We introduce a class Cn of n × n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of Cn, we show that if A ∈ Cn then A′ = RQ ∈ Cn, where A = QR is the QR decomposition of A. This allows one to implement t...

Classical ellipsoidal and sphero-conal harmonics are polynomial solutions of the Laplace equation that can be expressed in terms of Lamé polynomials. Generalized ellipsoidal and sphero-conal harmonics are polynomial solutions of the more general Dunkl equation that can be expressed in terms of Stieltjes polynomials. Niven’s formula connecting ellipsoidal and sphero-conal harmonics is generalize...

We constructmultiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n. In addition, we create the change-of-basis matrices between the generalized Tschebyscheff of the second kind polyno...

The notion of activities with respect to spanning trees in graphs was introduced by W.T. Tutte, and generalized to activities with respect to bases in matroids by H. Crapo. We present a further generalization, to activities with respect to arbitrary subsets of matroids. These generalized activities provide a unified view of several different expansions of the Tutte polynomial and the chromatic ...

Fast computation of two-dimensional (2-D) generalized discrete Fourier transforms (GDFTs) and generalized discrete Hartley transforms (GDHTs) are discussed in terms of a new method which is derived based on index permutation, linear congruences, and polynomial transforms. Further, the 2-D skew circular convolution computed by the proposed skew polynomial transform is also included. ( 2000 Elsev...

SUMMARY Applying a finite difference approximation to a biharmonic equation results in a very ill-conditioned system of equations. This paper examines the conjugate gradient method ' used in con junction with the generalized and approximate polynomial preconditionings for solving such linear systems. An approximate polynomial preconditioning is introduced, and is shown to be more efficient than...

A major general paradigm in cryptography is the following argument: Whatever an adversary could do in the real world, it could just as well do in the ideal world. The standard interpretation of “just as well” is that the translation from the real to the ideal world, usually called a simulator, is achieved by a probabilistic polynomial-time algorithm. This means that a polynomial blow-up of the ...

Conventional generalized polynomial chaos is known to fail for long time integration, loosing its optimal convergence behaviour and developing unacceptable error levels. The reason for this loss of convergence is the assumption that the probability density function is constant in time. By allowing a probability density function to evolve in time the optimal properties of polynomial chaos are re...