The Davidon formula and others of the “quasi-Newton” class, which are used in the unconstrained minimization of a function f f , provide a (generally) convergent sequence of approximations to the Hessian of f f . These formulas, however, require the independent calculation of the gradient of f f . In this paper, a set of new formulas is derived—using a previously described variational approach—which successively approximates the gradient as well as the Hessian, and uses only function values. These formulas are incorporated into an algorithm which, although still crude, works quite well for various standard test functions. Extensive numerical results are presented.