Behavior of fixed and critical points of the (alpha,c)-family of iterative methods

In this paper we study the dynamical behavior of the -family of iterative methods for solving nonlinear equations, when we apply the fixed point operator associated to this family on quadratic polynomials. This is a family of third-order iterative root-finding methods depending on two parameters; so, as we show throughout this paper, its dynamics is really interesting, but complicated. In fact, we have found in the real -plane a line in which the corresponding elements of the family have a lower number of free critical points. As this number is directly related with the quantity of basins of attraction, it is probable to find more stable behavior between the elements of the family in this region.Supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02. The first and fourth authors were also partially supported by P11B2011-30 (Universitat Jaume I), the second and third authors were also partially supported by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia SP20120474.Campos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; P. Vindel (2015). Behavior of fixed and critical points of the (alpha,c)-family of iterative methods. Journal of Mathematical Chemistry. 53(3):807-827. https://doi.org/10.1007/s10910-014-0465-3S807827533A.F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, vol. 132 (Springer, New York, 1991)B. Campos, A. Cordero, A. Magreñan, J.R. Torregrosa, P. Vindel, Study of a bi-parametric family of iterative methods, Abstra. Appl. Anal. (2014). doi: 10.1155/2014/141643B. Campos, A. Cordero, A. Magreñan, J.R. Torregrosa, P. Vindel, Bifurcations of the roots of a 6-degree symmetric polynomial coming from the fixed point operator of a class of iterative methods. in Proceedings of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE, ed. by J. Vigo-Aguiar (2014), pp. 253–264B. Campos, A. Cordero, J.R. Torregrosa, P. Vindel, Dynamics of the family of c-iterative methods. Int. J. Compt. Math. (2014). doi: 10.1080/00207160.2014.893608F. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. (2013). doi: 10.1155/2013/780153A. Cordero, J. García-Maimó, J.R. Torregrosa, M.P. Vassileva, P. Vindel, Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)A. Cordero, J.R. Torregrosa, P. Vindel, Dynamics of a family of Chebyshev–Halley type method. Appl. Math. Comput. 219, 8568–8583 (2013)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. J. Math. Chem. 49(7), 1384–1415 (2011)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: formalism and first applications to atomic problems. J. Math. Chem. 22, 107–116 (1997)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry. J. Math. Chem. 51(9), 2361–2385 (2013)K. Maleknejad, M. Alizadeh, An efficient numerical sheme for solving Hammerstein integral equation arisen in chemical phenomenon. Proc. Comput. Sci. 3, 361–364 (2011)J. Milnor, Dynamics in One Complex Variable (Princeton University Press, New Jersey, 2006

Iterative methods for nonlinear equations or systems and their applications 2014

Alicia Cordero and Juan R. Torregrosa were partially supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02.Torregrosa Sánchez, JR.; Argyros, IK.; Chun, C.; Cordero Barbero, A.; Soleymani, F. (2014). Iterative methods for nonlinear equations or systems and their applications 2014. Journal of Applied Mathematics. 2014. https://doi.org/10.1155/2014/293263S201

Iterative fixed-point methods for solving nonlinear problems: dynamics and applications

Alicia Cordero and Juan R. Torregrosa were partially supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02.Torregrosa Sánchez, JR.; Cordero Barbero, A.; Kanwar, V.; Kou, J. (2014). Iterative fixed-point methods for solving nonlinear problems: dynamics and applications. Abstract and Applied Analysis. 2014. https://doi.org/10.1155/2014/313061S201

Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators

[EN] In this paper, the convergence and dynamics of improved Chebyshev-Secant-type iterative methods are studied for solving nonlinear equations in Banach space settings. Their semilocal convergence is established using recurrence relations under weaker continuity conditions on first-order divided differences. Convergence theorems are established for the existence-uniqueness of the solutions. Next, center-Lipschitz condition is defined on the first-order divided differences and its influence on the domain of starting iterates is compared with those corresponding to the domain of Lipschitz conditions. Several numerical examples including Automotive Steering problems and nonlinear mixed Hammerstein-type integral equations are analyzed, and the output results are compared with those obtained by some of similar existing iterative methods. It is found that improved results are obtained for all the numerical examples. Further, the dynamical analysis of the iterative method is carried out. It confirms that the proposed iterative method has better stability properties than its competitors.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22.Kumar, A.; Gupta, DK.; Martínez Molada, E.; Hueso, JL. (2021). Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators. Numerical Algorithms. 86(3):1051-1070. https://doi.org/10.1007/s11075-020-00922-9S10511070863Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3-4), 433–445 (2001)Ezquerro, J.A., Grau-Sánchez, Miquel, Hernández, M.A.: Solving non-differentiable equations by a new one-point iterative method with memory. J. Complex. 28(1), 48–58 (2012)Ioannis , K.A., Ezquerro, J.A., Gutiérrez, J.M., hernández, M.A., saïd Hilout: On the semilocal convergence of efficient Chebyshev-Secant-type methods. J. Comput. Appl. Math. 235(10), 3195–3206 (2011)Hongmin, R., Ioannis, K.A.: Local convergence of efficient Secant-type methods for solving nonlinear equations. Appl. Math. comput. 218(14), 7655–7664 (2012)Ioannis, Ioannis K.A., Hongmin, R.: On the semilocal convergence of derivative free methods for solving nonlinear equations. J. Numer. Anal. Approx. Theory 41 (1), 3–17 (2012)Hongmin, R., Ioannis, K.A.: On the convergence of King-Werner-type methods of order $1+\sqrt {2}$ free of derivatives. Appl. Math. Comput. 256, 148–159 (2015)Kumar, A., Gupta, D.K., Martínez, E., Sukhjit, S.: Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces. J. Comput. Appl. Math. 330, 732–741 (2018)Grau-Sánchez, M., Noguera, M., Gutiérrez, J.M.: Frozen iterative methods using divided differences “à la Schmidt–Schwetlick”. J. Optim. Theory Appl. 160 (3), 931–948 (2014)Louis, B.R.: Computational Solution of Nonlinear Operator Equations. Wiley, New York (1969)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Parisa, B., Cordero, A., Taher, L., Kathayoun, M., Torregrosa, J.R.: Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics 87 (2), 913–938 (2017)Chun, C., Neta, B.: The basins of attraction of Murakami’s fifth order family of methods. Appl. Numer. Math. 110, 14–25 (2016)Magreñán, Á. A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)Ramandeep, B., Cordero, A., Motsa, S.S., Torregrosa, J.R.: Stable high-order iterative methods for solving nonlinear models. Appl. Math. Comput. 303, 70–88 (2017)Pramanik, S.: Kinematic synthesis of a six-member mechanism for automotive steering. Trans Ame Soc. Mech. Eng. J. Mech. Des. 124(4), 642–645 (2002

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

[EN] This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection-diffusion-reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive techniques into the splitting approach. The numerical analysis of the adapted iterative splitting schemes is considered and we develop the underlying error estimates for the application of the adaptive schemes. The performance of the method with respect to the accuracy and the acceleration is evaluated in different numerical experiments. We test the benefits of the adaptive splitting approach on highly nonlinear Burgers' and Maxwell-Stefan diffusion equations.This research was funded by German Academic Exchange Service grant number 91588469. We acknowledge support by the DFG Open Access Publication Funds of the Ruhr-Universität of Bochum, Germany and by Ministerio de Economía y Competitividad, Spain, under grant PGC2018-095896-B-C21-C22.Geiser, J.; Hueso, JL.; Martínez Molada, E. (2020). Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations. Mathematics. 8(3):1-22. https://doi.org/10.3390/math8030302S12283Auzinger, W., & Herfort, W. (2014). Local error structures and order conditions in terms of Lie elements for exponential splitting schemes. Opuscula Mathematica, 34(2), 243. doi:10.7494/opmath.2014.34.2.243Auzinger, W., Koch, O., & Quell, M. (2016). Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions. Numerical Algorithms, 75(1), 261-283. doi:10.1007/s11075-016-0206-8Descombes, S., & Massot, M. (2004). Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction. Numerische Mathematik, 97(4), 667-698. doi:10.1007/s00211-003-0496-3Descombes, S., Dumont, T., Louvet, V., & Massot, M. (2007). On the local and global errors of splitting approximations of reaction–diffusion equations with high spatial gradients. International Journal of Computer Mathematics, 84(6), 749-765. doi:10.1080/00207160701458716McLachlan, R. I., & Quispel, G. R. W. (2002). Splitting methods. Acta Numerica, 11, 341-434. doi:10.1017/s0962492902000053Trotter, H. F. (1959). On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4), 545-545. doi:10.1090/s0002-9939-1959-0108732-6Strang, G. (1968). On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis, 5(3), 506-517. doi:10.1137/0705041Jahnke, T., & Lubich, C. (2000). Bit Numerical Mathematics, 40(4), 735-744. doi:10.1023/a:1022396519656Nevanlinna, O. (1989). Remarks on Picard-Lindelöf iteration. BIT, 29(2), 328-346. doi:10.1007/bf01952687Farago, I., & Geiser, J. (2007). Iterative operator-splitting methods for linear problems. International Journal of Computational Science and Engineering, 3(4), 255. doi:10.1504/ijcse.2007.018264DESCOMBES, S., DUARTE, M., DUMONT, T., LOUVET, V., & MASSOT, M. (2011). ADAPTIVE TIME SPLITTING METHOD FOR MULTI-SCALE EVOLUTIONARY PARTIAL DIFFERENTIAL EQUATIONS. Confluentes Mathematici, 03(03), 413-443. doi:10.1142/s1793744211000412Geiser, J. (2008). Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations. Journal of Computational and Applied Mathematics, 217(1), 227-242. doi:10.1016/j.cam.2007.06.028Dimov, I., Farago, I., Havasi, A., & Zlatev, Z. (2008). Different splitting techniques with application to air pollution models. International Journal of Environment and Pollution, 32(2), 174. doi:10.1504/ijep.2008.017102Karlsen, K. H., Lie, K.-A., Natvig, J. ., Nordhaug, H. ., & Dahle, H. . (2001). Operator Splitting Methods for Systems of Convection–Diffusion Equations: Nonlinear Error Mechanisms and Correction Strategies. Journal of Computational Physics, 173(2), 636-663. doi:10.1006/jcph.2001.6901Geiser, J. (2010). Iterative operator-splitting methods for nonlinear differential equations and applications. Numerical Methods for Partial Differential Equations, 27(5), 1026-1054. doi:10.1002/num.20568Geiser, J., & Wu, Y. H. (2015). Iterative solvers for the Maxwell–Stefan diffusion equations: Methods and applications in plasma and particle transport. Cogent Mathematics, 2(1), 1092913. doi:10.1080/23311835.2015.1092913Geiser, J., Hueso, J. L., & Martínez, E. (2017). New versions of iterative splitting methods for the momentum equation. Journal of Computational and Applied Mathematics, 309, 359-370. doi:10.1016/j.cam.2016.06.002Boudin, L., Grec, B., & Salvarani, F. (2012). A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discrete & Continuous Dynamical Systems - B, 17(5), 1427-1440. doi:10.3934/dcdsb.2012.17.1427Duncan, J. B., & Toor, H. L. (1962). An experimental study of three component gas diffusion. AIChE Journal, 8(1), 38-41. doi:10.1002/aic.69008011

Impact on stability by the use of memory in Traub-type schemes

[EN] In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub's method, they have been designed using linear approximations or the Newton's interpolation polynomials. In both cases, the parameters use information from the current and the previous iterations, so they define a method with memory. Moreover, they achieve higher order of convergence than Traub's scheme without any additional functional evaluations. The real dynamical analysis verifies that the proposed methods with memory not only converge faster, but they are also more stable than the original scheme. The methods selected by means of this analysis can be applied for solving nonlinear problems with a wider set of initial estimations than their original partners. This fact also involves a lower number of iterations in the process.This research was partially supported by Ministerio de Ciencia, Innovacion y Universidades under grants PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE).Chicharro, FI.; Cordero Barbero, A.; Garrido, N.; Torregrosa Sánchez, JR. (2020). Impact on stability by the use of memory in Traub-type schemes. Mathematics. 8(2):1-16. https://doi.org/10.3390/math8020274S11682Shacham, M. (1989). An improved memory method for the solution of a nonlinear equation. Chemical Engineering Science, 44(7), 1495-1501. doi:10.1016/0009-2509(89)80026-0Balaji, G. V., & Seader, J. D. (1995). Application of interval Newton’s method to chemical engineering problems. Reliable Computing, 1(3), 215-223. doi:10.1007/bf02385253Shacham, M. (1986). Numerical solution of constrained non-linear algebraic equations. International Journal for Numerical Methods in Engineering, 23(8), 1455-1481. doi:10.1002/nme.1620230805Shacham, M., & Kehat, E. (1973). Converging interval methods for the iterative solution of a non-linear equation. Chemical Engineering Science, 28(12), 2187-2193. doi:10.1016/0009-2509(73)85008-0Amat, S., Busquier, S., & Plaza, S. (2010). Chaotic dynamics of a third-order Newton-type method. Journal of Mathematical Analysis and Applications, 366(1), 24-32. doi:10.1016/j.jmaa.2010.01.047Argyros, I. K., Cordero, A., Magreñán, Á. A., & Torregrosa, J. R. (2017). Third-degree anomalies of Traub’s method. Journal of Computational and Applied Mathematics, 309, 511-521. doi:10.1016/j.cam.2016.01.060Chicharro, F., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2013). Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computation, 219(12), 7023-7035. doi:10.1016/j.amc.2012.12.075Chicharro, F., Cordero, A., & Torregrosa, J. (2015). Dynamics and Fractal Dimension of Steffensen-Type Methods. Algorithms, 8(2), 271-279. doi:10.3390/a8020271Scott, M., Neta, B., & Chun, C. (2011). Basin attractors for various methods. Applied Mathematics and Computation, 218(6), 2584-2599. doi:10.1016/j.amc.2011.07.076Steffensen, J. F. (1933). Remarks on iteration. Scandinavian Actuarial Journal, 1933(1), 64-72. doi:10.1080/03461238.1933.10419209Wang, X., & Zhang, T. (2012). A new family of Newton-type iterative methods with and without memory for solving nonlinear equations. Calcolo, 51(1), 1-15. doi:10.1007/s10092-012-0072-2Džunić, J., & Petković, M. S. (2014). On generalized biparametric multipoint root finding methods with memory. Journal of Computational and Applied Mathematics, 255, 362-375. doi:10.1016/j.cam.2013.05.013Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2014). Multipoint methods for solving nonlinear equations: A survey. Applied Mathematics and Computation, 226, 635-660. doi:10.1016/j.amc.2013.10.072Campos, B., Cordero, A., Torregrosa, J. R., & Vindel, P. (2015). A multidimensional dynamical approach to iterative methods with memory. Applied Mathematics and Computation, 271, 701-715. doi:10.1016/j.amc.2015.09.056Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2019). Dynamics of iterative families with memory based on weight functions procedure. Journal of Computational and Applied Mathematics, 354, 286-298. doi:10.1016/j.cam.2018.01.019Chicharro, F. I., Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2017). King-Type Derivative-Free Iterative Families: Real and Memory Dynamics. Complexity, 2017, 1-15. doi:10.1155/2017/2713145Magreñán, Á. A., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2014). Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane. Mathematics and Computers in Simulation, 105, 49-61. doi:10.1016/j.matcom.2014.04.006Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-141. doi:10.1090/s0273-0979-1984-15240-6Magreñán, Á. A. (2014). A new tool to study real dynamics: The convergence plane. Applied Mathematics and Computation, 248, 215-224. doi:10.1016/j.amc.2014.09.061Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/78015

Stability anomalies of some jacobian-free iterative methods of high order of convergence

[EN] In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.This research was partially supported by Ministerio de Economia y Competitividad under grants PGC2018-095896-B-C22, Generalitat Valenciana PROMETEO/2016/089 and FONDOCYT 027-2018 and 029-2018, Dominican Republic.Cordero Barbero, A.; García-Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, MP. (2019). Stability anomalies of some jacobian-free iterative methods of high order of convergence. Axioms. 8(2):1-15. https://doi.org/10.3390/axioms8020051S11582Frontini, M., & Sormani, E. (2004). Third-order methods from quadrature formulae for solving systems of nonlinear equations. Applied Mathematics and Computation, 149(3), 771-782. doi:10.1016/s0096-3003(03)00178-4Homeier, H. H. . (2004). A modified Newton method with cubic convergence: the multivariate case. Journal of Computational and Applied Mathematics, 169(1), 161-169. doi:10.1016/j.cam.2003.12.041Aslam Noor, M., & Waseem, M. (2009). Some iterative methods for solving a system of nonlinear equations. Computers & Mathematics with Applications, 57(1), 101-106. doi:10.1016/j.camwa.2008.10.067Xiao, X., & Yin, H. (2015). A new class of methods with higher order of convergence for solving systems of nonlinear equations. Applied Mathematics and Computation, 264, 300-309. doi:10.1016/j.amc.2015.04.094Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.062Darvishi, M. T., & Barati, A. (2007). A third-order Newton-type method to solve systems of nonlinear equations. Applied Mathematics and Computation, 187(2), 630-635. doi:10.1016/j.amc.2006.08.080Sharma, J. R., Guha, R. K., & Sharma, R. (2012). An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numerical Algorithms, 62(2), 307-323. doi:10.1007/s11075-012-9585-7Narang, M., Bhatia, S., & Kanwar, V. (2016). New two-parameter Chebyshev–Halley-like family of fourth and sixth-order methods for systems of nonlinear equations. Applied Mathematics and Computation, 275, 394-403. doi:10.1016/j.amc.2015.11.063Behl, R., Sarría, Í., González, R., & Magreñán, Á. A. (2019). Highly efficient family of iterative methods for solving nonlinear models. Journal of Computational and Applied Mathematics, 346, 110-132. doi:10.1016/j.cam.2018.06.042Amorós, C., Argyros, I., González, R., Magreñán, Á., Orcos, L., & Sarría, Í. (2019). Study of a High Order Family: Local Convergence and Dynamics. Mathematics, 7(3), 225. doi:10.3390/math7030225Argyros, I., & González, D. (2015). Local Convergence for an Improved Jarratt-type Method in Banach Space. International Journal of Interactive Multimedia and Artificial Intelligence, 3(4), 20. doi:10.9781/ijimai.2015.344Sharma, J. R., & Gupta, P. (2014). An efficient fifth order method for solving systems of nonlinear equations. Computers & Mathematics with Applications, 67(3), 591-601. doi:10.1016/j.camwa.2013.12.004Cordero, A., Gutiérrez, J. M., Magreñán, Á. A., & Torregrosa, J. R. (2016). Stability analysis of a parametric family of iterative methods for solving nonlinear models. Applied Mathematics and Computation, 285, 26-40. doi:10.1016/j.amc.2016.03.021Cordero, A., Soleymani, F., & Torregrosa, J. R. (2014). Dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension? Applied Mathematics and Computation, 244, 398-412. doi:10.1016/j.amc.2014.07.010Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2009). A modified Newton-Jarratt’s composition. Numerical Algorithms, 55(1), 87-99. doi:10.1007/s11075-009-9359-zArgyros, I., & George, S. (2015). Ball Convergence for Steffensen-type Fourth-order Methods. International Journal of Interactive Multimedia and Artificial Intelligence, 3(4), 37. doi:10.9781/ijimai.2015.347Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/78015

Superiorization and Perturbation Resilience of Algorithms: A Continuously Updated Bibliography

This document presents a, (mostly) chronologically ordered, bibliography of scientific publications on the superiorization methodology and perturbation resilience of algorithms which is compiled and continuously updated by us at: http://math.haifa.ac.il/yair/bib-superiorization-censor.html. Since the beginings of this topic we try to trace the work that has been published about it since its inception. To the best of our knowledge this bibliography represents all available publications on this topic to date, and while the URL is continuously updated we will revise this document and bring it up to date on arXiv approximately once a year. Abstracts of the cited works, and some links and downloadable files of preprints or reprints are available on the above mentioned Internet page. If you know of a related scientific work in any form that should be included here kindly write to me on: [email protected] with full bibliographic details, a DOI if available, and a PDF copy of the work if possible. The Internet page was initiated on March 7, 2015, and has been last updated on March 12, 2020.Comment: Original report: June 13, 2015 contained 41 items. First revision: March 9, 2017 contained 64 items. Second revision: March 8, 2018 contained 76 items. Third revision: March 11, 2019 contains 90 items. Fourth revision: March 16, 2020 contains 112 item