PYTHON AS AN EFFECTIVE TOOL OF DEVELOPING DIGITAL RESOURCES FOR NUMERICAL SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

In this research paper a new modification of Gauss-Seidel method has been presented for solving the system of linear algebraic equations. The systems of linear algebraic equations have an important role in the field of science and engineering. This modification has been developed by using the procedure of Gauss-Seidel method and the concept of substitution techniques. Developed modification of Gauss-Seidel method is a fast convergent as compared to Gauss Jacobi’s method, Gauss-Seidel method and successive over-relaxation (SOR) method. It works on the diagonally dominant as well as positive definite symmetric systems of linear algebraic equations. Its solution has been compared with the Gauss Jacobi’s method, Gauss-Seidel method and Successive over-Relaxation method by taking different systems of linear algebraic equations and found that, it was reducing to the number of iterations and errors in each problem.

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

There are no strict rules for the numerical treatment of partial differential equations. We concentrate here on linear partial differential equations. As an example of elliptic partial differential equations the two-dimensional Poisson equation with Dirichlet boundary conditions is investigated. The application of the finite difference approximation transforms this equation into a set of algebraic equations which can be solved iteratively. The time dependent heat equation is an example of parabolic partial differential equations. The one-dimensional Dirichlet boundary value problem is transformed using the implicit and explicit Euler method into an inhomogeneous system of linear algebraic equations. Stability problems of the explicit Euler method are discussed and demonstrated numerically. The wave equation is a model for hyperbolic partial differential equations. It can be transformed into a system of algebraic equations using, for instance, the explicit Euler method. Again, stability problems connected with this method are discussed and demonstrated numerically. Finally, the quantum mechanical tunneling effect is studied numerically by solving the time dependent Schrodinger equation by means of a Crank-Nicholson scheme which transforms the Schrodinger equation into a system of linear algebraic equations with tridiagonal matrix.

Iterative refinement of solution with biparameter for solving ill-conditioned systems of linear algebraic equations

The article deals with high-performance information technology (HPC) for the problems of stress-strain analysis at all stages of the life cycle of buildings and structures: construction, operation and reconstruction. The results of numerical simulation of high buildings using software as a processor component based on a new hybrid algorithm for solving systems of linear algebraic equations [1] with a symmetric positive-definite matrix that combines computation on multi-core processors and graphs. It has been found that to accelerate the calculations, hybrid systems that combine multi-core CPUs with accelerator coprocessors, including GPUs, are promising [5]. To test the effectiveness of the proposed parallel algorithm for solving systems of linear algebraic equations [1], numerical experiments were carried out at the most dangerous loads of a 27-story building. Results of numerical researches with use for preprocessor (input of initial data) and postprocessor (output of results of calculations) of processing of the LIRA-SAPR software complex are presented [2, 4, 6]. The results of numerical studies of the behavior of structures of high buildings have shown a multiple reduction in the time of solving systems of linear algebraic equations with symmetric matrices on multiprocessor (multi-core) computers with graphical accelerators using the proposed hybrid algorithms [1]. High-performance technologies based on parallel calculations give more effect than more complex processes: modeling of life cycle of high buildings, bridges, especially complex structures of NPPs, etc. for static and dynamic loads, including emergencies in normal and difficult geological conditions, which make up 70% of Ukraine's territories.

Stable numerical methods for solving simultaneous systems of linear algebraic equations

The possibility of application of the interval analysis for data processing in the field of spectral analysis is considered. It is assumed that the data have interval uncertainty; therefore the problem of finding unknown concentrations is posed as a linear interval tolerance problem. The incompatibility of the interval system of linear algebraic equations is shown for the initial data using the apparatus of the recognizing functional. The relevance of the topic is due to the need for regularization of inconsistent interval systems of linear equations. The idea of S. P. Shary of a combined method for correcting a linear tolerance problem has been implemented. A new method for managing the solution by changing the linear algebraic equations interval system matrix elements radii has been developed. The research results can be used for example, to calculate the substance’s concentrations by measurement of the characteristic X-ray radiation.

A finite-element model developed for quasi-static processes describes irreversible processes of deformation and polarization occurring in polycrystalline ferroelectric media due to the effect of intense electric fields and mechanical loads. The paper presents external parameters such as strain and polarization as a sum of residual and reversible parts. Using the incremental theory, the virtual work law, and the constitutive relations for reversible and irreversible components, a system of linear algebraic equations was built for the increments of nodal values of the main variables, namely the displacement vector and the electric potential, during the transition from one equilibrium state to another.The constructed constitutive relations connect the reversible parts of the strain and polarization with the stresses and the electric field in the form of linear tensor equations. It is shown that the physical characteristics depend on the residual parameters so that the coefficients of elastic compliance and dielectric constant linearly depend on the principal values of residual strain, and the piezoelectric modules depend linearly on the module of residual polarization. The constitutive relations for the increments of the residual parameters are determined as element values for each finite element from the equations in differentials. Ultimately, the task is reduced to a system of linear algebraic equations, the matrix and right sides of which depend on the residual parameters and are determined at each equilibrium state. As a result, the non-linearity of the problem is replaced by solving a sequence of linear problems until the external loads reach their final values.The model is implanted into a finite-element complex, which allows us to determine the fields of residual strain and polarization, the physical characteristics of a partially polarized body, and local anisotropy for the case of complete and partial polarization.

Introduction. This study investigates the possibility of increasing the accuracy of numerically solving boundary value problems using the modified Bubnov-Galerkin method with a linear ordinary differential equation, where the coefficients and the right-hand side are continuous functions. The order of the differential equation n must be less than the number of coordinate functions m.Materials and Methods. A modified Petrov-Galerkin method was used to numerically solve the boundary value problem. It employs a system of linearly independent power-type basis functions on the interval [−1,1], each normalized by the unit Chebyshev norm. The system of linear algebraic equations includes only the linearly independent boundary conditions of the original problem.Results. For the first time, an integral quadrature formula with a 22nd order error was developed for a uniform grid. This formula is used to calculate the matrix elements and coefficients in the right-hand side of the system of linear algebraic equations, taking into account the scalar product of two functions based on the new quadrature formula. The study proves a theorem on the existence and uniqueness of a solution for boundary value problems with general non-separated conditions, provided that n linearly independent particular solutions of a homogeneous differential equation of order n are known.Discussion and Conclusion. The hydrodynamic problem in a viscous strong boundary layer with a third-order equation was precisely solved. The analytical solution was compared with its numerical counterpart, and the uniform norm of their difference did not exceed 5·10‒15. The formulas derived using the generalized Bubnov-Galerkin method may be useful for solving boundary value problems with linear ordinary differential equations of the third and higher orders.

The stress - strain state of a layer with a cylindrical cavity was investigated, when displacements are set at the boundaries of the layer and at the boundary of the cavity. On the cavity and on the boundaries of the layer, displacements are given.The solution of the spatial problem of the theory of elasticity is obtained by generalized Fourier method in relation to the system of Lamex equations in cylindrical coordinates connected with the cylinder, and Cartesian coordinates associated with the boundaries of the layer. Infinite systems of linear algebraic equations obtained as a result of satisfaction of boundary conditions, solved by the truncation method. As a result, movements and strains were obtained at different points of the elastic body.Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the method of reduction, which is proved by the high accuracy of the implementation of boundary conditions.A numerical analysis of the stress - strain state of the body at various distances from the cylindrical cavity to the boundaries of the layer is carried out.The greatest normal stresses occur on the surface of the cylindrical cavity and on the isthmus between the surface of the cylindrical cavity and the boundaries of the layer. As the boundary surfaces approach each other, the stresses increase.The greatest tangential stresses arise on the surface of the cylindrical cavity at φ = 5π / 16, φ = 11π / 16, φ = 21π / 16 and φ = 27π / 16, and with the approximation of the boundary surfaces to each other the stress increases.The greatest tangential stresses arise on the surface of the cylindrical cavity along the z axis at φ = 0 and at φ = p. When the boundaries of the layer approximation the cylindrical cavity decreases.The given analysis and algorithm of calculation can be used in the design of constructions, in the calculation schemes of which there is a layer with a cylindrical cavity and specified on the boundary surfaces by displacements.The stress - strain state of a layer with a cylindrical cavity was investigated, when displacements are set at the boundaries of the layer and at the boundary of the cavity. On the cavity and on the boundaries of the layer, displacements are given.The solution of the spatial problem of the theory of elasticity is obtained by generalized Fourier method in relation to the system of Lamex equations in cylindrical coordinates connected with the cylinder, and Cartesian coordinates associated with the boundaries of the layer. Infinite systems of linear algebraic equations obtained as a result of satisfaction of boundary conditions, solved by the truncation method. As a result, movements and strains were obtained at different points of the elastic body.Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the method of reduction, which is proved by the high accuracy of the implementation of boundary conditions.A numerical analysis of the stress - strain state of the body at various distances from the cylindrical cavity to the boundaries of the layer is carried out.The greatest normal stresses occur on the surface of the cylindrical cavity and on the isthmus between the surface of the cylindrical cavity and the boundaries of the layer. As the boundary surfaces approach each other, the stresses increase.The greatest tangential stresses arise on the surface of the cylindrical cavity at φ = 5π / 16, φ = 11π / 16, φ = 21π / 16 and φ = 27π / 16, and with the approximation of the boundary surfaces to each other the stress increases.The greatest tangential stresses arise on the surface of the cylindrical cavity along the z axis at φ = 0 and at φ = p. When the boundaries of the layer approximation the cylindrical cavity decreases.The given analysis and algorithm of calculation can be used in the design of constructions, in the calculation schemes of which there is a layer with a cylindrical cavity and specified on the boundary surfaces by displacements.

For the regularized solutions of systems of linear algebraic equations we find statistical estimates with respect to observations over the coefficient matrix of the system of equations. Under certain conditions it is proved that these estimates are asymptotically normal.

This paper proposes a distributed dynamics that solves the least-squares problem associated with a network system of linear algebraic equations. We consider static directed multi-agent networks. Each agent in the network has access to a private subset of the linear equations. Furthermore, we assume that agents cannot acquire any information about their "out-degrees" at any time. Under the strong connectivity condition on the underlying communication network, we show that the local estimated solution of each agent converges exponentially to the exact least-squares solution of the associated network system of linear algebraic equations.

An iterative solution process for systems of linear algebraic equations is proposed. It converges starting from any initial approximation and theoretically does not require preliminary transformation of the input data.