# Abstract

In matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost.
More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms.
We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions.
In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order

**Funding source: **National Science Foundation

**Award Identifier / Grant number: **ACI-1548562

**Funding statement: **This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562.
Via XSEDE, the authors made use of the TACC Stampede2 supercomputer.

# Acknowledgements

We would like to thank Devin Matthews, Toru Shiozaki, Hung Woei Neoh, and anonymous reviewers for helpful comments that served to improve this paper.

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**Received:**2019-04-28

**Revised:**2019-10-02

**Accepted:**2020-01-09

**Published Online:**2020-02-05

**Published in Print:**2021-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston