Matrix Problems Research Articles

2-D Hybrid Cylindrical FDTD Method With Unconditional Stability

A 2-D hybrid method combining cylindrical explicit and unconditionally stable finite-difference time-domain (CEUS-FDTD) with traditional cylindrical FDTD (C-FDTD) methods is proposed to analyze the electromagnetic cylindrical structures efficiently. The unconditional stability of the proposed method is achieved by eliminating the unstable modes which are obtained by solving the eigenvalue problem of the system matrix. However, solving global eigenvalue is time-consuming. To solve this, the domain is divided into the CEUS-FDTD region and C-FDTD region by analyzing the spectral radius of one single grid. All unstable modes can be found only in the CEUS-FDTD region. The electric fields on the boundary of the two regions are corrected without any approximation. The proposed method saves iteration time compared to C-FDTD and costs less time than CEUS-FDTD for the eigenvalue solution. Results show validity and efficiency.

Joint Communication and Sensing Design in Coal Mine Safety Monitoring: 3-D Phase Beamforming for RIS-Assisted Wireless Networks

This paper investigates the resource allocation of a reconfigurable intelligent surface (RIS)-aided joint communication and sensing (JCAS) system in a coal mine scenario. In the JCAS system, an RIS is implemented at the corner of the zigzag tunnels to improve the complicated wireless environment, where ground obstacles frequently block direct links. In addition, a wireless backhaul base station with a limited energy budget is deployed in the depth of the mine to sense the target area and provide internet of things (IoT) services and communication services for users. Furthermore, a data center is placed on the ground to analyze the obtained data and route the communication data. Under this deployment, a joint optimization problem of RIS phase shift matrix, RIS element switches, and area sensing time is proposed. We aim to maximize the successful sensed bits under total completion time, and maximum transmit power constraints. In order to solve this problem, an iterative algorithm is proposed. The successive convex approximation (SCA) based algorithm is used for the RIS phase shift matrix optimization subproblem. For the sensing time optimization subproblem, the quadratic approximation method is proposed to optimize the number of area perceptions. The coordinate descent method is utilized to optimize the RIS element switches. Simulation results show that the energy efficiency is improved by up to 38%, and 7% increases the specific data size compared with the benchmark solutions.

Extremely-Fast, Energy-Efficient Massive MIMO Precoding With Analog RRAM Matrix Computing

Signal processing in wireless communications, such as precoding, detection, and channel estimation, are basically about solving inverse matrix problems, which, however, are slow and inefficient in conventional digital computers, thus requiring a radical paradigm shift to achieve fast, real-time solutions. Here, for the first time, we apply the emerging analog matrix computing (AMC) to the linear precoding of massive MIMO. The real-valued AMC concept is extended to process complex-valued signals. In order to adapt the MIMO channel models to RRAM conductance mapping, a new matrix inversion circuit is developed. In addition, fully analog dataflow and optimized operational amplifiers are designed to support AMC precoding implementation. Simulation results show that the zero-forcing precoding is solved within 20 ns for a 16×128 MIMO system, which is two orders of magnitude faster than the conventional digital approach. Meanwhile, the energy efficiency is improved by 50×.

The capacity of Islamic senior high school students to comprehend mathematical ideas and solve matrix problems

Kemampuan pemahaman konsep dalam mata pelajaran matematika sangat diperlukan oleh setiap siswa. Tujuan penelitian ini untuk mendeskripsikan kemampuan pemahaman konsep matematis siswa MA dalam menyelesaikan soal matriks. Penelitian dilaksanakan pada 53 siswa kelas XI di MA Mambaul Ulum Banjarejo. Teknik pengumpulan data yang digunakan adalah tes dan wawancara. Hasil penelitian yakni Kemampuan pemahaman konsep matematis siswa dalam setiap indikator berbeda-beda. Persentase indikator kemampuan pemahaman konsep siswa pada indikator ke-1 yakni 72%, indikator ke-2 yakni 93%, indikator ke-3 yakni 83%, indikator ke-4 yakni 88%, indikator ke-5 yakni 60%, indikator ke-6 yakni 55% dan indikator ke-7 yakni 60%. Persentase indikator kemampuan pemahaman konsep matematis siswa yang paling tinggi adalah merepresentasikan suatu konsep dalam berbagai bentuk matematis. Sedangkan, persentase indikator kemampuan pemahaman konsep siswa yang paling rendah yakni menerapkan prosedur atau operasi matematika dalam menyelesaikan soal. Guru dalam proses pembelajaran disarankan untuk tidak hanya menekankan pada hafalan, akan tetapi pada pemahaman konsep siswa. The ability to understand concepts in mathematics is needed by every student. The purpose of this study was to describe the ability to understand mathematical concepts of MA students in solving matrix problems. The research was conducted on 53 students of class XI at MA Mambaul Ulum Banjarejo. Data collection techniques used were tests and interviews. The results of the research are the ability to understand students' mathematical concepts in each indicator is different. The percentage of students' concept understanding ability indicators on the 1st indicator is 72%, the 2nd indicator is 93%, the 3rd indicator is 83%, the 4th indicator is 88%, the 5th indicator is 60%, the 3rd indicator 6 is 55% and the 7th indicator is 60%. The highest percentage of student's ability to understand mathematical concepts is representing a concept in various mathematical forms. Meanwhile, the lowest percentage of students' concept comprehension indicators is applying mathematical procedures or operations in solving problems. Teachers in the learning process are advised not only to emphasize memorization but also to understand students' concepts.

Extremal problems for the eccentricity matrices of complements of trees

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$. The largest modulus of an eigenvalue is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs. First, we determine the unique tree whose complement has minimum (respectively, maximum) $\mathcal{E}$-spectral radius among the complements of trees. Then, we prove that the $\mathcal{E}$-eigenvalues of the complement of a tree are symmetric about the origin. As a consequence of these results, we characterize the trees whose complement has minimum (respectively, maximum) least $\mathcal{E}$-eigenvalues among the complements of trees. Finally, we discuss the extremal problems for the second largest $\mathcal{E}$-eigenvalue and the $\mathcal{E}$-energy of complements of trees and characterize the extremal graphs. As an application, we obtain a Nordhaus-Gaddum-type lower bounds for the second largest $\mathcal{E}$-eigenvalue and $\mathcal{E}$-energy of a tree and its complement.

The Sequence Recommendation Algorithm based on GRU and Attention

In recent years, with the booming development of social networks and e-commerce, users have increasingly convenient access to information, while a large amount of data continues to emerge, leading to more and more serious information overload. To alleviate the problem of information overload, recommendation systems have emerged to assist users in sifting through massive amounts of information to find content that meets their needs. Sequential recommendation, as a form of recommendation system, mainly analyzes the interaction behavior between users and items, models user characteristics, and then uses various methods to capture users' long-term and short-term preferences to recommend items of interest to users. Based on the perspective of user preference change over time, this paper provides an in-depth analysis of the current research progress and methods of user behavior sequence recommendation. At the same time, this paper proposes corresponding solution strategies for the problems of cold start, sparse matrix and noise interference faced by traditional recommendation systems. Finally, we will discuss the challenges and future research directions of recommendation systems to provide the theoretical basis for further improvement of recommendation systems.

Multiclass determination of drug residues in water and fish for bioaccumulation potential assessment

In this work, a wide-scope liquid chromatography-tandem mass spectrometry (LC-MS/MS) method for the quantitative determination of environmental levels of multiclass drugs and their metabolites in water and fish samples was developed. The method allowed the reliable determination of 44 drugs, covering a rather wide range of chemistries and physicochemical characteristics. In order to obtain a reliable and robust analytical protocol, different combinations of extraction and cleanup techniques were systematically examined. Aqueous samples were extracted using a simple Oasis HLB SPE enrichment protocol with pH-optimized sample percolation (pH 3). The extraction of cryo-homogenized biota samples was performed using double extraction with MeOH basified with 0.5% NH3, which allowed high extraction recoveries for all target analytes. The problem of the coextracted lipid matrix, which is known to be the key obstacle for reliable biota analysis, was systematically examined in a series of model cleanup experiments. A combination of cryo-precipitation, filtration, and HLB SPE cleanup was proposed as a protocol, which allowed reliable and robust analysis of all target compounds at low ng/g levels. At the final conditions, the method which was validated at three concentration levels showed high extraction recoveries (68–97%), acceptable matrix effects (12 to −32%), accuracies (81–129%), and reproducibilities (3–32%) for all analytes. The developed method was used to determine drug concentrations in river water and in feral freshwater fish, including whole fish and muscle tissue, from the Sava River (Croatia), in order to estimate their corresponding bioaccumulation potential. With respect to bioaccumulation potential in whole fish and fish muscle, the most relevant drugs were lisinopril, sertraline, terbinafine, torsemide, diazepam, desloratadine, and loratadine with estimated bioaccumulation factors ranging from 20 to 838 and from 1 to 431, respectively.

A Semismooth Newton-Type Method for the Nearest Doubly Stochastic Matrix Problem

We study a semismooth Newton-type method for the nearest doubly stochastic matrix problem where the nonsingularity of the Jacobian can fail. The optimality conditions for this problem are formulated as a system of strongly semismooth functions. We show that the nonsingularity of the Jacobian does not hold for this system. By exploiting the problem structure, we construct a modified two step semismooth Newton method that guarantees a nonsingular Jacobian matrix at each iteration, and that converges to the nearest doubly stochastic matrix quadratically. Funding: This work was supported by Canadian Network for Research and Innovation in Machining Technology. The research of H. Hu, H. Im, and H. Wolkowocz was supported by The Natural Sciences and Engineering Research Council of Canada. The research of X. Li was supported by the National Natural Science Foundation of China [No. 11601183] and Natural Science Foundation for Young Scientist of Jilin Province [No. 20180520212JH].

Global error bounds for the extended vertical linear complementarity problems of CKV-type matrices and CKV-type B-matrices

Global error bounds for the extended vertical linear complementarity problems of CKV-type matrices and CKV-type B-matrices

Design of OMC-Sagnac Loop Using PDMS and Different Package Structures to Improve Sensing Performance and Optimize the Ill-Conditioned Matrix

In the process of ocean exploration, highly accurate and sensitive measurements of seawater temperature and pressure significantly impact the study of seawater’s physical, chemical, and biological processes. In this paper, three different package structures, V-shape, square-shape, and semicircle-shape, are designed and fabricated, and an optical microfiber coupler combined Sagnac loop (OMCSL) is encapsulated in these structures with polydimethylsiloxane (PDMS). Then, the temperature and pressure response characteristics of the OMCSL, under different package structures, are analyzed by simulation and experiment. The experimental results show that structural change hardly affects temperature sensitivity, and square-shape has the highest pressure sensitivity. In addition, with an input error of 1% F.S., temperature and pressure errors were calculated, which shows that a semicircle-shape structure can increase the angle between lines in the sensitivity matrix method (SMM), and reduce the effect of the input error, thus optimizing the ill-conditioned matrix. Finally, this paper shows that using the machine learning method (MLM) effectively improves demodulation accuracy. In conclusion, this paper proposes to optimize the ill-conditioned matrix problem in SMM demodulation by improving sensitivity with structural optimization, which essentially explains the cause of the large errors for multiparameter cross-sensitivity. In addition, this paper proposes to use the MLM to solve the problem of large errors in the SMM, which provides a new method to solve the problem of the ill-conditioned matrix in SMM demodulation. These have practical implications for engineering an all-optical sensor that can be used for detection in the ocean environment.

Generalized fuzzy eigenvectors of real symmetric matrices1

In paper the generalized real eigenvalue and fuzzy eigenvector of a crisp real symmetric matrix with respect to another real symmetric matrix is studied. The original generalized fuzzy eigen problem is extended into a crisp generalized eigen problem of a real symmetric matrix with high orders using the arithmetic operation of LR fuzzy matrix and vector. Two cases are analysed: (a) the unknown eigenvalue λ is a non negative real number; (b) the unknown eigenvalue λ is a negative real number. Two computing models are established and an algorithm for finding the generalized fuzzy eigenvector of a real symmetric matrix is derived. Moreover, a sufficient condition for the existence of a strong generalized fuzzy eigenvector is given. Some numerical examples are shown to illustrated our proposed method.

Two inverse eigenvalue problems for matrices whose graphs are trees

Inverse eigenvalue problem refers to the problem of reconstructing a matrix of a desired structure from a prescribed eigendata. In this paper, we discuss two additive inverse eigenvalue problems for matrices whose graph is tree. In order to analyze the problems, the vertices of the given tree are labeled in a suitable way so that concrete recurrence relations between the characteristic polynomials of the leading principal submatrices of the matrix can be obtained in a simple form. The method of obtaining the entries of the required matrix is constructive and provides an algorithm for computing the same. We provide some numerical examples to illustrate the results. The computations are done using SCILAB by feeding the eigendata and the adjacency pattern of the tree as inputs.

Adapted AZNN methods for time-varying and static matrix problems

We present adapted Zhang neural networks (AZNN) in which the parameter settings for the exponential decay constant $\eta$ and the length of the start-up phase of basic ZNN are adapted to the problem at hand. Specifically, we study experiments with AZNN for time-varying square matrix factorizations as a product of time-varying symmetric matrices and for the time-varying matrix square roots problem. Differing from generally used small $\eta$ values and minimal start-up length phases in ZNN, we adapt the basic ZNN method to work with large or even gigantic $\eta$ settings and arbitrary length start-ups using Euler's low accuracy finite difference formula. These adaptations improve the speed of AZNN's convergence and lower its solution error bounds for our chosen problems significantly to near machine constant or even lower levels. Parameter-varying AZNN also allows us to find full rank symmetrizers of static matrices reliably, for example, for the Kahan and Frank matrices and for matrices with highly ill-conditioned eigenvalues and complicated Jordan structures of dimensions from $n = 2$ on up. This helps in cases where full rank static matrix symmetrizers have never been successfully computed before.

Efficient calculation of three-dimensional tensor networks

We have proposed an efficient algorithm to calculate physical quantities in the translational invariant three-dimensional tensor networks, which is particularly relevant to the study of the three-dimensional classical statistical models and the ($2+1$)-dimensional quantum lattice models. In the context of a classical model, we determine the partition function by solving the dominant eigenvalue problem of the transfer matrix, whose left and right dominant eigenvectors are represented by two projected entangled simplex states. These two projected entangled simplex states are not Hermitian conjugate to each other but are appropriately arranged so that their inner product can be computed much more efficiently than in the usual prescription. For the three-dimensional Ising model, the calculated internal energy and spontaneous magnetization agree with the published results in the literature. The possible improvement and extension to other models are also discussed.

Balancing permuted copies of multigraphs and integer matrices

Given a square matrix A over the integers, we consider the Z-module MA generated by the set of all matrices that are permutation-similar to A. Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices aI+bJ belonging to MA. We give a relatively fast method to compute a generator for such matrices, avoiding the need for a very large canonical form over Z. Several special cases are considered. In particular, the problem for symmetric matrices answers a question of Cameron and Cioabǎ on determining the eventual period for integers λ such that the λ-fold complete graph λKn has an edge-decomposition into a given (multi)graph.

Embeddability of centrosymmetric matrices capturing the double-helix structure in natural and synthetic DNA

In this paper, we discuss the embedding problem for centrosymmetric matrices, which are higher order generalizations of the matrices occurring in strand symmetric models. These models capture the substitution symmetries arising from the double helix structure of the DNA. Deciding whether a transition matrix is embeddable or not enables us to know if the observed substitution probabilities are consistent with a homogeneous continuous time substitution model, such as the Kimura models, the Jukes-Cantor model or the general time-reversible model. On the other hand, the generalization to higher order matrices is motivated by the setting of synthetic biology, which works with different sizes of genetic alphabets.

A sweeping algorithm of unstructured elements in discontinuous spectral element method for thermal radiation in irregular multidimensional geometries

The discontinuous spectral element method (DSEM) coupled with an element-by-element sweeping strategy is proposed as a solution to the radiative transfer equation (RTE) for participating media in irregular geometries. In the solving processing, a spatial domain of RTE is discretized by unstructured meshes; an angular direction of RTE is discretized as a set of contrast angles by structured quadrilateral meshes. Then, both spatial domain and angular direction are coupling solved by DSEM. However, this coupling solution leads to a large-scale matrix problem. To address this, the element-by-element sweeping strategy optimizes the ordering of unstructured spatial elements. Several examples are provided to demonstrate the effectiveness of DSEM with the element-by-element sweeping strategy. Compared to the original sweep strategy, the element-by-element approach significantly improves calculation efficiency and reduces computational time.

Mathematical Performance Assessment Instrument at SMK Mega Link Majene, Indonesia

One of the stages of assessing skills in mathematics is using a performance appraisal instrument, scoring guidelines, and analyzing the quality of the instrument. Mathematical performance assessment tests are carried out through practical methods to assess the learning process on matrix material. This research develops a math performance assessment test on maple application skills, especially on matrix material. The performance assessment test includes using the maple application to understand concepts and solving matrix problems that are carried out before and during the process of practical activities using the maple application. Based on the results of the research analysis obtained, aspects of how to open maple applications in general, there are few mistakes made by students. They include the use of pallets, typing directly, the use of linalg without mistakes, the use of the evaluatem command without, and the correctness of the work.

Generalized Langevin mode analysis (GLMA) for local density fluctuation of water in an inhomogeneous field of a biomolecule

An idea to analyze the fluctuational mode or/and spectrum of water molecules in an inhomogeneous environment around a biomolecule such as protein and DNA is proposed based on the generalized Langevin equation (GLE) for water interacting with a biomolecule. The idea is to solve an eigen-value problem of the collective frequency matrix of water, which has a physical meaning similar to the Hessian matrix in the case of a connected harmonic-oscillator. The diagonalization of the matrix gives the mode and frequency of the density fluctuation of water around the biomolecule. The treatment is a natural extension of that made for water in a homogeneous environment by Chong and Hirata in 1999, in which the authors have found the four modes for density fluctuation; one for an acoustic mode, and the other three for optical modes. It is the point of the new approach how the mode and frequency of the density fluctuation are modulated by the field from the biomolecule. The method provides a way to distinguish the mode of fluctuation with color.A few possible applications of the theoretical approach to problems of the life science including medicine are also proposed.

Inexact penalty decomposition methods for optimization problems with geometric constraints

This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication and decomposition strategy, the method presented here explicitly handles these difficult constraints, thus generating iterates which are feasible with respect to them, while the remaining (standard and supposingly simple) constraints are tackled by sequential penalization. Inexact optimization steps are proven sufficient for the resulting algorithm to work, so that it is employable even with difficult objective functions. The current work is therefore a significant generalization of existing papers on penalty decomposition methods. On the other hand, it is related to some recent publications which use an augmented Lagrangian idea to solve optimization problems with geometric constraints. Compared to these methods, the decomposition idea is shown to be numerically superior since it allows much more freedom in the choice of the subproblem solver, and since the number of certain (possibly expensive) projection steps is significantly less. Extensive numerical results on several highly complicated classes of optimization problems in vector and matrix spaces indicate that the current method is indeed very efficient to solve these problems.