171\,744, 172\,096, 172\,415, 172\,723, 173\,064, 173\,350, 173\,592, 173\,757, 173\,933, 174\,120, 174\,296, 174\,450, 174\,582, 174\,692, 174\,813, 174\,912, 175\,000, 175\,077, 175\,143, 175\,198, 175\,198\bigr\rbrack

\mathrm{FixedPointList}(\operatorname{\_1}↦\frac{\mathrm{d}\operatorname{\_1}}{\mathrm{d}x}, x^{10})

Out

\bigl\lbrack x^{10}, 10x^9, 90x^8, 720x^7, 5\,040x^6, 30\,240x^5, 151\,200x^4, 604\,800x^3, 1\,814\,400x^2, 3\,628\,800x, 3\,628\,800, 0, 0\bigr\rbrack

Properties #

\mathrm{FixedPointList}(\operatorname{\_1}↦\mathrm{ReplaceAll}(\operatorname{\_1}, \bigl\lbrack\mathrm{Rule}(a, b), \mathrm{Rule}(b, c), \mathrm{Rule}(c, d)\bigr\rbrack), a^2+c^3)

Out

\bigl\lbrack c^3+a^2, d^3+b^2, d^3+c^2, d^3+d^2, d^3+d^2\bigr\rbrack

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