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Inner

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Inner[f,list1,list2,g]
    is a generalization of Dot in which f plays the role of multiplication and g of addition.
Inner[f,list1,list2]
    uses Plus for g.
Inner[f,list1,list2,g,n]
    contracts index n of the first tensor with the first index of the second tensor.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Inner[f, {a, b}, {x, y}, g]
Out[1]= g[f[a, x], f[b, y]]

In[2]:= Inner[Times, {{a, b}, {c, d}}, {x, y}, Plus]
Out[2]= {a x + b y, c x + d y}

In[3]:= Inner[Times, {a1, a2, a3}, {b1, b2, b3}, Plus]
Out[3]= a1 b1 + a2 b2 + a3 b3

Implementation notes

Algorithm. builtin_inner is the generalisation of Dot that replaces the elementwise multiply with an arbitrary f and the summation with an arbitrary g: Inner[f, A, B, g] contracts the last index of A with the first index of B, combining matched leaves with f and reducing each contraction list with g (both default chains are built so that Inner[Times, A, B, Plus] reproduces Dot). g defaults to Plus when omitted. The contraction is performed recursively by the inner_A/inner_n1_A helpers (the n == 1 form contracts first-index-with-first-index directly), producing an unevaluated g[f[…], …] tree that is then run through evaluate. Non-List-structured operands, or mismatched contraction lengths, return NULL (leave unevaluated).

Data structures. Operands are walked as nested EXPR_FUNCTION trees keyed on A's head (the result head is taken from A); leaves are deep-copied into f-applications. No flat dense buffer is used — unlike Dot, Inner works on generic heads and arbitrary combiners.

  • Protected.
  • Like Dot, Inner effectively contracts the last index of the first tensor with the first index of the second tensor.
  • Applying Inner to a rank $r$ tensor and a rank $s$ tensor gives a rank $r+s-2$ tensor.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Inner[Times, {a, b}, {c, d}, Plus]
Out[1]= a c + b d

In[1]:= Inner[Times, {2, 3, 5}, {7, 11, 13}, Plus]
Out[1]= 112

In[1]:= Inner[f, {a, b}, {c, d}, g]
Out[1]= g[f[a, c], f[b, d]]

In[1]:= Inner[Times, {{1, 2}, {3, 4}}, {{5, 6}, {7, 8}}, Plus]
Out[1]= {{19, 22}, {43, 50}}

Notes

Inner[f, list1, list2, g] is the generalised dot product: f plays the role of elementwise multiplication and g the role of summation. With f = Times and g = Plus it reduces to ordinary Dot, so the matrix example above is just the matrix product. Supplying symbolic f and g exposes the contraction structure literally — g[f[a, c], f[b, d]] — which is useful for building custom tensor operations (max-plus algebra, fuzzy logic, polynomial convolutions, etc.). Inner contracts the last index of the first tensor with the first index of the second.