The computation of multiple roots of a Bernstein basis polynomial This paper describes the algorithms of Musser and Gauss for the computation of multiple roots of a theoretically exact Bernstein basis polynomial xcbx86 5 f(y) when the coefficients of  :29,form f(y) are corrupted by noise. The exact roots of f(y) can therefore be assumed to be simple, and thus the problem reduces to the calculation of multiple roots of a polynomial fxcbx9c(y) that is near f(y), such that the backward error is small. The algorithms require many greatest common divisor (GCD) computations and polynomial deconvolutions, both of which are implemented by a structurepreserving matrix method. The motivation of these algorithms arises from the unstructured and structured condition numbers of a multiple root of a polynomial. These condition numbers have an elegant interpretation in terms of the pejorative manifold of xcbx86 12 f(y), which allows the geometric significance of the GCD computations and polynomial deconvolutions to be considered. A variant of the Sylvester resultant matrix is used for the GCD computations because it yields better results than the standard :29,form of this matrix, and the polynomial deconvolutions can be computed in several different ways, sequentially or simultaneously, and with the inclusion or omission of the preservation of the structure of the coefficient matrix. It is shown that Gaussxe2x80x99 algorithm yields better results than Musserxe2x80x99s algorithm, and the reason for these superior results is explained.

A New Iterative Method for the Computation of the Solutions of Nonlinear Equations Recently, by Costabile, Gualtieri and Serra (1999), an iterative method was presented for the computation of zeros of  :18,72,functions. This method combines the assured convergence of the bisectionlike algorithms with a superlinear convergence speed which characterizes Newtonlike methods. The order of the method and the cost per iteration is exactly equivalent to the Newton method. In this paper we present a new iterative method for the computation of the zeros of  functions with the same properties of convergence as the method proposed by Costabile, Gualtieri and Serra (1999) but with order                                                                            :167,270,388,2                  :169,272,392,xe2x89x902.41 for  :18,72,functions. :188,Compared with the methods of order                                                                            :167,270,388,2                  presented by Traub (1964), our methods ensure global convergence. Then we consider a generalization of this procedure which gives a class of methods of order                                                                            :314,n2  4                 )2, where n is the degree of the approximating polynomial, with onepoint iteration functions with memory. Finally a number of numerical tests are performed. The numerical results seem to show that, at least on a set of problems, the new methods work better than the methods proposed, and, therefore, than both the Newton and Alefeld and Potra (1992) methods.