Fast automatically differentiable matrix functions and applications in molecular simulations

We describe efficient differentiation methods for computing Jacobians and gradients of a large class of matrix functions including the matrix logarithm $\log \left(A\right)$ and p-th roots $A^{\frac{1}{p}}$. We exploit contour integrals and conformal maps as described by Hale et al. (2008) [34] for evaluation and differentiation and analyze the computational complexity as well as numerical accuracy compared to high accuracy finite difference methods. As a demonstrator application we compute properties of structural defects in silicon crystals at positive temperatures, requiring efficient and accurate gradients of matrix trace-logarithms.