A family of iterative methods that uses divided differences of first and second orders

The family of fourth-order Steffensen-type methods proposed by Zheng et al. (Appl. Math. Comput. 217, 9592-9597 (2011)) is extended to solve systems of nonlinear equations. This extension uses multidimensional divided differences of first and second orders. For a certain computational efficiency index, two optimal methods are identified in the family. Semilocal convergence is shown for one of these optimal methods under mild conditions. Moreover, a numerical example is given to illustrate the theoretical results.Peer ReviewedPostprint (author's final draft

The 2024 Outline of Fungi and fungus-like taxa

With the simultaneous growth in interest from the mycological community to discover fungal species and classify them, there is also an important need to assemble all taxonomic information onto common platforms. Fungal classification is facing a rapidly evolving landscape and organizing genera into an appropriate taxonomic hierarchy is central to better structure a unified classification scheme and avoid incorrect taxonomic inferences. With this in mind, the Outlines of Fungi and fungus-like taxa (2020, 2022) were published as an open-source taxonomic scheme to assist mycologists to better understand the taxonomic position of species within the Fungal Kingdom as well as to improve the accuracy and consistency of our taxonomic language. In this paper, the third contribution to the series of Outline of Fungi and fungus-like taxa prepared by the Global Consortium for the Classification of Fungi and fungus-like taxa is published. The former is updated considering our previous reviews and the taxonomic changes based on recent taxonomic work. In addition, it is more comprehensive and derives more input and consensus from a larger number of mycologists worldwide. Apart from listing the position of a particular genus in a taxonomic level, nearly 1000 notes are provided for newly established genera and higher taxa introduced since 2022. The notes section emphasizes on recent findings with corresponding references, discusses background information to support the current taxonomic status and some controversial taxonomic issues are also highlighted. To elicit maximum taxonomic information, notes/taxa are linked to recognized databases such as Index Fungorum, Faces of Fungi, MycoBank and GenBank, Species Fungorum and others. A new feature includes links to Fungalpedia, offering notes in the Compendium of Fungi and fungus-like Organisms. When specific notes are not provided, links are available to webpages and relevant publications for genera or higher taxa to ease data accessibility. Following the recent synonymization of Caulochytriomycota under Chytridiomycota, with Caulochytriomycetes now classified as a class within the latter, based on formally described and currently accepted data, the Fungi comprises 19 Phyla, 83 classes, 1,220 families, 10,685 genera and ca 140,000 species. Of the genera, 39.5% are monotypic and this begs the question whether mycologists split genera unnecessarily or are we going to find other species in these genera as more parts of the world are surveyed? They are 433 speciose genera with more than 50 species. The document also highlights discussion of some important topics including number of genera categorized as incertae sedis status in higher level fungal classification. The number of species at the higher taxonomic level has always been a contentious issue especially when mycologists consider either a lumping or a splitting approach and herein we provide figures. Herein a summary of updates in the outline of Basidiomycota is provided with discussion on whether there are too many genera of Boletales, Ceratobasidiaceae, and speciose genera such as Colletotrichum. Specific case studies deal with Cortinarius, early diverging fungi, Glomeromycota, a diverse early divergent lineage of symbiotic fungi, Eurotiomycetes, marine fungi, Myxomycetes, Phyllosticta, Hymenochaetaceae and Polyporaceae and the longstanding practice of misapplying intercontinental conspecificity. The outline will aid to better stabilize fungal taxonomy and serves as a necessary tool for mycologists and other scientists interested in the classification of the Fungi

CMMSE-2019 mean-based iterative methods for solving nonlinear chemistry problems

[EN] The third-order iterative method designed by Weerakoon and Fernando includes the arithmetic mean of two functional evaluations in its expression. Replacing this arithmetic mean with different means, other iterative methods have been proposed in the literature. The evolution of these methods in terms of order of convergence implies the inclusion of a weight function for each case, showing an optimal fourth-order convergence, in the sense of Kung-Traub's conjecture. The analysis of these new schemes is performed by means of complex dynamics. These methods are applied on the solution of the nonlinear Colebrook-White equation and the nonlinear system of the equilibrium conversion, both frequently used in Chemistry.This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE) and Generalitat Valenciana PROMETEO/2016/089.Chicharro, FI.; Cordero Barbero, A.; Martínez, TH.; Torregrosa Sánchez, JR. (2020). CMMSE-2019 mean-based iterative methods for solving nonlinear chemistry problems. Journal of Mathematical Chemistry. 58(3):555-572. https://doi.org/10.1007/s10910-019-01085-2S555572583O. Ababneh, New Newton’s method with third order convergence for solving nonlinear equations. World Acad. Sci. Eng. Technol. 61, 1071–1073 (2012)S. Amat, S. Busquier, Advances in iterative methods for nonlinear equations, chapter 5. SEMA SIMAI Springer Series. (Springer, Berlin, 2016), vol. 10, pp. 79–111R. Behl, Í. Sarría, R. González, Á.A. Magreñán, Highly efficient family of iterative methods for solving nonlinear models. J. Comput. Appl. Math. 346, 110–132 (2019)B. Campos, J. Canela, P. Vindel, Convergence regions for the Chebyshev-Halley family. Commun. Nonlinear Sci. Numer. Simul. 56, 508–525 (2018)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 780513, 1–11 (2013)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Dynamics of iterative families with memory based on weight functions procedure. J. Comput. Appl. Math. 354, 286–298 (2019)C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. 161, 367–381 (1937)A. Constantinides, N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications (Prentice-Hall, Englewood Cliffs, 1999)A. Cordero, J. Franceschi, J.R. Torregrosa, A.C. Zagati, A convex combination approach for mean-based variants of Newton’s method. Symmetry 11, 1062 (2019)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)T. Lukić, N. Ralević, Geometric mean Newton’s method for simple and multiple roots. Appl. Math. Lett. 21, 30–36 (2008)A. Özban, Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004)M. Petković, B. Neta, L. Petković, J. Dz̆unić, Multipoint Methods for Solving Nonlinear Equations (Academic Press, Cambridge, 2013)E. Shashi, Transmission Pipeline Calculations and Simulations Manual, Fluid Flow in Pipes (Elsevier, London, 2015), pp. 149–234M.K. Singh, A.K. Singh, A new-mean type variant of Newton’s method for simple and multiple roots. Int. J. Math. Trends Technol. 49, 174–177 (2017)K. Verma, On the centroidal mean Newton’s method for simple and multiple roots of nonlinear equations. Int. J. Comput. Sci. Math. 7, 126–143 (2016)S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)Z. Xiaojian, A class of Newton’s methods with third-order convergence. Appl. Math. Lett. 20, 1026–1030 (2007