scholarly journals Gauss with Elementary Matrices in the SoftAge

2021 ◽  
Vol 3 (1) ◽  
pp. 7-11
Author(s):  
Márcio Antonio de Faria Rosa ◽  
Rafael Peres ◽  
Daniela Pereira Mendes Peres
Keyword(s):  
Gaussian Elimination ◽  
Good Strategy ◽  
Step Method ◽  
The Matrix ◽  
Mathematica Software ◽  
Math Problems

Software can be employed to solve math problems and we have to cooperate with them not passively but in an active way. In this article, we present situations in which the software user should be careful in order to avoid the use of the straight commands of the software that may put a matrix in its echelon form or give straight its rank. Wrong results appear in Wolfram’s Mathematica software if the matrix has algebraic entries. Using the Gaussian elimination step by step method with the help of elementary matrices is a good strategy for the software user in such cases.

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.

Characterisation of Ni-C films obtained in PVD/CVD process

Open Physics ◽  
2011 ◽  
Vol 9 (2) ◽  
Author(s):  
Justyna Kęczkowska
Keyword(s):  
Thin Films ◽  
Technological Process ◽  
Vapour Deposition ◽  
Chemical Vapour ◽  
Step Method ◽  
Raman Spectrometry ◽  
Two Phase ◽  
Cvd Method ◽  
The Matrix ◽  
Multiwall Nanotubes

AbstractThe work presents the results of the scanning electron microscopy (SEM) and Raman spectrometry studies of carbonaceous nanostructures containing nickel nanocrystallites. The films were obtained using a two-step method. In the first phase the Physical Vapour Deposition (PVD) method was applied, whereas in the second Chemical Vapour Deposition (CVD) method was used. The paper presents results for samples with various Ni content obtained with different parameters of the two-phase technological process. The research confirms that the thin films obtained by PVD method contain Ni nanocrystallites distributed in a carbonaceous matrix. The matrix is composed of various carbon allotropes (amorphous carbon, graphite, fullerene). The thin films made by CVD method make a matrix when multiwalled, carbonaceous nanotubes are obtained. Depending on the technological process parameters of each phase, we obtain multiwall nanotubes with a various degree of defects.

scholarly journals Partial Eigenvalue Assignment for Gyroscopic Second-Order Systems with Time Delay

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1235
Author(s):  
Hao Liu ◽  
Ranran Li ◽  
Yingying Ding
Keyword(s):  
Time Delay ◽  
Lower Order ◽  
Assignment Problem ◽  
Hadamard Product ◽  
Second Order ◽  
Step Method ◽  
Computational Costs ◽  
The Matrix ◽  
Second Order Systems ◽  
Eigenvalue Assignment

In this paper, the partial eigenvalue assignment problem of gyroscopic second-order systems with time delay is considered. We propose a multi-step method for solving this problem in which the undesired eigenvalues are moved to desired values and the remaining eigenvalues are required to remain unchanged. Using the matrix vectorization and Hadamard product, we transform this problem into a linear systems of lower order, and analysis the computational costs of our method. Numerical results exhibit the efficiency of our method.

Grain Refinement of Zn-50Al Alloy through the Addition of Zn–Al-Ti-B-C Master Alloy and Melt Thermal-Rate Treatment

2012 ◽  
Vol 562-564 ◽  
pp. 238-241
Author(s):  
Z.Q Wang ◽  
D.L Yang ◽  
Z.X Yang ◽  
H.R Geng
Keyword(s):  
Grain Refinement ◽  
Master Alloy ◽  
Treatment Process ◽  
Step Method ◽  
Master Alloys ◽  
The Matrix ◽  
Tib2 Particles ◽  
Refinement Effect ◽  
Al Matrix

In this paper, two types of Zn-Al-Ti-B-C master alloys were produced by a two-step method and were found to have good refinement effect for Zn-50Al alloy. SEM results show that TiC and TiB2 particles act as the nucleating center of α-Al grains in Zn-50Al alloy. The presence of TiAl3-xZnx phase in the matrix of Zn-Al-Ti-B-C master alloy was found to further enhance the refinement effect. The melt thermal-rate treatment process present good grain refinement effect for Zn-50Al alloy and it was further promoted by the addition of Zn-Al-Ti-B-C master alloy into Zn-Al matrix.

Nanocomposite Route to Ultra-sensitive Surface Enhanced Raman Scattering Substrates

2010 ◽  
Vol 1253 ◽  
Author(s):  
Abhijit Biswas ◽  
Ilker Bayer ◽  
Alexandru S. Biris
Keyword(s):  
Hot Spots ◽  
Direct Detection ◽  
Single Step ◽  
Metal Nanoparticle ◽  
Label Free ◽  
Step Method ◽  
Sers Substrates ◽  
The Matrix ◽  
Polymer Metal ◽  
Metal Nanocomposites

AbstractWe show a novel route to prepare SERS substrates, which is based on polymer–metal nanocomposites with a specific structure and composition just below the percolation threshold. The neighboring nanoparticles are still quite densely packed, but remain separated by narrow polymer gaps (<1 nm). Such a nanostructure allows the creation of densely packed hot spots where electromagnetic energy can be confined. The polymer–metal nanocomposites are fabricated by a simple and single-step method of electron-beam-assisted vapor-phase co-deposition. The preparation of the SERS substrates is based on a simple plasma-etching process, which removes the polymer structures that allow the formation of metal nanoparticle SERS nano-aggregates with very uniform and controllable inter-particle gaps. The method results in “ideal SERS hot spots” throughout the matrix. These hot spots can be created over very large areas. The prepared SERS substrates are promising candidates for the direct detection (label-free) and analysis of various biological and chemical samples.

Parallel Programming Of A Reservoir Simulator

1992 ◽  
pp. 1-13
Author(s):  
Mariyamni Awang
Keyword(s):  
Parallel Programming ◽  
Gaussian Elimination ◽  
Parallel Computer ◽  
Two Phase ◽  
Fully Implicit ◽  
The Matrix ◽  
Speed Up ◽  
Black Oil Model ◽  
Reservoir Simulator ◽  
Raphson Method

This study concerns applying parallel programming to reservoir simulation using a 32-Mbyte, 12-processor parallel computer. The effects of number of processes, granularity, load balancing and program structure were studied. The model simulated was a two-dimensionals, two-phase, black oil model with a fully-implicit formulation. The differenced equations were solved by the Newton-Raphson method and, Gaussian elimination was used to solve the Jacobian matrix. Matrix generation was parallelized using monitors as macros to synchronize calculation. The performance of the simulator was measured by the speed up. The speed ups of the matrix generation time increased almost linearly with increasing number of processes. For all of the models tested, the speed ups ranged from 3.5 to 4.0 for four processes and 7.0 to 7.9 for eight proceses.

scholarly journals ALLOCATION OF SECONDARY COSTS BY USING THE MATRIX FUNCTIONS OF SPREADSHEET PROGRAMS

2019 ◽  
Author(s):  
Lazar Radovanović ◽  
Irina Glotina ◽  
Teodor Petrović
Keyword(s):  
Cost Allocation ◽  
Direct Method ◽  
Linear Equations ◽  
Matrix Functions ◽  
Step Method ◽  
Allocation Model ◽  
The Matrix ◽  
The Cost ◽  
Starting Model ◽  
Correct Data

The paper describes and applies the method of linear equations, by using the matrixfunctions of spreadsheet programs, for the secondary cost allocation of fullyconditionedauxiliary cost centers. The aim of the paper is to demonstrate the possibilityof using matrix functions for cost allocation. A model has been formed based on thedata from a specific company, that has auxiliary, main and non-productive cost centers.The linear equations method is used to solve the problem of secondary cost allocation,by applying the corresponding matrix functions of a spreadsheet program.The goal of cost allocation of auxiliary cost centers to main cost centers, and later to thecost holders, is to calculate the exact cost, that is, the cost of products and services.The method of linear equations was chosen because a new model can be formed basedon the starting model, by changing the number of cost centers.This cost allocation model should encourage accountants and company management touse the more exact method of cost allocation instead of the simple direct method orcomplex step method. Matrix functions facilitate the method of linear equations,because they are quite simple to apply in models that can be easily adapted and appliedin practice later on.It has been concluded that this method can be easily described and applied, and theobtained results, with the correct data input and use of matrix functions, givecompletely accurate results, unlike other cost allocation methods.

scholarly journals Low rank approximation of positive semi-definite symmetric matrices using Gaussian elimination and volume sampling

2021 ◽  
Vol 62 ◽  
pp. C58-C71
Author(s):  
Markus Hegland ◽  
Frank De Hoog
Keyword(s):  
Point Processes ◽  
Symmetric Functions ◽  
Matrix Theory ◽  
Applied Mathematics ◽  
Gaussian Elimination ◽  
Approximation Error ◽  
Numerical Linear Algebra ◽  
Low Rank ◽  
Eigenvalue Decomposition ◽  
The Matrix

Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve systems with such a matrix can be very costly. A core idea to reduce computational complexity is to approximate the matrix by one with a low rank. The optimal and well understood choice is based on the eigenvalue decomposition of the matrix. Unfortunately, this is computationally very expensive. Cheaper methods are based on Gaussian elimination but they require pivoting. We show how invariant matrix theory provides explicit error formulas for an averaged error based on volume sampling. The formula leads to ratios of elementary symmetric polynomials on the eigenvalues. We discuss several bounds for the expected norm of the approximation error and include examples where this expected error norm can be computed exactly. References A. Dax. “On extremum properties of orthogonal quotients matrices”. In: Lin. Alg. Appl. 432.5 (2010), pp. 1234–1257. doi: 10.1016/j.laa.2009.10.034. M. Dereziński and M. W. Mahoney. Determinantal Point Processes in Randomized Numerical Linear Algebra. 2020. url: https://arxiv.org/abs/2005.03185. A. Deshpande, L. Rademacher, S. Vempala, and G. Wang. “Matrix approximation and projective clustering via volume sampling”. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm. SODA ’06. Miami, Florida: Society for Industrial and Applied Mathematics, 2006, pp. 1117–1126. url: https://dl.acm.org/doi/abs/10.5555/1109557.1109681. S. A. Goreinov, E. E. Tyrtyshnikov, and N. L. Zamarashkin. “A theory of pseudoskeleton approximations”. In: Lin. Alg. Appl. 261.1 (1997), pp. 1–21. doi: 10.1016/S0024-3795(96)00301-1. M. W. Mahoney and P. Drineas. “CUR matrix decompositions for improved data analysis”. In: Proc. Nat. Acad. Sci. 106.3 (Jan. 20, 2009), pp. 697–702. doi: 10.1073/pnas.0803205106. M. Marcus and L. Lopes. “Inequalities for symmetric functions and Hermitian matrices”. In: Can. J. Math. 9 (1957), pp. 305–312. doi: 10.4153/CJM-1957-037-9.

scholarly journals Analisis Kesalahan Siswa Berdasarkan Objek Matematika Menurut Soedjadi pada Materi Determinan dan Invers Matriks

2021 ◽  
Vol 10 (2) ◽  
pp. 235-244
Author(s):  
Restu Ayu Gustianingum ◽  
Kartini Kartini
Keyword(s):  
Qualitative Research ◽  
Data Collection ◽  
Error Analysis ◽  
Matrix Material ◽  
Inverse Matrix ◽  
Research Subjects ◽  
School Year ◽  
Online Test ◽  
The Matrix ◽  
Math Problems

AbstrakSebagian besar siswa terkadang membuat kesalahan dalam menyelesaikan soal-soal matematika baik yang disengaja maupun tidak disengaja. Penelitian ini bertujuan untuk menganalisis kesalahan-kesalahan yang dilakukan siswa dalam mengerjakan soal pada materi determinan dan invers matriks. Jenis penelitian ini adalah penelitian kualitatif. Teknik pengumpulan data yang digunakan adalah teknik tes dan wawancara secara daring. Subjek penelitian yaitu 30 siswa XI MIA 1 MAN 3 Kota Pekanbaru tahun pelajaran 2020/2021. Analisis kesalahan siswa dilihat berdasarkan objek matematika menurut Soedjadi yaitu fakta, konsep, prinsip, dan operasi. Hasil analisis kesalahan menunjukkan bahwa kesalahan paling banyak dilakukan siswa adalah kesalahan konsep dengan persentase sebesar 17,3%. Penyebab terjadinya kesalahan yang dilakukan siswa adalah siswa belum memahami konsep matriks, siswa lupa dengan konsep matriks dan kurang teliti dalam melakukan operasi perhitungan. Dalam pembelajaran, hendaknya guru tidak mengajarkan siswa untuk menghafalkan rumus namun lebih mengutamakan pemahaman konsep siswa. AbstractMost students sometimes make mistakes in solving math problems, either deliberately or unintentionally. This study aims to analyze the errors made by students in working on the questions on the determinant and inverse matrix material. This type of research is qualitative research. The data collection techniques used were online test and interview techniques. The research subjects were 30 students of XI MIA 1 MAN 3 Pekanbaru City in the 2020/2021 school year. Analysis of student errors is seen based on mathematical objects according to Soedjadi, namely facts, concepts, principles, and operations. The results of the error analysis showed that the most mistakes made by students were misconceptions with a percentage of 17.3%. The cause of the errors made by students is that students do not understand the concept of the matrix, students forget the concept of the matrix, and are not careful in performing calculation operations. In learning, the teacher should not teach students to memorize formulas but prioritize students' understanding of concepts.

scholarly journals The New Method of Analyzing the Continuous Curved Flat Bar Subject to Space Load

2019 ◽  
Vol 8 (2) ◽  
pp. 3620-3626
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Linear Analysis ◽  
New Method ◽  
Method Transfer ◽  
The Matrix ◽  
Math Problems

The report presents a new method for linear analysis of continuous curved flat bar, subject to any load in space. This method is a combination of improving the expression of the load and displacement at the two ends of the curved bar element of the Transfer Matrix Method and Finite Element Method (TMMFEM), called the Matrix Method transfer improvements. The research results are to build math problems and programming with Matlab, verify with the results according to SAP2000 oftware and « Strength of materials » documents

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