The secant method is one of the most popular methods for root finding. Standard text books in numerical analysis state that the secant method is super linear: the rate of convergence is set by the gold number. Nevertheless, this property holds only for simple roots. If the multiplicity of the root is larger than one, the convergence of the secant method becomes linear. This communication includ...

We study curves with linear series that are exceptional with regard to their secant planes. Working in the framework of an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve of genus g has no exceptional secant planes, in a very precise sense. We also address the problem of computing the number of linear series with exceptional secant planes in a one-para...

Let X ⊆ PN be a non-degenerate projective variety of dimension d. Then the sth secant variety of X, denoted σs(X), is the Zariski closure of the union of linear spans of s-tuples of points lying on X. The study of secant varieties has a long history. The interest in this subject goes back to the Italian school at the turn of the 20th century. This topic has received renewed interest over the pa...

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three with one exception, the secant variety of the 21-dimensional spinor variety in P, whose ideal is generated in degree four. We also discuss the coordinate ring of secant varietie...

For a variety X of dimension n in Pr, r n(k + 1) + k, the kth secant order of X is the number μk(X) of (k + 1)-secant k-spaces passing through a general point of the kth secant variety. We show that, if r > n(k + 1) + k, then μk(X) = 1 unless X is k-weakly defective. Furthermore we give a complete classification of surfaces X ⊂ Pr, r > 3k + 2, for which μk(X) > 1.

In this paper we study the dimension of secant varieties of Segre-Veronese varieties P × P embedded by the morphism given by O(1, 2). Given the dimensions m, n, we provide two functions s(m, n) and s(m, n), such that the s secant variety is nondefective, i.e. it has the expected dimension, if s ≤ s(m, n) or s ≥ s(m, n). Finally, we present a conjecturally complete list of defective secant varie...

We propose an approach to enhance the performance of a diagonal variant of secant method for solving large-scale systems of nonlinear equations. In this approach, we consider diagonal secant method using data from two preceding steps rather than a single step derived using weak secant equation to improve the updated approximate Jacobian in diagonal form. The numerical results verify that the pr...

An all-optical Orthogonal Frequency Division Multiplexing communication system at 2 THz is modeled using photonic crystal fiber of length 1km, and the performance of four carrier waveforms hyperbolic secant, square of hyperbolic secant, square and sinusoidal is evaluated using standard metrics such as eye diagram and bit error rate. From the above mentioned valuations, one can deduce the minima...

Some generalizations of the secant method to semismooth equations are presented. In the one-dimensional case the superlinear convergence of the classical secant method for general semismooth equations is proved. Moreover a new quadratically convergent method is proposed that requires two function values per iteration. For the n-dimensional cases, we discuss secant methods for two classes of com...

We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of smal...