A plethora of problems from diverse disciplines such as Mathematics, Mathematical: Biology, Chemistry, Economics, Physics, Scientific Computing and also Engineering can be formulated as an equation ...defined in abstract spaces using Mathematical Modelling. The solutions of these equations can be found in closed form only in special case. That is why researchers and practitioners utilize iterative procedures from which a sequence is being generated approximating the solution under some conditions on the initial data.
This type of research is considered most interesting and challenging. This is our motivation for the introduction of this special issue on Iterative Procedures.
In this paper, we consider two-step fourth-order and three-step sixth-order derivative free iterative methods and study their convergence in Banach spaces to approximate a locally-unique solution of ...nonlinear equations. Study of convergence analysis provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions using higher order derivatives. Furthermore, in quest of fast algorithms, a generalized
-step scheme with increasing convergence order 2
+ 2 is proposed and analyzed. Novelty of the
-step algorithm is that, in each step, order of convergence is increased by an amount of two at the cost of only one additional function evaluation. To maximize the computational efficiency, the optimal number of steps is calculated. Theoretical results regarding convergence and computational efficiency are verified through numerical experimentation.
In this study, we introduce a novel weight function-based eighth order derivative-free method for locating repeated roots of nonlinear equations. It is a three-step Steffensen-type scheme with first ...order divided differences in place of the first order derivatives. It is noteworthy that so far only few eighth order derivative free multiple root finding scheme exist in literature. Different nonlinear standard and applications based nonlinear functions are used to demonstrate the applicability of the suggested approach and to confirm its strong convergence tendency. Drawing basins of attraction on the graphical regions demonstrates how the offered family of approaches converge.
In this paper we establish the construction of a family of free derivative of point to point iterative processes, with quadratic convergence, from two known data in each previous iteration. Besides, ...we study the accessibility of this family by means of the basins of attraction and the convergence balls. We provide a local convergence analysis for the family of iterative processes free of derivatives, when the operator F is not necessarily Fréchet differentiable. The sufficient convergence conditions are weaker and more flexible than in earlier studies. An application is provided involving mixed Hammerstein nonlinear integral equation with application in real world problems. Finally, we show also a dynamical study and the convergence regions of some members of the family.
We present a derivative free two-step family of fourth order methods for solving systems of nonlinear equations using the well-known Traub–Steffensen method in the first step. In order to determine ...the local convergence order, we apply the first-order divided difference operator for functions of several variables and direct computation by Taylor’s expansion. Computational efficiencies of the methods of new family are considered and compared with existing methods of similar structure. It is showed that the new family is especially efficient in solving large systems. Four numerical examples are given to compare the proposed methods with existing methods and to confirm the theoretical results.
Many higher order multiple-root solvers that require derivative evaluations are available in literature. Contrary to this, higher order multiple-root solvers without derivatives are difficult to ...obtain, and therefore, such techniques are yet to be achieved. Motivated by this fact, we focus on developing a new family of higher order derivative-free solvers for computing multiple zeros by using a simple approach. The stability of the techniques is checked through complex geometry shown by drawing basins of attraction. Applicability is demonstrated on practical problems, which illustrates the efficient convergence behavior. Moreover, the comparison of numerical results shows that the proposed derivative-free techniques are good competitors of the existing techniques that require derivative evaluations in the iteration.
We present a family of multipoint methods without memory with sixth-order convergence for solving systems of nonlinear equations. The methods use first-order divided difference operators and are ...derivative free. Extending the work further, we explore a family of methods with memory with increasing order of convergence from the basic family without memory. The increase in convergence order is attained by varying a free parameter step to step using information available from the previous step. It is proved that the convergence order of the family with memory is increased from 6 to
, and in some special cases to
and
. Computational efficiency of the methods is discussed and compared with existing methods. Numerical examples, including those arise from integral equations and boundary value problems, are considered to verify the theoretical results. A comparison with the existing methods shows that the new methods are more efficient than existing ones and hence use the minimum computing time in multiprecision arithmetic.
It is generally desirable to construct a higher order iterative method for locating roots of multiplicity
m
that preserves its theoretical convergence in applications, but in reality, this is not the ...case in general. With this aim, a three-step Traub-Steffensen-type (derivative-free) iterative family is proposed and investigated for obtaining roots of multiplicity
m
of nonlinear equations. The iterative scheme uses four function evaluations per iteration and possesses eighth order of convergence, hence it is optimal. Special cases of the family are thoroughly investigated for their dynamical properties and numerical performance. Estimation of the convergence domain in the complex plane and comparison of the results with existing methods provide a substantial idea about the convergence of new methods. The findings of analysis are remarkable in the sense that the developed scheme exhibits superiority over the existing counterparts by displaying the wider convergence regions along with the rapid convergence rates. In addition, numerical accuracy is analyzed by locating the multiple zeros of selected nonlinear problems including some of the applied nature like Van der Waals equation and Eigen value problem. The performance analysis also shows additional properties of the family by demonstrating high precision in numerical computations while preserving the theoretical order of convergence.
The application of fractional calculus for solving nonlinear equations through iterative techniques is an emerging area of research. In recent times, several Newton-type methods have been proposed ...which particularly utilize the concept of fractional order derivatives. However, convergence of such methods essentially require the existence of at least first order derivative. Accordingly, in the case where derivative is not feasible to obtain, the derivative-free methods are of much significance. In this paper, a fractional Traub-Steffensen-type method is developed, the formulation of which is based on the idea of conformable fractional derivatives of order
α
∈
(
0
,
1
. In addition, the scheme is extended to its multi-dimensional case to solve the systems of equations. The proposed derivative-free scheme is further investigated for its dynamical aspects and convergence characteristics by varying the value of
α
in given range. In this concern, the convergence planes are presented in a well-defined region which in general provide the fundamental information about the stability of given method. Furthermore, the numerical performance is analyzed by locating the solutions of variety of nonlinear equations, including some applied problems.
We derive two iterative methods with memory for approximating a simple root of any nonlinear equation. For this purpose, we take two optimal methods without memory of order four and eight and convert ...them into the methods with memory without increasing any further function evaluation. These methods involve a self-accelerator (parameter) that depends upon the iteration index to increase the order of the optimal methods. Consequently, the efficiency of the new methods is considerably high as compared to the methods without memory. Some numerical examples are provided in support of the theoretical results.
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