scholarly journals A Simple Algorithm for Finding a Non-negative Basic Solution of a System of Linear Algebraic Equations

2021 ◽  
Vol 28 (3) ◽  
pp. 234-237
Author(s):  
Gleb D. Stepanov
Keyword(s):  
Simplex Method ◽  
Gaussian Elimination ◽  
Simple Algorithm ◽  
Basic Solution ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Linear Programming Problems ◽  
Two Stages ◽  
Computational Resources

This article describes an algorithm for obtaining a non-negative basic solution of a system of linear algebraic equations. This problem, which undoubtedly has an independent interest, in particular, is the most time-consuming part of the famous simplex method for solving linear programming problems.Unlike the artificial basis Orden’s method used in the classical simplex method, the proposed algorithm does not attract artificial variables and economically consumes computational resources.The algorithm consists of two stages, each of which is based on Gaussian exceptions. The first stage coincides with the main part of the Gaussian complete exclusion method, in which the matrix of the system is reduced to the form with an identity submatrix. The second stage is an iterative cycle, at each of the iterations of which, according to some rules, a resolving element is selected, and then a Gaussian elimination step is performed, preserving the matrix structure obtained at the first stage. The cycle ends either when the absence of non-negative solutions is established, or when one of them is found.Two rules for choosing a resolving element are given. The more primitive of them allows for ambiguity of choice and does not exclude looping (but in very rare cases). Use of the second rule ensures that there is no looping.

scholarly journals Solving Linear Programming Problems by Reducing to the Form with an Obvious Answer

2021 ◽  
Vol 28 (4) ◽  
pp. 434-451
Author(s):  
Gleb D. Stepanov
Keyword(s):  
Linear Programming ◽  
Simplex Method ◽  
Basic Solution ◽  
Algebraic Equations ◽  
Farkas Lemma ◽  
Second Stage ◽  
Positive Elements ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
The Right

The article considers a method for solving a linear programming problem (LPP), which requires finding the minimum or maximum of a linear functional on a set of non-negative solutions of a system of linear algebraic equations with the same unknowns. The method is obtained by improving the classical simplex method, which when involving geometric considerations, in fact, generalizes the Gauss complete exclusion method for solving systems of equations. The proposed method, as well as the method of complete exceptions, proceeds from purely algebraic considerations. It consists of converting the entire LPP, including the objective function, into an equivalent problem with an obvious answer. For the convenience of converting the target functional, the equations are written as linear functionals on the left side and zeros on the right one. From the coefficients of the mentioned functionals, a matrix is formed, which is called the LPP matrix. The zero row of the matrix is the coefficients of the target functional, $a_{00}$ is its free member. The algorithms are described and justified in terms of the transformation of this matrix. In calculations the matrix is a calculation table. The method under consideration by analogy with the simplex method consists of three stages. At the first stage the LPP matrix is reduced to a special 1-canonical form. With such matrices one of the basic solutions of the system is obvious, and the target functional on it is $ a_{00}$, which is very convenient. At the second stage the resulting matrix is transformed into a similar matrix with non-positive elements of the zero column (except $a_{00}$), which entails the non-negativity of the basic solution. At the third stage the matrix is transformed into a matrix that provides non-negativity and optimality of the basic solution. For the second stage the analog of which in the simplex method uses an artificial basis and is the most time-consuming, two variants without artificial variables are given. When describing the first of them, along the way, a very easy-to-understand and remember analogue of the famous Farkas lemma is obtained. The other option is quite simple to use, but its full justification is difficult and will be separately published.

Solving Systems of Linear Algebraic Equations with the Use of an Selective Regularization

2021 ◽  
pp. 11-15
Author(s):  
Vladimir N. Lutay
Keyword(s):  
Diagonal Element ◽  
Decomposition Process ◽  
Algebraic Equations ◽  
Original Matrix ◽  
Triangular Matrices ◽  
First Case ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Gauss Method ◽  
Scalar Products

The solution of systems of linear algebraic equations, which matrices can be poorly conditioned or singular is considered. As a solution method, the original matrix is decomposed into triangular components by Gauss or Chole-sky with an additional operation, which consists in increasing the small or zero diagonal terms of triangular matrices during the decomposition process. In the first case, the scalar products calculated during decomposition are divided into two positive numbers such that the first is greater than the second, and their sum is equal to the original one. In further operations, the first number replaces the scalar product, as a result of which the value of the diagonal term increases, and the second number is stored and used after the decomposition process is completed to correct the result of calculations. This operation increases the diagonal elements of triangular matrices and prevents the appearance of very small numbers in the Gauss method and a negative root expression in the Cholesky method. If the matrix is singular, then the calculated diagonal element is zero, and an arbitrary positive number is added to it. This allows you to complete the decomposition process and calculate the pseudo-inverse matrix using the Greville method. The results of computational experiments are presented.

A Linearized Two-Dimensional Theory for High-Speed Hydrofoils Near the Free Surface

1966 ◽  
Vol 10 (01) ◽  
pp. 25-48
Author(s):  
Richard P. Bernicker
Keyword(s):  
Direct Problem ◽  
High Speed ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Pressure Loading ◽  
Dimensional Theory ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Sectional Shape ◽  
Infinite Set

A linearized two-dimensional theory is presented for high-speed hydrofoils near the free surface. The "direct" problem (hydrofoil shape specified) is attacked by replacing the actual foil with vortex and source sheets. The resulting integral equation for the strength of the singularity distribution is recast into an infinite set of linear algebraic equations relating the unknown constants in a Glauert-type vorticity expansion to the boundary condition on the foil. The solution is achieved using a matrix inversion technique and it is found that the matrix relating the known and unknown constants is a function of depth of submergence alone. Inversion of this matrix at each depth allows the vorticity constants to be calculated for any arbitrary foil section by matrix multiplication. The inverted matrices have been calculated for several depth-to-chord ratios and are presented herein. Several examples for specific camber and thickness distributions are given, and results indicate significant effects in the force characteristics at depths less than one chord. In particular, thickness effects cause a loss of lift at shallow submergences which may be an appreciable percentage of the total design lift. The second part treats the "indirect" problem of designing a hydrofoil sectional shape at a given depth to achieve a specified pressure loading. Similar to the "direct" problem treated in the first part, integral equations are derived for the camber and thickness functions by replacing the actual foil by vortex and source sheets. The solution is obtained by recasting these equations into an infinite set of linear algebraic equations relating the constants in a series expansion of the foil geometry to the known pressure boundary conditions. The matrix relating the known and unknown constants is, again, a function of the depth of submergence alone, and inversion techniques allow the sectional shape to be determined for arbitrary design pressure distributions. Several examples indicate the procedure and results are presented for the change in sectional shape for a given pressure loading as the depth of submergence of the foil is decreased.

scholarly journals A note on adaptivity in factorized approximate inverse preconditioning

2020 ◽  
Vol 28 (2) ◽  
pp. 149-159
Author(s):  
Jiří Kopal ◽  
Miroslav Rozložník ◽  
Miroslav Tůma
Keyword(s):  
Large Scale ◽  
Condition Numbers ◽  
Algebraic Equations ◽  
Approximate Factorization ◽  
Approximate Inverse ◽  
Linear Algebraic Equations ◽  
Wide Range ◽  
The Matrix ◽  
Preconditioned Iterative ◽  
Principal Submatrices

AbstractThe problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

scholarly journals Diffraction of the field of vertical electric dipole on the spiral conductive sphere in the presence of a cone

2018 ◽  
Keyword(s):  
Hilbert Space ◽  
Electric Dipole ◽  
Infinite System ◽  
Matrix Operator ◽  
Algebraic Equations ◽  
Problem Statement ◽  
Vertical Electric Dipole ◽  
Linear Algebraic Equations ◽  
Method Of Regularization ◽  
The Matrix

The problem of diffraction of a vertical electric dipole field on a spiral conductive sphere and a cone has been solved. By the method of regularization of the matrix operator of the problem, an infinite system of linear algebraic equations of the second kind with a compact matrix operator in Hilbert space $\ell_2$ is obtained. Some limiting variants of the problem statement are considered.

scholarly journals USE OF THE LINK RANKING METHOD TO EVALUATE SCIENTIFIC ACTIVITIES OFSCIENTIFIC SPACE SUBJECTS

2020 ◽  
Author(s):  
A. Biloshchytskyi ◽  
А. Kuchansky ◽  
Yu. Andrashko ◽  
S. Biloshchytska
Keyword(s):  
Higher Education ◽  
Research Process ◽  
Algebraic Equations ◽  
Research Activity ◽  
Structural Units ◽  
Scientific Publications ◽  
Planning Stage ◽  
Research Results ◽  
Linear Algebraic Equations ◽  
The Matrix

A modification of the PageRank method based on link ranking is proposed to evaluate the research results of subjects of the scientific space, taking into account self-citation. The method of reducing the influence of self-citation on the final evaluation of the results of research activity of subjects of the scientific space is described. The evaluation of the results of research is calculated using the modified PR-q method, taking into account self-citation as a solution of a system of linear algebraic equations, matrix of which consists of coefficients determined by the number of citations of publications of one scientist in the publications of another scientist. The described method can be used for the task of evaluating the activity of the components of the scientific space: scientists, higher education institutions and their structural units. For the task of evaluating the research activity of subjects of the scientific space, a method based on link ranking (PageRank method for web pages) and taking into account the self-citation of scientists is proposed. The latter allows for an adequate assessment, taking into account the abuses associated with the authors excessive self-citation. The essence of the constructed method lies in the construction of a system of linear algebraic equations, whose coefficients of the matrix reflect the citations of some scientists by others in the citation network of scientific publications. The value of the coefficients of the matrix of such a system of linear algebraic equations is subject to certain restrictions, which allow to reduce the influence of the factor of excessive self-citation of the author on his overall assessment of research activity. The described method can be used to calculate the complex evaluation of the components of the scientific space: the scientist, the institution of higher education and its separate structural units. Evaluating research results provides an opportunity to verify the relevance of the research process to the goals identified at the planning stage and, if necessary, to adjust the progress of those studies. Also, the calculation of research evaluations of the components (objects and entities) of the scientific space is a powerful tool for managing research projects.

Calculation of the Magnetization of Permanent Magnets, Based on the Method for Solving the Inverse Problem Using Parallel Calculations

2021 ◽  
Vol 64 (1) ◽  
pp. 13-21
Author(s):  
Petr Denisov ◽  
◽  
Anna Balaban ◽  
Keyword(s):  
Inverse Problem ◽  
Permanent Magnets ◽  
Parallel Implementation ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Tikhonov Regularization Method ◽  
Field Pattern ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
The Matrix

The article proposes the modification of a technique for assessing the magnetization of permanent magnets from the known field pattern. The identification method is based on solving an ill-conditioned system of linear algebraic equations by the Tikhonov regularization method. The method of boundary integral equations based on scalar potentials is used to compile the matrix of coefficients. The article presents the algorithm that uses parallel computations when performing the most time-consuming operations to reduce the time for solving the inverse problem. In order to check the proposed method, a program was developed that allows to simulate the measurement process: to calculate the direct problem and find the magnetic induction at the points of the air gap, then introduce the error into the "measurement results" and solve the inverse problem. The results of nu-merical experiments that allow us to evaluate the advantages of parallel implementation using the capabilities of modern multi-core processors are presented.

Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

2013 ◽  
Vol 11 (8) ◽  
Author(s):  
Zahari Zlatev ◽  
Krassimir Georgiev
Keyword(s):  
Iterative Methods ◽  
Original Problem ◽  
Krylov Subspace ◽  
Gaussian Elimination ◽  
Algebraic Equations ◽  
Computational Process ◽  
Science And Engineering ◽  
Linear Algebraic Equations ◽  
Subspace Algorithms ◽  
Reliable Methods

AbstractMany problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important to select fast, robust and reliable methods for their solution, also in the case where fast modern computers are available. Since the coefficient matrices of the systems are normally sparse (i.e. most of their elements are zeros), the first requirement is to efficiently exploit the sparsity. However, this is normally not sufficient when the systems are very large. The computation of preconditioners based on approximate LU-factorizations and their use in the efforts to increase further the efficiency of the calculations will be discussed in this paper. Computational experiments based on comprehensive comparisons of many numerical results that are obtained by using ten well-known methods for solving systems of linear algebraic equations (the direct Gaussian elimination and nine iterative methods) will be reported. Most of the considered methods are preconditioned Krylov subspace algorithms.

The Recursive Algorithms of Yule-Walker Equation in Generalized Stationary Prediction

2013 ◽  
Vol 756-759 ◽  
pp. 3070-3073 ◽  
Author(s):  
Er Yan Zhang ◽  
Xiao Feng Zhu
Keyword(s):  
Toeplitz Matrix ◽  
Gaussian Elimination ◽  
Recursive Algorithm ◽  
Toeplitz Matrices ◽  
Recursive Formula ◽  
Coefficient Matrix ◽  
Recursive Algorithms ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Order Of Magnitude

Toeplitz matrix arises in a remarkable variety of applications such as signal processing, time series analysis, image processing. Yule-Walker equation in generalized stationary prediction is linear algebraic equations that use Toeplitz matrix as coefficient matrix. Making better use of the structure of Toeplitz matrix, we present a recursive algorithm of linear algebraic equations from by using Toeplitz matrix as coefficient matrix , and also offer the proof of the recursive formula. The algorithm, making better use of the structure of Toeplitz matrices, effectively reduces calculation cost. For n-order Toeplitz coefficient matrix, the computational complexity of usual Gaussian elimination is about , while this algorithm is about , decreasing of one order of magnitude.

scholarly journals Boubaker Wavelet Functions for Solving Higher Order Integro-Differential Equations

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Eman Hassan Ouda ◽  
Suha Shihab ◽  
Mohammed Rasheed
Keyword(s):  
Original Problem ◽  
General Procedure ◽  
Operational Matrix ◽  
Higher Order ◽  
Matrix Elements ◽  
Algebraic Equations ◽  
Orthonormal Wavelet ◽  
Easy Calculation ◽  
Linear Algebraic Equations ◽  
The Matrix

In the present paper, the properties of Boubaker orthonormal polynomials are used to construct new Boubaker wavelet orthonormal functions which are continuous on the interval [0, 1). Then, a Boubaker wavelet orthonormal operational matrix of the derivative is obtained with the new general procedure. The matrix elements can be expressed in a simple form that reduces the computational complexity. The collocation method of the Boubaker orthonormal wavelet functions together with the application of the derived operational matrix of the derivative are then utilized to transform the higher-order integro-differential equation into a solution of linear algebraic equations. As a result, the solution of the original problem reduces to the solution of a linear system of algebraic equations and can be sufficiently solved by an approximate technique. The main advantage of the suggested method is that the orthonormality property greatly simplifies the original problem and leads to easy calculation of the coefficients of expansion. Special attention is needed to perform the convergence analysis. The error is analyzed when a sufficiently smooth function is expanded in terms of the Boubaker orthonormal wavelet functions, then an estimation of the upper bound of the error is calculated. The results obtained by the technique in the current work are reported by solving some numerical examples and the accuracy is checked by comparing the results with the exact solution.

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