Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Wolfram`TensorNetworks`
EinsteinSummation  |
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accepts tensors of any kind, dense or sparse, including symbolic tensors.
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returns an expression with and left (and possibly a final ).
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Use (or ) to evaluate the inactive expression to a numerical or symbolic tensor.
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Indices may be any expressions. When the index spec is a string it is split on "," and "->" and each character of an input segment becomes one index, so the string form is convenient for single-character indices only; use the list form for multi-character or symbolic indices.
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Repeated indices within a single tensor are contracted first via before the cross-tensor is applied.
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Tensors whose index list is the empty list are treated as scalars and factored out as a multiplicative prefactor.
{}
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When an output index occurs more times than it appears in the input, the result is broadcast via repeated .
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issues , , or messages for index-spec/array-count mismatches, index-list/tensor-rank mismatches, dimension mismatches on contracted indices, or unreachable output indices.
EinsteinSummation::length
EinsteinSummation::shape
EinsteinSummation::dim
EinsteinSummation::output
Examples
(47)
Basic Examples
(9)
Multiply two matrices via the einsum-string form, then activate to a numerical result:
In[1]:=
{Amat,Bmat}={RandomReal[{-1,1},{2,3}],RandomReal[{-1,1},{3,4}]};
["ij,jk->ik",{Amat,Bmat}]
 | EinsteinSummation |
Out[1]=
TensorContract[{{0.14461,0.733859,0.557848},{0.806258,-0.231906,-0.102523}}{{-0.16682,-0.187316,0.252484,-0.720577},{0.986553,-0.370761,-0.760649,-0.58844},{0.595025,-0.968575,-0.494359,0.0681437}},{{2,3}}]
In[2]:=
 | ActivateTensors |
| EinsteinSummation |
Out[2]=
{{1.0318,-0.839492,-0.797474,-0.498021},{-0.424292,0.0342575,0.430649,-0.451495}}
Take the trace of a square matrix using a repeated index:
In[1]:=
Msq=RandomReal[{-1,1},{4,4}];
In[2]:=
| ActivateTensors |
| EinsteinSummation |
Out[2]=
1.05388
Form the outer product of two vectors:
In[1]:=
| ActivateTensors |
| EinsteinSummation |
Out[1]=
{{4,5},{8,10},{12,15}}
Define two random complex-valued tensors:
In[1]:=
A=SparseArray@RandomComplex[{-1-I,1+I},{2,3,2}];B=SparseArray@RandomComplex[{-1-I,1+I},{3,2}];
Execute a summation as
∑
j
A
ijk
B
jl
In[2]:=
| EinsteinSummation |
Out[2]=
TensorContractSparseArray
SparseArray
,{{2,4}}
 |  |
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Activate the outcome and get the final tensor:
In[3]:=
einsum=Activate@
[{{1,2,3},{2,4}},{A,B}]
 | EinsteinSummation |
Out[3]=
SparseArray
| |
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Check the result is the same as running summation manually:
In[4]:=
Table[Sum[A〚i,j,k〛B〚j,l〛,{j,3}],{i,2},{k,2},{l,2}]einsum
Out[4]=
True
Use string notation for compact expressions:
In[5]:=
Activate@
["ijk,jl",{A,B}]einsum
| EinsteinSummation |
Out[5]=
True
Use string notation for compact expressions:
In[1]:=
A=SparseArray@RandomComplex[{-1-I,1+I},{2,3,2}];B=SparseArray@RandomComplex[{-1-I,1+I},{3,2}];Activate@
["ijk,jl->lk",{A,B}]Activate@
[{{i,j,k},{j,l}}{l,k},{A,B}]
| EinsteinSummation |
| EinsteinSummation |
Out[1]=
True
Define two symbolic arrays of 3-rank and 2-rank with dimension and :
{,,}
n
1
n
2
n
3
{,}
n
2
n
4
In[1]:=
A=ArraySymbol["A",{n1,n2,n3}];B=ArraySymbol["B",{n2,n4}];
Execute a summation as
∑
j
A
ijk
B
jl
In[2]:=
| EinsteinSummation |
Out[2]=
TensorContract,{{2,4}}
3
A
2
B
Define two random complex-valued tensors:
In[1]:=
A=SparseArray@RandomComplex[{-1-I,1+I},{2,3}];B=SparseArray@RandomComplex[{-1-I,1+I},{2,3}];
Calculate the inner product of two tensors as
〈A,B〉=Tr[SuperDagger[A]B]=
∑
ij
*
A
ij
B
ij
In[2]:=
einsum=
@
[{{1,2},{1,2}},{Conjugate@A,B}]
| ActivateTensors |
| EinsteinSummation |
Out[2]=
-0.474878-0.0256801
Compare the result with the trace:
In[3]:=
einsumTr[ConjugateTranspose[A].B]
Out[3]=
True
Calculate the cross product via Levi-Civita =
(a×b)
i
∑
jk
ϵ
ijk
a
j
b
k
In[1]:=
{a,b}=RandomReal[{-1,1},{2,3}]
Out[1]=
{{-0.395861,0.032476,0.326959},{-0.36132,-0.175434,-0.964443}}
Excuse the cross product using EinsteinSummation:
In[2]:=
einsum=
@
[{{i,j,k},{j},{k}},{LeviCivitaTensor[3],a,b}]
| ActivateTensors |
| EinsteinSummation |
Out[2]=
{0.0260384,-0.499922,0.0811818}
Compare the result with manually summing over all indices:
In[3]:=
Table[Sum[LeviCivitaTensor[3]〚i,j,k〛a〚j〛b〚k〛,{j,3},{k,3}],{i,3}]einsum
Out[3]=
True
Double trace
∑
ij
T
ij,ij
In[1]:=
| EinsteinSummation |
Out[1]=
TensorContract,{{1,3},{2,4}}
4
T
Scope
(16)
Applications
(18)