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For solving large sparse systems of linear equations, we construct a paradigm of two‐step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive‐definite matrix class. This two‐step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two‐step matrix splitting iteration methods. This result provides systematic treatment for the two‐step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems. [ABSTRACT FROM AUTHOR]

In this paper, by using the geometrical classification of the types of linear equations or systems of two equations, we determine a method for solving systems of linear equations. The analysis specifically refers to systems of linear equations that reduce to a quadratic system of the second or third order in which all determinants of the system are zero. The methodological significance of the method is also discussed. [ABSTRACT FROM AUTHOR]

We investigate the randomized Kaczmarz method that adaptively updates the stepsize using readily available information for solving inconsistent linear systems. A novel geometric interpretation is provided which shows that the proposed method can be viewed as an orthogonal projection method in some sense. We prove that this method converges linearly in expectation to the unique minimum Euclidean norm least-squares solution of the linear system, and provide a tight upper bound for the convergence of the proposed method. Numerical experiments are also given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The aim of the present research article is to solve the system of linear equations using common fixed point theorems in the context of bicomplex valued metric spaces. To obtain our main objective, we introduce generalized contractive conditions in bicomplex valued metric spaces and establish common fixed point theorems for self mappings. We also give a significant example to demonstrate the legitimacy of the given theorems. [ABSTRACT FROM AUTHOR]

In order to improve the convergence property and computational behavior of the randomized extended Kaczmarz method, we propose a multi-step randomized extended Kaczmarz method, in which we repeatedly update the iterate several times at each iteration step, obtaining a nonstationary inner-outer iteration scheme for solving large-scale, sparse, and inconsistent system of linear equations. For this multi-step randomized extended Kaczmarz method, we prove its convergence, derive an upper bound for its convergence rate, and demonstrate that this upper bound can be smaller than that of the randomized extended Kaczmarz method for several typical choices of the numbers of inner iteration steps. Numerical experiments also show that the multi-step randomized extended Kaczmarz method can perform better than the randomized extended Kaczmarz method if we choose the numbers of inner iteration steps appropriately. [ABSTRACT FROM AUTHOR]

The concept of systems of linear equations (SLEs) is fundamental and core in linear algebra, a subject, which has many applications in a number of disciplines. Gaussian elimination is a versatile method, which can be used to solve almost all types of SLEs by using row-reductions. This study focused on exploring undergraduate students' conceptualizations of elementary row operations (EROs) as a means to solve SLEs. The purpose of this study was to explore undergraduate students' conceptualizations of row reductions and their applications to the solutions of systems of equations. The perspectives of the action-process-object-schema theoretical framework were used in analyzing data and discussing the findings. To explore the students' conceptualization of EROs, a descriptive research approach was followed. I considered a case study of 131 students registered for a mathematics for educators course, where linear algebra was one of the topics. The findings revealed that students attained the action conception of reducing a system with unique solutions but had challenges reducing and interpreting solutions to a system with nonunique solutions. The latter row-reduction implored process and object conceptions especially when variable elements in the augmented matrix were involved. As students find the learning of linear algebra difficult, this study contributes to the debate in literature on how to improve its teaching and make suggestions on the ways make more effective the learning of linear algebra. [ABSTRACT FROM AUTHOR]

We review and compare several representative and effective randomized projection iteration methods, including the randomized Kaczmarz method, the randomized coordinate descent method, and their modifications and extensions, for solving the large, sparse, consistent or inconsistent systems of linear equations. We also anatomize, extract, and purify the asymptotic convergence theories of these iteration methods, and discuss, analyze, and summarize their advantages and disadvantages from the viewpoints of both theory and computations. [ABSTRACT FROM AUTHOR]

Any set S of elements from an abelian group produces a graph with colored edges G(S), with its points the elements of S, and the edge between points P and Q assigned for its “color” the sum P+Q. Since any pair of identically colored edges is equivalent to an equation P+Q=P′+Q′, the geometric—combinatorial figure G(S) is thus equivalent to a system of linear equations. This article derives elementary properties of such “sum cographs”, including forced or forbidden configurations, and then catalogues the 54 possible sum cographs on up to 6 points. Larger sum cograph structures also exist: Points {Pi}in Zmclose up into a “Fibonacci cycle”–i.e. P0=1, P1=k, Pi+2=Pi+Pi+1for all integers i≥0, and then Pn=P0and Pn+1=P1–provided that m=Lnis a Lucas prime, in which case actually Pi=kifor all i≥0.

Telecommunications have penetrated deeply into all spheres of human activity. Due to the use of digital technologies, it has become possible to transmit and receive information of various kinds in one stream (not separately). Three geostationary satellites are sufficient to provide data transmission from anywhere in the world to another point using artificial Earth satellites. Data transmission systems using the satellite should contain at least two Earth Stations and a repeater of the communication satellite. There are no means of combating any types of artificial interference in the Earth Stations equipment. The jammer, an illegitimate user, can emit a signal from any point in space within a telecommunications satellite system. The paper provides an analysis of existing methods for determining the coordinates of a radio signal source, describes a method for determining the coordinates of an radio signal source (illegitimate user) in satellite telecommunications systems using a geostationary spacecraft. The development was based on a three-position virtual antenna array, the positions of which determined the base of an irregular tetrahedron, and the radio signal source itself was its vertex. In the course of the solution, canonical equations of straight lines in space were compiled, at the intersection point of which the source of the radio signal is located. The coordinates of the radio signal source were the roots of the system, when finding the unit vectors lying on these lines.

This manuscript contains several new notions including intuitionistic fuzzy  N b  metric space, intuitionistic fuzzy quasi- S b -metric space, intuitionistic fuzzy pseudo- S b -metric space, intuitionistic fuzzy quasi- N -metric space and intuitionistic fuzzy pseudo  N b  fuzzy metric space. We prove decomposition theorem and fixed-point results in the setting of intuitionistic fuzzy pseudo  N b  fuzzy metric space. Further, we provide several non-trivial examples to show the validity of introduced notions and results. At the end, we solve an integral equation, system of linear equations and nonlinear fractional differential equations as applications. [ABSTRACT FROM AUTHOR]

The results of a parallel implementation of a randomized vector algorithm for solving systems of linear equations are presented in the paper. The solution is represented in the form of a Neumann series. The stochastic method computes this series by sampling only random columns, avoiding multiplication of matrix by matrix and matrix by vector. We consider the case when the matrix is too large to fit in random-access memory (RAM). We use two approaches to solve this problem. In the first approach, the matrix is divided into parts that are distributed among MPI processes and stored in the available RAM of the cluster nodes. In the second approach, the entire matrix is stored on each node's hard drive, loaded into RAM, and processed in parts. Independent Monte Carlo experiments for random column indices are distributed among MPI processes or OpenMP threads for both approaches to matrix storage. The efficiency of parallel implementations is analyzed. Results are given for a system governed by dense matrices of size 10 4 and 10 5 . [ABSTRACT FROM AUTHOR]

Based on the method proposed for solving the so-called -systems of linear equations, it is proved that the orders of homogeneous invariant differential operators of smooth real functions of one variable take values from to , and the dimension of the space of all such operators does not exceed . A classification of invariant differential operators of order is obtained for and for for all orders from 4 to 10. Homogeneous invariant differential operators of the smallest order and the largest order are given, respectively, by the product of the first differentials () and the Wronskian (). The existence of nonzero homogeneous invariant differential operators of order for is proved. [ABSTRACT FROM AUTHOR]

In this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the cover order and the cover length, are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the matrix to obtain the cover order of any given matrix. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with non-negativity constraints on variables for both homogeneous and nonhomogeneous cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with non-negativity constraints on variables, including linear programming (LP) problems and non-negative least squares (NNLS) problems. For LP problems, the three possible behaviours of the solutions are studied through cover theory. On the other hand, we develop a method to obtain the cover length of the covered variable. In this process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem. [ABSTRACT FROM AUTHOR]

The Kaczmarz method is a classical while effective iteration method for solving very large-scale consistent systems of linear equations, and the randomized Kaczmarz method is an important and valuable variant of the Kaczmarz method. By theoretically analyzing and numerically experimenting several criteria typically adopted in the non-randomized and the randomized Kaczmarz method for selecting the working row, we derive sharper upper bounds for the convergence rates of some of the correspondingly induced Kaczmarz-type methods including those with respect to the maximal residual, maximal distance, and distance selection rules of the working row, and, for this whole suite of iteration methods consisting of the Kaczmarz methods with respect to the uniform, non-uniform, residual, distance, maximal residual, and maximal distance selection rules of the working row, we reveal their comparable relationships in terms of both mean-squared distance and mean-squared error, and show their computational effectiveness and numerical robustness based upon implementing a large number of test examples. Here the mean-squared distance is defined as the mean-value of the squared Euclidean norm of the current update increment of the iteration, and the mean-squared error is defined as the mean-value of the squared Euclidean norm of the current error that is the difference between the current iterate and the true solution of the target linear system. [ABSTRACT FROM AUTHOR]

Let M n , k , t be the inclusion matrix of k-subsets and t-subsets of a set of n elements. In this note, we give a new proof for the property that M n , k , t is either of full row rank or of full column rank. Our proof is based on a nontrivial use of system of linear equations. [ABSTRACT FROM AUTHOR]

The aim of the present research article is to solve the system of linear equations using common fixed point theorems in the context of bicomplex valued metric spaces. To obtain our main objective, we introduce generalized contractive conditions in bicomplex valued metric spaces and establish common fixed point theorems for self mappings. We also give a significant example to demonstrate the legitimacy of the given theorems.

In this manuscript, we introduce a three dimension metric type spaces so called $ J $-metric spaces. We prove the existence and uniqueness of a fixed point for self mappings in such spaces with different types of contractions. We use our result to prove the existence and uniqueness of a solution of the following fractional differential equations such as $ \mathcal{(P)}:\left\{ \begin{array}{ccl} D^{\lambda}x(t) & = & f(t,x(t)) = Fx(t) \;{\rm{ if }}\; t\in I_0 = (0,T] \\ x(0) & = & x(T) = r \\ \end{array} \right\} . $ Moreover, we present other applications to systems of linear equations and Fredholm type integral equation.

This study aims to establish a new fixed point theorem in the framework of $ S^{JS} $-metric spaces, recently introduced by Beg et al. We propose different principles of contraction using various techniques. The theorems obtained represent a new framework for other future work in the considered space. Also, we provide two applications of our results to linear system of equations and the following fractional differential equation $ \mathcal{(P)}:\left\{ \begin{array}{ccl} D^{\lambda}x(t) & = & f(t, x(t)) = Fx(t) \mbox{ if } t\in I_0 = (0, T] \\ x(0) & = & x(T) = r \ \end{array} \right\}. $ These applications show the effectiveness of our approach as a powerful tool for solving several types of differential equations.

This manuscript contains several new notions including intuitionistic fuzzy Nb metric space, intuitionistic fuzzy quasi-Sb-metric space, intuitionistic fuzzy pseudo-Sb-metric space, intuitionistic fuzzy quasi-N-metric space and intuitionistic fuzzy pseudo Nb fuzzy metric space. We prove decomposition theorem and fixed-point results in the setting of intuitionistic fuzzy pseudo Nb fuzzy metric space. Further, we provide several non-trivial examples to show the validity of introduced notions and results. At the end, we solve an integral equation, system of linear equations and nonlinear fractional differential equations as applications.

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The Jacobi iterative algorithm has the characteristic of low computational load, and multiple components of the solution can be solved independently. This paper applies these characteristics to the ternary optical computer, which can be used for parallel optimization because it has a large number of data bits and reconfigurable processor bits. Therefore, a new parallel design scheme is constructed to solve the problem of slow efficiency in solving large linear equations. And the elaborate experiment is used to verify. The experimental method is to simulate the calculation on the ternary optical computer experimental platform. Then, the resource consumption is numerically calculated and summarized to measure the feasibility of the parallel design. Eventually, the results show that the parallel design has obvious advantages in computing speed. The Jacobi iterative algorithm is optimized in parallel on ternary optical processor for the first time. There are two parallel highlights of the scheme. First, the n components are calculated in full parallel. Second, the modified signed-digit (MSD) multiplier based on the minimum module and one-step MSD adder are used to calculate each component to eliminate the impact of large amount of data on calculation time. The research provides a new method for fast solution of large linear equations. [ABSTRACT FROM AUTHOR]

In this manuscript, we introduce a three dimension metric type spaces so called J-metric spaces. We prove the existence and uniqueness of a fixed point for self mappings in such spaces with different types of contractions. We use our result to prove the existence and uniqueness of a solution of the following fractional differential equations such as ... Moreover, we present other applications to systems of linear equations and Fredholm type integral equation. [ABSTRACT FROM AUTHOR]

In this paper, a novel hyper-rectangle cover theory is developed. Two important concepts, the cover order and the cover length, are introduced. We construct a specific échelon form of the matrix in the same manner as that employed to determine the rank of the matrix to obtain the cover order of any given matrix. Using the properties of the cover order, we obtain the necessary and sufficient conditions for the existence and uniqueness of the solutions for linear equations system with non-negativity constraints on variables for both homogeneous and nonhomogeneous cases. In addition, we apply the cover theory to analyze some typical problems in linear algebra and optimization with non-negativity constraints on variables, including linear programming (LP) problems and non-negative least squares (NNLS) problems. For LP problems, the three possible behaviours of the solutions are studied through cover theory. On the other hand, we develop a method to obtain the cover length of the covered variable. In this process, we discover the relationship between the cover length determination problem and the NNLS problem. This enables us to obtain an analytical optimal value for the NNLS problem.

For the single-step matrix splitting iteration paradigm proposed and analyzed recently by Bai (Numer. Linear Algebra Appl. 2023; e2510), we further provide a simpler and more accurate convergence result when the coefficient matrix of the linear system is a nonsingular matrix. Moreover, we demonstrate its asymptotic convergence when the coefficient matrix is an H -matrix. These results then enrich the convergence theory and enlarge the application spectrum of this paradigm. [ABSTRACT FROM AUTHOR]

Students' creative thinking processes need to be developed to support them in solving complex mathematical problems, but this is not easy. This study aims to describe students' creative thinking skills in solving linear equation system problems based on brain dominance. This research is a qualitative descriptive research with a case study approach. Subjects were categorized into right brain dominance and left brain dominance by using a brain dominance questionnaire. The research instruments were in the form of tests and interview guidelines. Data validation uses method triangulation and data is analyzed through data reduction, data presentation, data verification, and drawing conclusions. The results showed that the subject of right brain dominance in fluency gave 3 responses, in flexible thinking the subject gave 3 different ideas, in original thinking the subject gave a unique response, and in elaboration the subject did not show a coherent answer. While the left brain dominant subject in fluent thinking (fluency) gave 4 responses, in flexible thinking the subject gave 3 general ideas, in original thinking the subject did not show a unique response, and in detailed thinking (elaboration) the subject showed a flurry of answers.

The article deals with the method of increasing the dynamic range of a standard analog-to-digital converter. The method is based on interpolation of signal sections, which are limited in amplitude, using undistorted sections. Interpolation is performed by solving a system of linear equations. The error estimation and factors limiting the application of the method is made. Simulation results are presented. The conclusion is made about the prospects of using the least squares method, since in this case the interpolation error is the smallest in the mean square sense.

For solving large-scale inconsistent systems of linear equations by iteration methods, with the application of the greedy randomized Kaczmarz method to a consistent augmented linear system, we find an appropriate balance between the updates of the two iteration vectors in the randomized extended Kaczmarz method, and, based on this balance, we propose a greedy randomized augmented Kaczmarz method. We prove the convergence of the greedy randomized augmented Kaczmarz method and derive an upper bound for its expected convergence rate. Numerical results show that the greedy randomized augmented Kaczmarz method can be much more effective than the randomized extended Kaczmarz method as well as the partially randomized extended Kaczmarz method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a normalization method for solving a system of linear equations over tropical semirings. We use a normalization method to construct an associated normalized matrix, which gives a technique for solving the system. If solutions exist, the method can also determine the degrees of freedom of the system. Moreover, we present a procedure to determine the column rank and the row rank of a matrix. Flowcharts for this normalization method and its applications are included as well. [ABSTRACT FROM AUTHOR]

Evaluating the security of a block cipher against impossible differential cryptanalysis, is an important aspect during the design process. The maximum length of impossible differentials is often used to evaluate this security. There have been many methods on giving upper bounds on the length of impossible differentials or finding longer impossible differentials. Two notable examples are the "Primitive Index" method proposed by Sun et al. at EUROCRYPT2016 and the MILP method proposed by Sasaki et al. at EUROCRYPT2017. However, these existing methods can only give upper bounds for some special SPN block ciphers or cannot give upper bounds due to the high time complexity. In this paper, we show that when ignoring the differential property of the underlying S-box, giving upper bounds on the length of impossible differentials is a linear problem. By using linear algebra, we propose the Expansion Index of the linear layer, with which we can give upper bounds on the length of impossible differentials for any SPN block cipher with the detail of the S-box omitted. The core of this method is establishing and solving systems of linear equations, thus the verification of a single differential has linear time complexity. What's more, to give upper bounds with this method, we only need to establish and solve systems for differentials whose input and output differences have only one active S-box, which greatly reduces its time complexity from O (2 t) to O(t) (here t denotes the number of S-boxes in the S-layer). The method in this paper is implemented in C and encapsulated into a tool freely available to readers. By applying our method on some SPN block ciphers, we give, for the first time, upper bounds on the length of impossible differentials for Midori, Skinny, CRYPTON, mCrypton, Minalpher. [ABSTRACT FROM AUTHOR]

The development of a structural-parametric approach to constructing analytical approximations is considered. Within the framework of the new method, which includes modified S-approximations and determination of the integral curves of the vector field, a method is proposed for finding the analytic continuation of the field in the case of profile (route) geophysical measurements. The results of a mathematical experiment using real data on an anomalous gravitational field from data of marine areal gravimetric and airborne gravimetric route surveys are presented. It is shown that the proposed analysis algorithm is promising for finding analytic continuations of geopotential fields within a certain neighborhood of the measurement profile when a sufficient volume of areal data is lacking. [ABSTRACT FROM AUTHOR]

This study explored the mental constructions displayed by pre-service mathematics teachers (PMTs) when applying Cramer’s rule. The aim was to reveal the character of mental constructions made around the nature of the solution set of equations and the role of parameters in the solution of equations with parametric coefficients. These mental constructions occur within the Action-Process-Object-Schema (APOS) theory. In this study, data was generated from 31 first year pre-service teachers by means of an activity sheet and interviews. The interviews were used to clarify pre-service teachers’ responses to tasks from the activity sheet. The findings revealed that many PMTs displayed procedural understanding of Cramer’s rule. It was also found that they understood what the solution of system of linear equations means. This meant that they were operating at the action stage, in terms of APOS. Additionally, it revealed that the lack of construction of related schemata negatively impacted the PMTs’ attempt to construct the necessary mental constructions. Therefore, the researchers provided the genetic decomposition (GD) for the use of Cramer’s rule to assist teacher educators to analyse the mental constructions of students. This study explored a new application of APOS theory. Analysing the mental constructions of students by means of research is intended to assist in designing alternative teaching strategies. In this way, this study makes a significant contribution to the solution of system of equations pedagogy.

The Gauss-Seidel and Kaczmarz methods are two classic iteration methods for solving systems of linear equations, which operate in column and row spaces, respectively. Utilizing the connections between these two methods and imitating the exact analysis of the mean-squared error for the randomized Kaczmarz method, we conduct an exact closed-form formula for the mean-squared residual of the iterate generated by the randomized Gauss-Seidel method. Based on this new formula, we further estimate an upper bound for the convergence rate of the randomized Gauss-Seidel method. Theoretical analysis and numerical experiments show that this bound measurably improves the existing ones. Moreover, these theoretical results are also extended to the more general extrapolated randomized Gauss-Seidel method. [ABSTRACT FROM AUTHOR]

The algebraic reconstruction technique (ART) and its generalization the block-iterative ART are often used to solve large scale linear systems Ax=b which arise in many applications, such as image processing. Two ingredients, i.e. stepsizes and blocks, play a key role in block-iterative ART. In this paper we address the problem of how to choose stepsizes and blocks of A so as to achieve optimal convergence rate of the block-iterative ART. We introduce a random cyclic selection method for selecting blocks for the block-iterative ART in order to reduce computational complexity. Our numerical experiments demonstrate the efficiency of our selections of stepsizes and blocks in the block-iterative ART. [ABSTRACT FROM AUTHOR]

We enrich the known results about coupled fixed points and coupled best proximity points. We generalize the notion of ordered pairs of cyclic contraction maps and we obtain sufficient conditions for the existence and uniqueness of best proximity points. We get a priori and a posteriori error estimates for the coupled fixed points and for the coupled best proximity points, provided that the underlying Banach space has modulus of convexity of power type in the case of best proximity points, obtained by sequences of successive iterations. We illustrate the main result with an example. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iterative algorithm and then prove the strong convergence of proposed algorithm in framework of CAT(0) space and Hilbert space for approximating the common fixed point of countably infinite family of nonexpansive mappings and minimizer of proper, convex, lower semicontinuous function. Then we implement the proposed algorithm to solve constrained minimization problem and system of linear equations. Our results genaralize the results of Phuengrattana et al. [36] and Suparatulatorn et al. [43]. [ABSTRACT FROM AUTHOR]

In this article, we propose a randomized Douglas-Rachford(DR) method for linear system. This algorithm is based on the cyclic DR method. We consider a linear system as a feasible problem of finding intersection of hyperplanes. In each iteration, the next iteration point is determined by a random DR operator. We prove the convergence of the iteration points based on expectation. And the variance of the iteration points declines to zero. The numerical experiment shows that the proposed algorithm performs better than the cyclic DR method. [ABSTRACT FROM AUTHOR]