Eigenvalue problem derivatives computation for a complex matrix using the adjoint method

Abstract

Derivatives of eigenvalues and eigenvectors with respect to design variables are required for gradient-based optimization in many engineering design problems. However, for the generalized and standard eigenvalue problems with general complex and non-Hermitian coefficient matrices, no method can accurately compute the eigenvalue and eigenvector derivatives while remaining efficient for large numbers of design variables. In this paper, we develop an adjoint method to compute complex eigenvalue and eigenvector derivatives with machine precision. For the special case when only the eigenvalue derivative is required, we propose a reverse algorithmic differentiation (RAD) formula using a newly developed dot product identity for complex functions. We verify the proposed method against the finite differences (FD) for a simple algebraic example with a 3-by-3 complex non-Hermitian matrix and a plane Poiseulle flow stability problem that is modeled as a generalized eigenvalue problem. The adjoint method is demonstrated to scale well with the number of design variables, matching the FD reference to about 5 to 7 digits.