Matrix Operations Research Articles

A STUDY ON THE STRUCTURE OF MATRICES RELATED TO THE VECH*, VECP*, AND VEC OPERATORS

The vec operator is an essential tool in matrix algebra that transforms a matrix into a column vector based on specific rules. This paper introduces two new operators, namely and , which take the main diagonal and supra-diagonal elements of the matrix, respectively. In this paper, we obtain the general form of the matrix , which transform to , with as a matrix of size . In addition, we also develop the general forms of matrices and , which transform into and into , with as a symmetric matrix of size . This study also explores the properties and relationships between these matrices and their relevance to duplication and commutation matrices, providing deeper insights into the structure and operations of matrices.

A quasi affine transformation evolution algorithm with evolution matrix selection operation for parameter estimation of proton exchange membrane fuel cells

Electrochemical energy conversion technologies include proton exchange membrane fuel cells (PEMFCs) where proton interchange is an alternative to diesel distributed generation, and PEMFCs are considered as a promising backup power source and a tool to regulate power consumption. Some of the major benefits of these PEMFCs especially in power system applications include low emission of carbon, fast load following capability, no noise and high start-up reliability. It is challenging to find the best PEMFC parameters because the model is complex and the problem is nonlinear; not all optimization algorithms can solve this problem. This paper presents a new approach that applies QUasi-Affine TRansformation Evolution algorithm with a new adaptation of Evolution Matrix and Selection operation (QUATRE-EMS) to determine optimal values of uncertain parameters in PEMFC stack references. The objective function of the optimization problem is defined as the sum of squared errors of the actual and predicted voltage data. The effectiveness of the proposed QUATRE-EMS algorithm is also checked through statistical analysis and the QUATRE-EMS variant is compared with other variants of DE optimization algorithms which are recently proposed in the state-of-the-art literature such as LSHADE, MadDE, CS-DE, LPalmDE, EDEV, jSO, SHADE, ISDE, and JADE. Results show that the QUATRE-EMS algorithm reduces SSE significantly, with an average SSE of 0.078492, which is 15% less than the best performing existing algorithms. QUATRE-EMS also achieved the lowest average values of absolute error, relative error and mean bias error among different PEMFC stack references, with accuracy improved by up to 20%. It was also computationally more efficient, cutting runtime in half compared to other methods. The results of these findings confirm the effectiveness and practicability of the QUATRE-EMS algorithm for improving the accuracy of BCS500W, NedStackPS6, SR12, H12, HORIZON, and Standard 250W PEMFC stack references.

A novel Q-neutrosophic soft under interval matrix setting and its applications

Decision-making theory serves as an effective framework to guide decision-makers in solving problems. One notable application of this theory is in the medical field, where it aids doctors in analyzing patient data to determine whether a patient is infected. To enhance this theory with more adaptable mathematical methods, we propose an expanded approach based on previously introduced matrixes of Q-neutrosophic soft under an Interval-valued setting (IV-Q-NSM). This represents a new finding of existing mathematical tools to address the two-dimensional uncertainty prevalent in various life domains. This work explores several algebraic properties and matrix operations associated with IV-Q-NSM. Subsequently, we introduce a new methodology for decision-making (DM) in medical diagnosis selection problems. This approach aims to provide a more flexible and comprehensive framework for evaluating complex medical data and improving diagnostic accuracy.

Hybrid affine cipher and eigenvector methods for cryptography

This paper systematically presents a specific application of linear algebra in information security and cryptography, highlighting the crucial role of matrix operations and linear techniques in the design and analysis of encryption algorithms. Specifically, we focus on a method that utilizes linear algebra to enhance the security and efficiency of modern cryptographic systems. A detailed illustrative example is provided to help readers better understand how these mathematical tools are applied in practice. Furthermore, we review existing encryption techniques and propose a new matrix-based scheme that leverages linear algebra to improve data security, optimize the encryption-decryption process, and ensure the integrity and confidentiality of information throughout transmission and storage.

Heterogeneous Edge Computing for Molecular Property Prediction with Graph Convolutional Networks

Graph-based neural networks have proven to be useful in molecular property prediction, a critical component of computer-aided drug discovery. In this application, in response to the growing demand for improved computational efficiency and localized edge processing, this paper introduces a novel approach that leverages specialized accelerators on a heterogeneous edge computing platform. Our focus is on graph convolutional networks, a leading graph-based neural network variant that integrates graph convolution layers with multi-layer perceptrons. Molecular graphs are typically characterized by a low number of nodes, leading to low-dimensional dense matrix multiplications within multi-layer perceptrons—conditions that are particularly well-suited for Edge TPUs. These TPUs feature a systolic array of multiply–accumulate units optimized for dense matrix operations. Furthermore, the inherent sparsity in molecular graph adjacency matrices offers additional opportunities for computational optimization. To capitalize on this, we developed an FPGA GFADES accelerator, using high-level synthesis, specifically tailored to efficiently manage the sparsity in both the graph structure and node features. Our hardware/software co-designed GCN+MLP architecture delivers performance improvements, achieving up to 58× increased speed compared to conventional software implementations. This architecture is implemented using the Pynq framework and TensorFlow Lite Runtime, running on a multi-core ARM CPU within an AMD/Xilinx Zynq Ultrascale+ device, in combination with the Edge TPU and programmable logic.

New $$\mathbb {A}$$-numerical Radius Equalities and Inequalities for Certain Operator Matrices and Applications

New $$\mathbb {A}$$-numerical Radius Equalities and Inequalities for Certain Operator Matrices and Applications

Lucas Operational Matrix Approach for Solving the Fractional Klein–Gordon Equation

Lucas Operational Matrix Approach for Solving the Fractional Klein–Gordon Equation

ALGORITHMIC DIFFERENCES OF COMPLETE AND PARTIAL ALGEBRAIC SYNTHESIS OF A FINITE STATE MACHINE WITH DATAPATH OF TRANSITIONS

Context. The problem of algorithmization the search for formal solutions of the problem of algebraic synthesis of a finite state ma-chine with datapath of transitions is considered. The concept of complete and partial solutions of this problem is proposed. The object of research is the automated synthesis of the finite state machine in the part of the function of transitions without taking into account the function of outputs. The basis of the algebraic implementation of the transition function is the author's approach to the transformation of state codes using a set of arithmetic and logical operations. The search for formal solutions to the problem of algebraic synthesis is a complex process that requires the use of appropriate methods and algorithms aimed at special coding of states and mapping of operations of transitions to individual state machine transitions. The use of partial solutions of the problem of algebraic synthesis can contribute to reducing the executing time of such algorithms and reducing the overall design time of digital control devices based on a finite state machine with an datapath of transitions. Objective. Development and research of algorithms for finding complete and partial solutions to the problem of algebraic synthesis of a finite state machine with datapath of transitions. Method. The research is based on the structure of a finite state machine with datapath of transitions. The synthesis of the circuit of the state machine involves the preliminary solution of the problem of algebraic synthesis. The result is the so-called formal solution of this problem, which contains two components. The first component is defined state codes, the second component is arithmetic and logic operations mapped to separate state machine transitions. Finite state machine can be synthesized in that case if transformation of given state codes in the process of making transitions is possible using a given set of operations. Verification of this possibility is carried out using the known matrix approach. It involves the formation and element-by-element mapping of two matrices – the matrix of transitions and the combined matrix of operations. As a result, a coverage matrix is formed, which reflects the possibility of implementing (covering) of state machine transitions by specified arithmetic and logic operations. Changing the way of encoding states or the set of operations can give different solutions to the problem of algebraic synthesis with a greater or lesser number of covered state machine transitions. Results. Using the example of an abstract graph-scheme of the control algorithm, it is demonstrated that the solution of the problem of algebraic synthesis of a finite state machine with datapath of transitions can be considered a situation when one or more state machine transitions cannot be implemented using a given set of operations. It is proposed to call such situation as a partial solution of the problem of algebraic synthesis. The implementation of all transitions by specified operations gives a complete solution of this problem, but the number of complete solutions is always much smaller than the number of partial solutions. Therefore, in the general case, the search for complete solutions takes much more time and, moreover, is not always possible. Conclusions. The design of a logical circuit of a finite state machine with datapath of transitions is possible in the case of a complete or partial solution of the problem of algebraic synthesis. In the case of a partial solution, those transitions that cannot be implemented by any of the available operations are implemented in a canonical way using a system of Boolean equations. The search for complete solutions generally takes more time than the search for partial solutions. This makes actual the development of algorithms and methods of synthesis of this class of finite state machine, based on the search for partial solutions of the problem of algebraic synthesis.

VECTOR-LOGIC FAULT SIMULATION

Context. The technological trends of Design&Test computing for the IT industry and academic science are determined by the following directions: in-memory computing, immersive computing, AI computing, focused on energy saving and reduction of computing time when providing services. A mechanism for simulating faults as addresses on smart data structures is proposed, which eliminates the algorithm for simulating input test sets to obtain a test map for logic functionality. The proposed mechanism is focused on the service of SoC IP-cores under the control of the IEEE 1500 standard, which can be perceived positively by engineers in the EDA market. Objective. The purpose of the research is time- and energy-saving mechanisms for simulating malfunctions, such as addresses, by using read-write transactions of in-memory computing to build a test map of any functionality on smart data structures. Method. Smart data structures are represented by a logical vector and its derivatives in the form of truth tables and matrices. The test map is a matrix whose coordinates are determined by the combinations of all logical faults that are tested on the binary sets of the comprehensive test. The construction of the test map is focused on the architecture of in-memory computing based on read-write transactions, which makes the simulation mechanism economical in terms of simulation time and energy consumption due to the absence of a central processor. A logical vector as a single component of input data does not require synthesis into a technologically permitted structure of elements. Synthesis of smart data structures based on four matrix operations creates a fault test map like addresses for any logic. Results. Deductive matrix vectors are effectively used to model faults as addresses in digital structures of any configuration, including convergent branches and feedback loops. The resulting test map is used to find the minimum fault-checking test of the input variables. The proposed fault simulation mechanism technologically easily fits into the architecture of in-memory computing and uses only read-write transactions. The vector logic engine can also be used to test graph structures that are described by a truth table or a logical vector. The truth table addresses used for fault simulation are effectively used for processorless processing of large data in the in-memory computing architecture. Conclusions. Scientific novelty – a vector-logic in-memory computing mechanism for building a test map is proposed, characterized by the construction of intelligent data structures that reset the fault modeling algorithm. The proposed mechanism has no analogues in the design & test industry in terms of simplicity and predictability of data structure sizes and the absence of a test set modeling algorithm. The practical significance is determined by the application of the mechanism for testing logical functionalities of any complexity to solve verification tasks. Prospects of the research – increasing the object of diagnosis to the scheme, i.e. building a test map of the scheme logical structure.

A camera for scanning objects in motion based on an integrated detection system working in on-chip TDI mode

The paper presents an X-ray camera for testing moving objects. A typical application of such cameras is scanning products on an industrial production line. Currently, the most popular device detecting radiation in this type of camera consists of a pixel line based on scintillator detectors. Unfortunately, increasing its resolution automatically involves reducing the pixel size and reducing the signal-to-noise ratio. This is where the time domain integration method comes in handy, increasing the resolution without degrading the signal-to-noise ratio. The camera presented in this paper is based on an application-specific integrated circuit dedicated to this purpose. The application-specific integrated circuit core is a pixel matrix operating in a single-photon counting mode. Its architecture was designed to implement the time domain integration method and construct high-resolution cameras with a large scanning area. The article also describes the hardware and software of the application-specific integrated circuit readout system.

Linear Detection and Circuit Structure Based on ZF, MMSE Matrix Iteration and Operations

The Multiple-Input Multiple-Output (MIMO) technology substantially boosts the reliability and data transfer rates of communication links by enabling the parallel transmission of multiple data signals over the same frequency bandwidth, utilizing multiple transmit and receive antennas.Despite these enhancements, traditional MIMO systems are approaching their theoretical throughput limits, which underscores the need for developing new algorithms to tackle computational complexity.This paper categorizes the primary technical approaches into three types: iterative approximation methods, matrix operation-based methods, and the emerging Joint channel Estimation and Data detection (JED) algorithms.It delves into the Neumann series approximation for efficient matrix inversion in large-scale MIMO systems, the Gauss-Seidel iterative approach for soft-output detection, and LDLT decomposition-based algorithms for matrix inversion efficiency. Additionally, the paper emphasizes the JED algorithm, such as PROX, which presents a low-complexity solution for large Single-Input Multiple-Output (SIMO) wireless systems.

BIEM_CH: An Efficient Algorithm for Dynamic Rupture Simulation of Complex Fault Systems with Unstructured Meshes and Half-Space Green’s Function

Abstract In this study, a fast 3D dynamic rupture simulation algorithm, named BIEM_CH (Boundary Integral Equation Method for Complex fault systems in Half-space), is presented. This algorithm, based on exact half-space Green’s functions, supports both structured and unstructured discretization schemes, allowing for the effective handling of a wide range of fault geometries, from simple to complex. Because of the semianalytical nature of the boundary integral equation method, the integral kernel (surface integral of the spatial derivatives of Green’s functions) and rupture processes can be computed separately, making BIEM_CH particularly suitable for applications requiring numerous forward simulations, such as dynamic source inversion, for which only the rupture process needs to be recalculated once the integral kernel is obtained. The performance of the algorithm has been significantly enhanced, achieving up to a hundredfold speed increase through the use of exact closed-form solutions for the time-domain half-space Green’s function and matrix operations leveraging graphical processing unit acceleration, resulting in dynamic rupture simulations that can be completed in a matter of seconds. Moreover, BIEM_CH maintains excellent stability when the mesh dimension does not exceed 375 m, irrespective of whether structured or unstructured discretization schemes are used. In addition, this algorithm demonstrates good agreement with other methods in benchmark exercises conducted by the Southern California Earthquake Center and the U.S. Geological Survey’s dynamic rupture code verification project.

EIF-SlideWindow: Enhancing Simultaneous Localization and Mapping Efficiency and Accuracy with a Fixed-Size Dynamic Information Matrix

This paper introduces EIF-SlideWindow, a novel enhancement of the Extended Information Filter (EIF) algorithm for Simultaneous Localization and Mapping (SLAM). Traditional EIF-SLAM, while effective in many scenarios, struggles with inaccuracies in highly non-linear systems or environments characterized by significant non-Gaussian noise. Moreover, the computational complexity of EIF/EKF-SLAM scales with the size of the environment, often resulting in performance bottlenecks. Our proposed EIF-SlideWindow approach addresses these limitations by maintaining a fixed-size information matrix and vector, ensuring constant-time processing per robot step, regardless of trajectory length. This is achieved through a sliding window mechanism centered on the robot’s pose, where older landmarks are systematically replaced by newer ones. We assess the effectiveness of EIF-SlideWindow using simulated data and demonstrate that it outperforms standard EIF/EKF-SLAM in both accuracy and efficiency. Additionally, our implementation leverages PyTorch for matrix operations, enabling efficient execution on both CPU and GPU. Additionally, the code for this approach is made available for further exploration and development.

Mixed singular value and phase majorization inequalities for accretive matrices

In this paper, we give a number of mixed singular value and phase majorization inequalities for a class of accretive matrices. The first group of inequalities is for a single matrix, the second group is for sums of matrices, and the third group is for matrix operations induced by network connections. The results in this paper provide improvements to some existing related inequalities.

A Robust Initial Basic Subset Selection Method for Outlier Detection Algorithms in Linear Regression

The main motivation of this study is to develop an efficient algorithm for diagnosing and detecting outliers in linear regression up to a reasonable level of contamination. The algorithm initially obtains a robust version of the hat matrix at the linear algebra level. The basic subset obtained in the first stage is improved through concentration steps as defined in the fast-LTS (Least Trimmed Squares) regression algorithm. The method can be plugged into another algorithm as a basic subset selection state. The algorithm is effective against outliers in both X and Y directions by a rate of 25%. The complexity of the algorithm increases linearly with the number of observations and parameters. The algorithm is quite fast as it does not require iterative calculations. The success of the algorithm against a specific contamination level is demonstrated through simulations.

LINEAR-THETA METHOD FOR THE DISCRETIZATION AND NUMERICAL SOLUTION OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH MULTIPLE RETARDATIONS

This study presents a special case of proximal point algorithm for solving linear programming problem (LPP). This method, also known as the Alternating Direction Method of multipliers (ADMM), was deployed because of its strong convergence properties of the method of multipliers, the decomposability property of dual ascent and the potential to solve large- scale structured optimization problems. The update formulas for the LPP were derived from the associated augmented Lagrangian with the primal and dual residuals also derived for the convergence of the algorithm. The Game theory was re-structured into a LPP amenable to the ADMM. Prisoner’s Dilemma in Game theory was tested with the ADMM provided the matrix operator is invertible to guarantee its convergence. Other Numerical examples were also tested and it was discovered that the developed algorithm performs faster than the conventional simplex method.

Symplectic contact analysis of a horizontally laminated magneto-electro-elastic finite specimen

This work formulates a symplectic framework for contact analysis of a horizontally laminated, magneto-electro-elastic, finite specimen. By incorporating dual variables and deriving the Hamiltonian operator matrix within the symplectic space, the constraints on the state variables are fulfilled through the property of symplectic orthogonality. The state variables between different layers are connected via the transfer matrices, and the Hamiltonian mixed energy variational principle is generalized for the laminated medium. Distinct from traditional methods, the symplectic framework offers a more realistic and efficient analysis approach for a finite-sized specimen, applicable to indenters with arbitrary profiles but predetermined contact region. The interfacial and boundary effects are explored, with remarkably accurate results as compared to the finite element simulations. This research not only enhances our understanding of the underlying phenomena but also lays a foundation for materials characterization using high-throughput testing techniques.

Fault analysis and elimination of motor sensors in pure electric vehicles

Sensor as the main component of the motor drive system is to ensure its stable and reliable operation of the core components, so to drive the system's normal operation must learn how to analyse and diagnose the motor sensor fault accurately. This paper first introduces different fault classification and basic diagnosis methods, uses the CNN algorithm to collect sensor information, and uses matrix, dot product and other mathematical operations to make deep diagnosis. The sensor's real-time working status and accurate fault feedback can be obtained. CNN algorithm is fully suitable for the research and application of motor sensor fault diagnosis and can provide the basis for fault diagnosis according to the change of its own operating data parameters, and achieve efficient analysis and elimination of existing faults, to a certain extent, it can provide reference for the replacement of motor sensors and provide analysis for personnel engaged in related diagnosis industries.

Some properties of the q-numerical radius

In the given study, we consider the q-numerical radius ω q ( ⋅ ) of operator matrices defined on a direct sum of Hilbert spaces and investigate the various inequalities involving these values. We also prove that sup n ∈ N ω q ( A n ) ⩽ ω q ( ⨁ n = 1 + ∞ A n ) ⩽ | q | + 2 1 − | q | 2 | q | sup n ∈ N ω q ( A n ) for all the q ∈ C ∖ { 0 } , | q | ⩽ 1 , thereby extending the well known equality regarding the numerical radii that occurs when we plug in q = 1. Subsequently, we give explicit formulae for computing ω q ( ⋅ ) for some special cases of operator matrices. Finally, we establish some analytical properties of ω q ( ⋅ ) regarded as a function in q.

Train a real-world local path planner in one hour via partially decoupled reinforcement learning and vectorized diversity

Deep Reinforcement Learning (DRL) has exhibited efficacy in resolving the Local Path Planning (LPP) problem. However, its practical application remains significantly constrained due to its limited training efficiency and generalization capability. To address these challenges, we propose a solution termed Color, which includes an Actor-Sharer-Learner (ASL) training framework designed to improve efficiency, and a fast yet diverse simulator named Sparrow aimed at elevating both efficiency and generalization. Specifically, the ASL employs a Vectorized Data Collection (VDC) mode to enhance data collection, decouples the model optimization from data collection to expedite data consumption, and partially connects the two procedures with a Time Feedback Mechanism (TFM) to evade data underuse or overuse. Meanwhile, the Sparrow simulator utilizes a 2-Dimensional (2D) grid-based world, simplified kinematics, matrix operation, and conversion-free data flow to achieve a lightweight design. The lightness facilitates vectorized diversity, allowing for rapid and diversified simulation across numerous copies of the vectorized environments, thereby significantly enhancing both efficiency and generalization capacity. Comprehensive experiments demonstrate that with merely one hour of simulation training, Color achieves impressive arrival rates of 84% and 90% on 32 simulated and 42 real-world LPP scenarios, respectively. The code and video of this paper are accessible on our website. 11https://github.com/XinJingHao/Color.