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Outer
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Outer

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Outer[f,list1,list2,…]

gives the generalized outer product of the listi, forming all possible combinations of the lowest‐level elements in each of them, and feeding them as arguments to f.

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Outer[f,list1,list2,…,n]

treats as separate elements only sublists at level n in the listi.

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Outer[f,list1,list2,…,n1,n2,…]

treats as separate elements only sublists at level ni in the corresponding listi.

Details

  • Outer[Times,list1,list2] gives an outer product.
  • The result of applying Outer to the tensors Ti1i2...ir and Uj1j2...js is the tensor Vi1i2...irj1j2...js with elements f[Ti1i2...ir,Uj1j2...js]. Applying Outer to two tensors of ranks r and s gives a tensor of rank r+s.
  • The heads of all listi must be the same, but need not necessarily be List. »
  • Outer[f] returns f[].
  • The listi need not necessarily be cuboidal arrays.
  • The specifications ni of levels must be positive integers, or Infinity.
  • If only a single level specification is given, it is assumed to apply to all the listi. If there are several ni, but fewer than the number of listi, the lowest‐level elements in the remaining listi will be used.
  • Outer can be used on SparseArray objects, returning a SparseArray object when possible. »

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-t7q
Out[1]=1

Outer product of vectors:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-dpv
Out[1]=1

Outer product of matrices:

In[2]:=2
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https://wolfram.com/xid/0ldlxz2-gn
Out[2]=2

Scope  (4)Survey of the scope of standard use cases

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-ly1
Out[1]=1

Treat nested lists as rank-1 vectors of sublists:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-wa
Out[1]=1

Arrays can be ragged:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-r48
Out[1]=1

Outer product of SparseArray objects:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-il4xj
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-gawx3t
Out[2]=2

Generalizations & Extensions  (1)Generalized and extended use cases

The head need not be List:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-pdv
Out[1]=1

Applications  (6)Sample problems that can be solved with this function

Word combinations:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-jzg
Out[1]=1

Function combinations:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-ekkchr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-erd9i
Out[2]=2

Complete bipartite graph:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-xt6
Out[1]=1

Lower-triangular matrix:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-yw1

Generate all possible binary trees with nodes from f and leaves from e to depth n:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-ipc
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-pij
Out[2]=2

Apply a function on a tensor product grid:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-jhp8zp
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-c6uxz4
Out[2]=2

Show a contour plot of the values and the grid:

In[3]:=3
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https://wolfram.com/xid/0ldlxz2-dekt0g
In[4]:=4
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https://wolfram.com/xid/0ldlxz2-jtpi58
Out[4]=4

Include coordinates:

In[5]:=5
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https://wolfram.com/xid/0ldlxz2-bst7pp

Make a piecewise polynomial that interpolates the data:

In[6]:=6
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https://wolfram.com/xid/0ldlxz2-caqvue
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0ldlxz2-fc0uj
Out[7]=7

Properties & Relations  (6)Properties of the function, and connections to other functions

The dimensions of the result are a concatenation of the dimensions of the inputs:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-juj
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-xzg
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ldlxz2-ynt
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ldlxz2-o1t
Out[4]=4

Outer[f] returns f[]:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-jt9pb7
Out[1]=1

Distribute forms the same combinations of all elements, but in a flat structure:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-p03ygy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-y35o87
Out[2]=2

KroneckerProduct is a flattened outer product of matrices:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-f4y13j
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-dtxazh
In[3]:=3
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https://wolfram.com/xid/0ldlxz2-djnzti
In[4]:=4
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https://wolfram.com/xid/0ldlxz2-cz1hki
Out[4]=4

Part effectively uses an outer product when given lists of parts at multiple levels:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-d20615
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-b4v3p4
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ldlxz2-fcm3gh
Out[3]=3

Table can also make a generalized outer product from lists:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-jaocql
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-u4wq9
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ldlxz2-74m5l
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

If backgrounds are inconsistent, a generalized outer product may not be sparse:

In[1]:=1
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https://wolfram.com/xid/0ldlxz2-bxkerf
In[2]:=2
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https://wolfram.com/xid/0ldlxz2-di58dx
Out[2]=2

You can convert it into a SparseArray by choosing a background:

In[3]:=3
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https://wolfram.com/xid/0ldlxz2-f7j01
Out[3]=3
Wolfram Research (1988), Outer, Wolfram Language function, https://reference.wolfram.com/language/ref/Outer.html (updated 2020).
✖
Wolfram Research (1988), Outer, Wolfram Language function, https://reference.wolfram.com/language/ref/Outer.html (updated 2020).

Text

Wolfram Research (1988), Outer, Wolfram Language function, https://reference.wolfram.com/language/ref/Outer.html (updated 2020).

✖
Wolfram Research (1988), Outer, Wolfram Language function, https://reference.wolfram.com/language/ref/Outer.html (updated 2020).

CMS

Wolfram Language. 1988. "Outer." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/Outer.html.

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Wolfram Language. 1988. "Outer." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/Outer.html.

APA

Wolfram Language. (1988). Outer. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Outer.html

✖
Wolfram Language. (1988). Outer. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Outer.html

BibTeX

@misc{reference.wolfram_2025_outer, author="Wolfram Research", title="{Outer}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/Outer.html}", note=[Accessed: 11-May-2025 ]}

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@misc{reference.wolfram_2025_outer, author="Wolfram Research", title="{Outer}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/Outer.html}", note=[Accessed: 11-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_outer, organization={Wolfram Research}, title={Outer}, year={2020}, url={https://reference.wolfram.com/language/ref/Outer.html}, note=[Accessed: 11-May-2025 ]}

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@online{reference.wolfram_2025_outer, organization={Wolfram Research}, title={Outer}, year={2020}, url={https://reference.wolfram.com/language/ref/Outer.html}, note=[Accessed: 11-May-2025 ]}