BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-89-1264
       ENTRY:: January 05, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: Chebyshev polynomials are not always optimal
        TYPE:: Technical Report
      AUTHOR:: Fischer, Bernd
      AUTHOR:: Freund, Roland
        DATE:: June 1989
       PAGES:: 15
    ABSTRACT:: We are concerned with the problem of finding among all
               polynomials of degree at most n and normalized to be 1 at c
               the one with minimal uniform norm on Epsilon. Here, Epsilon
               is a given ellipse with both foci on the real axis and c is a
               given real point not contained in Epsilon. Problems of this
               type arise in certain iterative matrix computations, and, in
               this context, it is generally believed and widely referenced
               that suitably normalized Chebyshev polynomials are optimal
               for such constrained approximation problems. In this note, we
               show that this is not true in general. Moreover, we derive
               sufficient conditions which guarantee that Chebyshev
               polynomials are optimal. Also, some numerical examples are
               presented.
       NOTES:: [Adminitrivia V1/RAM/19950105]
         END:: STAN//CS-TR-89-1264