Nest

f applied n times to x.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageNest✓↗

Nest(f, x, n)$f$ applied to $x$, $n$ times in a row.

Typed signatures

Type	Package	Applies
(function: any, x: any, n: integer) -> any	combinatorics	always

Domain: Collections

Details

• $f$ can be a Function literal or a plain symbol; an undeclared symbol stays an unevaluated call, the way $Nest(f, x, 3)$ shows below.
• See NestList for every intermediate step, and FixedPoint for iterating until the value stops changing instead of a fixed count.

Examples

Basic #

live ↗

In$\mathrm{Nest}(f, x, 3)$Out$f(f(f(x)))$

Repeated squaring: $((2^2)^2)^2$

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦\operatorname{\_1}^2, 2, 3)$Out$256$

Four nested applications of $f$

live ↗

In$\mathrm{Nest}(f, x, 4)$Out$f(f(f(f(x))))$

Three steps of $n \mapsto (n+1)^2$ from $1$: $1, 4, 25, 676$

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦(\operatorname{\_1}+1)^2, 1, 3)$Out$676$

Scope #

Zero applications leave x alone

live ↗

In$\mathrm{Nest}(f, x, 0)$Out$x$

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦2\operatorname{\_1}, 1, 10)$Out$1\,024$

Four square roots of $100$, left in radical form

live ↗

In$\mathrm{Nest}(\mathrm{Sqrt}, 100, 4)$Out$\sqrt{\sqrt{\sqrt{10}}}$

With a symbolic start, the full nesting is retained

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦(\operatorname{\_1}+1)^2, x, 5)$Out$(((((x+1)^2+1)^2+1)^2+1)^2+1)^2$

live ↗

In$\mathrm{Catch}(\mathrm{Nest}(\operatorname{\_1}↦\text{if }\operatorname{\_1}\gt10^6\text{ then }\mathrm{Throw}(\operatorname{\_1})\text{ else }\operatorname{\_1}^2, 2, 6))$Out$4\,294\,967\,296$

Applications #

Three Newton steps toward $\sqrt{2}$

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦\frac{\operatorname{\_1}+\frac{2}{\operatorname{\_1}}}{2}, 1, 3)$Out$\frac{577}{408}$

Builds a continued-fraction-like tower

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦\frac{1}{\operatorname{\_1}+1}, x, 5)$Out$\frac{1}{\frac{1}{\frac{1}{\frac{1}{1/(x+1)+1}+1}+1}+1}$

A power tower of $x$

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦x^{\operatorname{\_1}}, x, 6)$Out$x^{x^{x^{x^{x^{x^{x}}}}}}$

Newton's method for $\sqrt2$, converging quadratically to $1.41421\ldots$

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦\frac{\operatorname{\_1}+\frac{2}{\operatorname{\_1}}}{2}, 1, 5)$Out$\frac{886\,731\,088\,897}{627\,013\,566\,048}$

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦\mathrm{Dot}(\bigl\lbrack\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack\bigr\rbrack, \operatorname{\_1}), \bigl\lbrack0, 1\bigr\rbrack, 10)$Out$\bigl\lbrack55, 34\bigr\rbrack$

Properties #

Halving $20$ with floor six times drains it to $0$

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦\lfloor\frac{\operatorname{\_1}}{2}\rfloor, 20, 6)$Out$0$

Neat examples #

Wrapping in a pair three times gives a complete binary tree of $x$s

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦\bigl\lbrack\operatorname{\_1}, \operatorname{\_1}\bigr\rbrack, x, 3)$Out$\bigl\lbrack\bigl\lbrack\bigl\lbrack x, x\bigr\rbrack, \bigl\lbrack x, x\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack x, x\bigr\rbrack, \bigl\lbrack x, x\bigr\rbrack\bigr\rbrack\bigr\rbrack$

The reflected binary Gray code: each step appends a shifted reversal

live ↗

In$\mathrm{Nest}(\operatorname{\_1}↦\mathrm{Join}(\operatorname{\_1}, \mathrm{Length}(\operatorname{\_1})+\mathrm{Reverse}(\operatorname{\_1})), \bigl\lbrack0\bigr\rbrack, 4)$Out$\bigl\lbrack0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8\bigr\rbrack$

About

Examples	17
Checked against	Wolfram Language 17 agreemathlib4 5 agree
Crosswalk	Wolfram Language

Implementation

mappedmathlib4packages/oracle/src/mappings.ts

A negative or non-integer count fails to elaborate, where ours holds the call.

See also: NestList, FixedPoint