Automatic differentiation of the vector that solves a parametric linear system

AbstractWe consider a parametric linear system a(s)·x(s) = b(s) where a(s) is a regular matrix, b(s) is a vector, and s is a p-dimensional parameter. This equation represents an implicit definition of a function x. It is shown that automatic differentiation techniques can be used to compute derivatives of x for given parameter-value s. Especially we aim at the Jacobian matrix and the Hessian tensor of x. These quantities are of particular interest in sensitivity analysis and structural optimization