Stability analysis of a family of optimal fourth-order methods for multiple roots

[EN] Complex dynamics tools applied on the rational functions resulting from a parametric family of roots solvers for nonlinear equations provide very useful results that have been stated in the last years. These qualitative properties allow the user to select the most efficient members from the family of iterative schemes, in terms of stability and wideness of the sets of convergent initial guesses. These tools have been widely used in the case of iterative procedures for finding simple roots and only recently are being applied on the case of multiplicity m >1. In this paper, by using weight function procedure, we design a general class of iterative methods for calculating multiple roots that includes some known methods. In this class, conditions on the weight function are not very restrictive, so a large number of different subfamilies can be generated, all of them are optimal with fourth-order of convergence. Their dynamical analysis gives us enough information to select those with better properties and test them on different numerical experiments, showing their numerical properties.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P, Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.Zafar, F.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2019). Stability analysis of a family of optimal fourth-order methods for multiple roots. Numerical Algorithms. 81(3):947-981. https://doi.org/10.1007/s11075-018-0577-0S94798181

Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics

[EN] In this paper, we propose a family of iterative methods for finding multiple roots, with known multiplicity, by means of the introduction of four univariate weight functions. With the help of these weight functions, that play an important role in the development of higher order convergent iterative techniques, we are able to construct three-point eight-order optimal multiple-root finders. Also, numerical experiments have been applied to a number of test equations for different special schemes from this family satisfying the conditions given in the convergence analysis. We have also compared the basins of attraction of some proposed and known methods in order to check the wideness of the sets of converging initial points for each problem. (C) 2018 Elsevier B.V. All rights reserved.This research was partially supported by Ministerio de Economia y Competitividad, Spain MTM2014-52016-C2-2-P, MTM2015-64013-P and Generalitat Valenciana, Spain PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.Zafar, F.; Cordero Barbero, A.; Sultana, S.; Torregrosa Sánchez, JR. (2018). Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics. Journal of Computational and Applied Mathematics. 342:352-374. https://doi.org/10.1016/j.cam.2018.03.033S35237434

A class of four parametric with- and without memory root finding methods

[EN] In this paper, we have constructed a derivative¿free weighted eighth¿order iterative method with and without memory for solving nonlinear equations. This method is an optimal method as it satisfies the Kung¿Traub conjecture. We have used four accelerating parameters, a univariate and a multivariate weight function at the second and third step of the method, respectively. This method is converted into with¿memory method by approximating the parameters using Newton's interpolating polynomials of appropriate degree to increase the order of convergence to 15.51560 and the efficiency index is nearly two. Numerical comparison of our methods is done with the recent methods of respective domain.This research was partially supported by Ministerio de Economía y Competitividad MTM2014-52016-C2-2-P, Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.Zafar, F.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Rafi, A. (2019). A class of four parametric with- and without memory root finding methods. Computational and Mathematical Methods. 1-14. https://doi.org/10.1002/cmm4.1024S114Cordero, A., Junjua, M.-D., Torregrosa, J. R., Yasmin, N., & Zafar, F. (2018). Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices. Mathematical Problems in Engineering, 2018, 1-12. doi:10.1155/2018/8093673Zafar, F., Akram, S., Yasmin, N., & Junjua, M.-D. (2016). On the construction of three step derivative free four-parametric methods with accelerated order of convergence. Journal of Nonlinear Sciences and Applications, 09(06), 4542-4553. doi:10.22436/jnsa.009.06.92King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Herzberger, J. (1974). Über Matrixdarstellungen für Iterationsverfahren bei nichtlinearen Gleichungen. Computing, 12(3), 215-222. doi:10.1007/bf02293107Jay, L. O. (2001). Bit Numerical Mathematics, 41(2), 422-429. doi:10.1023/a:1021902825707Chun, C., & Neta, B. (2015). On the new family of optimal eighth order methods developed by Lotfi et al. Numerical Algorithms, 72(2), 363-376. doi:10.1007/s11075-015-0048-9Gdawiec, K. (2017). Fractal patterns from the dynamics of combined polynomial root finding methods. Nonlinear Dynamics, 90(4), 2457-2479. doi:10.1007/s11071-017-3813-

Synthesis of 5-Fluorouracil Cocrystals with Novel Organic Acids as Coformers and Anticancer Evaluation against HCT-116 Colorectal Cell Lines

5-Fluorouracil (5-FU) being a mainstream anticancer drug is under keen and detailed investigation for prodrugs formulations in order to minimize the associated side effects. Cocrystallization of 5-FU is an innovative technique for the synthesis of 5-FU prodrugs to improve its anticancer effectiveness. The present study is based on the synthesis of 5-FU supramolecular synthons with four coformers: succinic acid, cinnamic acid, malic acid, and benzoic acid utilizing acetone as a solvent. Solid state grinding followed by a slow evaporation solution method was applied. Colorless clear crystals were obtained in all the cases. The cocrystal formation was supported with the help of Fourier transform infrared (FTIR) spectroscopy and powder X-ray diffraction (PXRD). Through FTIR, the main peaks of interest in the spectrum of 5-FU were N–H (3409.02 cm–1) and carbonyl group (1647.80 cm–1), which were prominently shifted in all spectra of the cocrystals demonstrating the replacement as well as the development of already present interactions with the new ones. For 5-FU–cinnamic acid cocrystals, the anticipated peaks were observed at 1673.13 cm–1 (−C═O) and 3566.89 cm–1 (N–H) manifesting a significant change in comparison to 5-FU. Furthermore, with the help of PXRD characterization, the representative peak of 5-FU was recorded at 2θ = 28.80°. The shifting of this specific peak and development of many new ones in the spectra of cocrystals proved the development of new structural entities. Finally, the anticancer activity of all cocrystals was evaluated in comparison to that of API. All cocrystals manifest significantly greater growth inhibition potential than the main active pharmaceutical ingredient. 5-FU–Cinnamic acid (3C) was the one that proved to be the most potent anticancer agent at all four concentrations: 4.82% (12 μg/mL), 34.21% (25 μg/mL), 55.08% (50 μg/mL), and 67.29% (100 μg/mL). In short, this study proved to be a true example to enhance the anticancer potential of 5-FU following fairly easy fabrication requirements of the cocrystallization phenomenon. After the successful synthesis of these supramolecular synthons and subsequent enhancement of growth inhibition potential of 5-FU, these cocrystals can further be evaluated for in vivo trials and membrane crossing potentials in the future

Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters

[EN] In this paper, we propose a family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity with the introduction of two free parameters and three univariate weight functions. Also numerical experiments have applied to a number of academical test functions and chemical problems for different special schemes from this family that satisfies the conditions given in convergence result.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Zafar, F.; Cordero Barbero, A.; Quratulain, R.; Torregrosa Sánchez, JR. (2018). Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. Journal of Mathematical Chemistry. 56(7):1884-1901. https://doi.org/10.1007/s10910-017-0813-1S18841901567R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, On developing fourth-order optimal families of methods for multiple roots and their dynamics. Appl. Math. Comput. 265(15), 520–532 (2015)R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, V. Kanwar, An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algor. 71(4), 775–796 (2016)R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, An eighth-order family of optimal multiple root finders and its dynamics. Numer. Algor. (2017). doi: 10.1007/s11075-017-0361-6F.I. Chicharro, A. Cordero, J. R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. ID 780153 (2013)A. Constantinides, N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications (Prentice Hall PTR, New Jersey, 1999)J.M. Douglas, Process Dynamics and Control, vol. 2 (Prentice Hall, Englewood Cliffs, 1972)Y.H. Geum, Y.I. Kim, B. Neta, A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Appl. Math. Comput. 270, 387–400 (2015)Y.H. Geum, Y.I. Kim, B. Neta, A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 283, 120–140 (2016)J.L. Hueso, E. Martınez, C. Teruel, Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 53, 880–892 (2015)L.O. Jay, A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)S. Li, X. Liao, L. Cheng, A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)S.G. Li, L.Z. Cheng, B. Neta, Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)B. Liu, X. Zhou, A new family of fourth-order methods for multiple roots of nonlinear equations. Nonlinear Anal. Model. Control 18(2), 143–152 (2013)M. Shacham, Numerical solution of constrained nonlinear algebraic equations. Int. J. Numer. Method Eng. 23, 1455–1481 (1986)M. Sharifi, D.K.R. Babajee, F. Soleymani, Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)J.R. Sharma, R. Sharma, Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)F. Soleymani, D.K.R. Babajee, T. Lofti, On a numerical technique forfinding multiple zeros and its dynamic. J. Egypt. Math. Soc. 21, 346–353 (2013)F. Soleymani, D.K.R. Babajee, Computing multiple zeros using a class of quartically convergent methods. Alex. Eng. J. 52, 531–541 (2013)R. Thukral, A new family of fourth-order iterative methods for solving nonlinear equations with multiple roots. J. Numer. Math. Stoch. 6(1), 37–44 (2014)R. Thukral, Introduction to higher-order iterative methods for finding multiple roots of nonlinear equations. J. Math. Article ID 404635 (2013)X. Zhou, X. Chen, Y. Song, Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. J. Comput. Math. Appl. 235, 4199–4206 (2011)X. Zhou, X. Chen, Y. Song, Families of third and fourth order methods for multiple roots of nonlinear equations. Appl. Math. Comput. 219, 6030–6038 (2013

Some Generalized Error Inequalities and Applications

<p>Abstract</p> <p>We present a family of four-point quadrature rule, a generalization of Gauss-two point, Simpson's <inline-formula> <graphic file="1029-242X-2008-845934-i1.gif"/></inline-formula>, and Lobatto four-point quadrature rule for twice-differentiable mapping. Moreover, it is shown that the corresponding optimal quadrature formula presents better estimate in the context of four-point quadrature formulae of closed type. A unified treatment of error inequalities for different classes of function is also given.</p

doi:10.1155/2008/845934 Research Article Some Generalized Error Inequalities and Applications

We present a family of four-point quadrature rule, a generalization of Gauss-two point, Simpson’s 3/8, and Lobatto four-point quadrature rule for twice-differentiable mapping. Moreover, it is shown that the corresponding optimal quadrature formula presents better estimate in the context of four-point quadrature formulae of closed type. A unified treatment of error inequalities for different classes of function is also given. Copyright q 2008 F. Zafar and N. A. Mir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

A Generalzation of Ostrowski-gruss Type Inequality for First Dierentiable Mappings

[[abstract]]In this paper, we improve and further generalize some Ostrowski- Gruss type inequalities involving rst dierentiable functions and apply them to probability density functions, generalized beta random variable and special means

A Note on the Generalization of Some New Čebyšev Type Inequalities

[[abstract]]In this paper, we present a generalized Cebyšev type inequality for absolutely continuous functions whose derivatives belong to L(subscript p) [a, b], p>1. Applications for probabilty density functions are also given