lanczos-type algorithms are well known for their inherent instability. they typically breakdown occurs when relevant orthogonal polynomials do not exist. current approaches to curing breakdown rely on jumping over the non-existent polynomials to resume computation. this may have to be used many times during the solution process. we suggest an alternative to jumping, which consists of restarting...

Formal orthogonality on an algebraic variety for polynomials is deened and studied. Examples are given. Vector orthogonal polynomials of dimension d are proved to be orthogonal with respect to points on the unit circle.

Let c be a linear functional defined by its moments c(x) = ci for i = 0, 1, . . .. We proved that the nonlinear functional equations P (t) = c(P (x)P (x + t)) and P (t) = c(P (x)P (xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. Other types of nonlinear functional equations whose...

We give a framework for formal orthogonal polynomials with respect to an arbitrary moment matrix. When the moment matrix is Hankel, this simplifies to the classical framework. The relation with Padé approximation and with Krylov subspace methods is given. 1 Formal block orthogonal polynomials We consider a linear functional defined on the space of polynomials in two variables, defined by the mo...

The main focus of this paper is on efficiency analysis of two kinds of approximating functions (characteristic orthogonal polynomials and characteristic beam functions) that have been applied in the Rayleigh-Ritz method to determine the non-dimensional buckling and frequency parameters of an angle ply symmetric laminated composite plate with fully elastic boundaries. It has been observed that o...

the main purpose of this article is to present an approximate solution for the two-dimensional nonlinear volterra integral equations using legendre orthogonal polynomials. first, the two-dimensional shifted legendre orthogonal polynomials are defined and the properties of these polynomials are presented. the operational matrix of integration and the product operational matrix are introduced. th...

Gaussian and Chiral β-Ensembles, which generalise well known orthogonal (β = 1), unitary (β = 2), and symplectic (β = 4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like {β,N, n} ⇔ {4/β, n,N} for all β > 0, where N and n respectively denote the number of eigenvalues and products of characteristic polynomials. At the edge of the spectru...

We discuss the relationships among Jacobi matrices, orthogonal polynomials, spectral measure, moments, minors, Gaussian quadrature, resolvents and continued fractions in the simplest setting, namely the finite-dimensional one. The formal structure is essentially the same as that in the infinite-dimensional setting, where it leads into the rich analytic world of orthogonal polynomials on the rea...