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Variants of Steffensen-Secant method and applications. (English) Zbl 1200.65036

Let \(f:D \rightarrow\mathbb R\) be a sufficiently differentiable function with a simple root \(a \in D\), with \(D \subset\mathbb R\) an open set. The authors define a parametric variant of the Steffensen-secant method as follows. Let \(A_{n+1} = f(x_n)\) and \(B_{n+1} = f(x_n+\lambda_nA_{n+1})\). Also, let \(\bar{x_{n+1}} = x_n-\frac{\lambda_nA^2_{n+1}}{B_{n+1}-A_{n+1}}\), and \(C = f(\bar{x_{n+1}})\). Then
\[ x_{n+1} = x_n - \frac{\lambda_nA^3_{n+1}}{[B_{n+1}-A_{n+1}][A_{n+1}-C_{n+1}]}. \]
Note that this only requires three evaluations of the function at each step. The authors prove that one obtains at least cubic convergence.
For three judicious choices of \(\lambda_n\), the authors are able to prove \(\lim_{n\rightarrow \infty}\lambda_n = - 1/f'(a)\), and thus that the third-order asymptotic convergence constant is 0. This gives super cubic convergence for these choices of \(\lambda_n\).
The authors also present modifications of Steffensen-secant methods for multiple roots. With appropriate hypotheses, a modified parametric variant of the Steffensen-secant method is linearly convergent, and the modified variant is quadratically convergent.
In addition to proving the efficiency of their methods, the authors show by experimentation that their methods are nearly always faster and converge faster than previously proposed methods. They also apply their methods to the “multiple-shooting method” for solving boundary value problems.

MSC:

65H05 Numerical computation of solutions to single equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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