Numeric calculus of variation

Each Tensor has four attributes: the (real) tensor array and the (hyper-dual) variational arrays. To obtain the \(12\) - component of the gradient and the \(1223\) - component of the hessian, a tensor has to be created with the appropriate small-changes of the tensor components (dual arrays).

import numpy as np

import tensortrax as tr
from tensortrax import Tensor, Δ, Δδ, f, δ
from tensortrax.math import trace

δF_12 = np.array(
    [
        [0, 1, 0],
        [0, 0, 0],
        [0, 0, 0],
    ],
    dtype=float,
)

ΔF_23 = np.array(
    [
        [0, 0, 0],
        [0, 0, 1],
        [0, 0, 0],
    ],
    dtype=float,
)

x = np.eye(3) + np.arange(9).reshape(3, 3) / 10
F = Tensor(x=x, δx=δF_12, Δx=ΔF_23, Δδx=None)
I1_C = trace(F.T @ F)

The function as well as the gradient and hessian components are accessible with helpers.

ψ = f(I1_C)
P_12 = δ(I1_C)
A_1223 = Δδ(I1_C)

To obtain full gradients and hessians of scalar-valued functions in one function call, tensortrax provides helpers (decorators) which handle the multiple function calls.

fun = lambda F: trace(F.T @ F)

func = tr.function(fun)(x)
grad = tr.gradient(fun)(x)
hess = tr.hessian(fun)(x)

For tensor-valued functions, use jacobian() instead of gradient().

fun = lambda F: F.T @ F
jac = tr.jacobian(fun)(x)

Evaluate the gradient- as well as the hessian-vector(s)-product.

gvp = tr.gradient_vector_product(fun)(x, δx=x)
hvp = tr.hessian_vector_product(fun)(x, δx=x)
hvsp = tr.hessian_vectors_product(fun)(x, δx=x, Δx=x)