Steffensen's Method Research Articles

On a Halley-Steffensen method for approximating the solutions of scalar equations

In the present paper we show that the Steffensen method for solving the scalar equation \(f\left( x\right) =0\), applied to equation\[h\left( x\right) =\tfrac{f\left( x\right) }{\sqrt{f^{\prime}\left( x\right) }}=0,\]leads to bilateral approximations for the solution. Moreover, the convergence order is at least 3, i.e. as in the case of the Halley method.

A mesh-independence principle for operators equations and the Steffensen method

In this study we prove the mesh-independence principle via Steffensen’s method. This principle asserts that when Steffensen’s method is applied to a nonlinear equation between some Banach spaces, as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration. Local and semilocal convergence results as well as an error analysis for Steffensen’s method are also provided.

A constructive approach to the Schröder equation

Given a function ƒ, analytic at the origin and such that ƒ (0) = 0, ƒ′ (0) ≠ 0, the Schröder equation (Blanchard, 1984) reads β g( z) = g(ƒ( z)), where β = ƒ′ (0). In the present paper we introduce a formal computational algorithm to approximate the analytic function g. Our main tool consists of evaluating a generating function of certain, recursively defined, polynomials. Another application of our analysis is to convergence acceleration of functional iteration. As demonstrated by the author (1991), the analysis of a generalized Steffensen's method (1933) requires the consideration of certain determinants. In particular, it is required to show that they do not vanish. As an offshoot of our technique, we evaluate the exact value of these determinants.

A Fixed Point Method for Finding Percentage Points

SUMMARY Iterative methods for finding accurate estimates of percentage points are usually based on numerical root finding techniques applied to the distribution function. In this paper we take a different approach by creating an auxiliary function, whose fixed point is shown to be the desired percentage point, and then applying Steffenson's acceleration technique to find the fixed point. For the auxiliary function used, the conditions for convergence of Steffensen's method are mild and are satisfied by many percentage point problems arising in practice. The method is tested by calculating four-place x2 percentage points and two-place F percentage points which agree with those in published tables.

On the convergence of a class of generalized steffensen's iterative procedures and error analysis

We consider a class of generalized Steffensen's methods for solving nonlinear equations without any derivative. We establish successful the existence-convergence theorem of generalized Steffensen's iteration under the Kantorovich-Ostrowski's conditions and present an upper bound, but it is not optimal. These conclusions contain the special case of Steffensen's method [6]. In the mean time, we compare the accurate error bound of Newton's method with error bounds of generalized Steffensen's methods, we show that latter is less than former.

Iterative regularization of Steffensen's method

A regularized version of Steffensen's method is proposed for the minimization of functions on a set in a Hilbert space defined by inequality and equality constraints. We derive conditions linking the regularization parameters and the penalty coefficient with the numerical method parameter, which ensure strong convergence of the method to the normal solution.

Regularization of Steffensen's method with inexactly specified initial data

A minimization problem is considered for the case when the minimand function and the functions defining the set of equality and inequality constraints are known with an error. A regularized Steffensen's method for the solution of this problem is proposed. It is shown that, when the variation of the regularization and penalty parameters is compatible with the errors, the sequence generated by this method converges in norm to the minimum point with minimum norm.

Monotone enclosing of solutions of the Steffensen‐type

In this paper we consider two iterative methods of the Steffensen‐type. The iteration sequences which approximate the solution of f(x) = 0 from opposite sides are generated by Steffensen's method and a secant method, respectively. We show two enclosing theorems and establish orders of convergence. The iteration sequences converge with order two and three, respectively. Numerical examples complete the paper.

Results about monotone convergence of Steffensen-like-methods

Certain iterations are considered which are extensions of the Steffensen method to higher dimensions. Sequences of upper and lower bounds for the solution of nonlinear equations are obtained. It is shown under which conditions the sequences converge monotonically and the convergence is quadratic.

Monotonieeigenschaften des Steffensen-Verfahrens

Steffensen's method is slightly generalized by introducing a real parameter. In this way one can get different monotonicity properties, depending on the choice of this parameter. These monotonicity statements give the possibility of bracketing the solution of a given problem. In a special case they even ensure the convergence and the existence of a solution. Furthermore there are given sufficient conditions, which show that Steffensen's method converges at least as quickly as Newton's method. A numerical example shows the efficiency of the theorems and compares Steffensen's and Newton's method.

A Family of Functional Iterations and the Solution of Maximum Likelihood Estimating Equations

In this paper a family of functional iterations is introduced. One member of this family is the Newton-Raphson method and another member, obtained from a generalization of Steffensen's method to a system of equations, has been considered in [7]. The general member of the family is derived from a regulafalsi construction, due to Gauss, for a particular choice of points in the iteration. From the computational point of view, all the members of the family of iterations, except the Newton-Raphson method, have the property that the partial derivatives of the system of equations are used almost never if a computing device with unlimited precision is utilized. Further, the asymptotic speed of convergence for any member is at least of order two. In view of the difficulties of obtaining the functional form of the second order partials of the likelihood function for general linear and nonlinear simultaneous systems, the method proposed here may be recommended in the computation of full information maximum likelih Dod estimates. Even if the partials of the system of equations are easily calculated, then some member of the family may still lead to convergence if the Newton-Raphson method does not. Practically speaking, the proposed method can be used to determine an approximate solution and this approximate solution will be closer to the solution if the precision of the computations is higher.

Extension of steffensen's method for solving nonlinear operator equations

Extension of steffensen's method for solving nonlinear operator equations