scholarly journals Eigenvalues and Eigenvectors for 3×3 Symmetric Matrices: An Analytical Approach

2020 ◽  
pp. 106-118
Author(s):  
Abu Bakar Siddique ◽  
Tariq A. Khraishi
Keyword(s):  
Analytical Solutions ◽  
Analytical Approach ◽  
Numerical Solutions ◽  
Linear Equations ◽  
Symmetric Matrices ◽  
Matrix Equations ◽  
Eigenvalues And Eigenvectors ◽  
Engineering Problems ◽  
The Matrix ◽  
Research Problems

Research problems are often modeled using sets of linear equations and presented as matrix equations. Eigenvalues and eigenvectors of those coupling matrices provide vital information about the dynamics/flow of the problems and so needs to be calculated accurately. Analytical solutions are advantageous over numerical solutions because numerical solutions are approximate in nature, whereas analytical solutions are exact. In many engineering problems, the dimension of the problem matrix is 3 and the matrix is symmetric. In this paper, the theory behind finding eigenvalues and eigenvectors for order 3×3 symmetric matrices is presented. This is followed by the development of analytical solutions for the eigenvalues and eigenvectors, depending on patterns of the sparsity of the matrix. The developed solutions are tested against some examples with numerical solutions.

scholarly journals The Matrix Linear Unilateral and Bilateral Equations with Two Variables over Commutative Rings

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
N. S. Dzhaliuk ◽  
V. M. Petrychkovych
Keyword(s):  
Linear Equations ◽  
Standard Form ◽  
Commutative Rings ◽  
Matrix Equations ◽  
General Solutions ◽  
Particular Solutions ◽  
The Matrix ◽  
Bezout Domains

The method of solving matrix linear equations and over commutative Bezout domains by means of standard form of a pair of matrices with respect to generalized equivalence is proposed. The formulas of general solutions of such equations are deduced. The criterions of uniqueness of particular solutions of such matrix equations are established.

scholarly journals Matrix Diophantine equations over quadratic rings and their solutions

2020 ◽  
Vol 12 (2) ◽  
pp. 368-375
Author(s):  
N.B. Ladzoryshyn ◽  
V.M. Petrychkovych ◽  
H.V. Zelisko
Keyword(s):  
Matrix Equation ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Standard Form ◽  
Matrix Equations ◽  
System Of Linear Equations ◽  
Structure Of Solutions ◽  
Linear Diophantine Equations ◽  
The Matrix ◽  
Solution Methods

The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

scholarly journals A new analytical approach to investigate human gait dynamics

2019 ◽  
Vol 29 ◽  
pp. 02004
Author(s):  
Denisa Mihut ◽  
Nicolae Herisanu
Keyword(s):  
Exact Solution ◽  
Analytical Solutions ◽  
Analytical Approach ◽  
Reliable Method ◽  
Numerical Solutions ◽  
Human Gait ◽  
Control Parameters ◽  
Gait Dynamics ◽  
Convergence Control ◽  
Optimal Values

In this paper we propose a new analytical approach to the study of human gait dynamics. A new and reliable method, namely the Optimal Auxiliary Functions Method (OAFM) is employed to obtain explicit and accurate analytical solutions. The capabilities of this new method are successfully tested in the studyof human gait dynamics and an excellent agreement between analytical and numerical solutions is demonstrated. The accuracy of the analytical results is assured by the so-called convergence-control parameters, whose optimal values are rigorously identified in order to provide a fast convergence to the exact solution.

Numerical approach for partial eigenstructure assignment problems in singular vibrating structure using active control

2021 ◽  
pp. 014233122110646
Author(s):  
Peizhao Yu ◽  
Chuang Wang ◽  
Mengmeng Li
Keyword(s):  
Active Control ◽  
Numerical Solutions ◽  
Minimum Energy ◽  
Numerical Approach ◽  
Eigenstructure Assignment ◽  
Matrix Equations ◽  
Assignment Problems ◽  
Structure Information ◽  
Minimum Energy Consumption ◽  
The Matrix

In the paper, the partial eigenstructure assignment problems are investigated using acceleration–velocity–displacement active control in a singular vibrating structure. The problems are transformed into solving matrix equations using the receptance matrix method. Iterative sequences are constructed, and the iterative feasibility is presented for solving the matrix equations. The partial eigenvectors of the closed-loop system are reassigned by imposing modal constraints. An algorithm is proposed to get numerical solutions of the derived matrix equations. The initial value condition is discussed to obtain the minimum norm solution of the partial eigenstructure assignment problems. The designed acceleration–velocity–displacement active control can solve the partial eigenstructure assignment problems depending only on original vibrating structure information. The proposed numerical algorithm can obtain the minimum norms of controller gain, which implies minimum energy consumption. Numerical examples are given to illustrate the effectiveness of the proposed methods.

scholarly journals Analytical Solutions for Multiple Matrix in Fractured Reservoirs: Application to Conventional and Unconventional Reservoirs

SPE Journal ◽  
2013 ◽  
Vol 18 (05) ◽  
pp. 969-981 ◽  
Author(s):  
Mehmet A. Torcuk ◽  
Basak Kurtoglu ◽  
Najeeb Alharthy ◽  
Hossein Kazemi
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Control Volume ◽  
Fractured Reservoirs ◽  
Unconventional Reservoirs ◽  
Unconventional Reservoir ◽  
Flow Equations ◽  
Enhanced Production ◽  
Matrix Blocks ◽  
The Matrix

Summary In this paper, we present a new method to model heterogeneity and flow channeling in petroleum reservoirs—especially reservoirs containing interconnected microfractures. The method is applicable to both conventional and unconventional reservoirs where the interconnected microfractures form the major flow path. The flow equations, which could include flow contributions from matrix blocks of various size, permeability, and porosities, are solved by the Laplace-transform analytical solutions and finite-difference numerical solutions. The accuracy of flow from and into nanodarcy matrix blocks is of great interest to those dealing with unconventional reservoirs; thus, matrix flow equations are solved by use of both pseudosteady-state (PSS) and unsteady state (USS) formulations and the results are compared. The matrix blocks can be of different size and properties within the representative elementary volume (REV) in the analytical solutions, and within each control volume (CV) in the numerical solutions. Although the analytical solutions were developed for slightly compressible rock/fluid linear systems, the numerical solutions are general and can be used for nonlinear, multiphase, multicomponent flow problems. The mathematical solutions were used to analyze the longterm and short-term performances of two separate wells in an unconventional reservoir. It is concluded that matrix contribution to flow is very slow in a typical low-permeability unconventional reservoir and much of the enhanced production is from the fluids contained in the microfractures rather than in the matrix. In addition to field applications, the mathematical formulations and solution methods are presented in a transparent fashion to allow easy usage of the techniques for reservoir and engineering applications.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

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.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

1996 ◽  
Vol 28 (01) ◽  
pp. 114-165 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Complex Analysis ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Transform Method ◽  
Technical Problems ◽  
The Matrix ◽  
Transient Case

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

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