Iteration functions for approximating complex roots of cubic polynomials

König iteration functions are a generalization of Newton–Raphson method to determine roots of equations. These higher-degree rational functions possess additional fixed points, which are generally different from the desired roots. We first prove two new results: firstly, about these extraneous fixed points and, secondly, about the Julia sets of the König functions associated with the one-parameter family of quadratic polynomials. Then, after finding all the critical points of the König functions as applied to a one-parameter family of cubic polynomials, we examine the orbits of the ones available for convergence to an attracting periodic cycle, should such a cycle exist.

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On the additional fixed points of Schröder iteration functions associated with a one-parameter family of cubic polynomials

Computers & Graphics ◽

10.1016/s0097-8493(98)00071-5 ◽

1998 ◽

Vol 22 (5) ◽

pp. 629-634 ◽

Cited By ~ 6

Author(s):

V. Drakopoulos

Keyword(s):

Fixed Points ◽

Parameter Family ◽

Iteration Functions ◽

Cubic Polynomials

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A Neural Network-Based Approach for Approximating Arbitrary Roots of Polynomials

Mathematics ◽

10.3390/math9040317 ◽

2021 ◽

Vol 9 (4) ◽

pp. 317

Author(s):

Diogo Freitas ◽

Luiz Guerreiro Lopes ◽

Fernando Morgado-Dias

Keyword(s):

Iterative Methods ◽

Fundamental Problem ◽

Iterative Algorithms ◽

Rounding Errors ◽

Suggested Approach ◽

Simultaneous Iterative Methods ◽

Complex Roots ◽

Roots Of Polynomials ◽

Comparative Results ◽

The One

Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method.

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O31 Trajectories of anxiety in children young people and adults with rheumatic diseases in the wake of COVID-19: results from the COVID-19 European patient registry

Rheumatology ◽

10.1093/rheumatology/keab246.030 ◽

2021 ◽

Vol 60 (Supplement_1) ◽

Author(s):

Stephanie J W Shoop-Worrall ◽

Suzanne M M Verstappen ◽

Wendy Costello ◽

Saskya P Angevare ◽

Yosef Uziel ◽

...

Keyword(s):

Young People ◽

Rheumatic Diseases ◽

Model Fit ◽

Patient Registry ◽

Linear Quadratic ◽

High Anxiety ◽

Children And Young People ◽

Chi Squared ◽

The Uk ◽

Cubic Polynomials

Abstract Background/Aims Younger and older people with rheumatic diseases may experience increased anxiety during the COVID-19 pandemic, due to the uncertainty regarding their likelihood of contracting the virus, its complications alongside their existing condition and whether their immunosuppressive treatments pose additional risks. This study explored trajectories of anxiety in parents of children and young people (CYP) with rheumatic diseases and adults with rheumatic diseases in the six months following March 2020 during the COVID-19 pandemic. Methods CYP and adults recruited to the international COVID-19 European Patient Registry, a parent-led, online, self-referred prospective cohort recruiting participants globally, were selected if enrolled within 20th March to 17th April 2020. Anxiety scores (0-10, 10=Highest anxiety) were collected weekly for up to 28 weeks and denoted parent anxiety in the CYP cohort and self-reported anxiety in the adult cohort. Group-based trajectory models explored anxiety clusters using censored-normal models in the CYP and adult populations, separately. Linear, quadratic and cubic polynomials were tested within 1 to 10 clusters and optimal models selected based on a combination of model fit (BIC), parsimony and clinical plausibility. Demographic (country, age, gender) and clinical (diagnosis, disease control, respiratory comorbidity, immunosuppressive therapy) information and COVID-19 mitigation behaviours (isolation, distancing, none) were collected at initial enrolment and compared between clusters using Chi-squared, Fisher’s exact and Kruskal-Wallis tests. Results Among 498 CYP and 2640 adults, most were female (65%, 89%) and from the UK (50%, 84%), respectively. The most common diagnoses were polyarticular JIA (37%) and oligoarticular JIA (29%) among CYP and RA among the adults (63%). Respiratory comorbidities were uncommon in the CYP (10%) and adult (17%) cohorts, and most were taking any immunosuppressive therapies (85%, 87%), respectively. As of March 2020, 88% and 79% were self-isolating, respectively. In both the parents of CYP and adult cohorts, four trajectory clusters were identified with similar patterns: Persistent extremely high anxiety (32%, 17%), persistent high anxiety (43%, 41%), high anxiety that marginally improved (25%, 32%) and moderate anxiety that improved (11%, 10%). Among CYP, few characteristics distinguished the clusters. However, in the adult cohort, clusters with greater and more persistent anxiety were associated with higher levels of respiratory comorbidities, higher use of immunosuppressive therapies, higher initial levels of self-isolation and slightly older age than those with lower or improving anxiety over time. Conclusion This study reports four trajectories of anxiety during the COVID-19 pandemic that are consistent across parents of CYP with rheumatic diseases and among adults with these conditions. Despite relatively lower risks for CYP, parental anxiety regarding COVID-19 was high and not associated with characteristics of their child or of their child's disease. Among adults with rheumatic diseases, greater anxiety was associated with risk factors potentially associated with COVID-19 morbidity and mortality. Disclosure S.J.W. Shoop-Worrall: None. S.M.M. Verstappen: None. W. Costello: None. S.P. Angevare: None. Y. Uziel: None. C. Wouters: None. N. Wulffraat: Honoraria; Sobi. Grants/research support; Abbvie. R. Beesley: None.

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A New Family of High-Order Ehrlich-Type Iterative Methods

Mathematics ◽

10.3390/math9161855 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1855 ◽

Cited By ~ 1

Author(s):

Petko D. Proinov ◽

Maria T. Vasileva

Keyword(s):

Iterative Methods ◽

Local Convergence ◽

Semilocal Convergence ◽

High Order ◽

Convergence Theorems ◽

New Family ◽

Special Cases ◽

Iteration Functions ◽

Iteration Function ◽

Local Convergence Analysis

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.

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Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽

10.3390/axioms10030169 ◽

2021 ◽

Vol 10 (3) ◽

pp. 169

Author(s):

Avram Sidi

Keyword(s):

Nonlinear Equations ◽

Local Convergence ◽

Real Root ◽

Numerical Procedure ◽

Secant Method ◽

Convergence Theory ◽

Simple Complex ◽

Real Roots ◽

Complex Roots ◽

Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

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Locating complex eigenvalues for analytical eddy-current models used to detect flaws

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/compel-03-2019-0130 ◽

2019 ◽

Vol 38 (6) ◽

pp. 1800-1809

Author(s):

Grzegorz Tytko ◽

Łukasz Dawidowski

Keyword(s):

Eddy Current ◽

Design Methodology ◽

Complex Function ◽

Precise Determination ◽

Argument Principle ◽

Content Type ◽

Conductive Material ◽

Complex Roots ◽

Complex Eigenvalues

Purpose Discrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are complex numbers. Their computation consists of finding complex roots of a complex function that satisfies the electromagnetic interface conditions. The purpose of this paper is to present a method of computing complex eigenvalues that are roots of such a function. Design/methodology/approach The proposed approach involves precise determination of regions in which the roots are found and applying sets of initial points, as well as the Cauchy argument principle to calculate them. Findings The elaborated algorithm was implemented in Matlab and the obtained results were verified using Newton’s method and the fsolve procedure. Both in the case of magnetic and nonmagnetic materials, such a solution was the only one that did not skip any of the eigenvalues, obtaining the results in the shortest time. Originality/value The paper presents a new effective method of locating complex eigenvalues for analytical solutions of eddy current problems containing a conductive material with a hole.

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On Graeffe's Method for Complex Roots of Algebraic Equations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100002802 ◽

1924 ◽

Vol 22 (2) ◽

pp. 83-87 ◽

Cited By ~ 13

Author(s):

S. Brodetsky ◽

G. Smeal

Keyword(s):

Nineteenth Century ◽

Real Axis ◽

Practical Method ◽

Algebraic Equations ◽

Argand Diagram ◽

Considerable Difficulty ◽

The Real ◽

Real Roots ◽

Bipolar Coordinates ◽

Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

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