Stable numerical methods for solving simultaneous 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.

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.

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.

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 solution of systems of linear algebraic equations (SLAE) is one of the most common problems in applications. Such problems usually require large amounts of computing. Often, solving a system is an incorrect task when the matrix of its coefficients has a rectangular shape, degenerate or poorly conditioned. Today, without the use of modern means of information and communication technologies, it is generally impossible to imagine solution of a SLAU. The article substantiates the need to develop a digital resource for solving systems of linear algebraic equations of any complexity, with different types of coefficient matrix. It is shown that one of the most effective development tools is the Python programming language. Due to the presence of many packages, it has a sufficient set of tools for its implementation. The built-in functions of the mathematical packages NumPy and SciPy, designed to solve linear algebra problems, are considered. The possibilities of using the Matplotlib package to visualize the solution of SLAE and the Tkinter built-in package for developing the graphical interface of the resource being developed are shown.

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.

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.

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.

A discrete maximum principle is presented for linear algebraic systems with coefficients of arbitrary signs and the number of unknowns exceeding (in the general case) the number of equations. The presentation uses the formal language of linear algebra (without any notions from the theory of difference schemes). This formal algebraic approach allows one to reveal easily the existence or absence of data for realization of an extended maximum principle in any algorithm represented in the form of a system of linear algebraic equations. The theorems formulated here directly imply the absolute stability of solutions to systems of equations. Variants of homogeneous algebraic systems and systems of equations with a right-hand side are considered. Theoretical results are illustrated by examples of actual problems.

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

Recurrence solution of a block tridiagonal matrix equation with Neumann, Dirichlet, mixed or periodic boundary conditions

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.

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.

Article Systems of linear algebraic equations with random coefficients when the regularization matrices are used was published on January 1, 1993 in the journal Random Operators and Stochastic Equations (volume 1, issue 2).