Echelon Form Research Articles

Using Row Reduced Echelon Form in Balancing Chemical Equations

In an earlier paper published in the Journal of Natural Sciences Research in 2015 on how to balance chemical equations using matrix algebra, Gabriel and Onwuka showed how to reduce the resulting matrix to echelon form using elementary row operations. However, they did not show how elementary row operations can be used in reducing the resulting echelon matrix to row reduced echelon form. We show that the solution obtained is actually the nullspace of the matrix. Hence, the solution can be infinitely many. In addition, we show that instead of manually using row operations to reduce the matrix to row reduced echelon form, software environments like octave or Matlab can be used to reduce the matrix directly. In all the examples presented in this paper, we reduced all matrices to row reduced echelon form showing all row operations, which was not clearly stated in the Gabriel and Onwuka paper. Most importantly, with the availability of Mathematical software, we show that we do not need to carry out these row operations by brute force.

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

Gray Code for Row-Reduced Echelon Forms Over the Binary Field

We visit generator matrices A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ,··· of all k-dimensional subspaces of the n-dimensional vector space over the field with two elements such that two consecutive matrices A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> and A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i+1</sub> differ in exactly one bit.

SSMMATLAB: A Set ofMATLABPrograms for the Statistical Analysis of State Space Models

This article discusses and describes SSMMATLAB, a set of programs written by the author in MATLAB for the statistical analysis of state space models. The state space model considered is very general. It may have univariate or multivariate observations, time-varying system matrices, exogenous inputs, regression effects, incompletely specified initial conditions, such as those that arise with cointegrated VARMA models, and missing values. There are functions to put frequently used models, such as multiplicative VARMA models, VARMAX models in echelon form, cointegrated VARMA models, and univariate structural or ARIMA model-based unobserved components models, into state space form. There are also functions to implement the Hillmer-Tiao canonical decomposition and the smooth trend and cycle estimation proposed by Gómez (2001). Once the model is in state space form, other functions can be used for likelihood evaluation, model estimation, forecasting and smoothing. A set of examples is presented in the SSMMATLAB manual to illustrate the use of these functions.

Computing syzygies over [formula omitted], V a valuation domain

We give an algorithm for computing the V-saturation of any finitely-generated submodule of V[X1,…,Xk]m (k∈N, m∈N⁎), where V is a valuation domain. Our algorithm is based on a notion of “echelon form” which ensures its correctness. The proposed algorithm terminates when two (Hilbert) series on the quotient field and the residue field of V coincide. As application, our algorithm computes syzygies over V[X1,…,Xk].

Shape control of cable structures considering concurrent/sequence control

In this study, the control of the shape of pre-stressed cable structures and the effective control element were examined. The process of deriving the displacement control equations using the force method was explained, and the concurrent control scheme (CCS) and the sequence control scheme (SCS) were proposed. To explain the control scheme process, the quadrilateral cable net model was adopted and classified into a regular model and an irregular model for the analysis of the control results. In the control analysis of the regular model, the CCS and SCS analysis results proved reliable. For the SCS, the errors occur in the control stage and varied according to the control sequence. In the control analysis of the irregular model, the CCS analysis result also proved relatively reliable, and the SCS analysis result with the correction of errors in each stage was found nearly consistent with the target shape after the control. Finally, to investigate an effective control element, the Geiger cable dome was adopted. A set of non-redundant elements was evaluated in the reduced row echelon form of a coefficient matrix of control equations. Important elements for shape control were also evaluated using overlapping elements in the element sets, which were selected based on cable adjustments.

On static output‐feedback stabilization for multi‐input multi‐output positive systems

SummaryThis paper revisits the static output‐feedback stabilization problem for positive systems. We first point out that for a class of positive systems whose output matrix has a particular row echelon form, this problem can be completely solved via linear programming. By duality, the result is also valid when the column echelon form of the input matrix has a particular structure. Along this line, by augmenting the output matrix as well as the feedback gain matrix, an iterative convex optimization algorithm is developed for the more general case. Finally, we show that the proposed method has advantages over existing works via several numerical examples. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Speech Coding Based on Compressed Sensing and Sparse Representation

In this paper, we propose a novel speech coding scheme based on compressed sensing and sparse representation. Compressed sensing (CS) attracts great interest for its ability to utilize a few measurements to recover original signals. Measurements preserve part of speech features while projected by row echelon matrix. A dictionary is learned in order to contain redundant information about speech measurements. The synthesized speech is recovered from a sparse approximation of the corresponding measurement. A rear low-pass filter is adopted to improve the subject quality of synthesized speech. Results show that the proposed coding scheme has achieved average Mean Opinion Score (MOS) of the synthesized speech 3.083 in an appropriate bit rate (4.2 Kbps), which outperforms the quality of Code excited linear prediction (CELP).

Row Reduced Echelon Form for Solving Fully Fuzzy System with Unknown Coefficients

This study proposes a new method for finding a feasible fuzzy solution in positive Fully Fuzzy Linear System (FFLS), where the coefficients are unknown. The fully fuzzy system is transferred to linear system in order to obtain the solution using row reduced echelon form, thereafter; the crisp solution is restricted in obtaining the positive fuzzy solution. The fuzzy solution of FFLS is included crisp intervals, to assign alternative values of unknown entries of fuzzy numbers. To illustrate the proposed method, numerical examples are solved, where the entries of coefficients are unknown in right or left hand side, to demonstrate the contributions in this study.

Reduced row echelon form and non-linear approximation for subspace segmentation and high-dimensional data clustering

Given a set of data W={w1,…,wN}∈RD drawn from a union of subspaces, we focus on determining a nonlinear model of the form U=⋃i∈ISi, where {Si⊂RD}i∈I is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our approach is based on the binary reduced row echelon form of data matrix, combined with an iterative scheme based on a non-linear approximation method. We prove that, in absence of noise, our approach can find the number of subspaces, their dimensions, and an orthonormal basis for each subspace Si. We provide a comprehensive analysis of our theory and determine its limitations and strengths in presence of outliers and noise.

Asymptotic distributions for quasi-efficient estimators in echelon VARMA models

Two linear estimators for stationary invertible vector autoregressive moving average (VARMA) models in echelon form — to achieve parameter unicity (identification) — with known Kronecker indices are studied. It is shown that both estimators are consistent and asymptotically normal with strong innovations. The first estimator is a generalized-least-squares (GLS) version of the two-step ordinary least-squares (OLS) estimator studied in Dufour and Jouini (2005). The second is an asymptotically efficient estimator which is computationally much simpler than the Gaussian maximum-likelihood (ML) estimator which requires highly nonlinear optimization, and “efficient linear estimators” proposed earlier (Hannan and Kavalieris, 1984; Reinsel et al., 1992; Poskitt and Salau, 1995). It stands for a new relatively simple three-step estimator based on a linear regression involving innovation estimates which take into account the truncation error of the first-stage long autoregression. The complex dynamic structure of associated residuals is then exploited to derive an efficient covariance matrix estimator of the VARMA innovations, which is of order T−1 more accurate than the one by the fourth-stage of Hannan and Kavalieris’ procedure. Finally, finite-sample simulation evidence shows that, overall, the asymptotically efficient estimator suggested outperforms its competitors in terms of bias and mean squared errors (MSE) for the models studied.

The C-matrix: Augmentation and reduction in the analysis of chemical composition and structure

We propose a unified linear-algebraic approach that allows determining the three key matrices of chemical kinetics: molecular, stoichiometric and Horiuti matrix, all within a single procedure. This procedure is based on calculating a so called C-matrix, which is obtained from each matrix by augmenting it with a unit matrix and then reducing to Reduced Row Echelon Form (RREF). The procedure is rigorous and efficient to find answers to typical questions of chemical engineering and chemical kinetics, i.e., finding the independent components and reactions, expressing in them the dependent ones, and determining independent routes of complex chemical reactions as well. Contrary to the previous analysis procedures, the C-matrix contains all bookkeeping information on dependencies of components and elements, and their balances. The procedure can be used for developing new software tools in the analysis of complex chemical composition and networks.

Rank-profile revealing Gaussian elimination and the CUP matrix decomposition

Transforming a matrix over a field to echelon form, or decomposing the matrix as a product of structured matrices that reveal the rank profile, is a fundamental building block of computational exact linear algebra. This paper surveys the well-known variations of such decompositions and transformations that have been proposed in the literature. We present an algorithm to compute the CUP decomposition of a matrix, adapted from the LSP algorithm of Ibarra, Moran and Hui (1982), and show reductions from the other most common Gaussian elimination based matrix transformations and decompositions to the CUP decomposition. We discuss the advantages of the CUP algorithm over other existing algorithms by studying time and space complexities: the asymptotic time complexity is rank sensitive, and comparing the constants of the leading terms, the algorithms for computing matrix invariants based on the CUP decomposition are always at least as good except in one case. We also show that the CUP algorithm, as well as the computation of other invariants such as transformation to reduced column echelon form using the CUP algorithm, all work in place, allowing for example to compute the inverse of a matrix on the same storage as the input matrix.

Finding minimal generating set for metabolic network with reversible pathways

Elementary flux modes give a mathematical representation of metabolic pathways in metabolic networks satisfying the constraint of non-decomposability. The large cost of their computation shifts attention to computing a minimal generating set which is a conically independent subset of elementary flux modes. When a metabolic network has reversible reactions and also admits a reversible pathway, the minimal generating set is not unique. A theoretical development and computational framework is provided which outline how to compute the minimal generating set in this case. The method is based on combining existing software to compute the minimal generating set for a “pointed cone” together with standard software to compute the Reduced Row Echelon Form.

Rank subcodes in multicomponent network coding

A new class of subcodes in rank metric is proposed; based on it, multicomponent network codes are constructed. Basic properties of subspace subcodes are considered for the family of rank codes with maximum rank distance (MRD codes). It is shown that nonuniformly restricted rank subcodes reach the Singleton bound in a number of cases. For the construction of multicomponent codes, balanced incomplete block designs and matrices in row-reduced echelon form are used. A decoding algorithm for these network codes is proposed. Examples of codes with seven and thirteen components are given.

On the intersection of subgroups in free groups: Echelon subgroups are inert

A subgroup H of a free group F is called inert in F if for every . In this paper we expand the known families of inert subgroups. We show that the inertia property holds for 1-generator endomorphisms. Equivalently, echelon subgroups in free groups are inert. An echelon subgroup is defined through a set of generators that are in echelon form with respect to some ordered basis of the free group, and may be seen as a generalization of a free factor. For example, the fixed subgroups of automorphisms of finitely generated free groups are echelon subgroups. The proofs follow mostly a graph-theoretic or combinatorial approach.

On the Heights of Quaternionic Vectors

In this paper, we introduce a new numerical character, namely the height, for a quaternionic vector. According to this new concept (height), we give a complete classification of quaternionic vectors and enumerate all possible row echelon forms of the associated matrices for each type of the quaternionic vectors. By observing a link which the height has with the system of Cartesian frames, we give the character descriptions for several types of the systems of Cartesian frames.

A New Computational Method for Solving Fully Fuzzy Linear Systems of Triangular Fuzzy Numbers

In this paper, a new computational method is proposed to solve fully fuzzy linear systems (FFLS) of triangular fuzzy numbers based on the computation of row reduced echelon form for solving the crisp linear system of equations. The method is illustrated by solving three numerical examples. As compared to the existing methods, the proposed method is easy to understand and to apply for solving FFLS occurring in real life situations and scientific applications. The primary advantage of the proposed method is that, by using it, the consistency of the FFLS can be checked easily and nature of the solutions of an FFLS can also be obtained which may be unique or infinite.

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

Two Canonical VARMA Forms: Scalar Component Models Vis-à-Vis the Echelon Form

In this article we study two methodologies which identify and specify canonical form VARMA models. The two methodologies are: (1) an extension of the scalar component methodology which specifies canonical VARMA models by identifying scalar components through canonical correlations analysis; and (2) the Echelon form methodology, which specifies canonical VARMA models through the estimation of Kronecker indices. We compare the actual forms and the methodologies on three levels. Firstly, we present a theoretical comparison. Secondly, we present a Monte Carlo simulation study that compares the performances of the two methodologies in identifying some pre-specified data generating processes. Lastly, we compare the out-of-sample forecast performance of the two forms when models are fitted to real macroeconomic data.