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# Algebra Seminar: A Product Eigenvalue Algorithm for Unitary Hessenberg Matrices

2005-04-18

1:10p.m. Neill 106W

Roden David

A recent perspective in eigenvalue computations is to find the eigenvalues of a square matrix *A*, where *A* is given in the factored form:

*A = A*

_{1}A_{2}...A_{k}.
The idea is to work with the factors *A _{1}, A_{2}, ..., A_{k}* to compute the eigenvalues of

*A*without forming

*A*explicitly. This perspective is called the

*product eigenvalue problem.*

In this talk, we will present a new product eigenvalue algorithm for unitary upper Hessenberg matrices. We express the Hessenberg matrix *H* as a product of rotators *Q _{1}, Q_{2}, ..., Q_{k}*, and then seek the eigenvalues of

*H*by performing updates on its rotator factors. The algorithm is the QR algorithm, implemented to work with

*H*in the factored form. We will compare the advantages of the new algorithm in terms of storage cost and operation count. We will present some shift strategies to accelerate the convergence of the algorithm. Finally, we will report its performance on some test matrices.