Modified Potra-Pták multi-step schemes with accelerated order of convergence for solving sistems of nonlinear equations

[EN] In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Ptak method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3r + 6, r = 0, 1, 2, .... The sixth order method is the special case of this multi-step scheme for r = 0. The family gives a four-step ninth order method for r = 1. As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.This research was partially supported by Ministerio de Economia y Competitividad under grants MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.Arora, H.; Torregrosa Sánchez, JR.; Cordero Barbero, A. (2019). Modified Potra-Pták multi-step schemes with accelerated order of convergence for solving sistems of nonlinear equations. Mathematical and Computational Applications (Online). 24(1):1-15. https://doi.org/10.3390/mca24010003S115241Homeier, 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.2003.12.041Darvishi, M. T., & Barati, A. (2007). A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Applied Mathematics and Computation, 188(1), 257-261. doi:10.1016/j.amc.2006.09.115Cordero, 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.2010.11.006Grau-Sánchez, M., Grau, À., & Noguera, M. (2011). On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 236(6), 1259-1266. doi:10.1016/j.cam.2011.08.008Grau-Sánchez, M., Grau, À., & Noguera, M. (2011). Ostrowski type methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 218(6), 2377-2385. doi:10.1016/j.amc.2011.08.011Grau-Sánchez, M., Noguera, M., & Amat, S. (2013). On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. Journal of Computational and Applied Mathematics, 237(1), 363-372. doi:10.1016/j.cam.2012.06.005Sharma, J. R., & Arora, H. (2013). On efficient weighted-Newton methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 222, 497-506. doi:10.1016/j.amc.2013.07.066Cordero, 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.2007.01.062Chicharro, 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

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

Preliminary orbit determination of artificial satellites: a vectorial sixth-order approach

A modified classical method for preliminary orbit determination is presented. In our proposal, the spread of the observations is considerably wider than in the original method, as well as the order of convergence of the iterative scheme involved. The numerical approach is made by using matricial weight functions, which will lead us to a class of iterative methods with a sixth local order of convergence. This is a process widely used in the design of iterative methods for solving nonlinear scalar equations, but rarely employed in vectorial cases. The numerical tests confirm the theoretical results, and the analysis of the dynamics of the problem shows the stability of the proposed schemes.The authors thank the anonymous referees for their valuable comments and suggestions. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Andreu Estellés, C.; Cambil Teba, N.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2013). Preliminary orbit determination of artificial satellites: a vectorial sixth-order approach. Abstract and Applied Analysis. 2013. https://doi.org/10.1155/2013/960582S2013Fidkowski, K. J., Oliver, T. A., Lu, J., & Darmofal, D. L. (2005). p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. Journal of Computational Physics, 207(1), 92-113. doi:10.1016/j.jcp.2005.01.005Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0He, Y., & Ding, C. H. Q. (2001). The Journal of Supercomputing, 18(3), 259-277. doi:10.1023/a:1008153532043Revol, N., & Rouillier, F. (2005). Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library. Reliable Computing, 11(4), 275-290. doi:10.1007/s11155-005-6891-yBabajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., & Barati, A. (2008). A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule. Applied Mathematics and Computation, 200(1), 452-458. doi:10.1016/j.amc.2007.11.009Darvishi, 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.2006.08.080Darvishi, M. T., & Barati, A. (2007). Super cubic iterative methods to solve systems of nonlinear equations. Applied Mathematics and Computation, 188(2), 1678-1685. doi:10.1016/j.amc.2006.11.022Cordero, A., Martínez, E., & Torregrosa, J. R. (2009). Iterative methods of order four and five for systems of nonlinear equations. Journal of Computational and Applied Mathematics, 231(2), 541-551. doi:10.1016/j.cam.2009.04.015Babajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., Karami, A., & Barati, A. (2010). Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 233(8), 2002-2012. doi:10.1016/j.cam.2009.09.035Soleymani, F., Lotfi, T., & Bakhtiari, P. (2013). A multi-step class of iterative methods for nonlinear systems. Optimization Letters, 8(3), 1001-1015. doi:10.1007/s11590-013-0617-6Awawdeh, F. (2009). On new iterative method for solving systems of nonlinear equations. Numerical Algorithms, 54(3), 395-409. doi:10.1007/s11075-009-9342-8Babajee, 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:10.1155/2012/165452Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2012). Pseudocomposition: A technique to design predictor–corrector methods for systems of nonlinear equations. Applied Mathematics and Computation, 218(23), 11496-11504. doi:10.1016/j.amc.2012.04.081Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2013). Increasing the order of convergence of iterative schemes for solving nonlinear systems. Journal of Computational and Applied Mathematics, 252, 86-94. doi:10.1016/j.cam.2012.11.024Soleymani, F., & Stanimirović, P. S. (2013). A Higher Order Iterative Method for Computing the Drazin Inverse. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/708647Soleymani, F., Stanimirović, P. S., & Ullah, M. Z. (2013). An accelerated iterative method for computing weighted Moore–Penrose inverse. Applied Mathematics and Computation, 222, 365-371. doi:10.1016/j.amc.2013.07.039Sharma, 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/s11075-012-9585-7Sharma, J. R., & Arora, H. (2013). On efficient weighted-Newton methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 222, 497-506. doi:10.1016/j.amc.2013.07.066Abad, M. F., Cordero, A., & Torregrosa, J. R. (2013). Fourth- and Fifth-Order Methods for Solving Nonlinear Systems of Equations: An Application to the Global Positioning System. Abstract and Applied Analysis, 2013, 1-10. doi:10.1155/2013/586708Cordero, 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-zJarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Mathematics of Computation, 20(95), 434-434. doi:10.1090/s0025-5718-66-99924-8Cordero, 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.2007.01.062Chicharro, 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

A multigrid continuation method for elliptic problems with folds

We introduce a new multigrid continuation method for computing solutions of nonlinear elliptic eigenvalue problems which contain limit points (also called turning points or folds). Our method combines the frozen tau technique of Brandt with pseudo-arc length continuation and correction of the parameter on the coarsest grid. This produces considerable storage savings over direct continuation methods,as well as better initial coarse grid approximations, and avoids complicated algorithms for determining the parameter on finer grids. We provide numerical results for second, fourth and sixth order approximations to the two-parameter, two-dimensional stationary reaction-diffusion problem: Δu+λ exp(u/(1+au)) = 0. For the higher order interpolations we use bicubic and biquintic splines. The convergence rate is observed to be independent of the occurrence of limit points

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. 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