Contracting Symmetric Tensors Using Fewer Multiplications

摘要: We present more computationally-efficient algorithms for contracting symmetric tensors. Tensor contractions are reducible to matrix multiplication, but permutational symmetries of the data, which expressed by tensor representation, provide an opportunity efficient algorithms. Previously known methods have exploited only that yield identical computations directly evident in contraction expression. a new ‘symmetry preserving’ algorithm uses algebraic reorganization order exploit considerably symmetry computation than conventional approach. The requires fewer multiplications additions per multiplication previous approaches. applications this result include capability multiply vector, as well compute rank-2 vector outer product half number scalar multiplications, albeit with additions. preserving can also be adapted perform complex versions these operations, namely Hermitian and product, 3/4 overall operations. Consequently, operations needed direct eigenvalues is reduced same factor. Our antisymmetric case therefore applicable tensor-contraction employed quantum chemistry. For applications, notably coupled-cluster method, our yields highest potential speed-ups, since many higher-order reduction achieved enables equivalent cost. highlight three typical taken from different orders, achieves 2X, 4X, 9X improvements arithmetic cost over standard