F. Soleymani

Construction of Optimal Derivative-Free Techniques without Memory

Construction of iterative processes without memory, which are both optimal according to the Kung-Traub hypothesis and derivative-free, is considered in this paper. For this reason, techniques with four and five function evaluations per iteration, which reach to the optimal orders eight and sixteen, respectively, are discussed theoretically. These schemes can be viewed as the generalizations of the recent optimal derivative-free family of Zheng et al. in (2011). This procedure also provides an n-step family using function evaluations per full cycle to possess the optimal order 2n. The analytical proofs of the main contributions are given and numerical examples are included to confirm the outstanding convergence speed of the presented iterative methods using only few function evaluations. The second aim of this work will be furnished when a hybrid algorithm for capturing all the zeros in an interval has been proposed. The novel algorithm could deal with nonlinear functions having finitely many zeros in an interval.

Some Matrix Iterations for Computing Matrix Sign Function

Some iterative methods are introduced and demonstrated for finding the matrix sign function. It is analytically shown that the new schemes are asymptotically stable. Convergence analysis along with the error bounds of the main proposed method is established. Different numerical experiments are employed to compare the behavior of the new schemes with the existing matrix iterations of the same type.

Approximating the Matrix Sign Function Using a Novel IterativeMethod

This study presents a matrix iterative method for finding the sign of a square complex matrix. It is shown that the sequence of iterates converges to the sign and has asymptotical stability, provided that the initial matrix is appropriately chosen. Some illustrations are presented to support the theory.

A Class of Steffensen-Type Iterative Methods for Nonlinear Systems

A class of iterative methods without restriction on the computation of Frechet derivatives including multisteps for solving systems of nonlinear equations is presented. By considering a frozen Jacobian, we provide a class of m-step methods with order of convergence . A new method named as Steffensen-Schulz scheme is also contributed. Numerical tests and comparisons with the existing methods are included.

Computing Simple Roots by an Optimal Sixteenth-Order Class

The problem considered in this paper is to approximate the simple zeros of the function by iterative processes. An optimal 16th order class is constructed. The class is built by considering any of the optimal three-step derivative-involved methods in the first three steps of a four-step cycle in which the first derivative of the function at the fourth step is estimated by a combination of already known values. Per iteration, each method of the class reaches the efficiency index , by carrying out four evaluations of the function and one evaluation of the first derivative. The error equation for one technique of the class is furnished analytically. Some methods of the class are tested by challenging the existing high-order methods. The interval Newtons method is given as a tool for extracting enough accurate initial approximations to start such high-order methods. The obtained numerical results show that the derived methods are accurate and efficient.

A New High-Order Stable Numerical Method for Matrix Inversion

A stable numerical method is proposed for matrix inversion. The new method is accompanied by theoretical proof to illustrate twelfth-order convergence. A discussion of how to achieve the convergence using an appropriate initial value is presented. The application of the new scheme for finding Moore-Penrose inverse will also be pointed out analytically. The efficiency of the contributed iterative method is clarified on solving some numerical examples.

New Mono- and Biaccelerator Iterative Methods with Memory for Nonlinear Equations

Acceleration of convergence is discussed for some families of iterative methods in order to solve scalar nonlinear equations. In fact, we construct mono- and biparametric methods with memory and study their orders. It is shown that the convergence orders 12 and 14 can be attained using only 4 functional evaluations, which provides high computational efficiency indices. Some illustrations will also be given to reverify the theoretical discussions.

A Numerical Method for Computing the Principal Square Root of a Matrix

It is shown how the mid-point iterative method with cubical rate of convergence can be applied for finding the principal matrix square root. Using an identity between matrix sign function and matrix square root, we construct a variant of mid-point method which is asymptotically stable in the neighborhood of the solution. Finally, application of the presented approach is illustrated in solving a matrix differential equation.

Approximating the Inverse of a Square Matrix with Application in Computation of the Moore-Penrose Inverse

This paper presents a computational iterative method to find approximate inverses for the inverse of matrices. Analysis of convergence reveals that the method reaches ninth-order convergence. The extension of the proposed iterative method for computing Moore-Penrose inverse is furnished. Numerical results including the comparisons with different existing methods of the same type in the literature will also be presented to manifest the superiority of the new algorithm in finding approximate inverses.

Efficient iterative methods with and without memory possessing high efficiency indices

Two families of derivative-free methods without memory for approximating a simple zero of a nonlinear equation are presented. The proposed schemes have an accelerator parameter with the property that it can increase the convergence rate without any new functional evaluations. In this way, we construct a method with memory that increases considerably efficiency index from to . Numerical examples and comparison with the existing methods are included to confirm theoretical results and high computational efficiency.