An efficient two-parametric family with memory for nonlinear equations

A new two-parametric family of derivative-free iterative methods for solving nonlinear equations is presented. First, a new biparametric family without memory of optimal order four is proposed. The improvement of the convergence rate of this family is obtained by using two self-accelerating parameters. These varying parameters are calculated in each iterative step employing only information from the current and the previous iteration. The corresponding R-order is 7 and the efficiency index 7(1/3) = 1.913. Numerical examples and comparison with some existing derivative-free optimal eighth-order schemes are included to confirm the theoretical results. In addition, the dynamical behavior of the designed method is analyzed and shows the stability of the scheme.The second author wishes to thank the Islamic Azad University, Hamedan Branch, where the paper was written as a part of the research plan, for financial support.Cordero Barbero, A.; Lotfi, T.; Bakhtiari, P.; Torregrosa Sánchez, JR. (2015). An efficient two-parametric family with memory for nonlinear equations. Numerical Algorithms. 68(2):323-335. doi:10.1007/s11075-014-9846-8S323335682Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equation. J. Comput. Appl. Math. 252, 95–102 (2013)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Pseudocomposition: a technique to design predictor-corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218, 11496–11508 (2012)Džunić, J.: On efficient two-parameter methods for solving nonlinear equations. Numer. Algorithms. 63(3), 549–569 (2013)Džunić, J., Petković, M.S.: On generalized multipoint root-solvers with memory. J. Comput. Appl. Math. 236, 2909–2920 (2012)Petković, M.S., Neta, B., Petković, L.D., Džunić, J. (ed.).: Multipoint methods for solving nonlinear equations. Elsevier (2013)Sharma, J.R., Sharma, R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algorithms 54, 445–458 (2010)Soleymani, F., Shateyi, S.: Two optimal eighth-order derivative-free classes of iterative methods. Abstr. Appl. Anal. 2012(318165), 14 (2012). doi: 10.1155/2012/318165Soleymani, F., Sharma, R., Li, X., Tohidi, E.: An optimized derivative-free form of the Potra-Pták methods. Math. Comput. Model. 56, 97–104 (2012)Thukral, R.: Eighth-order iterative methods without derivatives for solving nonlinear equations. ISRN Appl. Math. 2011(693787), 12 (2011). doi: 10.5402/2011/693787Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Wang, X., Džunić, J., Zhang, T.: On an efficient family of derivative free three-point methods for solving nonlinear equations. Appl. Math. Comput. 219, 1749–1760 (2012)Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)Ortega, J.M., Rheinboldt, W.G. (ed.).: Iterative Solutions of Nonlinear Equations in Several Variables, Ed. Academic Press, New York (1970)Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)Blanchard, P.: Complex Analytic Dynamics on the Riemann Sphere. Bull. AMS 11(1), 85–141 (1984)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. arXiv: 1307.6705 [math.NA

Memory in a new variant of King's family for solving nonlinear systems

[EN] In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King's family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Kansal, M.; Cordero Barbero, A.; Bhalla, S.; Torregrosa Sánchez, JR. (2020). Memory in a new variant of King's family for solving nonlinear systems. Mathematics. 8(8):1-15. https://doi.org/10.3390/math8081251S11588Cordero, 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-zCordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2011). Efficient high-order methods based on golden ratio for nonlinear systems. Applied Mathematics and Computation, 217(9), 4548-4556. doi:10.1016/j.amc.2010.11.006Babajee, D. K. R., Cordero, A., Soleymani, F., & Torregrosa, J. R. (2012). On a Novel Fourth-Order Algorithm for Solving Systems of Nonlinear Equations. Journal of Applied Mathematics, 2012, 1-12. doi:10.1155/2012/165452Zheng, Q., Zhao, P., & Huang, F. (2011). A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs. Applied Mathematics and Computation, 217(21), 8196-8203. doi:10.1016/j.amc.2011.01.095Sharma, J., & Arora, H. (2013). An efficient derivative free iterative method for solving systems of nonlinear equations. Applicable Analysis and Discrete Mathematics, 7(2), 390-403. doi:10.2298/aadm130725016sSharma, J. R., Arora, H., & Petković, M. S. (2014). An efficient derivative free family of fourth order methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 235, 383-393. doi:10.1016/j.amc.2014.02.103Wang, X., Zhang, T., Qian, W., & Teng, M. (2015). Seventh-order derivative-free iterative method for solving nonlinear systems. Numerical Algorithms, 70(3), 545-558. doi:10.1007/s11075-015-9960-2Chicharro, F. I., Cordero, A., Garrido, N., & Torregrosa, J. R. (2020). On the improvement of the order of convergence of iterative methods for solving nonlinear systems by means of memory. Applied Mathematics Letters, 104, 106277. doi:10.1016/j.aml.2020.106277Petković, M. S., & Sharma, J. R. (2015). On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numerical Algorithms, 71(2), 457-474. doi:10.1007/s11075-015-0003-9Narang, M., Bhatia, S., Alshomrani, A. S., & Kanwar, V. (2019). General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 352, 23-39. doi:10.1016/j.cam.2018.10.048King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Hermite, M. C., & Borchardt, M. (1878). Sur la formule d’interpolation de Lagrange. Journal für die reine und angewandte Mathematik (Crelles Journal), 1878(84), 70-79. doi:10.1515/crelle-1878-18788405Petkovic, M., Dzunic, J., & Petkovic, L. (2011). A family of two-point methods with memory for solving nonlinear equations. Applicable Analysis and Discrete Mathematics, 5(2), 298-317. doi:10.2298/aadm110905021pCordero, 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.2007.01.062Awawdeh, F. (2009). On new iterative method for solving systems of nonlinear equations. Numerical Algorithms, 54(3), 395-409. doi:10.1007/s11075-009-9342-8Noor, M. A., Waseem, M., & Noor, K. I. (2015). New iterative technique for solving a system of nonlinear equations. Applied Mathematics and Computation, 271, 446-466. doi:10.1016/j.amc.2015.08.125Pramanik, S. (2002). Kinematic Synthesis of a Six-Member Mechanism for Automotive Steering. Journal of Mechanical Design, 124(4), 642-645. doi:10.1115/1.150337

Iterative methods with memory for solving systems of nonlinear equations using a second order approximation

[EN] Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of the set of convergent initial estimations. With the right choice of parameters, iterative methods without memory can increase their order of convergence significantly, becoming schemes with memory. In this work, starting from a simple method without memory, we increase its order of convergence without adding new functional evaluations by approximating the accelerating parameter with Newton interpolation polynomials of degree one and two. Using this technique in the multidimensional case, we extend the proposed method to systems of nonlinear equations. Numerical tests are presented to verify the theoretical results and a study of the dynamics of the method is applied to different problems to show its stability.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), Generalitat Valenciana PROMETEO/2016/089, and FONDOCYT 2016-2017-212 Republica Dominicana.Cordero Barbero, A.; Maimó, JG.; Torregrosa Sánchez, JR.; Vassileva, MP. (2019). Iterative methods with memory for solving systems of nonlinear equations using a second order approximation. Mathematics. 7(11):1-12. https://doi.org/10.3390/math7111069S112711Soleymani, F., Lotfi, T., Tavakoli, E., & Khaksar Haghani, F. (2015). Several iterative methods with memory using self-accelerators. Applied Mathematics and Computation, 254, 452-458. doi:10.1016/j.amc.2015.01.045Petković, M. S., & Sharma, J. R. (2015). On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numerical Algorithms, 71(2), 457-474. doi:10.1007/s11075-015-0003-9Narang, M., Bhatia, S., Alshomrani, A. S., & Kanwar, V. (2019). General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 352, 23-39. doi:10.1016/j.cam.2018.10.048Potra, F. A. (1982). An error analysis for the secant method. Numerische Mathematik, 38(3), 427-445. doi:10.1007/bf01396443Fatou, P. (1919). Sur les équations fonctionnelles. Bulletin de la Société mathématique de France, 2, 161-271. doi:10.24033/bsmf.998Cordero, 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.2007.01.062Campos, 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. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/78015

Widening basins of attraction of optimal iterative methods

[EN] In this work, we analyze the dynamical behavior on quadratic polynomials of a class of derivative-free optimal parametric iterative methods, designed by Khattri and Steihaug. By using their parameter as an accelerator, we develop different methods with memory of orders three, six and twelve, without adding new functional evaluations. Then a dynamical approach is made, comparing each of the proposed methods with the original ones without memory, with the following empiric conclusion: Basins of attraction of iterative schemes with memory are wider and the behavior is more stable. This has been numerically checked by estimating the solution of a practical problem, as the friction factor of a pipe and also of other nonlinear academic problems.This research was supported by Islamic Azad University, Hamedan Branch, Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Bakhtiari, P.; Cordero Barbero, A.; Lotfi, T.; Mahdiani, K.; Torregrosa Sánchez, JR. (2017). Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics. 87(2):913-938. https://doi.org/10.1007/s11071-016-3089-2S913938872Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Lett. 25, 2209–2217 (2012)Amat, S., Busquier, S., Bermúdez, C., Magreñán, Á.A.: On the election of the damped parameter of a two-step relaxed Newton-type method. Nonlinear Dyn. 84(1), 9–18 (2016)Chun, C., Neta, B.: An analysis of a family of Maheshwari-based optimal eighth order methods. Appl. Math. Comput. 253, 294–307 (2015)Babajee, D.K.R., Cordero, A., Soleymani, F., Torregrosa, J.R.: On improved three-step schemes with high efficiency index and their dynamics. Numer. Algorithms 65(1), 153–169 (2014)Argyros, I.K., Magreñán, Á.A.: On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)Petković, M., Neta, B., Petković, L., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, London (2013)Ostrowski, A.M.: Solution of Equations and System of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)Khattri, S.K., Steihaug, T.: Algorithm for forming derivative-free optimal methods. Numer. Algorithms 65(4), 809–824 (2014)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Cordero, A., Soleymani, F., Torregrosa, J.R., Shateyi, S.: Basins of Attraction for Various Steffensen-Type Methods. J. Appl. Math. 2014, 1–17 (2014)Devaney, R.L.: The Mandelbrot Set, the Farey Tree and the Fibonacci sequence. Am. Math. Mon. 106(4), 289–302 (1999)McMullen, C.: Families of rational maps and iterative root-finding algorithms. Ann. Math. 125(3), 467–493 (1987)Chicharro, F., Cordero, A., Gutiérrez, J.M., Torregrosa, J.R.: Complex dynamics of derivative-free methods for nonlinear equations. Appl. Math. Comput. 219, 70237035 (2013)Magreñán, Á.A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)Lotfi, T., Magreñán, Á.A., Mahdiani, K., Rainer, J.J.: A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: dynamic study and approach. Appl. Math. Comput. 252, 347–353 (2015)Chicharro, F.I., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, 1–11 (2013)Cordero, A., Lotfi, T., Torregrosa, J.R., Assari, P., Mahdiani, K.: Some new bi-accelerator two-point methods for solving nonlinear equations. Comput. Appl. Math. 35(1), 251–267 (2016)Cordero, A., Lotfi, T., Bakhtiari, P., Torregrosa, J.R.: An efficient two-parametric family with memory for nonlinear equations. Numer. Algorithms 68(2), 323–335 (2015)Lotfi, T., Mahdiani, K., Bakhtiari, P., Soleymani, F.: Constructing two-step iterative methods with and without memory. Comput. Math. Math. Phys. 55(2), 183–193 (2015)Cordero, A., Maimó, J.G., Torregrosa, J.R., Vassileva, M.P.: Solving nonlinear problems by Ostrowski–Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)Abad, M., Cordero, A., Torregrosa, J.R.: A family of seventh-order schemes for solving nonlinear systems. Bull. Math. Soc. Sci. Math. Roum. Tome 57(105), 133–145 (2014)Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)White, F.: Fluid Mechanics. McGraw-Hill, Boston (2003)Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)Soleymani, F., Babajee, D.K.R., Shateyi, S., Motsa, S.S.: Construction of optimal derivative-free techniques without memory. J. Appl. Math. (2012). doi: 10.1155/2012/49702

A study of the local convergence of a fifth order iterative method

[EN] We present a local convergence study of a fifth order iterative method to approximate a locally unique root of nonlinear equations. The analysis is discussed under the assumption that first order Frechet derivative satisfies the Lipschitz continuity condition. Moreover, we consider the derivative free method that obtained through approximating the derivative with divided difference along with the local convergence study. Finally, we provide computable radii and error bounds based on the Lipschitz constant for both cases. Some of the numerical examples are worked out and compared the results with existing methods.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C21-C22.Singh, S.; Martínez Molada, E.; Maroju, P.; Behl, R. (2020). A study of the local convergence of a fifth order iterative method. Indian Journal of Pure and Applied Mathematics. 51(2):439-455. https://doi.org/10.1007/s13226-020-0409-5S439455512A. Constantinides and N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications, Prentice Hall PTR, New Jersey, (1999).J. M. Douglas, Process Dynamics and Control, Prentice Hall, Englewood Cliffs, (1972).M. Shacham, An improved memory method for the solution of a nonlinear equation, Chem. Eng. Sci., 44 (1989), 1495–1501.J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New-York, (1970).J. R. Sharma and H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algorithms, 67 (2014), 917–933.I. K. Argyros, A. A. Magreńan, and L. Orcos, Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation, J. Math. Chem., 54 (2016), 1404–1416.E. L. Allgower and K. Georg, Lectures in Applied Mathematics, American Mathematical Society (Providence, RI) 26, 723–762.A. V. Rangan, D. Cai, and L. Tao, Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics, J. Comput. Phys., 221 (2007), 781–798.A. Nejat and C. Ollivier-Gooch, Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations, J. Comput. Phys., 227 (2008), 2366–2386.C. Grosan and A. Abraham, A new approach for solving nonlinear equations systems, IEEE Trans. Syst. Man Cybernet Part A: System Humans, 38 (2008), 698–714.F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algorithms, 54 (2010), 395–409.I. G. Tsoulos and A. Stavrakoudis, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Anal. Real World Appl., 11 (2010), 2465–2471.E. Martínez, S. Singh, J. L. Hueso, and D. K. Gupta, Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces, Appl. Math. Comput., 281 (2016), 252–265.S. Singh, D. K. Gupta, E. Martínez, and J. L. Hueso, Semi local and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition, Appl. Math. Comput., 276 (2016), 266–277.I. K. Argyros and S. George, Local convergence of modified Halley-like methods with less computation of inversion, Novi. Sad.J. Math., 45 (2015), 47–58.I. K. Argyros, R. Behl, and S. S. Motsa, Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative Algorithms 2015, 8, 1076–1087; doi:https://doi.org/10.3390/a8041076.A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett., 25 (2012), 2369–2374.I. K. Argyros and A. A. Magreñán, A study on the local convergence and dynamics of Chebyshev- Halley-type methods free from second derivative, Numer. Algorithms71 (2016), 1–23.M. Grau-Sánchez, Á Grau, asnd M. Noguera, Frozen divided difference scheme for solving systems of nonlinear equations, J. Comput. Appl. Math., 235 (2011), 1739–1743.M. Grau-Sánchez, M. Noguera, and S. Amat, On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. Appl. Math., 237 (2013), 363–372

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

Automatic implementation of material laws: Jacobian calculation in a finite element code with TAPENADE

In an effort to increase the versatility of finite element codes, we explore the possibility of automatically creating the Jacobian matrix necessary for the gradient-based solution of nonlinear systems of equations. Particularly, we aim to assess the feasibility of employing the automatic differentiation tool TAPENADE for this purpose on a large Fortran codebase that is the result of many years of continuous development. As a starting point we will describe the special structure of finite element codes and the implications that this code design carries for an efficient calculation of the Jacobian matrix. We will also propose a first approach towards improving the efficiency of such a method. Finally, we will present a functioning method for the automatic implementation of the Jacobian calculation in a finite element software, but will also point out important shortcomings that will have to be addressed in the future.Comment: 17 pages, 9 figure

Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part I: template-based generic programming

An approach for incorporating embedded simulation and analysis capabilities in complex simulation codes through template-based generic programming is presented. This approach relies on templating and operator overloading within the C++ language to transform a given calculation into one that can compute a variety of additional quantities that are necessary for many state-of-the-art simulation and analysis algorithms. An approach for incorporating these ideas into complex simulation codes through general graph-based assembly is also presented. These ideas have been implemented within a set of packages in the Trilinos framework and are demonstrated on a simple problem from chemical engineering