# Eigenvalues with absolute values close to 1

I have a generalized eigen-problem: $A\psi = \lambda B \psi$ with $A$ and $B$ are large (>1000) complex non-Hermitian matrices. I know that eigenvalues with largest and smallest absolute values can be found using iterative approaches like ARPACK. But I am only interested in those complex eigenvalues (not all of them) with absolute values closest to 1. It there any fast and parallel approach to do this?

• Are they dense or sparse matrices ? – percusse Nov 12 '16 at 9:43
• have you looked at the shift-invert mode of ARPACK? – GoHokies Nov 12 '16 at 19:04
• @GoHokies Yes, I know the shift-invert mode of ARPACK, but it is only useful for eigen values near some number. But here my question is about the eigen values with absolute values (not the value) around 1. – Yunpeng Wang Nov 13 '16 at 5:53
• @percusse Both the A and B matrices are dense. – Yunpeng Wang Nov 13 '16 at 5:54
• @YunpengWang shift-invert is used to find the eigenvalues nearest to the shift in absolute value. – GoHokies Nov 13 '16 at 8:49