The inexact secant method \([x_{k-1},x_{k};F]s_{k}=-F(x_k) +r_k\), \( x_{k+1}=x_k+s_k\), \( k=1,2,\ldots\), \( x_0,x_1 \in {\mathbb R}^n\) for solving the nonlinear system \( F(x)=0\) is considered, where \(F:{\mathbb R}^n \rightarrow {\mathbb R}^n\) is a nonlinear mapping.

We study the setting of the inexact Newton method, i.e., when the linear system (involving the divided differences) at each step is not solved exactly, and an error term \(r_k\) (called residual) is considered.

Under certain standard assumptions, we characterize the superlinear convergence and the r-convergence order of the secant method in terms of the residuals. We also give a sufficient result for linear convergence.


E. Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)


nonlinear system of equations; secant method; r-convergence order; inexact method; residual; error term; linear convergence.

Cite this paper as:

E. Cătinaş, A note on inexact secant methods, Rev. Anal. Numér. Théor. Approx., 25 (1996) nos. 1-2, pp. 33-41.


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