Using Taylor's series we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant relation we present a new BFGS method for solving unconstrained optimization problems. The proposed method make use of both gradient and function values while the usual secant relation uses only gradient values. U...

This paper describes an iterative method for factoring a polynomial that bears the same relation to Bairstow's method as the secant method in a single variable bears to Newton's method. Like the secant method, the generalized secant method requires only one function evaluation for each iteration, and like the secant method it converges to a simple factor with order (1+75)/2.

We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits, we also point out a relation between the existence of certain codes and non-defectiveness of certain higher secant varieties.

We show that the secant variety of the Segre variety gives useful information about the geometrical structure of an arbitrary multipartite quantum system. In particular, we investigate the relation between arbitrary bipartite and three-partite entangled states and this secant variety. We also discuss the geometry of an arbitrary general multipartite state.

We introduce a three-step Chebyshev–Secant-type method (CSTM) with high efficiency index for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for (CSTM)using recurrence relations. Numerical examples validating our theoretical results are also provided in this study. © 2011 Elsevier B.V. All rights reserved.

The derivative polynomials introduced by Knuth and Buckholtz in their calculations of the tangent and secant numbers are extended to a multivariable q– environment. The n-th q-derivatives of the classical q-tangent and q-secant are each given two polynomial expressions. The first polynomial expression is indexed by triples of integers, the second by compositions of integers. The functional rela...

The goal of this talk is to introduce secant varieies and show connections of secant varieties of Veronese variety to the Waring problem. 1 Secant varieties Let X ⊂ P be a variety over k, where k is algebraically closed field of characteristic 0 (or just assume k = C). Definition 1.1. For s ≥ 1 the s-th higher secant variety of X is

Historical Development of the BFGS Secant Method and Its Characterization Properties by Joanna Maria Papakonstantinou The BFGS secant method is the preferred secant method for finite-dimensional unconstrained optimization. The first part of this research consists of recounting the historical development of secant methods in general and the BFGS secant method in particular. Many people believe t...

We determine a Gröbner basis for the secant ideal of the toric ideal associated to the second hypersimplex ∆(2, n), with respect to any circular term order. The Gröbner basis of the secant ideal requires polynomials of odd degree up to n. This shows that the circular term order is 2-delightful, resolving a conjecture of Drton, Sturmfels, and the author. The proof uses Gröbner degenerations for ...