MapThread

f applied across the CORRESPONDING elements of each list, not each list in turn.

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

MapThread(f, lists){f(a1, b1, …), f(a2, b2, …), …} for lists = {{a1, a2, …}, {b1, b2, …}, …}

Typed signatures

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

Domain: Collections

Details

• Every row of lists must be the same length; a ragged input is left unevaluated.
• The 3-argument Wolfram form (a level spec) isn't implemented — only the top level threads.

Examples

Basic #

live ↗

In$\mathrm{MapThread}(f, \bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack f(a, c), f(b, d)\bigr\rbrack$

Pairs up corresponding elements of the lists

live ↗

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

Scope #

sums two rows elementwise

live ↗

In$\mathrm{MapThread}(\mathrm{Add}, \bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack10, 20, 30\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack11, 22, 33\bigr\rbrack$

Corresponding elements here are whole pairs, taken at the top level only

live ↗

In$\mathrm{MapThread}(f, \bigl\lbrack\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack u, v\bigr\rbrack, \bigl\lbrack w, x\bigr\rbrack\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack f(\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack u, v\bigr\rbrack), f(\bigl\lbrack c, d\bigr\rbrack, \bigl\lbrack w, x\bigr\rbrack)\bigr\rbrack$

Any number of lists, so $f$ takes four arguments

live ↗

In$\mathrm{MapThread}(f, \bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 9\bigr\rbrack, \bigl\lbrack10, 11, 12\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack f(1, 4, 7, 10), f(2, 5, 8, 11), f(3, 6, 9, 12)\bigr\rbrack$

Applications #

live ↗

In$\mathrm{MapThread}(()↦\operatorname{\_1}(\operatorname{\_2}), \bigl\lbrack\bigl\lbrack f, g, h\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack f(x), g(y), h(z)\bigr\rbrack$

Properties #

With Add, columnwise sums of the rows

live ↗

In$\mathrm{MapThread}(\mathrm{Add}, \bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack u, v, w\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack a+u+x, b+v+y, c+w+z\bigr\rbrack$

With List this transposes a rectangular array

live ↗

In$\mathrm{MapThread}(\mathrm{List}, \bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack u, v, w\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack a, u, x\bigr\rbrack, \bigl\lbrack b, v, y\bigr\rbrack, \bigl\lbrack c, w, z\bigr\rbrack\bigr\rbrack$

Pairs up functions with variables: $\partial_x x = 1$, $\partial_y xy = x$, $\partial_z xz = x$

live ↗

In$\mathrm{MapThread}(D, \bigl\lbrack\bigl\lbrack x, xy, xz\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack1, x, x\bigr\rbrack$

A single list reduces to ordinary mapping

live ↗

In$\mathrm{MapThread}(f, \bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack f(a), f(b), f(c)\bigr\rbrack$

Possible issues #

rows of different lengths are left unevaluated

live ↗

In$\mathrm{MapThread}(f, \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3\bigr\rbrack\bigr\rbrack)$Out$\mathrm{MapThread}(f, \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3\bigr\rbrack\bigr\rbrack)$

About

Examples	11
Checked against	Wolfram Language 11 agree
Crosswalk	Wolfram Language

See also: MapIndexed, Through