Ten methods to bound multiple roots of polynomials

Ten Methods to Bound Multiple Roots of Polynomials

Given a univariate polynomial P with a k-fold multiple root or a k-fold root cluster near some e z, we discuss various different methods to compute a disc near e z which either contains exactly or contains at least k roots of P . Many of the presented methods are known, some are new. We are especially interested in rigorous methods, that is taking into account all possible effects of rounding e...

Computing multiple roots of inexact polynomials

We present a combination of two novel algorithms that accurately calculate multiple roots of general polynomials. For a given multiplicity structure and initial root estimates, Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and calculates multiple roots simultaneously. To fulfill the input requirement of Algorithm I, we ...

Roots of Polynomials

A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product an(x − r1)(x − r2) · · · (x − rn), where an is the leading coefficient and r1, r2, . . . , rn are all of its n complex roots. We will look at how to find roots, or zeros, of polynomials in one variable. The solution of multi-variate polynomials can often be transformed ...

specialized methods to teach spelling: comparing three methods

چکیده: بررسی ادبیات مربوطه در کشور در زمینه یادگیری زبان انگلیسی نشان می‎دهد که علی‎رغم اهمیت املا در فرآیند یادگیری به طور عام و یادگیری زبان انگلیسی به طور خاص، این مولفه از جایگاهی متناسب با اهمیت آن برخوردار نیست و عمدتاً نادیده گرفته شده است. تحقیقات گسترده‎ای در خارج از کشور در مورد ماهیت این مولفه صورت گرفته است، در حالی که به جرأت می‎توان گفت در داخل کشور گامی در مورد درک ماهیت آن و فرآی...

Polynomials , roots , and interlacing