# zbMATH — the first resource for mathematics

Error bounds for the secant method. (English) Zbl 0757.65070
The paper studies an iterative procedure of the secant method in the form $$x_{n+1}=x_ n-\delta f(x_{n-1},x_ n)^{-1}f(x_ n)$$ for the solution of the equation $$f(x)=0$$, where $$f$$ is a nonlinear operator between Banach spaces $$E$$ and $$\hat E$$, $$x_{-1}$$ and $$x_ 0$$ are two points in the domain of $$f$$ and $$\delta f$$ is a consistent approximation of $$\nabla f$$.
A priori and a posteriori error estimates for the iterative procedure are provided. The estimates are eventually better than those presented in literature, under the same assumptions. A simple example is provided to show that the results are compared favourably with the available corresponding results.
##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators
Full Text:
##### References:
 [1] ARGYROS I. K.: Newton-like methods under mild differentiability conditions with error analysis. Bull. Austral. Math. Soc. Vol. 37, 2, 1987, 131-147. · Zbl 0629.65061 [2] ARGYROS I. K.: On Newton’s method and nondiscrete mathematical induction. Bull. Austral. Math. Soc. Vol. 38, 1988, 131-140. · Zbl 0642.65043 [3] DENNIS J. E.: Toward a unified convergence theory for Newton-like methods. Nonlinear Functional Analysis and Applications, L. B. Rail, Academic Press, New York, 1971. · Zbl 0276.65029 [4] GRAGG W. B., TAPIA R. A.: Optimal error bounds for the Newton-Kantorovich theorem. S.I.A.M. J. Numer. Anal. 11, 1, 1974, 10-13. · Zbl 0284.65042 [5] OSTROWSKI M. A.: Solution of equations in Euclidian and Banach spaces. Academic Press, New York, 1973. [6] POTRA F. A., PTÁK V.: Sharp error bounds for Newton’s process. Numer. Math. 34, 1980, 63-72. · Zbl 0434.65034 [7] POTRA F. A.: An error analysis for the Secant method. Numer. Math. 38, 1982, 427-445. · Zbl 0465.65033 [8] POTRA F. A.: Sharp error bounds for a class of Newton-like methods. Libertas Mathematica 5, 1985, 71-84. · Zbl 0581.47050 [9] POTRA F. A., PTÁK V.: Nondiscrete induction and iterative processes. Pitman Publ. Boston, 1984. · Zbl 0549.41001 [10] PTÁK V.: Nondiscrete mathematical induction and iterative existence proofs. Linear Algebra Appl. 13, 1976, 223-236. · Zbl 0323.46005 [11] SCHMIDT J. W.: Regula-Falsi Verfahren mit konsistenter Steigung und Majoranten Prinzip. Period. Math. Hungar. 5, 3, 1974, 187-193. · Zbl 0291.65017 [12] SERGEEV A. S.: On the method of chords Sibirsk. Mat. Z. 2, 1961, 282-289.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.