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The parameterized $SR$ algorithm for symplectic (butterfly) matrices

Author: H. Faßbender
Journal: Math. Comp. 70 (2001), 1515-1541
MSC (2000): Primary 65F15
Published electronically: June 27, 2000
MathSciNet review: 1836916
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The $SR$ algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix $B$ in butterfly form is uniquely determined by $4n-1$ parameters. Using these $4n-1$ parameters, we show how one step of the symplectic $SR$ algorithm for $B$ can be carried out in $\mathcal{O}(n)$ arithmetic operations compared to $\mathcal{O}(n^3)$ arithmetic operations when working on the actual symplectic matrix. Moreover, the symplectic structure, which will be destroyed in the numerical process due to roundoff errors when working with a symplectic (butterfly) matrix, will be forced by working just with the parameters.

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Additional Information

H. Faßbender
Affiliation: Zentrum Mathematik, Technische Universität München, 80290 München, Germany

Keywords: Symplectic matrix; eigenvalue problem; $SR$ algorithm
Received by editor(s): March 2, 1999
Received by editor(s) in revised form: November 17, 1999
Published electronically: June 27, 2000
Article copyright: © Copyright 2000 American Mathematical Society