Matrix Operations Research Articles

Reasoning with the inverse of 3D cardinal direction relations based on direction matrices

In order to improve the ability of intelligent reasoning and prediction of 3DR27 model which presented in our previous work, enhance the usability of this model, and better fulfill the demands of real applications for spatial database, we focused on the problem of reasoning with the inverse of 3D cardinal direction relations between spatial objects. In order to realize automated reasoning, an algorithm for computing the inverse of 3D cardinal direction relations based on matrices is proposed on the basis of the mapping between the 3D rectangular cardinal direction relations and 3D interval relation matrix. This algorithm improves the power of reasoning for this model by means of the excellent properties of matrix operations. Theorems are provided to prove formally that our algorithm is correct and complete. This study realized the automatic inference and calculation of the inverse of the 3D cardinal direction relations based on 3DR27 model and further improved the ability of spatial reasoning and spatial analysis of spatial database.

Adventitious Error and Its Implications for Testing Relations Between Variables and for Composite Measurement Outcomes

Wu and Browne (Psychometrika 80(3):571–600, 2015. https://doi.org/10.1007/s11336-015-9451-3; henceforth W &B) introduced the notion of adventitious error to explicitly take into account approximate goodness of fit of covariance structure models (CSMs). Adventitious error supposes that observed covariance matrices are not directly sampled from a theoretical population covariance matrix but from an operational population covariance matrix. This operational matrix is randomly distorted from the theoretical matrix due to differences in study implementations. W &B showed how adventitious error is linked to the root mean square error of approximation (RMSEA) and how the standard errors (SEs) of parameter estimates are augmented. Our contribution is to consider adventitious error as a general phenomenon and to illustrate its consequences. Using simulations, we illustrate that its impact on SEs can be generalized to pairwise relations between variables beyond the CSM context. Using derivations, we conjecture that heterogeneity of effect sizes across studies and overestimation of statistical power can both be interpreted as stemming from adventitious error. We also show that adventitious error, if it occurs, has an impact on the uncertainty of composite measurement outcomes such as factor scores and summed scores. The results of a simulation study show that the impact on measurement uncertainty is rather small although larger for factor scores than for summed scores. Adventitious error is an assumption about the data generating mechanism; the notion offers a statistical framework for understanding a broad range of phenomena, including approximate fit, varying research findings, heterogeneity of effects, and overestimates of power.

An approaching method based on integral for linear neutral delay differential equations by using Hermite polynomials

AbstractThis article presents a approximation method for linear neutral delay differential equations using Hermite polynomials. The method ensures the unknown function to be found by approximating the first derivative with the help of the finite Hermite series. This method reduces the problem to a linear algebraic system using the first derivative approach, matrix operations and collocation points. Also error of the solution is estimated by constructing an error problem. Numerical examples are solved to explain the method and error estimation technique. The results demonstrate the effectiveness and accuracy of the current study. Calculations have been made using MATLAB.

An efficient quantum algorithm for independent component analysis

Independent component analysis (ICA) is a fundamental data processing technique to decompose the captured signals into as independent as possible components. Computing the contrast function, which serves as a measure of the independence of signals, is vital and costs major computing resources in ICA. This paper presents a quantum algorithm that focuses on computing a specified contrast function on a quantum computer. Using the quantum acceleration in matrix operations, we efficiently deal with Gram matrices and estimate the contrast function with the complexity of O(ϵ1−2polylog⁡(N/ϵ1)) . This estimation subprogram, combined with the classical optimization framework, builds up our ICA algorithm, which exponentially reduces the complexity dependence on the data scale compared with algorithms using only classical computers. The outperformance is further supported by numerical experiments, while our algorithm is then applied for the separation of a transcriptomic dataset and for financial time series forecasting, to predict the Nikkei 225 opening index to show its potential application prospect.

Fast Sparse Discriminative K-Means for Unsupervised Feature Selection.

Embedded feature selection approach guides subsequent projection matrix (selection matrix) learning through the acquisition of pseudolabel matrix to conduct feature selection tasks. Yet the continuous pseudolabel matrix learned from relaxed problem based on spectral analysis deviates from reality to some extent. To cope with this issue, we design an efficient feature selection framework inspired by classical least-squares regression (LSR) and discriminative K-means (DisK-means), which is called the fast sparse discriminative K-means (FSDK) for the feature selection method. First, the weighted pseudolabel matrix with discrete trait is introduced to avoid trivial solution from unsupervised LSR. On this condition, any constraint imposed into pseudolabel matrix and selection matrix is dispensable, which is significantly beneficial to simplify the combinational optimization problem. Second, the l2,p -norm regularizer is introduced to satisfy the row sparsity of selection matrix with flexible p . Consequently, the proposed FSDK model can be treated as a novel feature selection framework integrated from the DisK-means algorithm and l2,p -norm regularizer to optimize the sparse regression problem. Moreover, our model is linearly correlated with the number of samples, which is speedy to handle the large-scale data. Comprehensive tests on various data terminally illuminate the effectiveness and efficiency of FSDK.

Bernstein Operational Matrix for Solving Variable Order Fractional Differential Equations with Generalized Caputo-Type

In this research, we present novel generalized derivative founded on the newly constructed (NGCFVODs) novel generalized Caputo Fractional variable order derivatives. Utilizing these operators, a numerical methodology has been formulated to resolve the Fractional Variable Order Differential Equations (FVODEs). We estimate the solution for (FVODEs) by employing bernstein polynomials as foundational vectors. We have further expanded the derivative operational matrix of bernstein Polynomials (bPs) to generalized derivative an operational matrix in sense of (NGCFVODs). The efficiency of developed numerical methodology is tested by a taking a various test examples. We also a compare results of our suggested approach with the methodologies existing in academic papers. In this study the Fractional variable order differential operator of new generalized Caputo was described by three categories: (i) various value in ρ and Fractional variable order a parameter, (ii) various value in a fractional parameters while Fractional variable order and ρ parameter are fixed, and (iii) various value in Fractional variable order parameter controlling fractional and ρ parameter.

A COMPARATIVE STUDY OF A CLASS OF LINEAR AND NONLINEAR PANTOGRAPH DIFFERENTIAL EQUATIONS VIA DIFFERENT ORTHOGONAL POLYNOMIAL WAVELETS

We propose a wavelet approach on different orthogonal polynomials for solving linear and nonlinear pantograph equations with stretch kind. The pantograph differential equation is a unique proportional delay functional differential equation class. It has been used to deal with numerous physics, mathematics, and engineering applications, such as quantum mechanics, control systems, electrodynamics, and number theory. This scheme is based on constructing the operational matrix for integration via different wavelets with their collocation nodes. This study aims to examine the numerical dynamics of the pantograph equation under stretch kind through different orthogonal polynomial wavelets. Illustrative examples are presented to highlight the flexibility of this scheme, and comparisons are made between the mentioned scheme and other existing schemes using tables and graphs. These numerical results correctly predict the applicability and effectiveness of the mentioned scheme.

An optimal sensor layout method based on noise reduction estimation for active road noise control

Active noise control (ANC) is an effective method to suppress vehicle road noise. The rapid selection of the reference sensor set and rational selection of its layout position are crucial for determining the vibration noise transmission paths with high contributions, which determines the noise reduction performance of the active road noise control (ARNC) system. The reference sensor set selected by the conventional multiple coherence function (MCOH) method does not always ensure the optimal noise reduction performance and the matrix singularity problem based on the Fisher information matrix (FIM) expansion method. To alleviate this problem, this paper proposes a method based on the overall level of noise reduction (OANR) and the FIM to select a set of reference sensors and determine their positions. This method uses an optimal causally constrained Wiener filter to estimate the theoretical OANR for each control area to ensure the reference sensor selected has the optimal noise reduction performance, and it expands efficiently based on the FIM method to select a set of reference sensors. Besides, the matrix singularity problems in the FIM-based expansion method are solved using noise frequency summation and matrix operation, which makes the method more universally applicable. Numerical simulations are performed to analyze and evaluate the noise reduction performance and selection efficiency of the proposed method under different conditions. In addition, a series of real vehicle ANC experiments are conducted. The results show that the reference sensor set selected based on the proposed method has an obvious advantage in noise reduction performance, and only four reference sensors are needed to achieve 4.63 dB(A) noise reduction in ARNC experiments. It fully confirms the effectiveness of the proposed method in real vehicle applications.

Cost-effectiveness analysis of optimal control strategies on the transmission dynamics of HIV and Varicella-Zoster co-infection

HIV and Varicella-Zoster co-infection has been threatens the lives and livelihood of millions of individuals all over the world. In various nations of the world the co-infection disease is presently a major health problem. Although, various researchers carried out studies related with HIV and Zoster single infections, no one has carried out a study on the co-infection disease. Therefore, investigating the HIV and Varicella-Zoster co-infection transmission dynamics using mathematical modeling approach is relatively necessary. The main objective of this study is to investigate the cost-effective analysis of the proposed optimal control strategies on the transmission dynamics of HIV and Varicella-Zoster co-infection in the community using compartmental modeling approach. Qualitatively, we have computed the model equilibrium points and analyze their local stabilities using Routh-Hurwitz stability criteria, the effective reproduction numbers by using the next generator matrix operator method, we re-formulated the optimal control problem of the complete co-infection model by implementing five time dependent optimal control strategies, analyzed the optimal control problem by applying the Pontryagin's Maximum/Minimum principle. Sensitivity analysis and numerical simulations have been carried out using the forward sensitivity index formula and the fourth order Runge-Kutta numerical methods respectively and verified the analytic results. Finally, we have performed the cost effective analysis for the proposed controlling strategies and from the result we have observed that implementing strategy four (treatment of Varicella-Zoster infected individuals to increase their recovery rate and recovery period) is the most cost effective-strategy as compared with other alternatives.

Neural networks for critical load factors of unstiffened plates according to the reduced stresses method of EN 1993-1-5

The critical load factor resulting from the complete stress field is the main input parameter for the verification against plate buckling according to the reduced stresses method (RSM) from Eurocode EN 1993-1-5. Currently, the code provides just a simple and often conservative interaction formula with a limited application range. Realistic values can only be obtained by numerical methods, like the FEM, which is tedious for many load cases. To enhance the ease of use of the RSM, simple-to-use but accurate methods are needed, which can be automated and implemented into EN 1993-1-5. The current research deals with this problem by applying artificial neural networks (ANN) to predict the critical load factors of unstiffened plates under longitudinal, shear, and patch loading stresses, including their interaction behaviour. ANNs were used because approximated solutions with polynomials (as actual in the standard) are not suitable due to the complexity of the problem. The developed ANNs are highly accurate (as shown by an independent verification), can be implemented into the Eurocode, and can easily be used, even without in-depth programming skills. Moreover, a simple Python code is provided to demonstrate the ease of use. Still, the code can also be implemented into other software packages (like Excel), as only basic matrix operations are needed. The interaction surfaces are visualized, gaining a deeper understanding of plates under multiaxial loading situations. Finally, recommendations are given for generating ANNs for plate buckling problems.

A Highly Accurate Computational Approach to Solving the Diffusion Equation of a Fractional Order

This study aims to present and apply an effective algorithm for solving the TFDE (Time-Fractional Diffusion Equation). The Chebyshev cardinal polynomials and the operational matrix for fractional derivatives based on these bases are relied on as crucial tools to achieve this objective. By employing the pseudospectral method, the equation is transformed into an algebraic linear system. Consequently, solving this system using the GMRES method (Generalized Minimal Residual) results in obtaining the solution to the TFDE. The results obtained are very accurate, and in certain instances, the exact solution is achieved. By solving some numerical examples, the proposed method is shown to be effective and yield superior outcomes compared to existing methods for addressing this problem.

An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations

In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme.

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

APPLICATIONS OF MATRICES IN COMPUTER GRAPHICS

In this paper, the usage of matrix in computer graphics is shown. A brief overview of geometric transformations in computer graphics is given. The "Matrix - Computer Graphics" application software is created for the representation and easier understanding of relations between geometric transformations and matrix calculus. We introduce and explore the basic matrix operation such as translation, rotation, scaling, reflection, shearing in 2D and 3D and at end we explain about the parallel and perspective view. These concepts commonly appear in video game graphics. The work has also shown some programming in using C/C++ code.

Non‐iterative stress‐displacement solution for surrounding rock mass with multiple non‐circular deep tunnels

AbstractA non‐iterative stress‐displacement solution is proposed in this paper to investigate surrounding rock mass with multiple non‐circular deep tunnels. Compared with the Schwarz alternating method, the proposed method only spends one‐time matrix operation, showing considerable efficiency and accuracy of the matrix solution. First, a matrix solution is formulated to obtain the conformal mapping function for the non‐circular tunnel. Then, in consideration of multiple boundary conditions, the generalized complex variable theory and fast Fourier transform (FFT) algorithm are used to formulate a matrix solution of potential functions. The stress and displacement fields around multiple non‐circular tunnels can be further determined from the matrix solution. A series of numerical examples are conducted to verify the proposed method, perform the convergency study, and discuss the effects of tunnel geometries, distances, and arrangements. The convergency study shows that the more accurate conformal mapping function requires more sampling points in the FFT. In addition, the high stress zone would be amplified if the arrangement of multiple tunnels makes the high stress zone adjacent or overlapped.

Air volume reconstruction and sensor optimization distribution in building intelligent ventilation network

Ensuring the accuracy and reliability of ventilation parameter monitoring is pivotal for the development of intelligent ventilation systems. To attain a visual representation of airflow, solving the challenge of airflow reconstruction necessitates the strategic use of a limited number of sensors. In addressing these concerns, this article introduces an optimization approach for ventilation airflow leveraging the Breadth-First Search (BFS) algorithm. Additionally, it proposes an optimization distribution method for mine ventilation sensors, grounded in the Independent Cut Set algorithm. Research has found that compared to the traditional PSO algorithm, the BFS algorithm produces a higher optimal air volume solution when optimizing the air volume; Comparatively, the proposed algorithm exhibits significantly shorter average running times than the Particle Swarm Optimization (PSO) algorithm. It boasts the highest average convergence rate, ensuring superior accuracy, and possesses a notable capability to escape local minima, facilitating the acquisition of optimal solutions. Leveraging the independent cut set algorithm optimizes the calculation process through matrix operations. Exploiting the properties of matrices allows for a more rapid and intuitive resolution of sensor localization problems.

Optimizing pantograph fractional differential equations: A Haar wavelet operational matrix method

In this study, we developed an operational matrix method of integration using Haar wavelets to solve both linear and nonlinear pantograph fractional differential equations by taking Atangana's beta derivative. The proposed method utilizes an operational matrix and truncated Haar wavelet series to transform linear and nonlinear problems into objective functions. These objective functions are then minimized using differential evolution optimization. Illustrative examples demonstrate convergence and validation of the proposed method. The proposed method yields exceptional numerical outcomes in terms of competitiveness and accuracy when compared to exact solutions. In addition, performance metrics, including the mean absolute deviation, root mean squared error, P2norm, P∞ norm, Theil's inequality coefficient, variance account for, coefficient of determination, and Nash-Sutcliffe efficiency, were calculated for a varying number of collocation points. The results indicate that the Haar wavelet operational matrix method is a straightforward and efficient method for solving pantograph fractional differential equations.

On coupled semilinear evolution systems: Techniques on fractional powers of 4×4$4\times 4$ matrices and applications

AbstractIn this paper, we provide several techniques to explicitly calculate fractional powers of operator matrices focusing on creating a theory that can be applied to distinct situations. To illustrate the abstract results developed, we consider its application in systems of coupled reaction–diffusion equations and in (strongly damped) wave equations. We also discuss how these techniques can be applied to higher order matrices and we specifically calculate the fractional powers of a operator matrix associated to a weakly coupled system of wave equation. In addition, we deal with the applicability of this analysis with respect to solvability, stabilization, regularity, smooth dynamics, and connection with evolutionary classic equation and its fractional counterpart.

Top-[formula omitted] discriminative feature selection with uncorrelated and [formula omitted]-norm equation constraints

Supervised feature selection (FS) as an interpretable dimensionality reduction technique has received increasing attention, where linear discriminative analysis (LDA)-based method can select informative features discriminatively and obtain promising performance. When original data has more features than samples, however, LDA-based method generally encounters degradation since the appearance of irreversible scatter matrix. This situation is called the small sample size (SSS) problem. To overcome it and enhance the discriminant power of selected feature subsets, in this paper, we design an elegant LDA-based FS model referred to as Top-k Discriminative FS (TDFS), which is constructed by seamlessly integrating the ℓ2,0-norm equation constraint into uncorrelated LDA model. More concretely, the ℓ2,0-norm equation constraint can explicitly characterize the number of selective features k to ensure the sparsity of projected matrix and select top features. The uncorrelated LDA model aims to improve discriminative ability based on uncorrelated data in projected subspace. Given the formidable nature of solving this non-convex model, a novel optimization algorithm is further developed and the SSS problem can be efficaciously addressed during the optimization process. We first decompose projection matrix into a discrete selection matrix and its corresponding nonzero projection matrix, then concurrently optimize above two matrices by employing a column-by-column update scheme, during which the reversibility of scatter matrix in selective feature subspace can be easily guaranteed to solve SSS problem. The extensive experiments on four synthetic data sets and eight real-world data sets show that the proposed method outperforms eight competitors validated by three classifiers. Moreover, although the theoretical analysis proves that our algorithm has quartic time complexity on the number of selected features k, the running time experiments verify that TDFS is still efficient and applicable in scenarios where only a small number of features need to be selected. From above perspectives, our algorithm shows desirable performance to achieve discriminative FS.

Dual space-based fuzzy graphs and orthogonal basis clustering for unsupervised feature selection

Unsupervised feature selection (UFS) takes an important position because gaining the class labels is laborious or even impossible. In the domain of UFS, clustering is a major means to exploit label information. The existing methods either could not model a clear clustering structure or could not utilize the local structure of data. Consequently, a new clustering method in combination with graph learning is proposed in this work. Specifically, for clustering, orthogonal basis clustering is introduced, where orthogonal constraints are imposed on the cluster center matrix and the clustering matrix. The clustering indicator matrix is also imposed by a non-negative constraint. A clearer clustering structure and more independent clustering centers are obtained through these constraints. For local preservation, given that traditional graphs for keeping the local manifold are faced with the problem of imbalanced neighbors, the fuzzy graph is introduced to acquire a robust structure, which is applied to both data space and feature space. The topological structure in these spaces can be well maintained. For the choice of salient features, ℓ2,0-norm regularization is imposed on the projection matrix. The object function is solved alternately. Then, a feature selection algorithm is designed. Experiments are designed and performed on nine real-world data sets. The results attest to the effectiveness of the proposed algorithm compared with other relevant algorithms.