Abstract
Two quadratically convergent gradient methods for minimizing an unconstrained function of several variables are examined. The heart of the Fletcher and Powell reformulation of Davidon's method is a variableH-matrix. The eigenvalues and eigenvectors of this matrix for a quadratic function are explored, leading to a proof that the gradient vectors at each step are mutually orthogonal. From this, a geometric interpretation of theH-matrix in terms of the projection of the gradient into a solution subspace is derived. These properties are then used to arrive at the main result, which states that, for a quadratic function, the direction vectors generated by the Davidon algorithm and the conjugate-gradient algorithm of Hestenes and Stiefel are scalar multiples of each other, provided the initial step each takes is in the direction of steepest descent. If one assumes no round-off error and a perfect one-dimensional search, the methods generate identical steps leading to the minimum.
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