Resultant

Abstract: Publisher Summary This chapter discusses the numerical condition related to polynomials Polynomials permeate much of classical numerical analysis, either in the role of approximators, or as gauge functions for a variety of numerical methods, or in the role of characteristic polynomials of one kind or another It seems appropriate, therefore, to study some of their basic properties as they relate to computation It is found that one particular aspect of polynomials, namely, the extent to which they, or quantities related to them, are sensitive to small perturbations Condition numbers, such as those proposed, cannot be expected to do more than convey general guidelines as to the susceptibility of the respective maps to small changes in their domains By their very definition, they reflect worst case situations and, therefore, are inherently conservative measures It is important to note that exponential growth of the condition is also observed for piecewise polynomial functions, if represented in terms of normalized B-splines The basis consisting of the Lagrange polynomials for Chebyshev nodes, therefore, is optimally conditioned among all Lagrangian bases, and indeed among all polynomial bases, in the sense of attaining the optimal growth rate

Abstract: In this article, we discuss the current status of polynomial factoring (root finding) algorithms with some historical and mathematical background including size limits, convergence, accuracy and speed. The methods of root approximation versus root refinement are also examined. We then focus on two improved general purpose computational techniques, and in particular the factorization algorithm by Lindsey-Fox (L-F), which makes use of the fast Fourier transform to factor polynomials with random coefficients of degrees as high as 1 million. Computer simulations give insight that result in significant improvements in traditional approaches to an ancient problem.