Implicit Deflation for Univariate Polynomial Root-finding

Journal of Computational and Applied Mathematics, 2003

We consider a modiÿcation of the Newton method for ÿnding a zero of a univariate function. The case of multiple roots is not treated. It is proven that the modiÿcation converges cubically. Per iteration it requires one evaluation of the function and two evaluations of its derivative. Thus, the modiÿcation is suitable if the calculation of the derivative has a similar or lower cost than that of the function itself. Classes of such functions are sketched and a numerical example is given.

Let F(x,u1,...,u') be a square-free polynomial which is monic w.r.t. x and let (s1,...,s') 2 C ' . If F(x,s1,...,s') is square-free then the roots of F w.r.t. x can be expanded into integral power series in u1 ¡ s1,...,u' ¡ s', and the power series roots can be calculated by well-known Newton's method. We use the floating-point number arithmetic to calculate the numerical coecients, and this paper investigates the numerical errors contained in the roots computed. We first express the power series root in terms of sub-polynomials of F(x,s1+v1,...,s'+v'), with (v1,...,v') a set of new variables, and clarify various properties of the power series roots. Then, we investigate the cancellation errors when the expansion point (s1,...,s') is close to a "singular point" and far from the origin, respectively. We find that, although the cancellation errors are not large usually, they become extremely large in some cases. In fact,...

Journal of Computational and Applied Mathematics, 1993

Carstensen, C. and M.S. PetkoviC, On iteration methods without derivatives for the simultaneous determination of polynomial zeros, Journal of Computational and Applied Mathematics 45 (1993) 251-266. Several algorithms for simultaneously approximating simple complex zeros of a polynomial are presented. These algorithms use Weierstrass' corrections and do not require any polynomial derivatives. It is shown that Nourein's method is, actually, regula falsi for Weierstrass' corrections. Convergence analysis and computational efficiency are given for the considered methods in complex and circular arithmetic. Special attention is paid to hybrid methods that combine the efficiency of floating-point arithmetic and the inclusion property of interval arithmetic.

Applied Mathematics and Computation, 2013

There are very few optimal fourth order methods for solving nonlinear algebraic equations having roots of multiplicity m. Here we compare 4 such methods, two of which require the evaluation of the ðm À 1Þ st root. We will show that such computation does not affect the overall cost of the method. Published by Elsevier Inc. The method is based on Newton's method for the function GðxÞ ¼ ffiffiffiffiffiffiffiffi ffi f ðxÞ m p which obviously has a simple root at a, the multiple root with multiplicity m of f ðxÞ.

2015

constructed an infinite sequence of Weierstrass-type iterative methods for approximating all zeros of a polynomial simultaneously. The first member of this sequence of iterative methods is the famous method of Weierstrass (1891) and the second one is the method of Nourein (1977). For a given integer N ≥ 1, the Nth method of this family has the order of convergence N + 1. Currently in the literature, there are only local convergence results for these methods. The main purpose of this paper is to present semilocal convergence results for the Weierstrass-type methods under computationally verifiable initial conditions and with computationally verifiable a posteriori error estimates.

Communications in Mathematical Physics, 1983

Using Newton's method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function. This article describes the behavior of Newton iteration for cubic polynomials. After a change of variables, these polynomials can be parametrized by a single complex parameter, and the Newton transformation has a single critical point other than its fixed points at the roots of the polynomial. We describe the behavior of the orbit of the free critical point as the parameter is varied. The Julia set, points where Newton's method fail to converge, is also pictured. These sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle.

2000

We describe the implementation and the use of the package MPSolve (Multiprecision Polynomial Solver) for the approximation of the roots of a univariate polynomial p(x) = n i=0 aix i . The package relies on an algorithm, based on simultaneous approximation techniques, that generates a sequence A1 ⊃ A2 ⊃ . . . ⊃ A k of nested sets containing the root neighborhood of p(x) (defined by the precision of the coefficients) and outputs Newton-isolated approximations of the roots. The algorithm, particularly suited to deal with sparse polynomials or polynomials defined by straight line programs, adaptively adjusts the working precision to the conditioning of each root. In this way, only the leading digits of the input coefficients sufficient to recover the requested information on the roots are actually involved in the computation. This feature makes our algorithm particularly suited to deal with polynomials arising from certain symbolic computations where the coefficients are typically integers with hundreds or thousands digits.

Numerical Algorithms, 2016

In this paper, we first present a family of iterative algorithms for simultaneous determination of all zeros of a polynomial. This family contains two well-known algorithms: Dochev-Byrnev's method and Ehrlich's method. Second, using Proinov's approach to studying convergence of iterative methods for polynomial zeros, we provide a semilocal convergence theorem that unifies the results of Proinov (Appl. Math. Comput. 284: 102-114, 2016) for Dochev-Byrnev's and Ehrlich's methods.

Fibonacci Quarterly

Journal of Computational and Applied Mathematics, 1991

We propose a hybrid method to determine all the zeros of a polynomial, simultaneously, by combining the single-root method and the simultaneous one. The present method has a high convergence order even for multiple roots by using the Pad6 approximation.