BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-69-144
       ENTRY:: November 27, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: The maximum and minimum of a positive definite quadratic
               polynomial on a sphere are convex functions of the radius
        TYPE:: Technical Report
      AUTHOR:: Forsythe, George E.
        DATE:: July 1969
       PAGES:: 10
    ABSTRACT:: It is proved that in euclidean n-space the maximum M($\rho$)
               and minimum m($\rho$) of a fixed positive definite quadratic
               polynomial Q on spheres with fixed center are both convex
               functions of the radius $\rho$ of the sphere. In the proof,
               which uses elementary calculus and a result of Forsythe and
               Golub, $m^" (\rho) and M^" (\rho)$ are shown to exist and lie
               in the interval [$2{\lambda}_1 ,2{\lambda}_n$], where
               ${\lambda}_i$ are the eigenvalues of the quadratic form of Q.
               Hence $m^" (\rho) > 0 and M^" (\rho) > 0$.
       NOTES:: [Adminitrivia V1/Prg/19951127]
         END:: STAN//CS-TR-69-144