We shall introduce and study quadratic entry locus varieties, a class of projective algebraic varieties whose extrinsic and intrinsic geometry is very rich. Let us recall that, for an irreducible non-degenerate variety X ⊂ P of dimension n ≥ 1, the secant defect of X , denoted by δ(X), is the difference between the expected dimension and the effective dimension of the secant variety SX ⊆ P of X...

We shall introduce and study quadratic entry locus varieties, a class of projective algebraic varieties whose extrinsic and intrinsic geometry is very rich. Let us recall that, for an irreducible non-degenerate variety X ⊂ P of dimension n ≥ 1, the secant defect of X , denoted by δ(X), is the difference between the expected dimension and the effective dimension of the secant variety SX ⊆ P of X...

New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant minors) are revisited to determine when they are sufficient to give scheme-theoretic defining equations. An algorithm to decompose a general ternary quintic as the sum of seven fifth powers is given as an illustration o...

Symmetric rank-one (SR1) is one of the competitive formulas among the quasi-Newton (QN) methods. In this paper, we propose some modified SR1 updates based on the modified secant equations, which use both gradient and function information. Furthermore, to avoid the loss of positive definiteness and zero denominators of the new SR1 updates, we apply a restart procedure to this update. Three new a...

Let X = P × P embedded with O(1, 2). We prove that its (n + 1)-secant variety σn+1(X) is a hypersurface, while it is expected that it fills the ambient space. The equation of σn+1(X) is the symmetric analog of the Strassen equation. When n = 4 the determinantal map takes σ5(X) to the hypersurface of Lüroth quartics, which is the image of the Barth map studied by LePotier and Tikhomirov. This hi...

We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre-Veronese embeddings of P1 × P1, P1 × P1 × P1, and P2 × P1, as well as for the variety F of incident point-line pairs in P2. For P2 × P1 and F the results are new, while the...

Diagonal updating scheme is among the cheapest Newton-like methods for solving system of nonlinear equations. Nevertheless, the method has some shortcomings. In this paper, we proposed an improved matrix-free secant updating scheme via line search strategies, by using the steps of backtracking in the Armijo-type line search as a step length predictor and Wolfe-Like condition as corrector. Our a...

This paper proposes a modification to the Hestenes-Stiefel (HS) method by devising spectral parameter using modified secant relation solve nonlinear least-squares problems. Notably, in implementation, proposed differs from existing approaches, that it does not require safeguarding strategy and its Hessian matrix is positive definite throughout iteration process. Numerical experiments are conduc...

New approximate secant equations are shown to result from the knowledge of (problem dependent) invariant subspace information, which in turn suggests improvements in quasi-Newton methods for unconstrained minimization. A new limitedmemory BFGS using approximate secant equations is then derived and its encouraging behaviour illustrated on a small collection of multilevel optimization examples. T...