Automatic differentiation

Warning

The user must provide the appropriate tensor symmetry information; otherwise, automatic differentiation may return unexpected values. In the following example, even with identical tensor values, the results vary depending on the Tensor type.

julia> A = rand(Mat{3,3})3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425
julia> S = A * A' # `S` is symmetric but not of the type `SymmetricSecondOrderTensor`3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 1.81438   1.253    0.894897
 1.253     1.05904  0.65243
 0.894897  0.65243  0.4475
julia> gradient(identity, S) ≈ one(FourthOrderTensor{3})true
julia> gradient(identity, symmetric(S)) ≈ one(SymmetricFourthOrderTensor{3})true

Tensorial.gradient — Function

gradient(f, x)
gradient(f, x, :all)

Compute the gradient of f with respect to x by the automatic differentiation. If pseudo keyword :all is given, the value of f(x) is also returned.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 0.325977  0.894245  0.953125
 0.549051  0.353112  0.795547
 0.218587  0.394255  0.49425

julia> gradient(tr, x)
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

julia> ∇f, f = gradient(tr, x, :all)
([1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0], 1.1733382401532275)

source

Tensorial.hessian — Function

hessian(f, x)
hessian(f, x, :all)

Compute the hessian of f with respect to x by the automatic differentiation. If pseudo keyword :all is given, the value of f(x) is also returned.

Examples

julia> x = rand(Vec{3})
3-element Vec{3, Float64}:
 0.32597672886359486
 0.5490511363155669
 0.21858665481883066

julia> hessian(norm, x)
3×3 Tensor{Tuple{3, 3}, Float64, 2, 9}:
  1.13603   -0.582196  -0.231782
 -0.582196   0.501079  -0.390397
 -0.231782  -0.390397   1.32626

julia> ∇∇f, ∇f, f = hessian(norm, x, :all)
([1.1360324375454411 -0.5821964220304534 -0.23178236037013888; -0.5821964220304533 0.5010791569244991 -0.39039709608344814; -0.23178236037013886 -0.39039709608344814 1.3262640626479867], [0.4829957515506539, 0.8135223859352438, 0.3238771859304809], 0.6749059962060727)

source