The Chebyshev and Stieltjes procedures are algorithms for computing recursion coefficients for polynomials that are orthogonal with respect to an inner product defined on (part of) the real axis. The Chebyshev procedure is an implementation of a map from moments to recursion coefficients of orthogonal...
Algorithms for the construction of the orthogonal polynomials for the new weight v in terms of those for the old weight w are presented. All the methods are based on modified moments. As applications we present Gaussian quadrature rules for integrals in which the integrant has singularities close...
In this work, we focus on the computation of the zeros of a monic Laguerre–Sobolev orthogonal polynomial of degreen. Taking into account the associated four–term recurrence relation, this problem can be formulated as a generalized eigenvalue problem, involving a lower bidiagonal matrix and a 2...
Unfortunately, I don't know how to do better than this in general. However, I do discover some special cases that have O(nlogn)O(nlogn) algorithms, where indeed handle the case x+1/x=(1+x2)/xx+1/x=(1+x2)/x. You might suspect that there is some magic theory behind this...
Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed. Special emphasis is given to finite fields, the integers, or algebraic extensions of the rationals, and to multivariate polynomials with integral coefficients. In particular, various squarefree de...
This textbook gives a well-balanced presentation of the classic procedures of polynomial algebra which are computationally relevant and some algorithms developed during the last decade. The first chapter discusses the construction and the representation of polynomials. The second chapter focuses on the com...
New algorithms for determining discrete and continuous symmetries of polynomials - also known as binary forms in classical invariant theory - are presented, and implemented in MAPLE. The results are based on a new, comprehensive theory of moving frames that completely characterizes the equivalence and...
This chapter shows how to construct all fields with a finite number of elements: they all have \(p^{n}\) elements where p can be any prime number and n can be any number \(\ge 1\) . For the construction we begin with the information on polynomials with coefficients in a field foun...
Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the half-plane property: a determinant construction (exploiting “energy” arguments), and a permanent construction (exploiting the Heilmann–Lieb theorem on matching polynomials). We conclude...
There has been a lot of activity in the past decade or so to find various formulas and/or algorithms to calculate quiver polynomials of certain Dynkin types, such as [4–10, 12, 16, 24, 25]. By now we can claim that there are effective methods to find any such quiver polynomial. ...