Thread

f applied elementwise across every list-headed argument, other arguments broadcast.

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Thread(f(a1, …, an))f threaded over every List-headed operand

Thread(f(a1, …, an), h)like Thread(f(...)), threading only over operands headed by h instead of List

Typed signatures

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

Domain: Collections

Details

• Operands not headed by h are broadcast unchanged to every threaded call.
• The operands are evaluated first, and a lazy ordered collection such as $Tuples(…)$ threads as the List it evaluates to. The threaded calls themselves are left as built, not evaluated.
• Left unevaluated when the h-headed operands don't all share one length.
• Many arithmetic heads already thread over lists automatically; Thread's value is threading a head that doesn't, such as Equal or a plain function.

Examples

Basic #

live ↗

In$\mathrm{Thread}(\bigl\lbrack1, 2, 3\bigr\rbrack=\bigl\lbrack1, 5, 3\bigr\rbrack)$Out$\bigl\lbrack\mathrm{True}, \mathrm{False}, \mathrm{True}\bigr\rbrack$

Turns an equation between lists into elementwise equations

live ↗

In$\mathrm{Thread}(\bigl\lbrack a, b, c\bigr\rbrack=\bigl\lbrack x, y, z\bigr\rbrack)$Out$\begin{cases}a=x\\b=y\\c=z\end{cases}$

Threading over Equal pushes the function inside both sides

live ↗

In$\mathrm{Thread}(\ln(x=y), \mathrm{Equal})$Out$\ln(x)=\ln(y)$

Scope #

A non-list operand is broadcast to every threaded call

live ↗

In$\mathrm{Thread}(\bigl\lbrack1, 2, 3\bigr\rbrack=1)$Out$\bigl\lbrack\mathrm{True}, \mathrm{False}, \mathrm{False}\bigr\rbrack$

A second argument threads over a head other than List

live ↗

In$\mathrm{Thread}(f(g(1, 2), g(3, 4)), g)$Out$g(f(1, 3), f(2, 4))$

Applications #

A scalar is paired with each element of the list

live ↗

In$\mathrm{Thread}(\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, 0\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack a, 0\bigr\rbrack, \bigl\lbrack b, 0\bigr\rbrack, \bigl\lbrack c, 0\bigr\rbrack\bigr\rbrack$

Rules between lists become a list of pairs

live ↗

In$\mathrm{Thread}(\mathrm{Rule}(\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack1, 2, 3\bigr\rbrack))$Out$\bigl\lbrack\mathrm{Rule}(a, 1), \mathrm{Rule}(b, 2), \mathrm{Rule}(c, 3)\bigr\rbrack$

Properties #

Lists of equal length pair up elementwise, like a transpose

live ↗

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

Possible issues #

live ↗

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

live ↗

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

About

Examples	10
Checked against	Wolfram Language 10 agree
Crosswalk	Wolfram Language

See also: MovingMap, Array