XII.—Studies in Practical Mathematics. I. The Evaluation, with Applications, of a Certain Triple Product Matrix

1938 ◽  
Vol 57 ◽  
pp. 172-181 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Matrix Multiplication ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Product Matrix ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
The Matrix ◽  
Practical Mathematics ◽  
General Operation

The solution of simultaneous linear algebraic equations, the evaluation of the adjugate or the reciprocal of a given square matrix, and the evaluation of the bilinear or quadratic form reciprocal to a given form, are all special cases of a certain general operation, namely the evaluation of a matrix product H′A-1K, where A is square and non-singular, that is, the determinant | A | is not zero. (Matrix multiplication is like determinant multiplication, but exclusively row-into-column. The matrix H′ is obtained from H by transposition, that is, by changing rows into columns.) The matrices H′ and K may be rectangular. If A is singular, the reciprocal A-1 does not exist; and in such a case the product H′(adj A) K may be required. Arithmetically, the only difference in the computation of H′A-1K and H(adj A) K is that in the latter case a final division of all elements by | A | is not performed.

scholarly journals Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

2021 ◽  
Vol 5 (1) ◽  
pp. 8
Author(s):  
Cundi Han ◽  
Yiming Chen ◽  
Da-Yan Liu ◽  
Driss Boutat
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Governing Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
The Matrix ◽  
The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

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.

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.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

Approximate solutions to the half-space integral transport equation near a plane boundary

1980 ◽  
Vol 58 (9) ◽  
pp. 1291-1310 ◽  
Author(s):  
Michael S. Milgram
Keyword(s):  
Transport Equation ◽  
Half Space ◽  
Matrix Multiplication ◽  
Solution Space ◽  
Plane Geometry ◽  
Plane Boundary ◽  
Infinite Plane ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Integral Transport

A set of functions spanning the solution space of the integral transport equation near a boundary in semi-infinite plane geometry is obtained and used to reduce the problem to that of a system of linear algebraic equations. Expressions for the boundary angular flux are obtained by matrix multiplication, and the theory is extended to adjacent half-space problems by matching the angular flux at the boundary. Thus a unified theory is obtained for well-behaved arbitrary sources in semi-infinite plane geometry. Numerical results are given for both Milne's problem and the problem of constant production in adjacent half-spaces, and albedo problems in semi-infinite geometry. The solutions for the flux density are best near the boundary, and for the angular flux are best for angles near the plane of the boundary; it is conjectured that the theory will prove most useful when extended to arrays of finite slabs.

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.

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