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.

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.

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.

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.

THE UNIVERSAL SOLUTIONS OF INTERVAL SYSTEMS OF LINEAR ALGEBRAICAL EQUATIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 477-485 ◽  
Author(s):  
L.T. ASCHEPKOV ◽  
D.V. DOLGY
Keyword(s):  
Linear Programming ◽  
Linear Equations ◽  
Constructive Methods ◽  
New Concepts ◽  
Systems Of Linear Equations ◽  
Interval Coefficients ◽  
Interval Systems ◽  
Universal Solutions

The new concepts of ε-solutions and universal solutions for systems of linear equations with interval coefficients are introduced. To find these solutions some constructive methods using linear programming technics are proposed.

scholarly journals CONSTRUCTION OF EXPERT-STATISTICAL MODELS FROM INCOMPLETE DATA

T-Comm ◽  
2021 ◽  
Vol 15 (6) ◽  
pp. 33-39
Author(s):  
Sergey I. Noskov ◽  
Keyword(s):  
Linear Regression ◽  
Statistical Models ◽  
Incomplete Data ◽  
Algebraic Equations ◽  
Linear Regression Equation ◽  
Railway Transport ◽  
Specific Question ◽  
Interval System ◽  
Linear Algebraic Equations ◽  
Initial Uncertainty

The article deals with the problem of constructing a linear regression model based on incomplete data containing gaps, using statistical and expert information. The reasons for the gaps in the data can be, in particular, a temporary malfunction (failure) of the measuring equipment when taking various technical characteristics, or negligence in the work of statistical services when fixing the reporting indicators. Very often, gaps arise when processing various kinds of sociological information in the form of questionnaires, when respondents refuse to answer a specific question (but answer others) or give an inadmissible, in particular, evasive answer. The approach proposed in the work involves filling the gaps with intervals, the boundaries of which are formed by experts, guided by both their experience and knowledge about the object of research, and using the well-known methods of point filling in the gaps. After that, the estimation of the parameters of the model, depending on the nature of the initial uncertainty in the data, is reduced to solving problems of linear or partially Boolean linear programming. The case is considered when the solution of the formalizing uncertainty in the initial data of the interval system of linear algebraic equations is not unique. The problem of constructing a linear regression equation for the influence of the volume of export of large-tonnage containers and the freight turnover of the PRC railway transport on the volume of import of large-capacity containers at the Zabaikalsk-Manchuria railway checkpoint is solved.

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.

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.

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