The generalized eigenproblem: pole-zero computation

A modification-decomposition (MD) method is used to compute linear system transfer function poles and zeros by transforming an N-dimensional generalized eigenvalue problem to an M-dimensional standard eigenvalue problem with M⩾ r, where r is the lesser of the ranks of the dynamic or nondynamic component matrix of the system. Hence, network eigenvalue problems normally solved by applying the QZ algorithm directly, or after deflation preprocessing, are solvable with the more efficient QR algorithm. It is shown that the flop (floating-point operations) count for MD-QR algorithms is always less than the flop count for the most efficient deflation-QZ algorithms. For r⩽N, the MD-QR algorithms are exceptionally efficient. Using a parameter matrix decomposition of the dynamic or nondynamic component matrix, the MD method gives physical insight, and it provides a general proof of manifold constraints relating network time constants and poles and zeros. From these relations, accurate dominant and subdominant pole approximations are derived. A general eigenvalue sensitivity formula and a very flexible method for computing eigenvectors is developed and applied to pole sensitivity computation