Solution Of Linear Equation Systems Research Articles

Parameters and Infinite Solutions of Linear Equation Systems

Resumen En el presente trabajo investigamos la utilidad didáctica de usar parámetros como variables de naturaleza dual (variables activas o inactivas), así como la actualización de cinco profesores mexicanos en este saber. En esta actualización se analizaron sistemas de ecuaciones lineales (SEL) con dos ecuaciones y tres incógnitas que tienen infinitas soluciones. Para ello propusimos, vía internet, cinco artefactos semióticos: 1. una definición, 2. la inclusión del parámetro en el SEL, 3. el cálculo de soluciones, 4. representaciones gráficas y 5. la aplicación a un problema en contexto. Los resultados revelaron la pertinencia de la introducción del parámetro mediante este proceso y, además, los profesores lograron un control sobre las soluciones válidas de entre las infinitas del sistema planteado. Para algunos profesores, la estrategia y el papel del parámetro fue sólo un método más que se agrega a los ya conocidos; sin embargo, para otros, permitió dar una explicación a la relación entre la variable y el parámetro, así como la posibilidad del uso controlado de las infinitas soluciones.

Discovery of new characterizations of parameter distributions in a semi- empirical mass model: an investigation by dividing an overdetermined system into numerous balanced subsystems* *Supported by the National Natural Science Foundation of China (11975209, U2032211, 12075287), the Physics Research and Development Program of Zhengzhou University (32410017), the Project of Youth Backbone Teachers of Colleges and Universities of Henan Province (2017GGJS008)

We propose and test a new method of estimating the model parameters of the phenomenological Bethe-Weizsäcker mass formula. Based on the Monte Carlo sampling of a large dataset, we obtain, for the first time, a Cauchy-type parameter distribution formed by the exact solutions of linear equation systems. Using the maximum likelihood estimation, the location and scale parameters are evaluated. The estimated results are compared with those obtained by solving overdetermined systems, e.g., the solutions of the traditional least-squares method. Parameter correlations and uncertainty propagation are briefly discussed. As expected, it is also found that improvements in theoretical modeling (e.g., considering microscopic corrections) decrease the parameter and propagation uncertainties.

On Results Between Matrix Division and Some other Matrix Operations

In this study, the relationships between matrix division and some matrix properties such as transpose, simplification, and expansion are discussed. Unless otherwise stated, our matrices are real number matrices. The relationship between our definitions of column co-divisor and row co-divisor of a matrix on another matrix is examined. Equality and differences of these connections are observed. The validity of the provided features is discussed under which conditions. These features have been determined. Examples are given for features that do not. Some of the obtained properties bring different solutions to the solution of linear equation systems, which are encountered in the literature and are more used. This led us to examine the relationship between the defined division operation and transpose. Investigations on this subject enabled the presentation of new definitions, lemmas and theorems. Examination of these offered a broader perspective on the given mathematical expressions.

SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR ANISOTROPIC PLATES AND SHELLS BY BOUNDARY ELEMENTS METHOD

Modern mechanical engineering sets the tasks of calculating thin-walled structures that combine lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of anisotropic materials and plastics seems justified. The problems of the theory of plates and shells belong to the class of boundary value problems, the analytical solution of which, due to various circumstances (nonlinearity of differential equations, complexity of geometry and boundary conditions, etc.), cannot bedetermined. Numerical methods help to solve this problem. Among numerical methods, undeservedly little attention is paid to the boundary element method. In this regard, the further development of indirect method of compensating loads for solving problems of the anisotropic plates and shells theory based on the applicationof exact fundamental solutions is relevant.The paper considers the application of the indirect boundary element method for solving of an anisotropic plates and shells nonlinear deformation problem. Since the kernels of the system of singular integral equations to which the solution of the problem is reduced are expressed in terms of the fundamental solution and itsderivatives, first of all, the article provides a method for determining the fundamental solutions to the problem of bending and the plane stress state of an anisotropic plate. The displacement vector is determined from the solution of linear equations system describing the bending and plane stress state of an anisotropic plate. The solution of the system is performed by the method of compensating loads, according to which the area representing the plan of the shallow shell is supplemented to an infinite plane, and on the contour that limits the area, compensating loads are applied to the infinite plate. Integral equations of indirect BEM are given. In this paper, the study of nonlinear deformation of anisotropic plates and shallow shells is carried out using the deflection load dependencies. The deflection at a given point on the median surface of the shell was taken as the leading parameter.

Hypothetical Learning Trajectory: To Determine The Solution of Linear Equation System in Elementary Row Operation

The background of this research is the learning obstacles that arise on the topic of determining the solution of a system of linear equations (SLE) in applying elementary row operations (ERO). From these findings, it is necessary to formulate a hypothetical learning trajectory (HLT) that can guide the flow of thinking in designing didactic situations in the learning process. HLT expect to anticipate learning obstacles (LO) that arise. This research is a preliminary design of a series of didactical design research. The steps taken in compiling the HLT are 1) reviewing the lesson plan, the methods used to convey the material to determine SST solutions, and the textbooks used; 2) reviewing students' thinking processes conceptually and procedurally regarding theoretically determining SLE solutions; 3) reviewing design plans didactic. HLT is formulated into five types of student responses. It integrates into didactic situations by explaining the relevance of the elimination method in high school with three operations on ERO and then by explaining matrix material before SPL material. From this didactic situation, it can be formulated in teaching material that can accommodate emerging LO.

Identification of Mathematics Prospective Teachers’ Conceptual Understanding in Determining Solutions of Linear Equation Systems

<p style="text-align:justify">This research is motivated by a linear equations system, which is the basis for studying necessary linear algebra materials, such as rank, range, linear independent/dependent, linear transformations, characteristic values and vectors. There are still prospective mathematics teachers who have difficulty solving linear equations system and understanding the form of row echelon and reduced row echelon forms. In this study, subjects were three prospective mathematics teachers from Swadaya Gunung Jati University Cirebon who were taking matrix algebra courses. This study aims to reveal the conceptual understanding of prospective mathematics teachers in determining the solution to systems of linear equations. The results show that there are still prospective mathematics teachers who only use memory about the properties and procedures in determining whether a matrix is said to be a row echelon form or a reduced row echelon form. Then, there is still weakness in building the algorithms' relationship due to the immature knowledge of the concepts. Researchers found that many prospective mathematics teachers were more comfortable solving problems that were performed procedurally. Further research is needed to determine how the mental construction process and mathematical conceptual knowledge of prospective mathematics teachers are through meaningful learning so that conceptual understanding is maximized.</p>

Geometric Interpretation and Manifold Structure of Markov Matrices

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Andreyevich Markov. Every Markov matrix gives a linear equation system, and the solution of this equation system gives us a subset of $\mathbb{R}_{n}^{n}$. This paper presents the new manifold structure on the set of the Markov matrices. In addition, this paper presents the set of Markov matrices is drawable, and this gives geometrical interpretation to Markov matrices. For the proof, we use the one-to-one corresponding among $n \times n$ Markov matrices, the solution of linear equation system from derived Markov property, and the set of $(n-1)$-polytopes.

Solving linear equation systems on noisy intermediate‐scale quantum computers

AbstractQuantum computing promises classically unparalleled benefits for various applications. Its properties are exploited in the Harrow‐Hassidim‐Lloyd (HHL) algorithm that, in conjunction with quantum phase estimation, is capable of constructing quantum states that are proportional to the solution of linear equation systems and does so exponentially faster than the fastest known classical algorithms. We explore this capability by computing the nodal displacements of a 1‐dimensional loaded cantilever, discretized by using the finite element method (FEM).

Individual amounts of Lewis and Brønsted acid sites on metal oxides from NH3 adsorption equilibrium: Case of TiO2 based solids

The present study is dedicated to the development of an original method for the measurement of the individual amounts QASi of the acid sites ASi (acidity of Lewis and Brønsted) present on three TiO2 based solids of increasing composition complexity: TiO2-P25, 6% WO3/TiO2-P25 and a sulfated 0.7 % V2O5/9% WO3/TiO2 NH3-SCR catalyst. The method is based on quantitative characterizations of the NH3 adsorption equilibrium (adsorption temperature Ta and pressure Pa) by the association of (a) the AEIR method providing the individual coverage θASi(Ta, Pa) of the adsorbed NH3 species on the ASi sites and (b) the total amount (in μmol/g) of the adsorbed NH3 species: QTNH3(Ta, Pa) by using a mass spectrometer. For a solid having n types of ASi sites, the QASi values are obtained from the numerical solution of linear equation systems (with at least n equations) obtained considering that QTNH3(Ta, Pa) is equal to the sum of the contribution of each adsorbed NH3 species: QASi θASi(Ta, Pa).This imposes the measurement of at least n QTNH3(Ta, Pa) in a Ta range preventing the contribution of parallel surface processes (i.e NH3 oxidation). On TiO2-P25, n = 3 (two Lewis and one weak Brønsted acid sites) and the QASi amounts are obtained from three QTNH3(Ta, Pa) values. The others solids having two Lewis (L1 and L2) and two Brønsted (B1 and B2) acid sites impose a series of m measurements of QTNH3(Ta, Pa) with m>> n: the QASi are obtained by optimization between theoretical and experimental QTNH3(Ta, Pa) curves such as 109, 202, 70 and 130 μmol/g for QL2, QL1, QB2 and QB1 respectively of 0.7 % V2O5/9% WO3/TiO2. It is shown that these four amounts of sites permit to conclude that the L2 Lewis acid site is the one forming the pivotal NH3ads-L2 species of the reaction by using nitrogen mass balances between the amounts of adsorbed NH3 species and the N2 production in the presence of NO.

Necessary and Sufficient Conditions for The Solutions of Linear Equation System

A Semiring is an algebraic structure (S,+,x) such that (S,+) is a commutative Semigroup with identity element 0, (S,x) is a Semigroup with identity element 1, distributive property of multiplication over addition, and multiplication by 0 as an absorbent element in S. A linear equations system over a Semiring S is a pair (A,b) where A is a matrix with entries in S and b is a vector over S. This paper will be described as necessary or sufficient conditions of the solution of linear equations system over Semiring S viewed by matrix X that satisfies AXA=A, with A in S. For a matrix X that satisfies AXA=A, a linear equations system Ax=b has solution x=Xb+(I-XA)h with arbitrary h in S if and only if AXb=b.

METHOD OF FAST MATRIX MULTIPLICATION UNDER ARM ARCHITECTURE USING SIMD INSTRUCTIONS

Background. Matrix multiplication is a rather complicated algorithm with a large number of operations. An additional problem is the nonlinear memory traversal of matrices. Matrix multiplication is widely used in various fields, such as neural networks, solutions of linear equation systems, matrix transformations, and so on. Therefore, it is important to develop a method of matrix multiplication, which will take into account the problems of the location of the matrices in memory, and will effectively manage the data when reused.Objective. The purpose of the paper is to develop a method of fast matrix multiplication of two matrices, as well as multiplying the matrix by the transposed matrix and by a list of vectors (including special case for only one vector), as well as to implement it as a function with optimization for ARM architecture processors. The function must be able to handle different types of data and submatrices. The integer result can be scaled.Methods. The main ideas of the developed method are simultaneous work with several rows/columns of input matrices and their splitting into blocks, which will allow the algorithm to run on the same memory for a while. The C programming language was chosen for implementation. SIMD instructions were used to increase productivity. We also need to properly organize the memory preloading for effective implementation under the ARM architecture.Results. A function that performs matrix multiplication by the developed method with the necessary parameters was implemented as a result of the study. Tests on various sizes and types have shown that the implemented function is faster than analogues from the OpenCV2 and Eigen 3 libraries. Testing was done using the vipmed utility for running and measuring features developed for enterprise use at VIT.Conclusions. The proposed matrix multiplication method gives the expected acceleration of matrix multiplication operations, has passed evaluation test for use and meets the target requirements. For further work, it is necessary to study in more detail the influence of the cache at different levels and compare with other existing libraries.

Improving Initial Guess for the Iterative Solution of Linear Equation Systems in Incompressible Flow

The pressure equation, generated while solving the incompressible Navier–Stokes equations with the segregated iterative algorithm such as PISO, produces a series of linear equation systems as the time step advances. In this paper, we target at accelerating the iterative solution of these linear systems by improving their initial guesses. We propose a weighted group extrapolation method to obtain a superior initial guess instead of a general one, the solution of the previous linear equation system. In this method, the previous solutions that are used to extrapolate the predicted solutions are carefully organized to address the oscillatory solution on each grid. The proposed method uses a weighted average of the predicted solutions as the new initial guess to avoid over extrapolating. Three numerical test results show that the proposed method can accelerate the iterative solution of most linear equation systems and reduce the simulation time up to 61.3%.

Pengembangan Catur SPLDV untuk Membantu Siswa dalam Menyelesaikan Sistem Persamaan Linier Dua Variabel

Learning mathematics aims to make students able to solve problems they will face in the future. This can be hampered if students have difficulty in learning. Difficulties experienced by students in solving system of linear equations material, namely difficulties to carry out procedures in determining the completion. In learning, the teacher has not used teaching aids to overcome student difficulties. Teaching aids can be an alternative solution to determine the completion of system of linear equations. This study aims to produce teaching aids that can help students to determine the solution. The type of research applied was a 4D model development research. To produce good teaching aids, valid tests, practical tests and effective tests were carried out. In this study, the produced teaching aid is called “Catur SPLDV”. The validity test result of the teaching aid was 3.59 with very valid criteria. The practicaliy test result of Catur SPLDV was 86.67%; very practical criteria. While the effective test result is 0.73 with high criteria analyzed using the normalized-gain (n-gain) test. Catur SPLDV is used to help students to determine the solution of linear equations system of two-variable using the concept of Cartesian coordinates and number patterns. It can be used while playing a game, and containing a little algebraic elements making it easier for students to determine the solution of system of linear equations with two-variable.

The characteristics of final mathematics test items based on classical test theory

Final Examination is the part of an evaluation that exists in school to assess student’s knowledge. This research aimed to show the characteristic of the final mathematics test in Junior High school, grade 8. This research was an explorative descriptive using a quantitative approach. Data collection was gathered using the documentation of mathematics’ test in Binjai, North Sumatera including student’s answers. The characteristics of test items were analyzed by expert judgment using the classical test theory approach. The average of content analysis was 0.81, there were three items without key answers so it can not be analyzed. The reliability average was 0.84 which has very good quality and difficulty index indicated the difficulty level of questions was on three categories (easy, moderate, and difficult). The difficulty is in the cartesian position, the relation of two sets, determining gradient, determining the solution of linear equation systems of two variables.

Реализация итерационных методов решения систем линейных уравнений в задачах математической физики на реконфигурируемых вычислительных системах

И.И. Левин, А.И. Дордопуло, А.В. Пелипец Научно-исследовательский институт многопроцессорных вычислительных систем Южного федерального университета (Таганрог, Российская Федерация) (347928 Таганрог, ул. Чехова, 2, ГСП-284, г. Таганрог, Ростовская область), E-mail: levin@mvs.tsure.ru, scorpio@mvs.tsure.ru, pelipets@mail.ru. I.I. Levin, A.I. Dordopulo, А.V. Pelipets Academician A.V. Kalyaev SRI Multiprocessor Computer System at Southern Federal University (2 Chekhov st., GSP-284, Taganrog, 347928, Russia) E-mail: levin@mvs.tsure.ru, scorpio@mvs.tsure.ru, pelipets@mail.ru

A doubly optimized solution of linear equations system expressed in an affine Krylov subspace

A mathematical procedure for finding a closed-form double optimal solution (DOS) of an n-dimensional linear equations system Bx=b is developed, which expresses the solution in an m-dimensional affine Krylov subspace with undetermined coefficients, and two optimization techniques are used to determine these coefficients in closed-form. To find the DOS, it is very time saving without the need of any iteration; in practice, we only need to invert an m×m matrix one time, where m≪n. The DOS is not exactly equal to the exact solution, but it can provide an acceptable approximate solution of linear equations system, whose applicable range is identified. Some properties are analyzed that the DOS is an exact solution of a projected linear system of Bx=b onto the affine Krylov subspace. The tests for large scale problems demonstrate the efficiency of DOS on non-sparse linear systems.

Using gauss - Jordan elimination method with CUDA for linear circuit equation systems

Many scientific and engineering problems can use a system of linear equations. In this study, solution of Linear Circuit Equation System (LCES) for an nxn matrix using Compute Unified Device Architecture (CUDA) is described. Solution of LCES is realized on Graphics Processing Unit (GPU) instead of Central Processing Unit (CPU). CUDA is a parallel computing architecture developed by NVIDIA. Linear Circuits include resistance, impedance, capacitance, dependent - independent current sources and DC, AC voltage source. In this study, solutions of circuits that include resistance, independent current sources and DC voltage source have analyzes. Circuit analysis frequently involves solution of linear simultaneous equations that are solved Gauss-Jordan Elimination Method in this study. Gauss-Jordan Elimination is a variant of Gaussian Elimination that a method of solving a linear system equations (Ax=B). Gauss-Jordan Elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. Gaussian Elimination has two parts. The first part (Forward Elimination) reduces a given system to triangular form. The second step uses back substitution to find the solution of the triangular echelon form system Because of elements of unknowns column matrix are dependent on each other, second step algorithm is not appropriate for parallel programming. Two parts of Gauss–Jordan Elimination are not like Gaussian Elimination's part so it is preferred. GPU implementation is more faster than solution of linear equation systems on CPU

A new Lanczos-based algorithm for simulating high-frequency two-dimensional electron spin resonance spectra

The Lanczos algorithm (LA) is a useful iterative method for the reduction of a large matrix to tridiagonal form. It is a storage efficient procedure requiring only the preceding two Lanczos vectors to compute the next. The quasi-minimal residual (QMR) method is a powerful method for the solution of linear equation systems, Ax = b. In this report we provide another application of the QMR method: we incorporate QMR into the LA to monitor the convergence of the Lanczos projections in the reduction of large sparse matrices. We demonstrate that the combined approach of the LA and QMR can be utilized efficiently for the orthogonal transformation of large, but sparse, complex, symmetric matrices, such as are encountered in the simulation of slow-motional 1D- and 2D-electron spin resonance (ESR) spectra. Especially in the 2D-ESR simulations, it is essential that we store all of the Lanczos vectors obtained in the course of the LA recursions and maintain their orthogonality. In the LA-QMR application, the QMR weight matrix mitigates the problem that the Lanczos vectors lose orthogonality after many LA projections. This enables substantially more Lanczos projections, as required to achieve convergence for the more challenging ESR simulations. It, therefore, provides better accuracy for the eigenvectors and the eigenvalues of the large sparse matrices originating in 2D-ESR simulations than does the previously employed method, which is a combined approach of the LA and the conjugate-gradient (CG) methods, as evidenced by the quality and convergence of the 2D-ESR simulations. Our results show that very slow-motional 2D-ESR spectra at W-band (95 GHz) can be reliably simulated using the LA-QMR method, whereas the LA-CG consistently fails. The improvements due to the LA-QMR are of critical importance in enabling the simulation of high-frequency 2D-ESR spectra, which are characterized by their very high resolution to molecular orientation.

Parallel architecture for the solution of linear equations systems based on division free Gaussian elimination method implemented in FPGA

This paper presents a parallel architecture for the solution of linear equations systems based on the Division Free Gaussian Elimination Method. This architecture was implemented in a Field Program...

Enhanced-approximation linear solution technique (EALST)

The enhanced discretization and solution techniques are among the advanced computational methods we rely on in simulation and modeling of complex flow problems, including those with moving boundaries and interfaces. The set of enhanced discretization and solution techniques includes those based on enhancement in spatial discretization, enhancement in time discretization, and enhancement in iterative solution of nonlinear and linear equation systems. The enhanced-approximation linear solution technique (EALST) was introduced to increase the performance of the iterative technique used in solution of the linear equation systems when some parts of the computational domain may offer more of a challenge for the iterative method than the others. The EALST can be used for computations based on semi-discrete or space–time formulations.