BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-64-9
       ENTRY:: January 19, 1996
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: The QD-algorithm as a method for finding the roots of a
               polynomial equation when all roots are positive
        TYPE:: Technical Report
      AUTHOR:: Andersen, Christian
        DATE:: June 1964
       PAGES:: 86
    ABSTRACT:: The Quotient-Difference (QD)-scheme, symmetric functions and
               some results from the theory of Hankel determinants are
               treated. Some well known relations expressing the elements of
               the QD-scheme by means of the Hankel determinants are
               presented. The question of convergence of the columns of the
               QD-scheme is treated. An exact expression for $q_{n}^{k}$ is
               developed for the case of different roots. It is proved that
               the columns of the QD-scheme will converge not only in the
               well known case of different roots, but in all cases where
               the roots are positive. A detailed examination of the
               convergence to the smallest root is presented. An exact
               expression for $q_{n}^{N}$ is developed. This expression is
               correct in all cases of multiple positive roots. It is shown
               that the progressive form of the QD-algorithm is only 'mildly
               unstable'. Finally, some ALGOL programs and some results
               obtained by means of these are given.
       NOTES:: [Adminitrivia V1/Prg/19960119]
         END:: STAN//CS-TR-64-9