Fast automatically differentiable matrix functions and applications in molecular simulations

Abstract

We describe efficient differentiation methods for computing Jacobians and gradients of a large class of matrix functions including the matrix logarithm $\log (A)$ and $p$-th roots $A^{\frac{1}{p}}$. We exploit contour integrals and conformal maps as described by (Hale et al., SIAM J. Numer. Anal. 2008) 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.