### Wednesday, October 29

## IP1

Geometrical Approaches to the Algebraic Eigenvalue Problem

8:30 AM-9:30 AM

*Chair: Ilse Ipsen, North Carolina State University*

*Room: Ballroom 3*

Nearness of a matrix to a multiple eigenvalue is indicated by a small cosine between left and right eigenvectors, as for example is computed by MATLAB's condeig routine or the RCONDE variable in LAPACK. What can we supply the user who wishes to *exactly* obtain the nearest matrix with a multiple eigenvalue or more generally the nearest matrix (or pencil) with a particular canonical form? Such problems have been motivated from problems in systems and control, and have been studied by many experts in numerical linear algebra. We are working towards solving these problems so that the information could be available easily to users of library software.

In this talk, the speaker will indicate how new geometrical approaches are being successfully applied towards the solution of this problem. His proposed solution combines the power of a new understanding of the perturbation theory of so-called "staircase algorithms" with manifold techniques for optimization on the Grassmann and Stiefel manifolds. By way of background, the speaker will discuss how manifold approaches provide a unifying taxonomy for any eigenvalue algorithm, linear or nonlinear, involving orthogonality constraints.

The talk surveys a number of recent developments representing joint work with Arias, Demmel, Elmroth, Kagstrom, Ma, Ripper, and Smith and will consider some open problems.

**Alan Edelman**

*Department of Mathematics*

*Massachusetts Institute of Technology*

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*MMD, 6/26/97*