On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004) [2], Bosarge and Falb (1969) [9], Dennis (1971) [10], Ezquerro et al. (2010) [11], Hernandez et al. (2005, 2000) [13,12], Kantorovich and Akilov (1982) [14], Laasonen (1969) [15], Ortega and Rheinboldt (1970) [16], Parida and Gupta (2007) [17], Potra (1982, 1984-1985, 1985) [18-20], Proinov (2009, 2010) [21,22], Schmidt (1978) [23], Wolfe (1978) [24] and Yamamoto (1987) [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.

[1]  Petko D. Proinov.  New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems , 2010, J. Complex..

[2]  D. K. Gupta,et al.  Recurrence relations for a Newton-like method in Banach spaces , 2007 .

[3]  Pentti Laasonen,et al.  Ein überquadratisch konvergerter iterativer Algorithmus , 1969 .

[4]  Katta G. Murty,et al.  On KΔ , 1986, Discret. Appl. Math..

[5]  F. Potra.  On an iterative algorithm of order 1.839… for solving nonlinear operator equations ∗) , 1985 .

[6]  M. A. Wolfe.  Extended iterative methods for the solution of operator equations , 1978 .

[7]  Ioannis K. Argyros,et al.  A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space , 2004 .

[8]  Peter Falb,et al.  A multipoint method of third order , 1969 .

[9]  Petko D. Proinov.  General local convergence theory for a class of iterative processes and its applications to Newton's method , 2009, J. Complex..

[10]  Ioannis K. Argyros,et al.  Weaker conditions for the convergence of Newton's method , 2012, J. Complex..

[11]  J. W. Schmidt,et al.  Untere Fehlerschranken für Regula-falsi-Verfahren , 1978 .

[12]  Ioannis K. Argyros,et al.  Convergence conditions for secant-type methods , 2010 .

[13]  Vicente F. Candela,et al.  A class of quasi-Newton generalized Steffensen methods on Banach spaces , 2002 .

[14]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[15]  Y. Cho,et al.  Numerical Methods for Equations and its Applications , 2012 .

[16]  J. Dennis.  Toward a Unified Convergence Theory for Newton-Like Methods , 1971 .

[17]  Miguel Ángel Hernández,et al.  Solving a special case of conservative problems by Secant-like methods , 2005, Appl. Math. Comput..

[18]  tetsuro Yammoto,et al.  A convergence theorem for Newton-like methods in Banach spaces , 1987 .

[19]  F. Potra.  On the convergence of a class of newton-like methods , 1982 .

[20]  S. Amat,et al.  Adaptive Approximation of Nonlinear Operators , 2004 .

[21]  I. Argyros.  Computational Theory of Iterative Methods , 2014 .

[22]  M. J. Rubio,et al.  Secant-like methods for solving nonlinear integral equations of the Hammerstein type , 2000 .

[23]  Ioannis K. Argyros.  New sufficient convergence conditions for the secant method , 2005 .