Perturbation Mappings in Polynomiography

In the paper, a modification of rendering algorithm of polynomiograph is presented. Polynomiography is a method of visualization of complex polynomial root finding process and it has applications among other things in aesthetic pattern generation. The proposed modification is based on a perturbation mapping, which is added in the iteration process of the root finding method. The use of the perturbation mapping alters the shape of the polynomiograph, obtaining in this way new and diverse patterns. The results from the paper can further enrich the functionality of the existing polynomiography software

On improved three-step schemes with high efficiency index and their dynamics

This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT Republica Dominicana.Babajee, DKR.; Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR. (2014). On improved three-step schemes with high efficiency index and their dynamics. Numerical Algorithms. 65(1):153-169. https://doi.org/10.1007/s11075-013-9699-6S153169651Pang, J.S., Chan, D.: Iterative methods for variational and complementary problems. Math. Program. 24(1), 284–313 (1982)Sun, D.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91(1), 123–140 (1996)Chun, C., Neta, B.: A new sixth-order scheme for nonlinear equations. Appl. Math. Lett. 25, 185–189 (2012)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)Neta, B.: A new family of high-order methods for solving equations. Int. J. Comput. Math. 14, 191–195 (1983)Neta, B.: On Popovski’s method for nonlinear equations. Appl. Math. Comput. 201, 710–715 (2008)Chun, C., Neta, B.: Some modifications of Newton’s method by the method of undeterminate coefficients. Comput. Math. Appl. 56, 2528–2538 (2008)Chun, C., Lee, M.Y., Neta, B., Dzunic, J.: On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth-order convergence. J. Comput. Appl. Math. 235, 3189–3194 (2011)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: A family of modified Ostrowski’s methods with optimal eighth order of convergence. Appl. Math. Lett. 24, 2082–2086 (2011)Heydari, M., Hosseini, S.M., Loghmani, G.B.: On two new families of iterative methods for solving nonlinear equations with optimal order. Appl. Anal. Dis. Math. 5, 93–109 (2011)Neta, B., Petkovic, M.S.: Construction of optimal order nonlinear solvers using inverse interpolation. Appl. Math. Comput. 217, 2448–2445 (2010)Sharifi, M., Babajee, D.K.R., Soleymani, F.: Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)Soleymani, F., Karimi Vanani, S., Khan, M., Sharifi, M.: Some modifications of King’s family with optimal eighth order of convergence. Math. Comput. Model. 55, 1373–1380 (2012)Soleymani, F., Karimi Vanani, S., Jamali Paghaleh, M.: A class of three-step derivative-free root solvers with optimal convergence order. J. Appl. Math. 2012, Article ID 568740, 15 pp. (2012). doi: 10.1155/2012/568740Soleymani, F., Sharifi, M., Mousavi, B.S.: An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. J. Optim. Theory Appl. 153, 225–236 (2012)Stewart, B.D.: Attractor basins of various root-finding methods. M.S. Thesis, Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA (2001)Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Scientia 10, 3–35 (2004)Amat, S., Busquier, S., Plaza, S.: Dynamics of the King and Jarratt iterations. Aequ. Math. 69, 212–223 (2005)Amat, S., Busquier, S., Plaza, S.: Chaotic dynamics of a third-order Newton type method. J. Math. Anal. Appl. 366, 24–32 (2010)Neta, B., Chun, C., Scott, M.: A note on the modified super-Halley method. Appl. Math. Comput. 218, 9575–9577 (2012)Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)Ardelean, G.: A comparison between iterative methods by using the basins of attraction. Appl. Math. Comput. 218, 88–95 (2011)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Babajee, D.K.R.: Analysis of higher order variants of Newton’s method and their applications to differential and integral equations and in ocean acidification. Ph.D. Thesis, University of Mauritius (2010

Polynomiography for the Polynomial Infinity Norm via Kalantari's Formula and Nonstandard Iterations

In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-Euler-Schroder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantari's recent results in finding the maximum modulus of a complex polynomial based on Newton's process with the Picard iteration to other MMP-processes with various non-standard iterations

Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications

There is a huge group of algorithms described in the literature that iteratively find solutions of a given equation. Most of them require tuning. The article presents root-finding algorithms that are based on the Newton-Raphson method which iteratively finds the solutions, and require tuning. The modification of the algorithm implements the best position of particle similarly to the particle swarm optimisation algorithms. The proposed approach allows visualising the impact of the algorithm's elements on the complex behaviour of the algorithm. Moreover, instead of the standard Picard iteration, various feedback iteration processes are used in this research. Presented examples and the conducted discussion on the algorithm's operation allow to understand the influence of the proposed modifications on the algorithm's behaviour. Understanding the impact of the proposed modification on the algorithm's operation can be helpful in using it in other algorithms. The obtained images also have potential artistic applications

High Performance Multidimensional Iterative Processes for Solving Nonlinear Equations

[ES] En gran cantidad de problemas de la matemática aplicada, existe la necesidad de resolver ecuaciones y sistemas no lineales, dado que numerosos problemas, finalmente, se reducen a estos. Conforme aumenta la dificultad de los sistemas, la obtención de la solución analítica se vuelve más compleja. Además, con el aumento de las herramientas computacionales, las dimensiones de los problemas a resolver han crecido de manera exponencial, por lo que se vuelve más necesario obtener una aproximación a la solución de manera sencilla y que no requiera mucho tiempo y coste computacional. Esta es una de las razones por las que los métodos iterativos han aumentado su importancia en los últimos años, ya que se han diseñado multitud de procesos con el fin de que converjan rápidamente a la solución y, de esta forma, poder resolver problemas que con las herramientas clásicas resultaría más costoso. La presente Tesis Doctoral, se centra en estudiar y diseñar numerosos métodos iterativos que mejoren a los esquemas clásicos en cuanto a su orden de convergencia, accesibilidad, cantidad de soluciones que obtienen o aplicabilidad a problemas con características especiales, como la no diferenciabilidad o la multiplicidad de las raíces. Entre los procesos que se estudian en esta memoria, se pueden encontrar desde una familia de métodos multipaso óptimos para la resolución de ecuaciones, hasta una familia paramétrica libre de derivadas de esquemas con función peso a la que se introduce memoria para la resolución de sistemas no lineales. Se destacan otros métodos en esta memoria como esquemas iterativos que obtienen raíces con diversas multiplicidades para ecuaciones y procesos que aproximan raíces de forma simultánea, tanto para ecuaciones como para sistemas, y, tanto para raíces simples como para múltiples. Además, parte de esta memoria se centra en cómo realizar el análisis dinámico para métodos iterativos con memoria que resuelven sistemas de ecuaciones no lineales, a la par que se realiza dicho estudio para diversos esquemas iterativos conocidos. Este análisis dinámico permite visualizar y analizar los posibles comportamientos de los procesos iterativos en función de las aproximaciones iniciales. Los resultados anteriormente descritos forman parte de esta Tesis Doctoral para la obtención del título de Doctora en Matemáticas.[CA] En gran quantitat de problemes de la matemàtica aplicada, existeix la necessitat de resoldre equacions i sistemes no lineals, atés que nombrosos problemes, finalment, es redueixen a aquests. Conforme augmenta la dificultat dels sistemes, l'obtenció de la solució analítica es torna més complexa. A més, amb l'augment de les eines computacionals, les dimensions dels problemes a resoldre han crescut de manera exponencial, per la qual cosa es torna més necessari obtindre una aproximació a la solució de manera senzilla i que no requerisca molt temps i cost computacional. Aquesta és una de les raons per les quals els mètodes iteratius han augmentat la seua importància en els últims anys, ja que s'han dissenyat multitud de processos amb la finalitat que convergisquen ràpidament a la solució i, d'aquesta manera, poder resoldre problemes que amb les eines clàssiques resultaria més costós. La present Tesi Doctoral, es centra en estudiar i dissenyar nombrosos mètodes iteratius que milloren als esquemes clàssics en quant al seu ordre de convergència, accessibilitat, quantitat de solucions que obtenen o aplicabilitat a problemes amb característiques especials, com la no diferenciabilitat o la multiplicitat de les arrels. Entre els processos que s'estudien en aquesta memòria, es poden trobar des d'una família de mètodes multipas òptims per a la resolució d'equacions, fins a una família paramètrica lliure de derivades de esquemes amb funció pes a la que s'introdueix memòria per a la resolució de sistemes no lineals. Es destanquen altres mètodes en aquesta memòria com esquemes iteratius que obtenen arrels amb diverses multiplicitats per a equacions i processos que aproximen arrels de manera simultània, tant per a equacions com per a sistemes, i, tant per a arrels simples com per a múltiples. A més, part d'aquesta memòria es centra en com realitzar l'anàlisi dinàmic per a mètodes iteratius amb memòria que resolen sistemes d'equacions no lineals, al mateix temps que es realitza aquest estudi per a diversos esquemes iteratius coneguts. Aquest anàlisi dinàmic permet visualitzar i analitzar els possibles comportaments dels mètodes iteratius en funció de les aproximacions inicials. Els resultats anteriorment descrits formen part d'aquesta Tesi Doctoral per a l'obtenció del títol de Doctora en Matemàtiques.[EN] In a large number of problems in applied mathematics, there is a need to solve nonlinear equations and systems, since many problems eventually are reduced to these. As the difficulty of the systems increases, obtaining the analytical solution becomes more complex. Furthermore, with the growth of computational tools, the dimensions of the problems to be solved have increased exponentially, making it more essential to obtain an approximation to the solution in a simple way that does not require significant time and computational cost. That is one of the reasons why iterative methods have increased their importance in recent years, as a multitude of schemes have been designed to converge rapidly to the solution and, in this way, to be able to solve problems that would be more arduous to solve using classical tools. This Doctoral Thesis focuses on the study and design of numerous iterative methods that improve classical schemes in terms of their order of convergence, accessibility, number of solutions obtained or applicability to problems with special characteristics, such as non-differentiability or multiplicity of roots. The procedures studied in this report range from a family of optimal multi-step methods for solving equations, to a parametric derivative-free family of weight function schemes, to which memory is introduced for solving nonlinear systems. Additional procedures are described in this report such as iterative schemes that obtain roots with different multiplicities for equations and methods that approximate roots simultaneously for equations as well as for systems, and for simple as well as for multiples roots. In addition, part of this report focuses on how to perform the dynamical analysis for iterative schemes with memory that solve systems of nonlinear equations, as well as this study is carried out for different known iterative procedures. This dynamical analysis allows us to visualise and analyse the possible behaviours of the iterative methods depending on the initial approximations. The results described above form part of this Doctoral Thesis to obtain the title of Doctor in Mathematics.Triguero Navarro, P. (2023). High Performance Multidimensional Iterative Processes for Solving Nonlinear Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/19426