124,832 research outputs found

    Some iterative methods for solving a system of nonlinear equations

    Get PDF
    AbstractIn this paper, we suggest and analyze two new two-step iterative methods for solving the system of nonlinear equations using quadrature formulas. We prove that these new methods have cubic convergence. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. These new iterative methods may be viewed as an extension and generalizations of the existing methods for solving the system of nonlinear equations

    A New Accelerated Third-Order Two-Step Iterative Method for Solving Nonlinear Equations

    Get PDF
    In the paper a new two-step iterative method has been suggested for estimating nonlinear problems by using quadrature formula. We have also proved that the proposed two-step iterative method has third-order of convergence. For the duration of study, it has been detected that the third-order iterative Method faster than existing cubic Methods such as variant of Newton Raphson Method and Halley Method. Numerous numerical illustrations are specified to demonstrate the efficiency and the performance of the new two-step method. Henceforth, the third-order iterative Method has been considered as an imperative enhancement and refinement to find the root of nonlinear equations. Keyword: Non-linear equations, cubic methods, order of convergence, error and accuracy

    Numerical iterative methods for nonlinear problems.

    Get PDF
    The primary focus of research in this thesis is to address the construction of iterative methods for nonlinear problems coming from different disciplines. The present manuscript sheds light on the development of iterative schemes for scalar nonlinear equations, for computing the generalized inverse of a matrix, for general classes of systems of nonlinear equations and specific systems of nonlinear equations associated with ordinary and partial differential equations. Our treatment of the considered iterative schemes consists of two parts: in the first called the ’construction part’ we define the solution method; in the second part we establish the proof of local convergence and we derive convergence-order, by using symbolic algebra tools. The quantitative measure in terms of floating-point operations and the quality of the computed solution, when real nonlinear problems are considered, provide the efficiency comparison among the proposed and the existing iterative schemes. In the case of systems of nonlinear equations, the multi-step extensions are formed in such a way that very economical iterative methods are provided, from a computational viewpoint. Especially in the multi-step versions of an iterative method for systems of nonlinear equations, the Jacobians inverses are avoided which make the iterative process computationally very fast. When considering special systems of nonlinear equations associated with ordinary and partial differential equations, we can use higher-order Frechet derivatives thanks to the special type of nonlinearity: from a computational viewpoint such an approach has to be avoided in the case of general systems of nonlinear equations due to the high computational cost. Aside from nonlinear equations, an efficient matrix iteration method is developed and implemented for the calculation of weighted Moore-Penrose inverse. Finally, a variety of nonlinear problems have been numerically tested in order to show the correctness and the computational efficiency of our developed iterative algorithms

    Third-order iterative methods with applications to Hammerstein equations: A unified approach

    Get PDF
    AbstractThe geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in [S. Amat, S. Busquier, J.M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1) (2003) 197–205]. This family includes, as particular cases, some of the most famous third-order iterative methods: Chebyshev methods, Halley methods, super-Halley methods, C-methods and Newton-type two-step methods. The aim of the present paper is to analyze the convergence of this family for equations defined between two Banach spaces by using a technique developed in [J.A. Ezquerro, M.A. Hernández, Halley’s method for operators with unbounded second derivative. Appl. Numer. Math. 57(3) (2007) 354–360]. This technique allows us to obtain a general semilocal convergence result for these methods, where the usual conditions on the second derivative are relaxed. On the other hand, the main practical difficulty related to the classical third-order iterative methods is the evaluation of bilinear operators, typically second-order Fréchet derivatives. However, in some cases, the second derivative is easy to evaluate. A clear example is provided by the approximation of Hammerstein equations, where it is diagonal by blocks. We finish the paper by applying our methods to some nonlinear integral equations of this type

    Three-step iterative methods with eighth-order convergence for solving nonlinear equations

    Get PDF
    A family of eighth-order iterative methods for solution of nonlinear equations is presented. We propose an optimal three-step method with eight-order convergence for finding the simple roots of nonlinear equations by Hermite interpolation method. Per iteration of this method requires two evaluations of the function and two evaluations of its first derivative, which implies that the efficiency index of the developed methods is 1.682. Some numerical examples illustrate that the algorithms are more efficient and performs better than the other methods

    Development of iterative techniques for the solution of unsteady compressible viscous flows

    Get PDF
    Efficient iterative solution methods are being developed for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. Thus, the extra work required by iterative schemes can also be designed to perform efficiently on current and future generation scalable, missively parallel machines. An obvious candidate for iteratively solving the system of coupled nonlinear algebraic equations arising in CFD applications is the Newton method. Newton's method was implemented in existing finite difference and finite volume methods. Depending on the complexity of the problem, the number of Newton iterations needed per step to solve the discretized system of equations can, however, vary dramatically from a few to several hundred. Another popular approach based on the classical conjugate gradient method, known as the GMRES (Generalized Minimum Residual) algorithm is investigated. The GMRES algorithm was used in the past by a number of researchers for solving steady viscous and inviscid flow problems with considerable success. Here, the suitability of this algorithm is investigated for solving the system of nonlinear equations that arise in unsteady Navier-Stokes solvers at each time step. Unlike the Newton method which attempts to drive the error in the solution at each and every node down to zero, the GMRES algorithm only seeks to minimize the L2 norm of the error. In the GMRES algorithm the changes in the flow properties from one time step to the next are assumed to be the sum of a set of orthogonal vectors. By choosing the number of vectors to a reasonably small value N (between 5 and 20) the work required for advancing the solution from one time step to the next may be kept to (N+1) times that of a noniterative scheme. Many of the operations required by the GMRES algorithm such as matrix-vector multiplies, matrix additions and subtractions can all be vectorized and parallelized efficiently

    An efficient two-parametric family with memory for nonlinear equations

    Full text link
    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

    Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: application to PDEs and ODEs

    Get PDF
    This paper proposes a multi-step iterative method for solving systems of nonlinear equations with a local convergence order of 3m - 4, where in (>= 2) is the number of steps. The multi-step iterative method includes two parts: the base method and the multi-step part. The base method involves two function evaluations, two Jacobian evaluations, one LU decomposition of a Jacobian, and two matrix-vector multiplications. Every stage of the multi-step part involves the solution of two triangular linear systems and one matrix-vector multiplication. The computational efficiency of the new method is better than those of previously proposed methods. The method is applied to several nonlinear problems resulting from discretizing nonlinear ordinary differential equations and nonlinear partial differential equations. (C) 2015 Elsevier Ltd. All rights reserved.Postprint (updated version
    corecore