Szegö polynomials, i.e., polynomials orthogonal with respect to a measure on the unit cir-eIe in "the eomplex plane, ean be viewed as the charaeteristic polynomials of a struetured Hessenberg matrix that is determined by the recurrence coeffieients of the polynomialsA This is in analogy with the relationship of polynomials orthogonal with respect to a mea-_ sure on the real axis and Jacobi matrices. Similarly, computational problems involving Szegö poly~omials ean be developed by reeasting the

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... roblems in terms of the structured Hessenberg matrix. _ The struetured Hessenberg matriees can be viewed as submatriees of a larger unitary Hessenberg matrix. The unitary Hessenberg matrix itself arises in the special ease that the Szegö polynomials are orthogonal with respect to a singularmeasure. Unitary Hessenberg matrices have a structure that is quite amenable to exploitation in eigenvalue . computations, and a variety of efficient algorithms have recently been developed for solving unitary Hessenberg eigenproblems. We will give an overview of these points, and give particular attention to Gragg's unitary Hessenberg QR algorithm. We'll outline a derivation of the UHQR algorithm using a device for describing the efficient implementation of single-bulge chasing procedures on unitary Hessenberg matrices. We then show how our device for deriving the UHQR algorithm can be used to derive an efficient implementation of a Francis QR step on (real) orthogonal Hessenberg matrices. In particular, we will see that the double-bulge chasing sweep that arises from the Francis shift strategy can be implemented by interleaving three single-bulge chasing sweeps. The resulting OHQR algorithm avoids the additional storage and computation assoeiated with the complex arithmetic that is required when the single-shift UHQR algorithm is applied to a real matrix. We will also outline same current• work on how the QR algorithm ean be efficiently applied to a submatrix of a unitary Hessenberg matrix. The resulting algorithm, which ia. being developed in collaboration with William Gragg and Chunyang He, provides a nevw; approach to computing the zeros of an arbitrary Szegö polynomial. © The sensitivity of least squares polynomial approximation Bernhard Beckermann (Villeneuve cl' Ascq) Joint work with Ed Saff (Tampa) We consider the least squares problem of finding the coefficients with respect to a poly-