Tensorial

Statically sized tensors and related operations for Julia

Tensorial provides useful tensor operations (e.g., contraction; tensor product, ⊗; inv; etc.) written in the Julia programming language. The library supports arbitrary size of non-symmetric and symmetric tensors, where symmetries should be specified to avoid wasteful duplicate computations. The way to give a size of the tensor is similar to StaticArrays.jl, and symmetries of tensors can be specified by using @Symmetry. For example, symmetric fourth-order tensor (symmetrizing tensor) is represented in this library as Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}}. Einstein summation macro and automatic differentiation functions are also provided.

Speed

a = rand(Vec{3})                         # vector of length 3
A = rand(SecondOrderTensor{3})           # 3x3 second order tensor
S = rand(SymmetricSecondOrderTensor{3})  # 3x3 symmetric second order tensor
B = rand(Tensor{Tuple{3,3,3}})           # 3x3x3 third order tensor
AA = rand(FourthOrderTensor{3})          # 3x3x3x3 fourth order tensor
SS = rand(SymmetricFourthOrderTensor{3}) # 3x3x3x3 symmetric fourth order tensor (symmetrizing tensor)

See here for above aliases.

The benchmarks are generated by runbenchmarks.jl on the following system:

julia> versioninfo()
Julia Version 1.6.3
Commit ae8452a9e0 (2021-09-23 17:34 UTC)
Platform Info:
  OS: macOS (x86_64-apple-darwin19.5.0)
  CPU: Intel(R) Core(TM) i7-7567U CPU @ 3.50GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, skylake)

Installation

pkg> add Tensorial

Cheat Sheet

# tensor aliases
rand(Vec{3})                        # vector
rand(Mat{2,3})                      # matrix
rand(SecondOrderTensor{3})          # 3x3 second-order tensor (this is the same as the Mat{3,3})
rand(SymmetricSecondOrderTensor{3}) # 3x3 symmetric second-order tensor (3x3 symmetric matrix)
rand(FourthOrderTensor{3})          # 3x3x3x3 fourth-order tensor
rand(SymmetricFourthOrderTensor{3}) # 3x3x3x3 symmetric fourth-order tensor

# identity tensors
one(SecondOrderTensor{3,3})        # second-order identity tensor
one(SymmetricSecondOrderTensor{3}) # symmetric second-order identity tensor
one(FourthOrderTensor{3})          # fourth-order identity tensor
one(SymmetricFourthOrderTensor{3}) # symmetric fourth-order identity tensor (symmetrizing tensor)

# zero tensors
zero(Mat{2,3}) == zeros(2,3)
zero(SymmetricSecondOrderTensor{3}) == zeros(3,3)

# random tensors
rand(Mat{2,3})
randn(Mat{2,3})

# macros (same interface as StaticArrays.jl)
@Vec [1,2,3]
@Vec rand(4)
@Mat [1 2
      3 4]
@Mat rand(4,4)
@Tensor rand(2,2,2)

# statically sized getindex by `@Tensor`
x = @Mat [1 2
          3 4
          5 6]
@Tensor(x[2:3, :])   === @Mat [3 4
                               5 6]
@Tensor(x[[1,3], :]) === @Mat [1 2
                               5 6]

# contraction and tensor product
x = rand(Mat{2,2})
y = rand(SymmetricSecondOrderTensor{2})
x ⊗ y isa Tensor{Tuple{2,2,@Symmetry{2,2}}} # tensor product
x ⋅ y isa Tensor{Tuple{2,2}}                # single contraction (x_ij * y_jk)
x ⊡ y isa Real                              # double contraction (x_ij * y_ij)

# det/inv for 2nd-order tensor
A = rand(SecondOrderTensor{3})          # equal to one(Tensor{Tuple{3,3}})
S = rand(SymmetricSecondOrderTensor{3}) # equal to one(Tensor{Tuple{@Symmetry{3,3}}})
det(A); det(S)
inv(A) ⋅ A ≈ one(A)
inv(S) ⋅ S ≈ one(S)

# inv for 4th-order tensor
AA = rand(FourthOrderTensor{3})          # equal to one(Tensor{Tuple{3,3,3,3}})
SS = rand(SymmetricFourthOrderTensor{3}) # equal to one(Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}})
inv(AA) ⊡ AA ≈ one(AA)
inv(SS) ⊡ SS ≈ one(SS)

# Einstein summation convention
A = rand(Mat{3,3})
B = rand(Mat{3,3})
(@einsum (i,j) -> A[i,k] * B[k,j]) == A ⋅ B
(@einsum A[i,j] * B[i,j]) == A ⊡ B

# Automatic differentiation
gradient(tr, rand(Mat{3,3})) == one(Mat{3,3}) # Tensor -> Real
gradient(identity, rand(SymmetricSecondOrderTensor{3})) == one(SymmetricFourthOrderTensor{3}) # Tensor -> Tensor

Other tensor packages

• Einsum.jl
• TensorOprations.jl
• Tensors.jl

Inspiration

• StaticArrays.jl
• Tensors.jl