Petko D. Proinov
In this paper we develop a unified theory for cone metric spaces over a solid vector space. As an application of the new theory we present full statements of the iterated contraction principle and the Banach contraction principle in cone metric spaces over a solid vector space.
Petko D. Proinov
In this paper, we prove some general convergence theorems for the Picard iteration in cone metric spaces over a solid vector space. As an application, we provide a detailed convergence analysis of the Weierstrass iterative method for computing all zeros of a polynomial simultaneously. These results improve and generalize existing ones in the literature.
Petko D. Proinov
In this paper we present a new semilocal convergence theorem from data at one point for the Weierstrass iterative method for the simultaneous computation of polynomial zeros. The main result generalizes and improves all previous ones in this area.
Petko D. Proinov
In this paper, we establish a general semilocal convergence theorem (with computationally verifiable initial conditions and error estimates) for iterative methods for simultaneous approximation of polynomial zeros. As application of this theorem, we provide new semilocal convergence results for Ehrlich's and Dochev-Byrnev's root-finding methods. These results improve the results of Petković, Herceg and Ilić [Numer. Algorithms 17 (1998) 313--331] and Proinov [C.~R. Acad. Bulg. Sci. 59 (2006) 705--712]. We also prove that Dochev-Byrnev's method (1964) is identical to Pre{\v s}ić-Tanabe's method (1972).
Petko D. Proinov
In this paper we study some iterative methods for simultaneous approximation of polynomial zeros. We give new semilocal convergence theorems with error bounds for Ehrlich's and Nourein's iterations. Our theorems generalize and improve recent results of Zheng and Huang [J. Comput. Math. 18 (2000), 113--122], Petković and Herceg [J. Comput. Appl. Math. 136 (2001), 283--307] and Nedić [Novi Sad J. Math. 31 (2001), 103--111]. We also present a new sufficient condition for simple zeros of a polynomial.
Petko D. Proinov, Maria T. Vasileva
Kyurkchiev and Andreev (1985) 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 $N \ge 1$, the $N$th 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.
Petko D. Proinov, Maria T. Vasileva
We study a family of high order Ehrlich-type methods for approximating all zeros of a polynomial simultaneously. Let us denote by $T^{(1)}$ the famous Ehrlich method (1967). Starting from $T^{(1)}$, Kjurkchiev and Andreev (1987) have introduced recursively a sequence ${(T^{(N)})_{N=1}^\infty}$ of iterative methods for simultaneous finding polynomial zeros. For given $N \ge 1$, the Ehrlich-type method $T^{(N)}$ has the order of convergence ${2 N + 1}$. In this paper, we establish two new local convergence theorems as well as a semilocal convergence theorem (under computationally verifiable initial conditions and with a posteriori error estimate) for the Ehrlich-type methods $T^{(N)}$. Our first local convergence theorem generalizes a result of Proinov (2015) and improves the result of Kjurkchiev and Andreev (1987). The second local convergence theorem generalizes another recent result of Proinov (2015), but only in the case of maximum-norm. Our semilocal convergence theorem is the first result in this direction.
Petko D. Proinov
The construction of initial conditions of an iterative method is one of the most important problems in solving nonlinear equations. In this paper, we obtain relationships between different types of initial conditions that guarantee the convergence of iterative methods for simultaneous finding all zeros of a polynomial. In particular, we show that any local convergence theorem for a simultaneous method can be converted into a convergence theorem with computationally verifiable initial conditions which is of practical importance. Thus, we propose a new approach for obtaining semilocal convergence results for simultaneous methods via local convergence results.