A heuristic verification of the degree of the approximate GCD of two univariate polynomials

We consider the problem of computing verified real interval perturbations of the coefficients of two univariate polynomials such that there exist corresponding perturbed polynomials which have an exact greatest common divisor (GCD) of a given degree k. Based on the certification of the rank deficiency of a submatrix of the Bezout matrix of two univariate polynomials, we propose an algorithm to compute verified real perturbations. Numerical experiments show the performance of our algorithm.     

Computation of the GCD of polynomials of several variables on a MPC

The article is devoted to computer calculations with polynomials of several variables, in particular, to the construction of sets of large-block polynomial operations and to the concretization of the generalized algorithm of Euclid for computing the greatest common divisor (GCD) of two polynomials of several variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 148–155, 1976.     

An improved approximate method for computing eigenvector derivatives

An improved approximate method for computing eigenvector derivatives has been developed. In this formulation an eigenvector derivative is assumed to be spanned by a set of truncated normal modes augmented by a residual static mode. The coefficients in the expansion are computed by a Bubnov-Galerkin method. The formulation has been implemented as a set of Direct Matrix Abstraction Programming alters for . Numerical examples show the method provides sufficient accuracy and reduced computation time when compared to the exact solution.     

An unconditionally convergent method for computing zeros of splines and polynomials

We present a simple and efficient method for computing zeros of spline functions. The method exploits the close relationship between a spline and its control polygon and is based on repeated knot insertion. Like Newton's method it is quadratically convergent, but the new method overcomes the principal problem with Newton's method in that it always converges and no starting value needs to be supplied by the user.

     

Using Chebyshev polynomials and approximate inverse triangular factorizations for preconditioning the conjugate gradient method

In order to precondition a sparse symmetric positive definite matrix, its approximate inverse is examined, which is represented as the product of two sparse mutually adjoint triangular matrices. In this way, the solution of the corresponding system of linear algebraic equations (SLAE) by applying the preconditioned conjugate gradient method (CGM) is reduced to performing only elementary vector operations and calculating sparse matrix-vector products. A method for constructing the above preconditioner is described and analyzed. The triangular factor has a fixed sparsity pattern and is optimal in the sense that the preconditioned matrix has a minimum K-condition number. The use of polynomial preconditioning based on Chebyshev polynomials makes it possible to considerably reduce the amount of scalar product operations (at the cost of an insignificant increase in the total number of arithmetic operations). The possibility of an efficient massively parallel implementation of the resulting method for solving SLAEs is discussed. For a sequential version of this method, the results obtained by solving 56 test problems from the Florida sparse matrix collection (which are large-scale and ill-conditioned) are presented. These results show that the method is highly reliable and has low computational costs.     

Permutable polynomials for several variables

Summary The Russian mathematician P. L. Chebyshev defined and studied a class of polynomials of one variable. These polynomials have many in teresting properties including commutativity and closure with respect to composition. In this article we show how to generalize this property to several variables. Special attention is given to the case of three variables. Results concerning how to compute the polynomials, their value at certain points, closed forms, recurrence relations, and generating functions are presented.     

An approximate method of computing the magneto-acoustic heating of electrically conducting bodies

We discuss a method of approximate computation of the noncontact magneto-acoustic heating of nonmagnetic viscoelastic matter, based on the asymptotic separation of the initial equations of the theory of magnetoelasticity and the expansion of the solutions of the resulting sequence of problems in a series of eigenfunctions of the classical problems of electrodynamics and the dynamic theory of elasticity. The asymptotic parameter ε<1 for the problem was taken to be the criterion ε=Co. Rm forRm≤1 and ε=Co forRm≥1. We obtain expressions for the average power of the Joule losses and losses due to internal friction. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37. 1994. pp. 70–73.     

The Laguerre polynomials in several variables

In this paper, we give some relations between multivariable Laguerre polynomials and other well-known multivariable polynomials. We get various families of multilinear and multilateral generating functions for these polynomials. Some special cases are also presented.     

A new algorithm for computing orthogonal polynomials

The aim of this paper is to present an algorithm for computing orthogonal polynomials. The fast Fourier transform completes the connection given by Draux between the three-term recurrence relationship and the Euclidian algorithm. Applications to Hankel systems and numerical examples illustrate our purpose.     

The norm of minimal polynomials on several intervals

Using works of Franz Peherstorfer, we examine how close the $n$th Chebyshev number for a set $E$ of finitely many intervals can get to the theoretical lower limit $2\mathrm{cap}{\left(E\right)}^n$.     

Generating approximate parametric roots of parametric polynomials

Built upon a ground field is the parametric field, the Puiseux field, of semi-terminating formal fractional power series. A parametric polynomial is a polynomial with coefficients in the parametric field, and roots of parametric polynomials are parametric. For a parametric polynomial with nonterminating parametric coefficients and a target accuracy, using sensitivity of the Newton Polygon process, a complete set of approximate parametric roots, each meeting target accuracy, is generated. All arguments are algebraic, from the inside out, self-contained, penetrating, and uniform in that only the Newton Polygon process is used, for both preprocessing and intraprocessing. A complexity analysis over ground field operations is developed; setting aside root generation for ground field polynomials, but bounding such, polynomial bounds are established in the degree of the parametric polynomial and the target accuracy.     

On the cost of computing roots of polynomials

Recently Smale has obtained probabilistic estimates of the cost of computing a zero of a polynomial using a global version of Newton's method. Roughly speaking, his result says that, with the exception of a set of polynomials where the method fails or is very slow, the cost grows as a polynomial in the degree. He also asked whether similar results hold for PL homotopy methods. This paper gives such a result for a special algorithm of the PL homotopy type devised by Kuhn. Its main result asserts that the cost of computing some zero of a polynomial of degreen to an accuracy of ε (measured by the number of evaluations of the polynomial) grows no faster than O(n ³ log₂(n/ε)). This is a worst case analysis and holds for all polynomials without exception. This work was supported, in part, by National Science Foundation Grant MCS79-10027 and, in part, by a fellowship of the Guggenheim Foundation.     

GRPIA: a new algorithm for computing interpolation polynomials

Numerical Algorithms - Let x0,x1, ? , xn, be a set of n + 1 distinct real numbers (i.e., xm ≠ xj, for m ≠ j) and let ym,k, for m = 0, 1, ? , n, and k = 0, 1, ? , rm,...     

On the orthogonal polynomials in several variables

We study the connection between orthogonal polynomials in several variables and families of commuting symmetric operators of a special form.     

Irreducibility results for compositions of polynomials in several variables

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.