Publications on the project
Publications on the project
[1] P. D. Proinov,
On the local convergence of Gargantini-Farmer-Loizou method for simultaneous approximation of multiple polynomial zeros,
Journal of Nonlinear Sciences and Applications 11 (2018), No. 9, 1045-1055.
http://dx.doi.org/10.22436/jnsa.011.09.03
Abstract
The paper deals with a well known iterative method for simultaneous computation of all zeros (of known multiplicities) of a polynomial with coefficients in a valued field. This method was independently introduced by Farmer and Loizou [M. R. Farmer, G. Loizou, Math. Proc. Cambridge Philos. Soc., 82 (1977), 427–437] and Gargantini [I. Gargantini, SIAM J. Numer. Anal., 15 (1978), 497–510]. If all zeros of the polynomial are simple, the method coincides with the famous Ehrlich’s method [L. W. Ehrlich, Commun. ACM, 10 (1967), 107–108]. We provide two types of local convergence results for the Gargantini-Farmer-Loizou method. The first main result improves the results of [N. V. Kyurkchiev, A. Andreev, V. Popov, Ann. Univ. Sofia Fac. Math. Mech., 78 (1984), 178–185] and [A. I. Iliev, C. R. Acad. Bulg. Sci., 49 (1996), 23–26] for this method. Both main results of the paper generalize the results of Proinov [P. D. Proinov, Calcolo, 53 (2016), 413–426] for Ehrlich’s method. The results in the present paper are obtained by applying a new approach for convergence analysis of Picard type iterative methods in finite-dimensional vector spaces.
[2] P. D. Proinov, S. I. Ivanov,
Convergence analysis of Sakurai–Torii–Sugiura iterative method for simultaneous approximation of polynomial zeros,
Journal of Computational and Applied Mathematics 357 (2019), 56-70.
https://doi.org/10.1016/j.cam.2019.02.021
Abstract
In 1991, T. Sakurai, T. Torii and H. Sugiura presented a fourth-order iterative algorithm for finding all zeros of a polynomial simultaneously. In this paper, we provide a detailed convergence analysis (local and semilocal) of this method. The new results improve and complement existing results due to Petković et al. (2003) and Petković (2008). Numerical examples are given to show the applicability of our semilocal convergence results.
[3] S. I. Cholakov,
Local and semilocal convergence of Wang-Zheng’s method for simultaneous finding polynomial zeros,
Symmetry (2019), Article Nо. 736, 15 pages.
https://doi.org/10.3390/sym11060736
Abstract
In 1984, Wang and Zheng (J. Comput. Math. 1984, 1, 70–76) introduced a new fourth order iterative method for the simultaneous computation of all zeros of a polynomial. In this paper, we present new local and semilocal convergence theorems with error estimates for Wang–Zheng’s method. Our results improve the earlier ones due to Wang and Wu (Computing 1987, 38, 75–87) and Petković, Petković and Rančić (J. Comput. Appl. Math. 2007, 205, 32–52).
[4] P.D. Proinov, M.T. Vasileva,
On the convergence of high-order Gargantini–Farmer–Loizou type iterative methods for simultaneous approximation of polynomial zeros,
Applied Mathematics and Computation 361 (2019), 202–214.
https://doi.org/10.1016/j.amc.2019.05.026
Abstract
In 1984, Kyurkchiev et al. constructed an infinite sequence of iterative methods for simultaneous approximation of polynomial zeros (with known multiplicity). The first member of this sequence of iterative methods is the famous root-finding method derived independently by Farmer and Loizou (1977) and Gargantini (1978). For every given positive integer N , the Nth method of this family has the order of convergence 2 N + 1. In this paper, we prove two new local convergence results for this family of iterative methods. The first one improves the result of Kyurkchiev et al. (1984). We end the paper with a comparison of the computational efficiency, the convergence behavior and the computational order convergence of different methods of the family.
[5] P.D. Proinov, M.D. Petkova,
Local and semilocal convergence of a family of multi-point Weierstrass-type root-finding methods,
Mediterranean Journal of Mathematics, 17 (2020), No. 4, Article No. 107, 20 pages.
https://doi.org/10.1007/s00009-020-01545-z
Full text can be found at https://rdcu.be/b46OW
Abstract
Weierstrass (1891) introduced his famous iterative method for numerical finding all zeros of a polynomial simultaneously. Kyurkchiev and Ivanov (1984) constructed a family of multi-point root-finding methods which are based on the Weierstrass method. The purpose of this research is threefold: (i) to develop a new simple approach for the study
of the local convergence of the multi-point simultaneous iterative methods; (ii) to present a new local convergence result for this family which improves in several directions the result of Kyurkchiev and Ivanov; (iii) to provide semilocal convergence results for Kyurkchiev-Ivanov's family of iterative methods.
[6] P. D. Proinov,
Fixed point theorems for generalized contractive mappings in metric spaces,
Journal of Fixed Point Theory and Applications 22 (2020), No. 1, Article No. 21, 27 pages.
https://doi.org/10.1007/s11784-020-0756-1
Full text can be found at https://rdcu.be/ca2az
Abstract
Let T be a self-mapping on a complete metric space (X, d). In this paper, we obtain new fixed point theorems assuming that T satisfies a contractive-type condition of the form
φ(d(Tx, Ty)) <= ψ(d(x, y)) or T satisfies a generalized contractive-type condition of the form φ(d(Tx, Ty)) <= ψ(m(x, y)), where φ, ψ: (0, \infty) \to R and m(x, y) is defined by
m(x, y) = max{d(x, y), d(x, Tx), d(y, Ty), [d(x, Ty) + d(y, Tx)]/2}.
In both cases, the results extend and unify many earlier results.
Among the other results, we prove that recent fixed point theorems of Wardowski (1912) and Jleli and Samet (2014) are equivalent to a special case of the well-known fixed point theorem of Skof (1977).
[7] P.I. Marcheva, S.I. Ivanov,
Convergence Analysis of a Modified Weierstrass Method for the Simultaneous Determination of Polynomial Zeros,
Symmetry 12 (2020), Article Nо. 1408, 19 pages.
https://doi.org/10.3390/sym12091408
Abstract
In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical importance because of its computationally verifiable initial condition and error estimate. Numerical experiments to show the applicability of our semilocal theorem are also presented. We finish this study with a theoretical and numerical comparison between the modified Weierstrass method and the classical Weierstrass method.
[8] P.D. Proinov, M.T. Vasileva,
Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously,
Symmetry 12 (2020), Article No. 1801, 25 pages.
https://doi.org/10.3390/sym12111801
Abstract
In 1977, Nourein (Intern. J. Comput. Math. 6:3, 1977) constructed a fourth-order iterative method for finding all zeros of a polynomial simultaneously. This method is also known as Ehrlich’s method with Newton’s correction because it is obtained by combining Ehrlich’s method (Commun. ACM 10:2, 1967) and the classical Newton’s method. The paper provides a detailed local convergence analysis of a well-known but not well-studied generalization of Nourein’s method for simultaneous finding of multiple polynomial zeros. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein’s method. Each of the new semilocal convergence results improves the result of Petković, Petković and Rančić (J. Comput. Appl. Math. 205:1, 2007) in several directions. The paper ends with several examples that show the applicability of our semilocal convergence theorems.
[9] P.D. Proinov,
Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros,
Symmetry 13 (2021), Article No. 371, 29 pages.
https://doi.org/10.3390/sym13030371
Abstract
In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.
[10] P.D. Proinov, M.D. Petkova,
On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros,
Mathematics, (2021), Article No. 1640, 16 pages.
https://doi.org/10.3390/math9141640
Abstract
In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an
approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.
[11] P.D. Proinov, M.T. Vasileva,
A New Family of High-Order Ehrlich-Type Iterative Methods,
Mathematics 9 (2021), Article No. 1855, 25 pages.
https://doi.org/10.3390/math9161855
Abstract
One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a
consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.
[12] P. D. Proinov, S. I. Ivanov
Semilocal convergence of Sakurai-Torii-Sugiura method for simultaneous approximation of polynomial,
in: Y. Simsek, ed., Proceedings Book of MICOPAM2018, Antalya, Turkey, 2018, ISBN 978-86-6016-036-4, pp. 94-98.
https://scholar.google.bg/scholar?oi=bibs&cluster=5861520848124309104&btnI=1&hl=bg
Abstract
In this talk, we provide a new semilocal convergence theorem for a fourth-order iterative method for the simultaneous approximation of polynomial zeros due to Sakurai, Torii and Sugiura [1]. This theorem improves and complements the existing result of Petković, Rančić and Milošević [2]. Two numerical examples are given to show some practical applications of our result.
[13] P. D. Proinov, S. I. Ivanov
Local and semilocal convergence of an accelerated Sakurai-Torii-Sugiura method with Newton's correction,
in: M. Bayram and A. Secer, ed., ICAAMM2019-Proceedings Book, Istanbul, Turkey, 2019, ISBN 978-605-69181-0-0, pp. 31-36.
Abstract
In this note, we provide local and semilocal convergence theorems for a new fifth-order method for simultaneous approximation of all the zeros of a polynomial. The new method is obtained by combining the famous Newton's method with a forth-order method due to Sakurai, Torii and Sugiura. We end the note with two numerical examples that show the applicability of our semilocal convergence result.
[14] P. D. Proinov, S. I. Ivanov
On the local convergence of Sakurai-Torii-Sugiura method for simultaneous approximation of polynomial zeros,
AIP Conference Proceedings 2116 (2019), Article Nо. 450027, 3 pages.
https://doi.org/10.1063/1.5114494
Abstract
In this talk, we will discuss new results on the convergence of a fourth-order iterative method for approximation of all zeros of a polynomial simultaneously due to Sakurai, Torii and Sugiura [1]. We provide two local convergence results for this method. The first one improves the result of Petković [2].