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Tensor type

Type parameters

All tensors are represented by a type Tensor{S, T, N, L} where each type parameter represents following:

  • S: The size of Tensors which is specified by using Tuple (e.g., 3x2 tensor becomes Tensor{Tuple{3,2}}).
  • T: The type of element which must be T <: Real.
  • N: The number of dimensions (the order of tensor).
  • L: The number of independent components.

Basically, the type parameters N and L do not need to be specified for constructing tensors because it can be inferred from the size of tensor S.

Symmetry

If possible, specifying the symmetry of the tensor is good for performance since Tensorial.jl provides the optimal computations. The symmetries can be applied using Symmetry in type parameter S (e.g., Symmetry{Tuple{3,3}}). @Symmetry macro can omit Tuple like @Symmetry{2,2}. The following are examples to specify symmetries:

  • $A_{(ij)}$ with 3x3: Tensor{Tuple{@Symmetry{3,3}}}
  • $A_{(ij)k}$ with 3x3x2: Tensor{Tuple{@Symmetry{3,3}, 2}}
  • $A_{(ijk)}$ with 3x3x3: Tensor{Tuple{@Symmetry{3,3,3}}}
  • $A_{(ij)(kl)}$ with 3x3x3x3: Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}}

where the bracket $()$ in indices denotes the symmetry.