Array

f applied over every point of an n-dimensional index range.

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

Array(f, n)$\{f(1), \ldots, f(n)\}$.

Array(f, {n1, …, nk})the $n_1 \times \cdots \times n_k$ array with $f$ applied to every index tuple.

Array(f, n, r)like $Array(f, n)$, but the index range starts at $r$ instead of 1.

Array(f, {n1, …, nk}, {r1, …, rk})like the multi-dimensional form, with each dimension's index range starting at its own origin.

Array(f, n, r, h)like $Array(f, n, r)$, wrapped in $h$ at every level instead of $List$.

Typed signatures

Type	Package	Applies
((integer*) any -> any, any, any?, symbol?) -> collection	combinatorics	always

Domain: Collections

Details

• The scalar forms of $n$ and $r$ are shorthand for $\{n\}$ and every dimension sharing origin $r$.
• See Tabulate for a lazy array read without materialising it.

Examples

1 more kept as data, checked against the oracles

Basic #

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In$\mathrm{Array}(f, 5)$Out$\bigl\lbrack f(1), f(2), f(3), f(4), f(5)\bigr\rbrack$

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In$\mathrm{Array}(\operatorname{\_1}↦\operatorname{\_1}^2, 5)$Out$\bigl\lbrack1, 4, 9, 16, 25\bigr\rbrack$

Applies $f$ to the indices $1..10$

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In$\mathrm{Array}(f, 10)$Out$\bigl\lbrack f(1), f(2), f(3), f(4), f(5), f(6), f(7), f(8), f(9), f(10)\bigr\rbrack$

A list of dimensions builds a nested array of two-index calls

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In$\mathrm{Array}(f, \bigl\lbrack3, 2\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack f(1, 1), f(1, 2)\bigr\rbrack, \bigl\lbrack f(2, 1), f(2, 2)\bigr\rbrack, \bigl\lbrack f(3, 1), f(3, 2)\bigr\rbrack\bigr\rbrack$

A third argument shifts the starting index to $0$

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In$\mathrm{Array}(f, 10, 0)$Out$\bigl\lbrack f(0), f(1), f(2), f(3), f(4), f(5), f(6), f(7), f(8), f(9)\bigr\rbrack$

Origins can differ per dimension

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In$\mathrm{Array}(f, \bigl\lbrack2, 3\bigr\rbrack, \bigl\lbrack0, 4\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack f(0, 4), f(0, 5), f(0, 6)\bigr\rbrack, \bigl\lbrack f(1, 4), f(1, 5), f(1, 6)\bigr\rbrack\bigr\rbrack$

A pure function gives the values $k^2+1$

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In$\mathrm{Array}(\operatorname{\_1}↦\operatorname{\_1}^2+1, 10)$Out$\bigl\lbrack2, 5, 10, 17, 26, 37, 50, 65, 82, 101\bigr\rbrack$

A two-argument function encodes row and column in the digits

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In$\mathrm{Array}((\operatorname{\_1}, \operatorname{\_2})↦10\operatorname{\_1}+\operatorname{\_2}, \bigl\lbrack3, 4\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack11, 12, 13, 14\bigr\rbrack, \bigl\lbrack21, 22, 23, 24\bigr\rbrack, \bigl\lbrack31, 32, 33, 34\bigr\rbrack\bigr\rbrack$

Scope #

A list of dimensions builds a multi-dimensional array

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In$\mathrm{Array}(f, \bigl\lbrack2, 2\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack f(1, 1), f(1, 2)\bigr\rbrack, \bigl\lbrack f(2, 1), f(2, 2)\bigr\rbrack\bigr\rbrack$

A third argument pins where the index range starts

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In$\mathrm{Array}(f, 3, 0)$Out$\bigl\lbrack f(0), f(1), f(2)\bigr\rbrack$

A fourth argument wraps the result in that head instead of List

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In$\mathrm{Array}(f, 3, 1, g)$Out$g(f(1), f(2), f(3))$

An undefined head stays symbolic

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In$\mathrm{Array}(a, \bigl\lbrack3, 2\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack a(1, 1), a(1, 2)\bigr\rbrack, \bigl\lbrack a(2, 1), a(2, 2)\bigr\rbrack, \bigl\lbrack a(3, 1), a(3, 2)\bigr\rbrack\bigr\rbrack$

Three dimensions, each with its own origin including negatives

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In$\mathrm{Array}(f, \bigl\lbrack4, 3, 2\bigr\rbrack, \bigl\lbrack0, -1, -2\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\bigl\lbrack f(0, -1, -2), f(0, -1, -1)\bigr\rbrack, \bigl\lbrack f(0, 0, -2), f(0, 0, -1)\bigr\rbrack, \bigl\lbrack f(0, 1, -2), f(0, 1, -1)\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack f(1, -1, -2), f(1, -1, -1)\bigr\rbrack, \bigl\lbrack f(1, 0, -2), f(1, 0, -1)\bigr\rbrack, \bigl\lbrack f(1, 1, -2), f(1, 1, -1)\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack f(2, -1, -2), f(2, -1, -1)\bigr\rbrack, \bigl\lbrack f(2, 0, -2), f(2, 0, -1)\bigr\rbrack, \bigl\lbrack f(2, 1, -2), f(2, 1, -1)\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack f(3, -1, -2), f(3, -1, -1)\bigr\rbrack, \bigl\lbrack f(3, 0, -2), f(3, 0, -1)\bigr\rbrack, \bigl\lbrack f(3, 1, -2), f(3, 1, -1)\bigr\rbrack\bigr\rbrack\bigr\rbrack$

A fourth argument heads each level, here $g$ around rows and then the whole

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In$\mathrm{Array}(f, \bigl\lbrack3, 3\bigr\rbrack, 1, g)$Out$g(g(f(1, 1), f(1, 2), f(1, 3)), g(f(2, 1), f(2, 2), f(2, 3)), g(f(3, 1), f(3, 2), f(3, 3)))$

The Hilbert matrix with entries $\frac1{i+j-1}$

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In$\mathrm{Array}((\operatorname{\_1}, \operatorname{\_2})↦\frac{1}{\operatorname{\_1}+\operatorname{\_2}-1}, \bigl\lbrack3, 3\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, \frac{1}{2}, \frac{1}{3}\bigr\rbrack, \bigl\lbrack\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\bigr\rbrack, \bigl\lbrack\frac{1}{3}, \frac{1}{4}, \frac{1}{5}\bigr\rbrack\bigr\rbrack$

A constant function fills the grid

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In$\mathrm{Array}(()↦x, \bigl\lbrack5, 5\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack x, x, x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x, x\bigr\rbrack\bigr\rbrack$

Argument order is bound by the function's parameters, same Hilbert matrix

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In$\mathrm{Array}((k, j)↦\frac{1}{j+k-1}, \bigl\lbrack3, 3\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, \frac{1}{2}, \frac{1}{3}\bigr\rbrack, \bigl\lbrack\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\bigr\rbrack, \bigl\lbrack\frac{1}{3}, \frac{1}{4}, \frac{1}{5}\bigr\rbrack\bigr\rbrack$

Neat examples #

A table of powers $i^j$

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In$\mathrm{Array}(\mathrm{Power}, \bigl\lbrack5, 5\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 1, 1, 1, 1\bigr\rbrack, \bigl\lbrack2, 4, 8, 16, 32\bigr\rbrack, \bigl\lbrack3, 9, 27, 81, 243\bigr\rbrack, \bigl\lbrack4, 16, 64, 256, 1\,024\bigr\rbrack, \bigl\lbrack5, 25, 125, 625, 3\,125\bigr\rbrack\bigr\rbrack$

A triangular shape from rows of growing length

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In$\mathrm{Array}(\operatorname{\_1}↦\mathrm{Array}(()↦x, \operatorname{\_1}), \bigl\lbrack6\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack x\bigr\rbrack, \bigl\lbrack x, x\bigr\rbrack, \bigl\lbrack x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x, x, x\bigr\rbrack\bigr\rbrack$

About

Examples	20
Checked against	Wolfram Language 19 agree
Crosswalk	Wolfram Language

See also: Tabulate