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

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. Some aspects of the perturbation theory for eigenvalues of unitary matrices are considered. Making use of the close relation between unitary and Hermitian eigenvalue problems a Courant-Fischer-type theorem for unitary matrices is derived and an inclusion theorem analogue to the Kahan theorem for Hermitian matrices is presented. Implications for the special case of unitary Hessenberg matrices are discussed. Key words. unitary eigenvalue problem, perturbation theory AMS(MOS) subject classifications. 15A18, 65F99 1. Introduction. New numerical methods to compute eigenvalues of unitary matrices have been developed during the last ten years. Unitary QR-type methods [19, 9], a divide-and-conquer method [20, 21], a bisection method [10], and some special methods for the real orthogonal eigenvalue problem [1, 2] have been presented. Interest in this task arose from problems in signal processing [11, 29, 33], in Gaussian quadrature on the unit circle [18], and in trigonometric approximation...

Fast, efficient, and reliable algorithms for up- and downdating discrete leastsquares approximations of a real-valued function given at arbitrary distinct nodes in [0; 2ß) by trigonometric polynomials are presented. A combination of the up- and downdating algorithms yields a sliding window scheme. The algorithms are based on schemes for the solution of (inverse) unitary eigenproblems and require only O(mn) arithmetic operations as compared to O(mn 2 ) operations needed for algorithms that ignore the structure of the problem. Numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general QR decomposition methods for the least-squares problem. Key words. trigonometric approximation, unitary Hessenberg matrix, Schur parameter, Szego polynomial, updating, downdating, sliding window scheme 1 Introduction A problem in signal processing is the approximation of a function known only at some ...

This paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions. In particular, the inverse eigenvalue problems for unitary Hessenberg matrices and for Schur parameter pencils are considered. The Szego recursion is known to be identical to the Arnoldi process and can be seen as an algorithm for solving an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils. It is shown that solving this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only O(mn) arithmetic operations as compared with O(mn 2 ) operations needed for algorithms that ignore the structure of the problem. Key words. inverse unitary eigenvalue problem, Arnoldi process, Szego polynomials...