Secant-type Methods Research Articles

On a unified convergence analysis for Newton-type methods solving generalized equations with the Aubin property

A plethora of applications from diverse disciplines reduce to solving generalized equations involving Banach space valued operators. These equations are solved mostly iteratively, when a sequence is generated approximating a solution provided that certain conditions are valid on the starting point and the operators appearing on the method. Secant-type methods are developed whose specializations reduce to well known methods such as Newton, modified Newton, Secant, Kurchatov and Steffensen to mention a few. Unified local as well as semi-local analysis of these methods is presented using the celebrated contraction mapping principle in combination with the Aubin property of a set valued operator, and generalized continuity assumption on the operators on these methods. Numerical applications complement the theory.

Secant-inexact projection algorithms for solving a new class of constrained mixed generalized equations problems

In this paper, a new version of a secant-type method for solving constrained mixed generalized equations is addressed. The method is a combination of the secant method applied to generalized equations with the conditional gradient method. We use the contraction mapping principle to establish the convergence results. Moreover, by assuming the Lipschitz condition on the gradient and the metric regularity property, we show that the sequence generated by the proposed algorithm is well-defined and locally convergent for a solution with linear or superlinear rate.

Derivative-Free Conformable Iterative Methods for Solving Nonlinear Equations

In this manuscript, we use approximations of conformable derivatives for designing iterative methods to solve nonlinear algebraic or trascendental equations. We adapt the approximation of conformable derivatives in order to design conformable derivative-free iterative schemes to solve nonlinear equations: Steffensen and Secant-type methods. To our knowledge, these are the first conformable derivative-free schemes in the literature, where the Steffensen conformable method is also optimal; moreover, the Secant conformable scheme is also a procedure with memory. A convergence analysis is made, preserving the order of classical cases, and the numerical performance is studied in order to confirm the theoretical results. It is shown that these methods can present some numerical advantages versus their classical partners, with wide sets of converging initial estimations.

A significant improvement of a family of secant-type methods

Secant-type iterative methods are usually used to solve nonlinear systems of equations where the operator involved is nondifferentiable or its derivative is costly. From a uniparametric family of secant-type methods, that is reduced to the secant method and Newton’s method for some particular values of the parameter involved, with operational cost similar to that of the secant and Newton methods and superlinear convergence, we construct a biparametric family of iterative methods with quadratic convergence, study their local convergence, their efficiency and, based on the dynamics of the methods, their accessibility.

Extending the solvability of equations using secant-type methods in Banach space

We extend the solvability of equations dened on a Banach space using numerically ecient secant-type methods. The convergence domain of these methods is enlarged using our new idea of restricted convergence region. By using this approach, we obtain a more precise location where the iterates lie than in earlier studies leading to tighter Lipschitz constants. This way the semi-local convergence produces weaker sucient convergence criteria and tighter error bounds than in earlier works. These improvements are also obtained under the same computational eort, since the new Lipschitz constants are special cases of the old ones.

Extended Convergence of a Two-Step-Secant-Type Method Under a Restricted Convergence Domain

We present a local as well as a semi-local convergence analysis of a two-step secant-type method for solving nonlinear equations involving Banach space valued operators. By using weakened Lipschitz and center Lipschitz conditions in combination with a more precise domain containing the iterates, we obtain tighter Lipschitz constants than in earlier studies. This technique lead to an extended convergence domain, more precise information on the location of the solution and tighter error bounds on the distances involved. These advantages are obtained under the same computational effort, since the new constants are special cases of the old ones used in earlier studies. The new technique can be used on other iterative methods. The numerical examples further illustrate the theoretical results.

Further Acceleration of Secant-Type Methods for Solving Nonlinear Equations

Further Acceleration of Secant-Type Methods for Solving Nonlinear Equations

Generalized Inverses Estimations by Means of Iterative Methods with Memory

A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the Moore–Penrose inverse and Drazin inverse are obtained. The convergence and the order of convergence is presented in each case. Some numerical tests allowed us to confirm the theoretical results and to compare the performance of our method with other known ones. With these results, the iterative methods with memory appear for the first time for estimating the solution of a nonlinear matrix equations.

Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces

The semilocal convergence of double step Secant method to approximate a locally unique solution of a nonlinear equation is described in Banach space setting. Majorizing sequences are used under the assumption that the first-order divided differences of the involved operator satisfy the weaker Lipschitz and the center-Lipschitz continuity conditions. A theorem is established for the existence-uniqueness region along with the estimation of error bounds for the solution. Our work improves the results derived in Ren and Argyros (2015) in more stringent Lipschitz and center Lipschitz conditions and gives finer majorizing sequences. Also, an example is worked out where the conditions of Ren and Argyros (2015) fail but our’s work. Numerical examples including nonlinear elliptic differential equations and integral equations are worked out. It is found that our conditions enlarge the convergence domain of the solution. Finally, taking a nonlinear system of m equations, the Efficiency Index (EI) and the Computational Efficiency Index (CEI) of double step Secant method are computed and its comparison with respect to other similar existing iterative methods are summarized in the tabular forms.

Accelerating order of convergence using secant type methods

By amalgamating a secant type method and a Newton type method, another method of order 4 has been derived. The method has been supported by examples and has been compared with existing similar methods.

A UNIFIED CONVERGENCE ANALYSIS FOR SECANT-TYPE METHODS

We present a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost our semilocal convergence criteria can be weaker; the error bounds more precise and in the local case the convergence balls can be larger and the error bounds tighter than in earlier studies such as [1-3,7-14,16,20,21] at least for the cases of Newton's method and the secant method. Numerical examples are also presented to illustrate the theoretical results obtained in this study.

Two-step secant type method with approximation of the inverse operator

The two-step secant type method with a consistent approximation of the inverse operator for solving nonlinear equations is proposed. The local convergence of the proposed method is studied and the quadratic convergence order is established. A numerical experiment is made on the test problems.

Directional Secant-Type Methods for Solving Equations

A semilocal convergence analysis for directional Secant-type methods in multidimensional space is provided. Using weaker hypotheses than the ones exploited by An and Bai, we provide a semilocal convergence analysis with the following advantages: weaker convergence conditions, larger convergence domain, finer error estimates on the distances involved, and more precise information on the location of the solution. A numerical example, where our results apply to solve an equation but not the ones of An and Bai, is also provided. In a second example, we show how to implement the method.

Local convergence of efficient Secant-type methods for solving nonlinear equations

A local convergence analysis for an efficient Secant-type method (STM) for solving nonlinear equations is given using both the Lipschitz continuous and center-Lipshchitz continuous divided differences of order one. An estimate of the radius of the convergence ball of (STM) is provided, the error estimate matching its convergence order is established. Numerical examples validating the theoretical results are also provided in the concluding section of this study.

Secant-type methods and nondiscrete induction

The celebrated nondiscrete mathematical induction has been used to improve error bounds of distances involved in the discrete case but not the sufficient convergence conditions for Secant---type methods. We show that using the same information as before, the following advantages can be obtained: weaker sufficient convergence conditions; tighter error bounds on the distances involved and a more precise information on the location of the solution. Numerical examples validating the theoretical conclusions are also provided in this study.

Three-point methods with and without memory for solving nonlinear equations

A new family of three-point derivative free methods for solving nonlinear equations is presented. It is proved that the order of convergence of the basic family without memory is eight requiring four function-evaluations, which means that this family is optimal in the sense of the Kung–Traub conjecture. Further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory. The self-correcting parameter is calculated applying the secant-type method in three different ways and Newton’s interpolatory polynomial of the second degree. The corresponding R-order of convergence is increased from 8 to 4 ( 1 + 5 / 2 ) ≈ 8.472 , 9, 10 and 11. The increase of convergence order is attained without any additional function calculations, providing a very high computational efficiency of the proposed methods with memory. Another advantage is a convenient fact that these methods do not use derivatives. Numerical examples and the comparison with existing three-point methods are included to confirm theoretical results and high computational efficiency.

On the semilocal convergence of efficient Chebyshev–Secant-type methods

We introduce a three-step Chebyshev–Secant-type method (CSTM) with high efficiency index for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for (CSTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study.

Optimal Secant-Type Methods for Operator Equations

We prove an existence and uniqueness theorem for operator equations in Banach spaces with (generally non-differentiable) operators whose divided differences are Lipschitz continuous on operator's domain. The theorem makes possible to apply the concept of entropy optimality of iterative methods introduced in our earlier work to the class of secant-type methods. Using this concept, we show that it is feasible to get a method that needs the same information (one value of the operator) per iteration but exhibits a faster convergence than the secant and secant-update methods.

Local convergence of a secant type method for solving least squares problems

Local convergence of a secant type iterative method for approximating a solution of nonlinear least squares problems is investigated in this paper. The radius of convergence is determined as well as usable error estimates. Numerical examples are also provided.

Convergence conditions for secant-type methods

We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.