In this paper we suggest a hybrid method for computing the smallest eigenvalue of a symmetric and positive definite Toeplitz matrix which takes advantage of two types of methods, Newton’s method for the characteristic polynomial and projection methods based on rational interpolation of the secular equation.

Newton’s formula for constructing the interpolation polynomial is well–known. It makes use of divided differences. It was generalized around 1971–1973 by Mühlbach for interpolation by a linear family of functions forming a complete Chebyshev system. This generalization rests on a generalization of divided differences due to Popoviciu. In this paper, it is shown that Mühlbach’s formula is relate...

Several representations for the interpolating polynomial exist: Lagrange, Newton, orthogonal polynomials etc. Each representation is characterized by some basis functions. In this paper we investigate the transformations between the basis functions which map a specific representation to another. We show that for this purpose the LU and the QR decomposition of the Vandermonde matrix play a cruci...

Given 2n distinct points z1, z′ 1, z2, z ′ 2, . . . , zn, z ′ n (in this order) on the unit circle, and n points w1, . . . , wn on the unit circle, we show how to construct a Blaschke product B of degree n such that B(zj) = wj for all j and, in addition, B(z′ j) = B(z ′ k) for all j and k. Modifying this example yields a Blaschke product of degree n− 1 that interpolates the zj ’s to the wj ’s. ...

-Protective systems require a faithful reproduction of primary current on the Current Transformer (CT) secondary side. Saturation is a common problem in a steel core CT. Saturation problem causes dreadful effects in the protection systems. This paper presents a simple solution towards detection and compensation of current transformer saturation problems. This paper describes a method in which C...

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss-Lobatto-Legendre-Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a useroriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach al...

For diffeomorphisms of smooth compact finite-dimensional manifolds, we consider the problem of how fast the number of periodic points with period n grows as a function of n. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for C or sm...

For diffeomorphisms of smooth compact finite-dimensional manifolds, we consider the problem of how fast the number of periodic points with period n grows as a function of n. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for C2 or s...

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [−1,1]. It is proved that the constructed QFs converge for any singular point x not coinciding with t...

The trigonometric interpolants to a periodic function f in equispaced points converge if f is Dini-continuous, and the associated quadrature formula, the trapezoidal rule, converges if f is continuous. What if the points are perturbed? With equispaced grid spacing h, let each point be perturbed by an arbitrary amount ≤ αh, where α ∈ [0, 1/2) is a fixed constant. The Kadec 1/4 theorem of samplin...