scholarly journals SECTIONING OF HIGH DIMENSIONAL BANDED MATRICES

2014 ◽  
pp. 14-21
Author(s):  
Dmytro Fedasyuk ◽  
Pavlo Serdyuk ◽  
Yuriy Semchyshyn
Keyword(s):  
Linear Equations ◽  
Computing Time ◽  
Mathematical Physics ◽  
High Dimensional ◽  
Systems Of Equations ◽  
Algebraic Equations ◽  
Banded Matrices ◽  
Linear Algebraic Equations ◽  
Systems Of Linear Equations ◽  
The Subject

Solving high dimensional systems of linear algebraic equations is of use to many problems of mathematical physics, in particular, it is one of the main subgoals at solving systems of equations in partial derivatives. Distributed solving of high dimensional systems of linear equations allows to reduce computing time, especially in cases when these matrices can not be kept in one computer's RAM. The subject of this study is the search of optimal high dimensional matrices sectioning algorithms for distributed solving systems of linear algebraic equations.

Combined method for correction interval systems of linear algebraic equations

Metrologiya ◽  
2021 ◽  
pp. 17-39
Author(s):  
A. N. Bazhenov ◽  
A. Yu. Telnova
Keyword(s):  
Linear Equations ◽  
Interval Uncertainty ◽  
Algebraic Equations ◽  
System Matrix ◽  
Combined Method ◽  
Interval System ◽  
Linear Algebraic Equations ◽  
Linear Interval ◽  
Systems Of Linear Equations ◽  
Interval Systems

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 Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

Activities: Medical Applications of Systems of Linear Equations

1996 ◽  
Vol 89 (5) ◽  
pp. 398-410
Author(s):  
David Jabon ◽  
Gail Nord ◽  
Bryce W. Wilson ◽  
Penny Colfman ◽  
John Nord
Keyword(s):  
Small Group ◽  
Group Work ◽  
Linear Equations ◽  
Medical Application ◽  
Medical Applications ◽  
Systems Of Equations ◽  
Small Group Work ◽  
Geometric Concepts ◽  
Systems Of Linear Equations

This activity is aimed at enriching the algebra strand by bringing to the classroom a medical application of systems of linear equations. These problems go beyond the traditional interpretation of systems of equations as intersections of lines and planes and furnish a nice mix of algebraic and geometric concepts. The activity is designed for small-group work.

scholarly journals On the Solubility of Linear Algebraic Equations (Contd.)

1913 ◽  
Vol 12 ◽  
pp. 137-138
Author(s):  
John Dougall
Keyword(s):  
Linear Equations ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Similar Reasoning ◽  
Null Solution ◽  
Linear Algebraic Equations ◽  
The Given ◽  
Homogeneous Linear

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.

scholarly journals ANALYSIS OF MULTI-THREADED MARKOV SYSTEMS

2021 ◽  
Vol 5 (4) ◽  
pp. 70-78
Author(s):  
Lev Raskin ◽  
Larysa Sukhomlyn ◽  
Dmytro Sagaidachny ◽  
Roman Korsun
Keyword(s):  
Markov Property ◽  
Computational Method ◽  
High Dimensional ◽  
Algebraic Equations ◽  
Parametric Synthesis ◽  
Markov Systems ◽  
Linear Algebraic Equations ◽  
Computational Implementation ◽  
System States ◽  
Analytical Relations

Known technologies for analyzing Markov systems use a well-operating mathematical apparatus based on the computational implementation of the fundamental Markov property. Herewith the resulting systems of linear algebraic equations are easily solved numerically. Moreover, when solving lots of practical problems, this numerical solution is insufficient. For instance, both in problems of structural and parametric synthesis of systems, as well as in control problems. These problems require to obtain analytical relations describing the dependences of probability values of states of the analyzed system with the numerical values of its parameters. The complexity of the analytical solution of the related systems of linear algebraic equations increases rapidly along with the increase in the system dimensionality. This very phenomenon manifests itself especially demonstratively when analyzing multi-threaded queuing systems.  Accordingly, the objective of this paper is to develop an effective computational method for obtaining analytical relations that allow to analyze high-dimensional Markov systems. To analyze such systems this paper provides for a decomposition method based on the idea of phase enlargement of system states. The proposed and substantiated method allows to obtain analytical relations for calculating the distribution of Markov system states.  The method can be effectively applied to solve problems of analysis and management in high-dimensional Markov systems. An example has been considered

scholarly journals On the economical solution method for a system of linear algebraic equations

2004 ◽  
Vol 2004 (4) ◽  
pp. 377-410 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Vadim A. Krysko ◽  
Anton V. Krysko
Keyword(s):  
Differential Equations ◽  
Boundary Value ◽  
Linear Equations ◽  
Three Dimensional ◽  
Boundary Element Methods ◽  
Minimal Amount ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Point Of Intersection ◽  
Stationary Heat Transfer

The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped inℝ3is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error ofO(hx12+hx22). The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

Unsteady double-diffusive buoyancy-induced flow in a triangular solar collector shape enclosure with corrugated bottom wall

2017 ◽  
Vol 27 (6) ◽  
pp. 1282-1303 ◽  
Author(s):  
M.M. Rahman ◽  
Sourav Saha ◽  
Satyajit Mojumder ◽  
Khan Md. Rabbi ◽  
Hasnah Hasan ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Solar Collector ◽  
Linear Equations ◽  
Bottom Wall ◽  
Algebraic Equations ◽  
Weighted Residual Method ◽  
Content Type ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Induced Flow

Purpose The purpose of this investigation is to determine the nature of the flow field, temperature distribution and heat and mass transfer in a triangular solar collector enclosure with a corrugated bottom wall in the unsteady condition numerically. Design/methodology/approach Non-linear governing partial differential equations (i.e. mass, momentum, energy and concentration equations) are transformed into a system of integral equations by applying the Galerkin weighted residual method. The integration involved in each of these terms is performed using Gauss’ quadrature method. The resulting non-linear algebraic equations are modified by the imposition of boundary conditions. Finally, Newton’s method is used to modify non-linear equations into the linear algebraic equations. Findings Both the buoyancy ratio and thermal Rayleigh number play an important role in controlling the mode of heat transfer and mass transfer. Originality/value Calculations are performed for various thermal Rayleigh numbers, buoyancy ratios and time periods. For each specific condition, streamline contours, isotherm contours and iso-concentration contours are obtained, and the variation in the overall Nusselt and Sherwood numbers is identified for different parameter combinations.

The 3-D distribution of sources of optical scattering computed from complex-amplitude far-field data

1976 ◽  
Vol 54 (19) ◽  
pp. 1925-1936 ◽  
Author(s):  
D. K. Lam ◽  
H. G. Schmidt-Weinmar ◽  
A. Wouk
Keyword(s):  
Gaussian Elimination ◽  
Singular Systems ◽  
Linear Equations ◽  
Computing Time ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Optical Scattering ◽  
Far Field ◽  
First Approximation ◽  
Systems Of Linear Equations

We present a fast computer algorithm to solve the scalar inverse scattering problem numerically by inverting a linear transformation which maps a 3-D distribution of scattering sources into the angular distribution of the resultant scattered far field. We show how an approximate solution to the problem can be found in discrete form which leads to non-singular systems of linear equations of a type whose matrix can be inverted readily by fast algorithms.The method uses Born's first approximation and is valid for a slowly varying refractive index: the resultant numerical problem can be solved by a fast algorithm which reduces computing time by ~10−7, storage requirement by ~10−5, as compared with Gaussian elimination applied to 125 000 sample points. With this algorithm, computerized 3-D reconstruction becomes feasible.

scholarly journals Development of a finite-element-based design sensitivity analysis for buckling and postbuckling of composite plates

1995 ◽  
Vol 1 (3) ◽  
pp. 255-274 ◽  
Author(s):  
Ruijiang Guo ◽  
Aditi Chattopadhyay
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Linear Equations ◽  
Composite Plates ◽  
Material Model ◽  
Design Sensitivity ◽  
Algebraic Equations ◽  
Sensitivity Derivatives ◽  
Analysis Technique ◽  
Linear Algebraic Equations

A finite element based sensitivity analysis procedure is developed for buckling and postbuckling of composite plates. This procedure is based on the direct differentiation approach combined with the reference volume concept. Linear elastic material model and nonlinear geometric relations are used. The sensitivity analysis technique results in a set of linear algebraic equations which are easy to solve. The procedure developed provides the sensitivity derivatives directly from the current load and responses by solving the set of linear equations. Numerical results are presented and are compared with those obtained using finite difference technique. The results show good agreement except at points near critical buckling load where discontinuities occur. The procedure is very efficient computationally.

Configuration of 3D Indoor Positioning System Based on Bluetooth Beacons

2021 ◽  
Vol 27 (1) ◽  
pp. 32-40
Author(s):  
V. M. Grinyak ◽  
Keyword(s):  
Measurement Errors ◽  
Theoretical Calculations ◽  
Tracking System ◽  
Linear Equations ◽  
Estimation Procedure ◽  
Accuracy Evaluation ◽  
Algebraic Equations ◽  
Positioning Accuracy ◽  
Model Interpretation ◽  
Linear Algebraic Equations

This paper devoted to research of indoors navigation problems under poor or insufficient quality of satellite navigational data environment. The problem of object positioning in 3D space by Bluetooth devices located indoors forming a multi-position tracking system is considered in this research. Emphasized that in order to succeed for such system it is required to pre-estimate distinctive accuracy. The proposed model interpretation of the positioning problem as the system of linear equations. The classic model interpretation for method of least squares is used for resolution. General problem of linearization around reference resolution is the locality of its features. There are three concepts of problems solvability, such as fundamental solvability (observability), solvability in conditions of instrumental measurement errors and solvability under conditions of finite accuracy of computation on a computer. The first aspect of solvability is interpreted by the completeness of the rank of the corresponding system of linear algebraic equations, the second and third ones represents by the conditionality of the problem and the convergence of the iterative estimation procedure. The conducted experiments show that for the positioning problem the attributes of the linearized model are accurate enough to represent the original nonlinear problem. Such interpretation allows to build theoretical accuracy estimation priors for object coordinates evaluations and to identify the areas with insufficient positioning accuracy. In this paper there are results of expected accuracy evaluation for various system patterns with full-scale experiments proving the theoretical calculations. Experiments for problems with using SKYLAB Beacon VG01 Bluetooth transmitters and smartphone HUAWEI WAS-LX1 are presented and confirmed that math model with linear approximation defined by authors is usable for solving indoors navigation problems using Bluetooth signal. So, for good enough quantity and appropriate location of the tracks the achievable positioning accuracy could be as good as 0.2—0.3 meters for all three coordinates. Such accuracy allows to navigate small hovering objects such as drones. In general, it looks promising to use Bluetooth trackers for solving positioning problems for indoors environments.

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