## Tensorial.jl

Statically sized tensors and related operations for Julia
Author KeitaNakamura
Popularity
25 Stars
Updated Last
6 Days Ago
Started In
January 2021

# Tensorial

Statically sized tensors and related operations for Julia

Tensorial provides useful tensor operations, such as contraction, tensor product (`⊗`), and inversion (`inv`), implemented in the Julia programming language. The library supports tensors of arbitrary size, including both symmetric and non-symmetric tensors, where symmetries can be specified to avoid redundant computations. The approach for defining the size of a tensor is similar to that used in StaticArrays.jl, and tensor symmetries can be specified using the `@Symmetry` macro. For instance, a symmetric fourth-order tensor (a symmetrized tensor) is represented in this library as `Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}}`. The library also includes an Einstein summation macro `@einsum` and functions for automatic differentiation, such as `gradient` and `hessian`.

## 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.

Operation `Tensor` `Array` speed-up
Single contraction
`a ⋅ a` 1.537 ns 16.943 ns ×11.0
`A ⋅ a` 1.538 ns 80.647 ns ×52.4
`S ⋅ a` 1.545 ns 79.630 ns ×51.5
Double contraction
`A ⊡ A` 2.730 ns 17.909 ns ×6.6
`S ⊡ S` 1.704 ns 19.099 ns ×11.2
`B ⊡ A` 4.886 ns 183.789 ns ×37.6
`AA ⊡ A` 7.035 ns 193.607 ns ×27.5
`SS ⊡ S` 3.589 ns 192.727 ns ×53.7
Tensor product
`a ⊗ a` 2.035 ns 53.872 ns ×26.5
Cross product
`a × a` 2.035 ns 53.872 ns ×26.5
Determinant
`det(A)` 1.541 ns 223.537 ns ×145.1
`det(S)` 1.542 ns 227.196 ns ×147.3
Inverse
`inv(A)` 5.432 ns 544.122 ns ×100.2
`inv(S)` 3.872 ns 552.627 ns ×142.7
`inv(AA)` 854.691 ns 1.727 μs ×2.0
`inv(SS)` 310.218 ns 1.728 μs ×5.6

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```