Higher-order derivatives of the QR and of the real symmetric eigenvalue decomposition in forward and reverse mode algorithmic differentiation
It provides a systematic method for derivative computation in numerical programs involving these matrix decompositions, which is useful for optimization and sensitivity analysis in scientific computing.
This work presents algorithms for computing higher-order derivatives of QR and real symmetric eigenvalue decompositions using forward and reverse mode algorithmic differentiation, combining Taylor polynomial arithmetic and matrix calculus.
We address the task of higher-order derivative evaluation of computer programs that contain QR decompositions and real symmetric eigenvalue decompositions. The approach is a combination of univariate Taylor polynomial arithmetic and matrix calculus in the (combined) forward/reverse mode of Algorithmic Differentiation (AD). Explicit algorithms are derived and presented in an accessible form. The approach is illustrated via examples.