BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//NA-M-80-08
       ENTRY:: January 28, 1996
ORGANIZATION:: Stanford University, Department of Computer Science, Numerical
               Analysis Project
       TITLE:: Rational Chebyshev approximation on the unit disk
        TYPE:: Manuscript
      AUTHOR:: Trefethen, Lloyd N.
        DATE:: October 1980
       PAGES:: 46
    ABSTRACT:: In a recent paper we showed that error curves in polynomial
               Chebyshev approximation of analytic functions on the unit
               disk tend to approximate perfect circles about the origin.
               Making use of a theorem of Caratheodory and Fejer, we derived
               in the process a method for calculating near-best
               approximations rapidly by finding the principal singular
               value and corresponding singular vector of a complex Hankel
               matrix. This paper extends these developments to the problem
               of Chebyshev approximation by rational functions, where
               non-principal singular values and vectors of the same matrix
               turn out to be required. The theory is based on certain
               extensions of the Caratheodory-Fejer result which are also
               currently finding application in the fields of digital signal
               processing and linear systems theory.

               It is shown among other things that if f($\epsilon z$) is
               approximated by a rational function of type (m,n) for
               $\epsilon$ > 0, then under weak assumptions the corresponding
               error curves deviate from perfect circles of winding number M
               + N + 1 by a relative magnitude O(${\epsilon}^{m+n+2}$) as
               $\epsilon\ \rightarrow\ 0$. The "CF approximation" that our
               method computes approximates the true best approximation to
               the same high relative order. A numerical procedure for
               computing such approximations is described and shown to give
               results that confirm the asymptotic theory. Approximation of
               $e^z$ on the unit disk is taken as a central computational
               example.
       NOTES:: [Adminitrivia V1/Prg/19960128]
         END:: STAN//NA-M-80-08