Choice Of Initial Approximations Research Articles

Some Aspects of Identification of Dynamic Objects under Incorrect Observation Conditions

We solve the problems of numerical-analytical representation of the solution of an equation describing the dynamic object and its measurable output, as well as optimal computation of the values of continuous linear functionals (numerical characteristics) of measurable functions based on incorrect data containing not only fluctuation error but also singular interference. The method provides the maximum possible decomposition of computational procedures, does not require to carry out traditional linearization operations and the choice of initial approximations, and also is not related to the computation of spectral coefficients in finite linear combinations (with given basic functions) that describe integral curves, measurable functions, and singular interference.

Computational geometry as a tool for studying root-finding methods

We present an efficient method from Computational geometry, a branch of computer science devoted to the study of algorithms, for mathematical visualization of a third order root solver. For many decades the quality of iterative methods for solving nonlinear equations were analyzed only by using numerical experiments. The disadvantage of this approach is the inconvenient fact that convergence behavior strictly depends on the choice of initial approximations and the structure of functions whose zeros are sought, which often makes the convergence analysis very hard and incomplete. For this reason in this paper we apply dynamic study of iterative processes relied on basins of attraction, a new and powerful methodology developed at the beginning of the 21th century. This approach provides graphic visualization of the behavior of convergent sequences and, consequently, offers considerably better insight into the quality of applied root solvers, especially into the domain of convergence. For demonstration, we present dynamic study of one parameter family of Halley?s type introduced in the first part of the paper. Characteristics of this family are discussed by basins of attractions for various values of the involved parameter. Special attention is devoted to clusters of polynomial roots, one of the most difficult problems in the topic. The analysis of the methods and presentation of basins of attractions are performed by the computer algebra system Mathematica.

Restoration of Orientation Distribution Function Using Texture Components with Radial Normal Distributions

The methods for restoration of the orientation distribution function (ODF) from experimental pole figures have been compared for materials possessing a low symmetry of specimen (by example of the median section of the hot-pressed band from Mg–4.5% Nd magnesium alloy), namely, the method of texture components using radial normal distributions on SO(3) with different spreading and the method of superposition of a large number of normal distributions with equivalent small spreading. Both approaches have demonstrated similar ODFs. In this case, the former method, which is less sensitive to measurement errors of pole figures, is based on nonlinear optimization with a complex choice of initial approximations of the parameters of the model. The latter approach is more sensitive and easier to use.

Mathematical modelling of the reaction of condensation telomerization and the investigation of the model

A mathematical model of the distribution of mixture components in the equilibrium condensation telomerization is developed depending on the ratio of the amounts of monomers and telogen, as well as the number of HX as regulating parameters, the computer implementation of the model is carried out, and its study is carried out by numerical simulation. The model is based on the well-known schematic diagram of the flow of the condensation telomerization process under the assumption of equal reactivity of the same functional groups (Flory principle). Based on the analysis of the flow pattern of the process, 6 structural elements are identified, reproducible at each stage associated with an increase in the degree of polymerization based on 4 basic components. It is proved that the equilibrium concentrations of these elements, depending on the polymerization degree, depend on the equilibrium concentration of products with a degree of polymerization 1 and are described by infinite geometric progression with the same denominator. According to the physical content of the task, this progression must be convergent. Equations of material balance of components are contained in the form of a system with 4 equations containing infinite sums. It is possible to minimize these sums using the properties of geometric progressions and to obtain a closed system with 4 nonlinear equations for the equilibrium concentrations of the base components. The Monte Carlo method is used to study the features of the numerical solution of the system of equations of the model. It is found that with a random choice of initial approximations of solutions from an admissible region, the system contains 4 roots, of which 2 contain positive and negative components and are false, and 2 have completely positive components. A valid criterion for finding a real root has a physical meaning based on the calculation of the denominator of a geometric progression. The possibilities of practical use of the model are discussed

A note on some recent methods for solving nonlinear equations

In this note we present some comments on the recent results concerning iterative methods for solving nonlinear equations. In the first part we compare a new method [N. Ujević, A method for solving nonlinear equations, Appl. Math. Comput. 174 (2006) 1416–1426] with the Newton and Ostrowski method, including an extensive analysis of numerical results together with the choice of initial approximations, and the computational efficiency of the considered methods. In the second part we give a list of recently derived root-finding methods and we study the question of priority; namely, we find out that all reviewed methods were already derived at the beginning of the 60s of the last century, one of them dates from the 18th century. Some remarks on the comparison analysis and the application of iteration methods are included.

Necessary conditions of the isolated zero-points with nonzero degree and choice of initial approximations for the secant method in the n-dimensional case

Necessary conditions of isolated zero-points with nonzero degree (in particular, point of local minimum or maximum either saddlepoint) in R n which is an extension of the rule that the first derivative of a function changes a sign in every neighborhood of an isolated point of local minimum or maximum in JR1 and connected with this choice of initial approximations for the secant method are given and proved. Most of the statements are given through using the basic topological facts such as degree, the fiberings of a finite dimensional sphere, homotopy, quotient topology, etc.

The optimal choice of initial approximations to solutions of regularized equations in the theory of branching

In [1, 2], V. A. Trenogin and the author proposed a method for constructing regularizing equations (R.E.) for ill-posed problems in the theory of branching [3, 4]. Here we consider the problem of choosing initial approximations, that are optimal in a certain sense, to simple solutions of R.E. in a neighborhood of a branching point.