THE SYMPLECTIC EIGENVALUE PROBLEM, THE BUTTERFLY FORM, THE SR ALGORITHM, AND THE LANCZOS METHOD

We discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices. Any 2n x 2n sy mplectic matrix can be reduced to this condensed form which contains 8 n - 4 nonzero entries and is determined by 4n - 1 parameters. The symp lectic eigenvalue problem can be solved using the SR algorithm based o n this condensed form. The SR algorithm preserves this form and can be modified to work only with the 4n - 1 parameters instead of the 4n(2) matrix elements. The reduction of symplectic matrices to symplectic b utterfly form has a close analogy to the reduction of arbitrary matric es to Hessenberg form. A Lanczos-Iike algorithm for reducing a symplec tic matrix to butterfly form is also presented. (C) 1998 Elsevier Scie nce Inc. All rights reserved.