The parameterized SR algorithm for symplectic (butterfly) matrices

The SR algorithm is a structure-preserving algorithm for computing the spec trum of symplectic matrices. Any symplectic matrix can be reduced to symple ctic butterfly form. A symplectic matrix B in butterfly form is uniquely de termined by 4n - 1 parameters. Using these 4n - 1 parameters, we show how o ne step of the symplectic SR algorithm for B can be carried out in O(n) ari thmetic operations compared to O(n(3)) arithmetic operations when working o n the actual symplectic matrix. Moreover, the symplectic structure, which w ill be destroyed in the numerical process due to roundoff errors when worki ng with a symplectic (butterfly) matrix, will be forced by working just wit h the parameters.