Quadratic Termination Properties of Minimization Algorithms I. Statement and Discussion of Results

There is a family of gradient algorithms (Broyden, 1970) thatincludes many useful methods for calculating the least valueof a function F(x), and some of these algorithms have been extendedto solve linearly constrained problems (Fletcher, 1971). Somenew and fundamental properties of these algorithms are given,in the case that F(x) is a positive definite quadratic function.In particular these properties are relevant to the case whenonly some of the iterations of an algorithm make a completelinear search. They suggest that Goldfarb's (1969) algorithmfor linearly constrained problems has excellent quadratic terminationproperties, and it is proved that these properties are betterthan has been stated in previously published papers. Also anew technique is identified for unconstrained minimization withoutlinear searches.     

On the A-Acceptability of Rational Approximations that Interpolate the Exponential Function

Rational approximations to exp (x), which use all the availablecoefficients to satisfy interpolation conditions (which maybe Hermitian) are considered. It is shown that the only A-acceptableapproximations of this kind are the functions of Ehle &Picel (1975) and Norsett (1975). These results are used to verifya conjecture of Ehle (1976) on the characterization of the A-acceptableorder-constrained Chebyshev approximations to exp (x).