Ivan Petković

On an application of symbolic computation and computer graphics to root-finders

The contemporary powerful mathematical software enables a new approach to handling and manipulating complex mathematical expressions and other mathematical objects. Particularly, the use of symbolic computation leads to new contribution to constructing and analyzing numerical algorithms for solving very difficult problems in applied mathematics and other scientific disciplines. In this paper we are concerned with the problem of determining multiple zeros when the multiplicity is not known in advance, a task that is seldom considered in literature. By the use of computer algebra system Mathematica, we employ symbolic computation through several programs to construct and investigate algorithms which both determine a sought zero and its multiplicity. Applying a recurrent formula for generating iterative methods of higher order for solving nonlinear equations, we construct iterative methods that serve (i) for approximating a multiple zero of a given function f when the order of multiplicity is unknown and, simultaneously, (ii) for finding exact order of multiplicity. In particular, we state useful cubically convergent iterative sequences that find the exact multiplicity in a few iteration steps. Such approach, combined with a rapidly convergent method for multiple zeros, provides the construction of efficient composite algorithms for finding multiple zeros of very high accuracy. The properties of the proposed algorithms are illustrated by several numerical examples and basins of attraction.

Computational efficiency of some combined methods for polynomial equations

Abstract The iterative methods for the simultaneous determination of all simple complex zeros of algebraic polynomials, based on the fixed point relation of Ehrlich’s type, are considered. Using the iterative correction appearing in the Jarratt method of the fourth order, it is proved that the convergence rate of the modified Ehrlich method is increased from 3 to 6. This acceleration of the convergence is obtained with few additional numerical operations which means that the proposed combined method possesses very high computational efficiency. Moreover, the convergence rate can be further accelerated using the Gauss–Siedel approach (single-step or serial mode). A great part of the paper is devoted to the computational aspects of the discussed methods, including numerical examples. A comparison procedure shows that the new iterative method is more efficient than existing methods in the considered class.

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newtons type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratts corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.

Symbolic computation and computer graphics as tools for developing and studying new root-finding methods

Abstract Many very difficult problems in applied mathematics and other scientific disciplines cannot be solved without powerful computational systems, such as symbolic computation and computer graphics. In this paper we construct two new families of the fourth order iterative methods for finding a multiple real or complex zero of a given function. For developing these methods, a recurrent formula for generating iterative methods of higher order for solving nonlinear equations is applied and implemented by symbolic computation through several programs in computer algebra system Mathematica . Symbolic computation was the only tool for solving the considered complex problem since it provides handling and manipulating complex mathematical expressions and other mathematical objects. The properties of the proposed rapidly convergent methods are illustrated by several numerical examples. To examine the convergence behavior of the presented methods, we also give the dynamic study of these methods using basins of attraction. Such a methodology, besides a visualization of iterative processes, deliveries very important features on iterations including running CPU time and average number of iterations, as a function of starting points. The program for plotting basins of attraction in Mathematica is included.

Interval matrix models of design iteration

Engineering design and development of new products, which are either industrial products, technical innovations, hardware or software, often contain a very complex set of relationships among many coupled tasks. Controlling, redesigning and identifying features of these tasks can be usefully performed by a suitable model based on the design structure matrix in an iteration procedure. The proposed interval matrix model of design iteration controls and predicts slow and rapid convergence of iteration work on tasks within a project. A new model is based on Perron-Frobenius theorem and interval linear algebra where intervals and interval matrices are employed instead of real numbers and real matrices. In this way a more relaxed quantitative estimation of tasks is achieved and the presence of undetermined quantities is allowed to a certain extent. The presented model is demonstrated in the example of simplified software development process.

ON THE CHARACTERIZATION OF TASKS MODELED BY INTERVAL DESIGN STRUCTURE MATRIX ON DOMAIN-DRIVEN DESIGN SOFTWARE DEVELOPMENT

Development and design of new products of various kinds often contain a very complex set of relationships among many coupled tasks. Ranking, controlling and redesigning the features of these tasks can be usefully performed by a suitable model based on the design structure matrix in an iteration procedure. The proposed interval approach of design iteration controls and predicts the convergence speed of iteration work on tasks within a project. Interval method is based on Perron-Frobenius theorem and interval linear algebra where intervals and interval matrices are employed instead of real numbers and real matrices. In this way, a more relaxed quantitative estimation of tasks is achieved and the presence of undetermined quantities is allowed to a certain extent. The presented model is demonstrated in the example of simplified domain-driven design process, an approach to software development.