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<DIV><FONT face=Arial size=2><FONT face="Times New Roman" size=3>Si avvisa che
il Il Prof. Joab R. Winkler dell'Universita` di Sheffield <BR>terra`
Mercoledi` 21 alle ore 15.00 in Aula Seminari un seminario dal <BR>titolo
"THE COMPANION AND SYLVESTER RESULTANT MATRICES FOR<BR> BERNSTEIN
POLYNOMIALS". </FONT></FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2><FONT face="Times New Roman" size=3>THE COMPANION
AND SYLVESTER RESULTANT MATRICES FOR<BR>
--------------------------------------------------<BR> <BR>
BERNSTEIN
POLYNOMIALS<BR>
---------------------<BR> <BR> <BR> Joab R. Winkler, The
University of Sheffield, United
Kingdom<BR> <BR> <BR> <BR>Resultants have a rich history and
their theoretical properties<BR>have been investigated extensively. It is
usually assumed that<BR>the polynomials are expressed in the power basis, but
this basis<BR>is not the natural representation of curves and surfaces in
geometric<BR>modelling systems. The practical application of resultants
to<BR>geometric computations requires, therefore, that they be developed
for<BR>the Bernstein basis, such that the power basis is not used. <BR>I will
address this issue by considering the companion and Sylvester<BR>resultant
matrices for Bernstein polynomials. <BR><BR> <BR>Several condition numbers
of resultant matrices are considered, and<BR>it is shown that the ideal
condition number is difficult to<BR>compute, and practical constraints dictate
that a sub-optimal<BR>condition number be used. Computational results for
the<BR>companion matrix resultant are presented, and it is shown that<BR>the
Bernstein form of the companion matrix resultant is numerically<BR>superior to
its power basis equivalent. The Sylvester resultant<BR>matrix is fundamentally
different because a unique condition number<BR>for this matrix cannot be
defined, and it is shown that this is<BR>due to the structure of the matrix.
Finally, the transformation<BR>of the companion and Sylvester resultant matrices
between the<BR>power and Bernstein bases is considered, and it is shown that
the<BR>transformation of the companion matrix resultant is
ill-conditioned,<BR>even for polynomials of low degree. It is concluded that
the<BR>resultant of two polynomials should always be computed when<BR>they are
expressed in the Bernstein basis rather than the power basis. </FONT><BR></DIV>
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