Outer

The generalized outer product: f applied to every pair drawn from two lists.

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

Outer(f, list1, list2)the array (descending to the atoms of nested lists) whose $(i, j)$ entry is $f$ applied to $list1$'s $i$-th and $list2$'s $j$-th element.

Typed signatures

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

Domain: Collections

Details

• $Outer(List, a, b)$ is the Cartesian product of $a$ and $b$, each pair wrapped in its own List.

Examples

1 more kept as data, checked against the oracles

Basic #

The outer product of two vectors, a $3 \times 2$ matrix

live ↗

In$\mathrm{Outer}(\mathrm{Multiply}, \bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack1, 2\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack, \bigl\lbrack3, 6\bigr\rbrack\bigr\rbrack$

Symbolic entries

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In$\mathrm{Outer}(\mathrm{Multiply}, \bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack x, y\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack2x, 2y\bigr\rbrack, \bigl\lbrack3x, 3y\bigr\rbrack\bigr\rbrack$

Every pair from the two lists, as a $2\times3$ table

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In$\mathrm{Outer}(f, \bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack f(a, x), f(a, y), f(a, z)\bigr\rbrack, \bigl\lbrack f(b, x), f(b, y), f(b, z)\bigr\rbrack\bigr\rbrack$

With Multiply this is the outer product, a multiplication table

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In$\mathrm{Outer}(\mathrm{Multiply}, \bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack a, b, c\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack2a, 2b, 2c\bigr\rbrack, \bigl\lbrack3a, 3b, 3c\bigr\rbrack, \bigl\lbrack4a, 4b, 4c\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{Outer}(\mathrm{Multiply}, \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack2a, 2b\bigr\rbrack, \bigl\lbrack2c, 2d\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3a, 3b\bigr\rbrack, \bigl\lbrack3c, 3d\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack4a, 4b\bigr\rbrack, \bigl\lbrack4c, 4d\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack$

Scope #

Any binary function: an addition table

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In$\mathrm{Outer}(\mathrm{Add}, \bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack10, 20\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack11, 21\bigr\rbrack, \bigl\lbrack12, 22\bigr\rbrack\bigr\rbrack$

A table of powers

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In$\mathrm{Outer}(\mathrm{Power}, \bigl\lbrack2, 3\bigr\rbrack, \bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack2, 4, 8\bigr\rbrack, \bigl\lbrack3, 9, 27\bigr\rbrack\bigr\rbrack$

$List$ gives every pair, like a Cartesian product

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In$\mathrm{Outer}(\mathrm{List}, \bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack x, y\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\bigl\lbrack a, x\bigr\rbrack, \bigl\lbrack a, y\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack b, x\bigr\rbrack, \bigl\lbrack b, y\bigr\rbrack\bigr\rbrack\bigr\rbrack$

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In$\mathrm{Outer}(\mathrm{Multiply}, \mathrm{s1}, \mathrm{s2})$Out$\mathrm{Outer}(\mathrm{Multiply}, \mathrm{s1}, \mathrm{s2})$

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In$\mathrm{Outer}(\mathrm{Multiply}, \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack d, k\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack d, k\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack2a, 2b, 2c\bigr\rbrack, \bigl\lbrack2d, 2k\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3a, 3b, 3c\bigr\rbrack, \bigl\lbrack3d, 3k\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack4a, 4b, 4c\bigr\rbrack, \bigl\lbrack4d, 4k\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack$

Properties #

The shape comes from the list lengths, even when all entries are equal

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In$\mathrm{Outer}(f, \bigl\lbrack x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack f(x, x), f(x, x), f(x, x), f(x, x)\bigr\rbrack, \bigl\lbrack f(x, x), f(x, x), f(x, x), f(x, x)\bigr\rbrack, \bigl\lbrack f(x, x), f(x, x), f(x, x), f(x, x)\bigr\rbrack\bigr\rbrack$

With List, every pairing is listed, nested by first list

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In$\mathrm{Outer}(\mathrm{List}, \bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack x, y\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\bigl\lbrack a, x\bigr\rbrack, \bigl\lbrack a, y\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack b, x\bigr\rbrack, \bigl\lbrack b, y\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack c, x\bigr\rbrack, \bigl\lbrack c, y\bigr\rbrack\bigr\rbrack\bigr\rbrack$

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In$\mathrm{Outer}(f, \mathrm{l1}, \mathrm{l2})$Out$\mathrm{Outer}(f, \mathrm{l1}, \mathrm{l2})$

Possible issues #

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In$\mathrm{Outer}(f, \mathrm{s1}, \mathrm{s2})$Out$\mathrm{Outer}(f, \mathrm{s1}, \mathrm{s2})$

About

Examples	15
Checked against	Wolfram Language 15 agree
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