## Background

Traditional methods of solving polynomial equations of high degree with
multiple roots, for example, those based on the Newton-Raphson method, yield
unsatisfactory answers. Roundoff errors are sufficient to
cause incorrect solutions, even if all the roots are simple, and these problems are
compounded when the polynomial has multiple roots because they cause
a multiple root to break up into simple roots. It is known that
simple clustering of the computed roots
does not necessarily yield a satisfactory solution. Further complications occur
when the coefficients of the polynomial are subject to error because there now
exists a (potentially infinite) family of solutions, each of which is valid
provided that the backward error of each computed root is less than the error in
the data.

The course will consist of lectures and laboratory classes on a new
technique for computing the roots of a polynomial whose coefficients are
subject to error, when the computed roots must satisfy stated conditions.
This technique uses structured matrix methods on the
Sylvester resultant matrix in order to perform a sequence of approximate
greatest common divisor (GCD)
computations.

## Aims

The aims of the Summer School are:

- To provide examples from geometric modelling, systems theory and other
disciplines in which computations on polynomials are required.
- To provides lectures and computer classes on a new and successful method of
solving polynomial equations of high degree with multiple roots when the
coefficients are subject to error.
- To provide lectures on structured matrix methods for approximate
greatest common divisor computations.

In order to emphasise these aims of the Summer School, the theory of the
methods will be presented for Bernstein basis polynomials (which is applicable
to geometric modelling), and the power (monomial) basis, which is the most
popular polynomial basis and used in many other applications, including systems
theory and signal processing.

## Technical Program

The topics covered in the Summer School include:

- Problems that yield polynomial equations of high degree
- Reasons for the failure of classical methods for solving polynomial
equations of high degree
- Structured and unstructured condition numbers of roots of polynomials
- Structure preserving matrix methods with respect to:
- The Sylvester resultant matrix and its subresultants
- Approximate greatest common divisor computations
- Equality constrained least squares problems

- The method of non-linear least squares

## Invited speakers

The invited speakers are

- Professor Rob Corless (University of Western Ontario, Canada)
slides
- Professor Gershon Elber (The Technion, Haifa, Israel)
slides
- Professor Gene Golub (Stanford University, USA)
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- Dr Ivan Markovsky (Southampton University, UK)
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- Professor Zhonggang Zeng (Northeastern Illinois University, USA)
slides