A New Derivative-Free Quasi-Secant Algorithm For Solving Non-Linear Equations
Most of the nonlinear equation solvers do not converge always or they use the derivatives of the function to approximate the
root of such equations. Here, we give a derivative-free algorithm that guarantees the convergence. The proposed two-step method, which
is to some extent like the secant method, is accompanied with some
numerical examples. The illustrative instances manifest that the rate of convergence in proposed algorithm is more than the quadratically
iterative schemes.
[1] K.E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., John
Wiley & Sons, Singapore, 1988.
[2] R.P. Brent, Algorithms for Minimization without Derivatives, Prentice
Hall, Englewood Cliffs, NJ, 1973.
[3] C. Chun, Some fourth-order iterative methods for solving nonlinear
equations, Applied Mathematics and Computation 195 (2008), 454-459.
[4] M. Grau and M. Noguera, A variant of Cauchy-s method with accelerated
fifth-order convergence, Applied Mathematics Letters 17 (2004) 509-517.
[5] M. Grau-Snchez, Improvements of the efficiency of some three-step
iterative like-Newton methods, Numerical Mathematics 107 (2007) 131-146.
[6] Z. Hui, L. De-Sheng and L. Yu-Zhong, A new method of secant-like for nonlinear equations, Communications for Nonlinear Sciences and
Numerical Simulation 14 (2009) 2923-2927.
[7] J. Kou and X. Wang, Y. Li, Some eighth-order root-finding threestep
methods, Communications for Nonlinear Sciences and Numerical Simulation, In Press (2009).
[8] J. Kou and X. Wang, Some improvements of Ostrowski-s method, Applied
Mathematics Letters, In Press (2009).
[9] A.M. Ostrowski, Solution of equations in Euclidean and Banach space,
Academic Press, New York 1973.
[10] T. Sauer, Numerical Analysis, Addison Wesley Publication, USA, 2005.
[11] B. I. Yun and M.S. Petkovi, A quadratically convergent iterative method
for nonlinear equations, In Press. (2009).
[1] K.E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., John
Wiley & Sons, Singapore, 1988.
[2] R.P. Brent, Algorithms for Minimization without Derivatives, Prentice
Hall, Englewood Cliffs, NJ, 1973.
[3] C. Chun, Some fourth-order iterative methods for solving nonlinear
equations, Applied Mathematics and Computation 195 (2008), 454-459.
[4] M. Grau and M. Noguera, A variant of Cauchy-s method with accelerated
fifth-order convergence, Applied Mathematics Letters 17 (2004) 509-517.
[5] M. Grau-Snchez, Improvements of the efficiency of some three-step
iterative like-Newton methods, Numerical Mathematics 107 (2007) 131-146.
[6] Z. Hui, L. De-Sheng and L. Yu-Zhong, A new method of secant-like for nonlinear equations, Communications for Nonlinear Sciences and
Numerical Simulation 14 (2009) 2923-2927.
[7] J. Kou and X. Wang, Y. Li, Some eighth-order root-finding threestep
methods, Communications for Nonlinear Sciences and Numerical Simulation, In Press (2009).
[8] J. Kou and X. Wang, Some improvements of Ostrowski-s method, Applied
Mathematics Letters, In Press (2009).
[9] A.M. Ostrowski, Solution of equations in Euclidean and Banach space,
Academic Press, New York 1973.
[10] T. Sauer, Numerical Analysis, Addison Wesley Publication, USA, 2005.
[11] B. I. Yun and M.S. Petkovi, A quadratically convergent iterative method
for nonlinear equations, In Press. (2009).
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55094", author = "F. Soleymani and M. Sharifi", title = "A New Derivative-Free Quasi-Secant Algorithm For Solving Non-Linear Equations", abstract = "Most of the nonlinear equation solvers do not converge always or they use the derivatives of the function to approximate the
root of such equations. Here, we give a derivative-free algorithm that guarantees the convergence. The proposed two-step method, which
is to some extent like the secant method, is accompanied with some
numerical examples. The illustrative instances manifest that the rate of convergence in proposed algorithm is more than the quadratically
iterative schemes.", keywords = "Non-linear equation, iterative methods, derivative-free, convergence.", volume = "3", number = "7", pages = "488-3", }