Deflation schemes for the generalized nonlinear eigenvalue problem

Abstract Deflation is an artifice to remove from a search subspace of interest already evaluated eigenpairs. It is a necessary step in solving structure vibration or even the complete simulation of time-dependent problems with a large number of degrees of freedom, which in our particular case is in the frame of a generalized modal analysis related to high-order nonlinear eigenvalue problems. We start with the linear algebra outline of the complex symmetric, nonlinear eigenvalue problem for structures submitted to viscous damping. However, we soon evolve to a more general development for complex, nonsymmetric, general matrices that may be expressed as closed-form analytical functions or as a power series of the eigenvalues -- as long as the matrices are simultaneously diagonalizable. After a short literature review of deflation techniques for linear and some restricted nonlinear cases, we propose a few families of deflation schemes applicable to problems that may not have mechanical meaning, thus in terms of sheer linear algebra. We show that a general deflation scheme works as penalty functions accrued to the original problem to prevent already obtained results from being re-evaluated. We illustratively assess the issues of combining the proposed deflation schemes with a generalized, inverse-free Krylov subspace iteration technique, as applied to nonlinear symmetric problems generated randomly. The numerical application to more general, nonsymmetric eigenvalue problems is also presented.