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Classification of dynamical switching regimes in a threelayered ferromagnetic nanopillar governed by spinpolarized injection current and external magnetic field. I. Longitudinal anisotropy
Computer Research and Modeling, 2016, v. 8, no. 4, pp. 605620Views (last year): 2. Citations: 6 (RSCI).The mathematical model of the magnetic memory cell MRAM with the inplane anisotropy axis parallel to the edge of a free ferromagnetic layer (longitudinal anisotropy) has been constructed using approximation of uniform magnetization. The model is based on the Landau–Lifshits–Gilbert equation with the injectioncurrent term in the Sloncžewski–Berger form. The set of ordinary differential equations for magnetization dynamics in a threelayered Co/Cu/Cu valve under the control of external magnetic field and spinpolarized current has been derived in the normal coordinate form. It was shown that the set of equations has two main stationary points on the anisotropy axis at any values of field and current. The stationary analysis of them has been performed. The algebraic equations for determination of additional stationary points have been derived. It has been shown that, depending on the field and current magnitude, the set of equations can have altogether two, four, or six stationary points symmetric in pairs relatively the anisotropy axis. The bifurcation diagrams for all the points have been constructed. The classification of the corresponding phase portraits has been performed. The typical trajectories were calculated numerically using Runge–Kutta method. The regions, where stable and unstable limit cycles exist, have been determined. It was found that the unstable limit cycles exist around the main stable equilibrium point on the axis that coincides with the anisotropy one, whereas the stable cycles surround the unstable additional points of equilibrium. The area of their existence was determined numerically. The new types of dynamics, such as accidental switching and noncomplete switching, have been found. The threshold values of switching current and field have been obtained analytically. The estimations of switching times have been performed numerically.

On the convergence of the implicit iterative linebyline recurrence method for solving difference elliptical equations
Computer Research and Modeling, 2017, v. 9, no. 6, pp. 857880Views (last year): 15. Citations: 1 (RSCI).In the article a theory of the implicit iterative linebyline recurrence method for solving the systems of finitedifference equations which arise as a result of approximation of the twodimensional elliptic differential equations on a regular grid is stated. On the one hand, the high effectiveness of the method has confirmed in practice. Some complex test problems, as well as several problems of fluid flow and heat transfer of a viscous incompressible liquid, have solved with its use. On the other hand, the theoretical provisions that explain the high convergence rate of the method and its stability are not yet presented in the literature. This fact is the reason for the present investigation. In the paper, the procedure of equivalent and approximate transformations of the initial system of linear algebraic equations (SLAE) is described in detail. The transformations are presented in a matrixvector form, as well as in the form of the computational formulas of the method. The key points of the transformations are illustrated by schemes of changing of the difference stencils that correspond to the transformed equations. The canonical form of the method is the goal of the transformation procedure. The correctness of the method follows from the canonical form in the case of the solution convergence. The estimation of norms of the matrix operators is carried out on the basis of analysis of structures and element sets of the corresponding matrices. As a result, the convergence of the method is proved for arbitrary initial vectors of the solution of the problem.
The norm of the transition matrix operator is estimated in the special case of weak restrictions on a desired solution. It is shown, that the value of this norm decreases proportionally to the second power (or third degree, it depends on the version of the method) of the grid step of the problem solution area in the case of transition matrix order increases. The necessary condition of the method stability is obtained by means of simple estimates of the vector of an approximate solution. Also, the estimate in order of magnitude of the optimum iterative compensation parameter is given. Theoretical conclusions are illustrated by using the solutions of the test problems. It is shown, that the number of the iterations required to achieve a given accuracy of the solution decreases if a grid size of the solution area increases. It is also demonstrated that if the weak restrictions on solution are violated in the choice of the initial approximation of the solution, then the rate of convergence of the method decreases essentially in full accordance with the deduced theoretical results.

Computer studies of polynomial solutions for gyrostat dynamics
Computer Research and Modeling, 2018, v. 10, no. 1, pp. 725Views (last year): 15.We study polynomial solutions of gyrostat motion equations under potential and gyroscopic forces applied and of gyrostat motion equations in magnetic field taking into account Barnett–London effect. Mathematically, either of the above mentioned problems is described by a system of nonlinear ordinary differential equations whose right hand sides contain fifteen constant parameters. These parameters characterize the gyrostat mass distribution, as well as potential and nonpotential forces acting on gyrostat. We consider polynomial solutions of Steklov–Kovalevski–Gorjachev and Doshkevich classes. The structure of invariant relations for polynomial solutions shows that, as a rule, on top of the fifteen parameters mentioned one should add no less than twenty five problem parameters. In the process of solving such a multiparametric problem in this paper we (in addition to analytic approach) apply numeric methods based on CAS. We break our studies of polynomial solutions existence into two steps. During the first step, we estimate maximal degrees of polynomials considered and obtain a nonlinear algebraic system for parameters of differential equations and polynomial solutions. In the second step (using the above CAS software) we study the solvability conditions of the system obtained and investigate the conditions of the constructed solutions to be real.
We construct two new polynomial solutions for Kirchhoff–Poisson. The first one is described by the following property: the projection squares of angular velocity on the nonbaracentric axes are the fifth degree polynomials of the angular velocity vector component of the baracentric axis that is represented via hypereliptic function of time. The second solution is characterized by the following: the first component of velocity conditions is a second degree polynomial, the second component is a polynomial of the third degree, and the square of the third component is the sixth degree polynomial of the auxiliary variable that is an inversion of the elliptic Legendre integral.
The third new partial solution we construct for gyrostat motion equations in the magnetic field with Barnett–London effect. Its structure is the following: the first and the second components of the angular velocity vector are the second degree polynomials, and the square of the third component is a fourth degree polynomial of the auxiliary variable which is found via inversion of the elliptic Legendre integral of the third kind.
All the solutions constructed in this paper are new and do not have analogues in the fixed point dynamics of a rigid body.

Quadratic Padé Approximation: Numerical Aspects and Applications
Computer Research and Modeling, 2019, v. 11, no. 6, pp. 10171031Padé approximation is a useful tool for extracting singularity information from a power series. A linear Padé approximant is a rational function and can provide estimates of pole and zero locations in the complex plane. A quadratic Padé approximant has square root singularities and can, therefore, provide additional information such as estimates of branch point locations. In this paper, we discuss numerical aspects of computing quadratic Padé approximants as well as some applications. Two algorithms for computing the coefficients in the approximant are discussed: a direct method involving the solution of a linear system (wellknown in the mathematics community) and a recursive method (wellknown in the physics community). We compare the accuracy of these two methods when implemented in floatingpoint arithmetic and discuss their pros and cons. In addition, we extend Luke’s perturbation analysis of linear Padé approximation to the quadratic case and identify the problem of spurious branch points in the quadratic approximant, which can cause a significant loss of accuracy. A possible remedy for this problem is suggested by noting that these troublesome points can be identified by the recursive method mentioned above. Another complication with the quadratic approximant arises in choosing the appropriate branch. One possibility, which is to base this choice on the linear approximant, is discussed in connection with an example due to Stahl. It is also known that the quadratic method is capable of providing reasonable approximations on secondary sheets of the Riemann surface, a fact we illustrate here by means of an example. Two concluding applications show the superiority of the quadratic approximant over its linear counterpart: one involving a special function (the Lambert $W$function) and the other a nonlinear PDE (the continuation of a solution of the inviscid Burgers equation into the complex plane).

Theoretical substantiation of the mathematical techniques for joint signal and noise estimation at rician data analysis
Computer Research and Modeling, 2016, v. 8, no. 3, pp. 445473Views (last year): 2. Citations: 2 (RSCI).The paper provides a solution of the twoparameter task of joint signal and noise estimation at data analysis within the conditions of the Rice distribution by the techniques of mathematical statistics: the maximum likelihood method and the variants of the method of moments. The considered variants of the method of moments include the following techniques: the joint signal and noise estimation on the basis of measuring the 2nd and the 4th moments (MM24) and on the basis of measuring the 1st and the 2nd moments (MM12). For each of the elaborated methods the explicit equations’ systems have been obtained for required parameters of the signal and noise. An important mathematical result of the investigation consists in the fact that the solution of the system of two nonlinear equations with two variables — the sought for signal and noise parameters — has been reduced to the solution of just one equation with one unknown quantity what is important from the view point of both the theoretical investigation of the proposed technique and its practical application, providing the possibility of essential decreasing the calculating resources required for the technique’s realization. The implemented theoretical analysis has resulted in an important practical conclusion: solving the twoparameter task does not lead to the increase of required numerical resources if compared with the oneparameter approximation. The task is meaningful for the purposes of the rician data processing, in particular — the image processing in the systems of magneticresonance visualization. The theoretical conclusions have been confirmed by the results of the numerical experiment.

Numerical solution of Urysohn type nonlinear second kind integral equations by successive quadratures using embedded Dormand and Prince scheme 5(4)
Computer Research and Modeling, 2020, v. 12, no. 2, pp. 275300We present the iterative algorithm that solves numerically both Urysohn type Fredholm and Volterra nonlinear onedimensional nonsingular integral equations of the second kind to a specified, modest userdefined accuracy. The algorithm is based on descending recursive sequence of quadratures. Convergence of numerical scheme is guaranteed by fixedpoint theorems. Picard’s method of integrating successive approximations is of great importance for the existence theory of integral equations but surprisingly very little appears on numerical algorithms for its direct implementation in the literature. We show that successive approximations method can be readily employed in numerical solution of integral equations. By that the quadrature algorithm is thoroughly designed. It is based on the explicit form of fifthorder embedded Runge–Kutta rule with adaptive stepsize selfcontrol. Since local error estimates may be cheaply obtained, continuous monitoring of the quadrature makes it possible to create very accurate automatic numerical schemes and to reduce considerably the main drawback of Picard iterations namely the extremely large amount of computations with increasing recursion depth. Our algorithm is organized so that as compared to most approaches the nonlinearity of integral equations does not induce any additional computational difficulties, it is very simple to apply and to make a program realization. Our algorithm exhibits some features of universality. First, it should be stressed that the method is as easy to apply to nonlinear as to linear equations of both Fredholm and Volterra kind. Second, the algorithm is equipped by stopping rules by which the calculations may to considerable extent be controlled automatically. A compact C++code of described algorithm is presented. Our program realization is selfconsistent: it demands no preliminary calculations, no external libraries and no additional memory is needed. Numerical examples are provided to show applicability, efficiency, robustness and accuracy of our approach.

Reduction of decision rule of multivariate interpolation and approximation method in the problem of data classification
Computer Research and Modeling, 2016, v. 8, no. 3, pp. 475484Views (last year): 5.This article explores a method of machine learning based on the theory of random functions. One of the main problems of this method is that decision rule of a model becomes more complicated as the number of training dataset examples increases. The decision rule of the model is the most probable realization of a random function and it's represented as a polynomial with the number of terms equal to the number of training examples. In this article we will show the quick way of the number of training dataset examples reduction and, accordingly, the complexity of the decision rule. Reducing the number of examples of training dataset is due to the search and removal of weak elements that have little effect on the final form of the decision function, and noise sampling elements. For each $(x_i,y_i)$th element sample was introduced the concept of value, which is expressed by the deviation of the estimated value of the decision function of the model at the point $x_i$, built without the $i$th element, from the true value $y_i$. Also we show the possibility of indirect using weak elements in the process of training model without increasing the number of terms in the decision function. At the experimental part of the article, we show how changed amount of data affects to the ability of the method of generalizing in the classification task.

Mirror descent for constrained optimization problems with large subgradient values of functional constraints
Computer Research and Modeling, 2020, v. 12, no. 2, pp. 301317The paper is devoted to the problem of minimization of the nonsmooth functional $f$ with a nonpositive nonsmooth Lipschitzcontinuous functional constraint. We consider the formulation of the problem in the case of quasiconvex functionals. We propose new strategies of stepsizes and adaptive stopping rules in Mirror Descent for the considered class of problems. It is shown that the methods are applicable to the objective functionals of various levels of smoothness. Applying a special restart technique to the considered version of Mirror Descent there was proposed an optimal method for optimization problems with strongly convex objective functionals. Estimates of the rate of convergence for the considered methods are obtained depending on the level of smoothness of the objective functional. These estimates indicate the optimality of the considered methods from the point of view of the theory of lower oracle bounds. In particular, the optimality of our approach for Höldercontinuous quasiconvex (sub)differentiable objective functionals is proved. In addition, the case of a quasiconvex objective functional and functional constraint was considered. In this paper, we consider the problem of minimizing a nonsmooth functional $f$ in the presence of a Lipschitzcontinuous nonpositive nonsmooth functional constraint $g$, and the problem statement in the cases of quasiconvex and strongly (quasi)convex functionals is considered separately. The paper presents numerical experiments demonstrating the advantages of using the considered methods.

The stable estimation of intensity of atmospheric pollution source on the base of sequential function specification method
Computer Research and Modeling, 2009, v. 1, no. 4, pp. 391403The approach given in this work helps to organize the operative control over action intensity of pollution emissions in atmosphere. The approach allows to sequential estimate of unknown intensity of atmospheric pollution source on the base of concentration measurements of impurity in several stationary control points is offered in the work. The inverse problem was solved by means of the stepbystep regularization and the sequential function specification method. The solution is presented in the form of the digital filter in terms of Hamming. The fitting algorithm of regularization parameter r for function specification method is described.
Keywords: atmospheric pollution, digital filter.Views (last year): 2. 
Finding equilibrium in twostage traffic assignment model
Computer Research and Modeling, 2021, v. 13, no. 2, pp. 365379Authors describe a twostage traffic assignment model. It contains of two blocks. The first block consists of a model for calculating a correspondence (demand) matrix, whereas the second block is a traffic assignment model. The first model calculates a matrix of correspondences using a matrix of transport costs (it characterizes the required volumes of movement from one area to another, it is time in this case). To solve this problem, authors propose to use one of the most popular methods of calculating the correspondence matrix in urban studies — the entropy model. The second model describes exactly how the needs for displacement specified by the correspondence matrix are distributed along the possible paths. Knowing the ways of the flows distribution along the paths, it is possible to calculate the cost matrix. Equilibrium in a twostage model is a fixed point in the sequence of these two models. In practice the problem of finding a fixed point can be solved by the fixedpoint iteration method. Unfortunately, at the moment the issue of convergence and estimations of the convergence rate for this method has not been studied quite thoroughly. In addition, the numerical implementation of the algorithm results in many problems. In particular, if the starting point is incorrect, situations may arise where the algorithm requires extremely large numbers to be computed and exceeds the available memory even on the most modern computers. Therefore the article proposes a method for reducing the problem of finding the equilibrium to the problem of the convex nonsmooth optimization. Also a numerical method for solving the obtained optimization problem is proposed. Numerical experiments were carried out for both methods of solving the problem. The authors used data for Vladivostok (for this city information from various sources was processed and collected in a new dataset) and two smaller cities in the USA. It was not possible to achieve convergence by the method of fixedpoint iteration, whereas the second model for the same dataset demonstrated convergence rate $k^{1.67}$.
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