clarke@csri.toronto.edu (Jim Clarke) (01/25/89)
THEORY SEMINAR - Thursday, February 9, 3 p.m. in Room GB 244
(GB = Galbraith Building, 35 St. George Street)
Wayne Eberly
University of Toronto
"Efficient Decompositions of Matrix Algebras"
We consider efficient algorithms for the wedderburn decomposition of
Finite-dimensional matrix algebras over large extensions of Q.
We begin by noting two earlier treatments of this computational problem.
Gabriel presented a solution of the problem over R and C in the mid 1960's
and early 1970's. More recently, Friedl and R'onyai gave solutions for the
problems over number fields and finite fields. Unfortunately, the solution
over R and C requires the use of exact arithmetic over these fields - and
cannot be implemented correctly using standard numerical representations.
The solution over number fields can be implemented using exact arithmetic,
but is not applicable to the problems which arise most often in theoretical
physics and chemistry.
We overcome these difficulties by using a nonstandard representation of
large extensions of Q. While this still does not allow us to represent ar-
bitrary real or complex numbers exactly, we show that it is general enough
to permit efficient exact solutions of decomposition problems with physical
applications.
--
Jim Clarke -- Dept. of Computer Science, Univ. of Toronto, Canada M5S 1A4
(416) 978-4058
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