The symplectic eigenvalue problem, the butterfly form, the SR algorithm, and the Lanczos method

Abstract

We discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices. Any 2 n × 2 n symplectic matrix can be reduced to this condensed form which contains 8 n − 4 nonzero entries and is determined by 4 n − 1 parameters. The symplectic eigenvalue problem can be solved using the SR algorithm based on this condensed form. The SR algorithm preserves this form and can be modified to work only with the 4 n − 1 parameters instead of the 4 n 2 matrix elements. The reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to Hessenberg form. A Lanczos-like algorithm for reducing a symplectic matrix to butterfly form is also presented.