### Optimization and Systems Theory Seminar

Friday, Nov. 5, 1999, 11.00-12.00, Room 3721, Lindstedtsv. 25

**Professor Lars Eldén**

Department of Mathematics

Linköping University

Linköping, Sweden

E-mail: laeld@mai.liu.se

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Solving constrained linear algebra problems using a differential
geometric approach

In recent years development has taken place, where certain constrained
problems in linear algebra are solved numerically using a differential
geometric approach. Orthonormality constraints can be thought of as
restrictions that the solution lies on the Stiefel or Grassmann
manifold. Newton's method and the conjugate gradient method are
formulated on the manifold, and are used in the actual numerical
procedure.
We will illustrate the ideas using the quadratically constrained least
squares problem min ||A x - b|| subject to x^{T}x = 1, which
occurs in the numerical solution of ill-conditioned linear problems,
cf. also trust region methods in optimization. Numerical examples
will be given showing that the Newton-Stiefel method converges faster
than the standard approach. If time permits we will describe how
manifold theory can be used in the analysis of a generalization of the
abovementioned problem, the orthogonal Procrustes problem.

Calendar of seminars

*Last update: October 22, 1999 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*