A family of root-finding methods with accelerated convergence

A parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given.     

The simultaneous determination of all zeros of a polynomial

A class of adaptive iterative methods of higher order for the simultaneous determination of all zeros of a polynomial is constructed. These methods preserve their order of convergence also in the case of multiple roots. Numerical examples are included.     

Higher-order simultaneous methods for the determination of polynomial multiple zeros

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.     

Numerical solution of initial-value problems by collocation methods using generalized piecewise functions

A general class of piecewise functions is described which leads to the same order of convergence of collocation methods as piecewise polynomials. This order only depends on the collocation points used.     

Efficient Jarratt-like methods for solving systems of nonlinear equations

We present the iterative methods of fourth and sixth order convergence for solving systems of nonlinear equations. Fourth order method is composed of two Jarratt-like steps and requires the evaluations of one function, two first derivatives and one matrix inversion in each iteration. Sixth order method is the composition of three Jarratt-like steps of which the first two steps are that of the proposed fourth order scheme and requires one extra function evaluation in addition to the evaluations of fourth order method. Computational efficiency in its general form is discussed. A comparison between the efficiencies of proposed techniques with existing methods of similar nature is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in the examples. It is shown that the present methods are more efficient than their existing counterparts, particularly when applied to the large systems of equations.     

A hybrid optimization method and its role in computer-aided design

The variable metric methods have been analyzed and exploited in the past to achieve a superlinear rate of convergence in unconstrained optimization. Recently, the update formulas used in these methods have been extended for constrained optimization. The update formulas construct approximate second order derivatives using first order information. These are referred to as constrained variable metric (CVM) methods. Higher order of convergence is often accompanied by a smaller domain of convergence. To overcome this limitation of CVM methods, a hybrid optimization algorithm is presented in this paper. It uses a cost function bounding concept initially and a CVM method in later stages of the search process. In addition, the algorithm uses an active set strategy and is globally convergent. An improved active set strategy is suggested. Besides developing a superlinear optimization algorithm, an efficient programming structure for computer-aided design of engineering systems is suggested and implemented. A number of mathematical programming problems, and small and large scale engineering design problems are solved to test numerical aspects of the algorithm. The algorithm has performed extremely well on the test problems.     

On a second order method for the simultaneous inclusion of polynomial complex zeros in rectangular arithmetic

Starting from separated rectangles in the complex plane which contain polynomial complex zeros, an iterative method of second order for the simultaneous inclusion of these zeros is formulated in rectangular arithmetic. The convergence and a condition for convergence are considered. Applying Gauss-Seidel approach to the proposed method, two accelerated interval methods are formulated. TheR-order of convergence of these methods is determined. An analysis of the convergence order is given in the presence of rounding errors. The presented methods are illustrated numerically in examples of polynomial equations.     

On theR-order of some accelerated methods for the simultaneous finding of polynomial zeros

A Gauss-Seidel procedure is applied to increase the convergence of a basic fourth order method for finding polynomial complex zeros. Further acceleration of convergence is performed by using Newton's and Halley's corrections. It is proved that the lower bounds of theR-order of convergence for the proposed serial (single-step) methods lie between 4 and 7. Computational efficiency and numerical examples are also given.     

A family of Newton-type methods for solving nonlinear equations

A family of Newton-type methods free from second and higher order derivatives for solving nonlinear equations is presented. The order of the convergence of this family depends on a function. Under a condition on this function this family converge cubically and by imposing one condition more on this function one can obtain methods of order four. It has been shown that this family covers many of the available iterative methods. From this family two new iterative methods are obtained. Numerical experiments are also included.     

Derivative Free Inclusion Methods for Polynomial Zeros

Two iterative methods for the simultaneous inclusion of complex zeros of a polynomial are presented. Both methods are realized in circular interval arithmetic and do not use polynomial derivatives. The first method of the fourth order is composed as a combination of interval methods with the order of convergence two and three. The second method is constructed using double application of the inclusion method of Weierstrass’ type in serial mode. It is shown that its R-order of convergence is bounded below by the spectral radius of the corresponding matrix. Numerical examples illustrate the convergence rate of the presented methods     

Efficient two-step derivative-free iterative methods with memory and their dynamics

In this paper, we present a family of three-parameter derivative-free iterative methods with and without memory for solving nonlinear equations. The convergence order of the new method without memory is four requiring three functional evaluations. Based on the new fourth-order method without memory, we present a family of derivative-free methods with memory. Using three self-accelerating parameters, calculated by Newton interpolatory polynomials, the convergence order of the new methods with memory are increased from 4 to 7.0174 and 7.5311 without any additional calculations. Compared with the existing methods with memory, the new method with memory can obtain higher convergence order by using relatively simple self-accelerating parameters. Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods.     

Efficient iterative algorithms for bounding the inverse of a matrix

In this paper we are considering iterative methods for bounding the inverse of a matrix, which make use of interval arithmetic. We present a class of methods as a combination of ordinary Schulz's methods for only approximating the inverse matrix (see [3]) and of well-known interval Schulz's methods (see [1]). Two convergence theorems are proved. Our methods are shown to be asymptotically of the same order of convergence as the ordinary Schulz's methods being part of them. Therefore we are getting considerably more efficient interval methods by our approach than by the classical interval Schulz's methods in [1] or [5]. A numerical example is given.     

基于维度分区的果蝇优化新算法

为提高果蝇算法的收敛稳定性,提出了一种基于维度分区的果蝇优化新算法。将果蝇种群均分为两组:跟随果蝇和搜索果蝇。跟随果蝇在全局最优果蝇附近实现精细化局部搜索,而搜索果蝇则将位置向量的每个维度搜索范围划分为若干个区间,通过比较各个区间的最优位置来更新果蝇位置。为加快算法收敛速度,若某搜索果蝇在连续若干次迭代过程中 均 表现最差,则在当前最优果蝇位置附近产生该果蝇的新位置。针对8种典型函数的仿真实验表明:与传统算法相比, 所提算法所需参数较少,收敛稳定性高,并且在收敛精度及收敛速度等方面具有明显优势。     

A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function

A new family of four-step optimal multipoint iterative methods of order sixteen for solving nonlinear equations are developed along with their convergence properties. Numerical experiments with comparison to some existing methods are demonstrated to support the underlying theory.     

Isomorphic iterative methods in solving singularly perturbed elliptic difference equations

A new approach for the efficient numerical solution of Singular Perturbation (SP) second order boundary-value problems based on Gradient-type methods is introduced. Isomorphic implicit iterative schemes in conjuction with the Extended to the Limit sparse factorization procedures [9] are used for solving SP second order elliptic equations in two and three-space dimensions. Theoretical results on the convergence rate of these first-degree iterative methods for three-space variables are presented. The application of the new methods on characteristic SP boundary-value problems is discussed and numerical results are given.     

An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points

In this paper, we not only develop an optimal class of three-step eighth-order methods with higher order weight functions employed in the second and third sub-steps, but also investigate their dynamics underlying the purely imaginary extraneous fixed points. Their theoretical and computational properties are fully described along with a main theorem stating the order of convergence and the asymptotic error constant as well as extensive studies of special cases with rational weight functions. A number of numerical examples are illustrated to confirm the underlying theoretical development. Besides, to show the convergence behaviour of global character, fully explored is the dynamics of the proposed family of eighth-order methods as well as an existing competitive method with the help of illustrative basins of attraction.     

Efficient algorithms for the inclusion of the inverse matrix using error-bounds for hyperpower methods

By exploiting generalized error-bounds for the well-known hyperpower methods for approximating the inverse of a matrix we derive inclusion methods for the inverse matrix. These methods make use of interval operations in order to give guaranteed inclusions whenever the convergence of the applied hyperpower method can be shown. The efficiency index of some of the new methods is greater than that of the optimal methods in [2] or [5]. A numerical example is given.     

The numerical solution of higher index differential–algebraic equations by 4-point spline collocation methods

We study the order, stability, and convergence properties of 4-point spline collocation methods if applied to differential/algebraic systems with index greater than or equal one. These methods do not in general attain the same order of accuracy for higher index differential/algebraic systems as they do for index-1 systems and for purely differential systems. We show that the 4-point spline collocation methods applied to differential/algebraic systems with index-ν are stable and the order of convergence is 8???ν, ν?=?2(1)7. For both index-1 and purely differential systems the order is seven. Finally, some numerical experiments are presented that illustrate the theoretical results.     

Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations

Numerical methods preserving a conserved quantity for stochastic differential equations are considered. A class of discrete gradient methods based on the skew-gradient form is constructed, and the sufficient condition of convergence order 1 in the mean-square sense is given. Then a class of linear projection methods is constructed. The relationship of the two classes of methods for preserving a conserved quantity is proved, which is, the constructed linear projection methods can be considered as a subset of the constructed discrete gradient methods. Numerical experiments verify our theory and show the efficiency of proposed numerical methods.     

Order barriers for the B-convergence of ROW methods

The convergence of ROW methods is studied when these methods are applied to the stiff model equation of Prothero and Robinson. It turns out that there are barriers of the attainable order of convergence. Furthermore, the existence of ROW methods is shown the accuracy of which increases asymptotically with the stiffness of the model. The theoretical results are demonstrated by numerical examples.