[CvGmt News] avviso di seminario

[Giulia Curciarello]

Si avvisa che il Il Prof. Joab R. Winkler dell'Universita` di Sheffield  
terra` Mercoledi` 21 alle ore 15.00 in Aula Seminari  un seminario dal 
titolo "THE COMPANION AND SYLVESTER RESULTANT MATRICES FOR
 BERNSTEIN POLYNOMIALS". 

THE COMPANION AND SYLVESTER RESULTANT MATRICES FOR
  --------------------------------------------------
 
                BERNSTEIN POLYNOMIALS
                ---------------------
 
 
 Joab R. Winkler, The University of Sheffield, United Kingdom
 
 
 
Resultants have a rich history and their theoretical properties
have been investigated extensively. It is usually assumed that
the polynomials are expressed in the power basis, but this basis
is not the natural representation of curves and surfaces in geometric
modelling systems. The practical application of resultants to
geometric computations requires, therefore, that they be developed for
the Bernstein basis, such that the power basis is not used. 
I will address this issue by considering the  companion and Sylvester
resultant matrices for Bernstein polynomials. 

 
Several condition numbers of resultant matrices are considered, and
it is shown that the ideal condition number is difficult to
compute, and practical constraints dictate that a sub-optimal
condition number be used. Computational results for the
companion matrix resultant are presented, and it is shown that
the Bernstein form of the companion matrix resultant is numerically
superior to its power basis equivalent. The Sylvester resultant
matrix is fundamentally different because a unique condition number
for this matrix cannot be defined, and it is shown that this is
due to the structure of the matrix. Finally, the transformation
of the companion and Sylvester resultant matrices between the
power and Bernstein bases is considered, and it is shown that the
transformation of the companion matrix resultant is ill-conditioned,
even for polynomials of low degree. It is concluded that the
resultant of two polynomials should always be computed when
they are expressed in the Bernstein basis rather than the power basis. 


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