# Speeding up sums involving 16x16 matrices and 16x16x16x16 antisymmetric tensor

I need to perform the following contraction involving four 16x16 matrices and a tensor $$\theta$$ that is 16x16x16x16.

T4 = Table[Simplify[Sum[VL[[m, m1]] VL[[n, n1]] VL[[p, p1]] VL[[q, q1]] θ4[[m1, n1,p1, q1]], {m1, 16}, {n1, 16}, {p1, 16}, {q1,
16}]], {m, 16}, {n,16}, {p, 16}, {q, 16}]


where \theta and VL are given as follows.

θ4=SparseArray[{{15, 3, 4, 5} -> g1, {15, 3, 5, 4} -> -g1, {15,4, 3, 5} -> -g1, {15, 4, 5, 3} -> g1, {15, 5, 3, 4} -> g1, {15, 5, 4, 3} -> -g1, {16, 6, 7, 8} -> g2, {16, 6, 8, 7} -> -g2, {16, 7, 6, 8} -> -g2, {16, 7, 8, 6} ->g2, {16, 8, 6, 7} -> g2, {16, 8, 7, 6} -> -g2, {15, 9, 10, 11} -> gt1, {15, 9, 11, 10} -> -gt1, {15, 10, 9, 11} -> -gt1, {15, 10, 11, 9} -> gt1, {15, 11, 9, 10} -> gt1, {15, 11, 10, 9} -> -gt1, {16, 12, 13, 14} -> gt2, {16, 12, 14, 13} -> -gt2, {16, 13, 12, 14} -> -gt2, {16, 13, 14, 12} -> gt2, {16, 14, 12, 13} -> gt2, {16, 14, 13, 12} -> -gt2},{16,16,16,16}];

VL={{Cosh[τ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -Sinh[τ], 0}, {0, Cosh[ϕ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -Sinh[ϕ]}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0,  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,  0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {-Sinh[τ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Cosh[τ], 0}, {0, -Sinh[ϕ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Cosh[ϕ]}};


This should be a simple operation, but it takes ages (tens of minutes) and I have to abort the operation every single time. I don't know whether the difficulty is due to the large size of the matrices (but this shouldn't be since the matrices are sparse in entries). Thanks in advance for any possible help !

• An explicit loop using Sum[] or similar constructs is not efficient. Use Dot[ ] instead for contractions. And SparseArray[] representation eventually. – Michael Weyrauch Nov 21 '19 at 5:45

You can use TensorContract instead of Sum:

r = Activate @ TensorContract[
Inactive[TensorProduct][VL, VL, VL, VL, θ4],
{{2, 9}, {4, 10}, {6, 11}, {8, 12}}
]; //AbsoluteTiming


{0.419766, Null}

Using Activate with Inactive is a tip from @jose.

• Many thanks. That works out great ! – user195583 Nov 22 '19 at 4:33

Using Dot[] appears to be about a factor of 3 faster than Carl Wolls solution (although I doubt that the timing for such short times is precise)

   vlT = Transpose[VL]
r1 = Flatten[
VL.Flatten[
VL.\[Theta]4.vlT, {{2}, {1}, {4}, {3}}].vlT, {{2}, {1}, {4}, {3}}];


The timings on my computer are 0.000207 for the Dot[] solution and 0.0006595 for Carl's. Both results agree. (I converted VL to a SparseArray[] as well.)

• At least for numerical tensors, using Transpose[#, {2, 1, 4, 3}] & instead of Flatten[#, {{2}, {1}, {4}, {3}}] &  might be a bit faster. – Henrik Schumacher Nov 21 '19 at 9:43
• Interesting. I thought that they internally both use the same code. I will try it out – Michael Weyrauch Nov 21 '19 at 10:50