Constructing tensors

Tensors can be created in multiple ways but they usually include running a function on tensor types of which there are two kinds, Tensor{order, dim, T} for non-symmetric tensors and SymmetricTensor{order, dim, T} for symmetric tensors. The parameter order is an integer of value 1, 2 or 4, excluding 1 for symmetric tensors. The second parameter dim is an integer which corresponds to the dimension of the tensor and can be 1, 2 or 3. The last parameter T is the number type that the tensors contain, i.e. Float64 or Float32.

Tensors.Tensor — Type

Tensor{order,dim,T<:Real}

Tensor type supported for order ∈ (1,2,4) and dim ∈ (1,2,3).

Examples

julia> Tensor{1,3,Float64}((1.0, 2.0, 3.0))
3-element Tensor{1,3,Float64,3}:
 1.0
 2.0
 3.0

Tensors.SymmetricTensor — Type

SymmetricTensor{order,dim,T<:Real}

Symmetric tensor type supported for order ∈ (2,4) and dim ∈ (1,2,3). SymmetricTensor{4} is a minor symmetric tensor, such that A[i,j,k,l] == A[j,i,k,l] and A[i,j,k,l] == A[i,j,l,k].

Examples

julia> SymmetricTensor{2,2,Float64}((1.0, 2.0, 3.0))
2×2 SymmetricTensor{2,2,Float64,3}:
 1.0  2.0
 2.0  3.0

Zero tensors

A tensor with only zeros is created using the function zero, applied to the type of tensor that should be created:

julia> zero(Tensor{1, 2})
2-element Tensor{1,2,Float64,2}:
 0.0
 0.0

By default, a tensor of Float64s is created, but by explicitly giving the T parameter this can be changed:

julia> zero(SymmetricTensor{4, 2, Float32})
2×2×2×2 SymmetricTensor{4,2,Float32,9}:
[:, :, 1, 1] =
 0.0  0.0
 0.0  0.0

[:, :, 2, 1] =
 0.0  0.0
 0.0  0.0

[:, :, 1, 2] =
 0.0  0.0
 0.0  0.0

[:, :, 2, 2] =
 0.0  0.0
 0.0  0.0

A Julia Array with zeroed tensors can be created with zeros, with the tensor type and dimensions of the array as arguments:

julia> zeros(Tensor{2,2}, 2, 3)
2×3 Array{Tensor{2,2,Float64,4},2}:
 [0.0 0.0; 0.0 0.0]  [0.0 0.0; 0.0 0.0]  [0.0 0.0; 0.0 0.0]
 [0.0 0.0; 0.0 0.0]  [0.0 0.0; 0.0 0.0]  [0.0 0.0; 0.0 0.0]

Constant tensors

A tensor filled with ones is created using the function ones, applied to the type of tensor that should be created:

julia> ones(Tensor{2,2})
2×2 Tensor{2,2,Float64,4}:
 1.0  1.0
 1.0  1.0

By default, a tensor of Float64s is created, but by explicitly giving the T parameter this can be changed, like for zero.

Note

The function ones has double meaning: it can create a tensor filled with ones (as described above) or create a Julia Array with identity tensors. Thus, to create an Array with tensors filled with ones, instead use array comprehension:

julia> [ones(Tensor{2,2}) for i in 1:2, j in 1:3]
2×3 Array{Tensor{2,2,Float64,4},2}:
 [1.0 1.0; 1.0 1.0]  [1.0 1.0; 1.0 1.0]  [1.0 1.0; 1.0 1.0]
 [1.0 1.0; 1.0 1.0]  [1.0 1.0; 1.0 1.0]  [1.0 1.0; 1.0 1.0]

Random tensors

A tensor with random numbers is created using the function rand, applied to the type of tensor that should be created:

julia> rand(Tensor{2, 3})
3×3 Tensor{2,3,Float64,9}:
 0.590845  0.460085  0.200586
 0.766797  0.794026  0.298614
 0.566237  0.854147  0.246837

By specifying the type, T, a tensor of different type can be obtained:

julia> rand(SymmetricTensor{2,3,Float32})
3×3 SymmetricTensor{2,3,Float32,6}:
 0.0107703  0.305865  0.2082
 0.305865   0.405684  0.257278
 0.2082     0.257278  0.958491

Identity tensors

Identity tensors can be created for orders 2 and 4. The components of the second order identity tensor $\mathbf{I}$ are defined as $I_{ij} = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. The fourth order identity tensor $\mathsf{I}$ is the resulting tensor from taking the derivative of a second order tensor $\mathbf{A}$ with itself:

\[\mathsf{I} = \frac{\partial \mathbf{A}}{\partial \mathbf{A}} \Leftrightarrow I_{ijkl} = \frac{\partial A_{ij}}{\partial A_{kl}} = \delta_{ik} \delta_{jl}.\]

The symmetric fourth order tensor, $\mathsf{I}^\text{sym}$, is the resulting tensor from taking the derivative of a symmetric second order tensor $\mathbf{A}^\text{sym}$ with itself:

\[\mathsf{I}^\text{sym} = \frac{\partial \mathbf{A}^\text{sym}}{\partial \mathbf{A}^\text{sym}} \Leftrightarrow I^\text{sym}_{ijkl} = \frac{\partial A^\text{sym}_{ij}}{\partial A^\text{sym}_{kl}} = \frac{1}{2} (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}).\]

Identity tensors are created using the function one, applied to the type of tensor that should be created:

julia> one(SymmetricTensor{2, 2})
2×2 SymmetricTensor{2,2,Float64,3}:
 1.0  0.0
 0.0  1.0

A Julia Array with identity tensors can be created with ones, with the tensor type and dimensions of the array as arguments:

julia> ones(Tensor{2,2}, 2, 2)
2×2 Array{Tensor{2,2,Float64,4},2}:
 [1.0 0.0; 0.0 1.0]  [1.0 0.0; 0.0 1.0]
 [1.0 0.0; 0.0 1.0]  [1.0 0.0; 0.0 1.0]

Base vectors

Tensors.basevec — Function

basevec(::Type{Vec{dim, T}})
basevec(::Type{Vec{dim, T}}, i)
basevec(::Vec{dim, T})
basevec(::Vec{dim, T}, i)

Return a tuple with the base vectors corresponding to the dimension dim and type T. An optional integer i can be used to extract the i:th base vector. The alias eᵢ can also be used, written e\_i<TAB>.

Examples

julia> eᵢ(Vec{2, Float64})
([1.0, 0.0], [0.0, 1.0])

julia> eᵢ(Vec{2, Float64}, 2)
2-element Tensor{1,2,Float64,2}:
 0.0
 1.0

From arrays / tuples

Tensors can also be created from a tuple or an array with the same number of elements as the number of independent indices in the tensor. For example, a first order tensor (vector) in two dimensions is here created from a vector of length two:

julia> Tensor{1,2}([1.0,2.0])
2-element Tensor{1,2,Float64,2}:
 1.0
 2.0

Below, a second order symmetric tensor in two dimensions is created from a tuple. Since the number of independent indices in this tensor is three, the length of the tuple is also three. For symmetric tensors, the order of the numbers in the input tuple is column by column, starting at the diagonal.

julia> SymmetricTensor{2,2}((1.0,2.0,3.0))
2×2 SymmetricTensor{2,2,Float64,3}:
 1.0  2.0
 2.0  3.0

From a function

A tensor can be created from a function f(indices...) -> v which maps a set of indices to a value. The number of arguments of the function should be equal to the order of the tensor.

julia> SymmetricTensor{2,2,Float64}((i,j) -> i + j)
2×2 SymmetricTensor{2,2,Float64,3}:
 2.0  3.0
 3.0  4.0

For symmetric tensors, the function is only called for the lower triangular part.

Diagonal tensors

A diagonal second order tensor can be created by either giving a number or a vector that should appear on the diagonal:

julia> diagm(Tensor{2,2}, 2.0)
2×2 Tensor{2,2,Float64,4}:
 2.0  0.0
 0.0  2.0

julia> diagm(SymmetricTensor{2,3}, [1.0, 2.0, 3.0])
3×3 SymmetricTensor{2,3,Float64,6}:
 1.0  0.0  0.0
 0.0  2.0  0.0
 0.0  0.0  3.0

Converting to tensors

Sometimes it is necessary to convert between standard Julia Array's and Tensor's. When the number type is a bits type (like for floats or integers) this is conveniently done by the reinterpret function. For example, a 2×5 Julia Array can be translated to a vector of Vec{2} with the following code

julia> data = rand(2, 5)
2×5 Array{Float64,2}:
 0.590845  0.566237  0.794026  0.200586  0.246837
 0.766797  0.460085  0.854147  0.298614  0.579672

julia> tensor_data = reinterpret(Vec{2, Float64}, vec(data))
5-element reinterpret(Tensor{1,2,Float64,2}, ::Array{Float64,1}):
 [0.5908446386657102, 0.7667970365022592]
 [0.5662374165061859, 0.4600853424625171]
 [0.7940257103317943, 0.8541465903790502]
 [0.20058603493384108, 0.2986142783434118]
 [0.24683718661000897, 0.5796722333690416]

The data can also be reinterpreted back to a Julia Array

julia> data = reshape(reinterpret(Float64, tensor_data), (2, 5))
2×5 reshape(reinterpret(Float64, reinterpret(Tensor{1,2,Float64,2}, ::Array{Float64,1})), 2, 5) with eltype Float64:
 0.590845  0.566237  0.794026  0.200586  0.246837
 0.766797  0.460085  0.854147  0.298614  0.579672