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

We discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices. Any 2n × 2n symplectic matrix can be reduced to this condensed form which contains 8n − 4 nonzero entries and is determined by 4n − 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 4n − 1 parameters instead of the 4n2 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.