Rectangular Matrix Research Articles

A reciprocity-aware, low-rank regularization for multidimensional deconvolution

A novel factorization-based, low-rank regularization method is introduced to solve multidimensional deconvolution (MDD) problems in the frequency domain. In this approach, each frequency component of the unknown wavefield is represented as a complex-valued square matrix and approximated as the product of a rectangular matrix and its transpose. The benefit of such a parameterization is twofold. First, each frequency matrix is guaranteed to be symmetric. From a physical standpoint, the recovered wavefield inherently adheres to the principle of reciprocity; in other words, the response to a receiver from a virtual source does not change when the two are swapped. Therefore, we refer to this low-rank parameterization as reciprocity-aware in that it respects the underlying physics of wave propagation associated with the MDD solution. Second, the size of the unknown matrix is greatly reduced compared with that of the original wavefield of interest (and halved compared with the existing factorization-based low-rank approximations). We further show that the associated objective function can be successfully optimized using the accelerated proximal gradient algorithm and present a robust strategy to define the initial guess of the solution. Numerical examples on synthetic and field data demonstrate the effectiveness of the proposed method in compressing the retrieved Green’s function while preserving its accuracy.

Research on the Influence of Matrix Shape on Percolation Threshold Values for Current Flow Conducted Using the Monte Carlo Simulation Method

In this study, in order to determine the effect of matrices’ shape on the percolation threshold values, computer simulations were performed using the Monte Carlo method for a 200 × 200 square-shaped matrix and rectangular matrices containing the same number of nodes as the square matrix. Based on the simulations, the average values of the percolation thresholds and standard deviations for the current flow along and across the matrices were determined. It was determined that for a square-shaped matrix, the average values of the percolation thresholds in both directions of the current flow were the same. Extending the rectangular matrix while reducing its height causes the average value of the percolation threshold in the direction of the current flow along the matrix to increase from 0.592740 to 0.759847, while in the transverse direction, it decreases from 0.592664 to 0.403614. The values of the classical asymmetry coefficients of the probability distributions of the percolation thresholds for both directions of the current flow were determined. Histograms of the probability distributions of the percolation threshold values for a square-shaped matrix and rectangular matrices were made and compared with the normal distributions. It was found that the occurrence of two percolation thresholds in rectangular layers should be considered when analyzing the electrical conductivity measurements of nanocomposite thin films.

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.

Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse

We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting an m×m positive definite matrix. The DOIA is a fast, convergent, iterative algorithm. Some properties and the estimation of residual error of the DOIA are given to prove the absolute convergence. Numerical tests demonstrate the usefulness of the double-optimal solution (DOS) and DOIA in solving square or nonsquare linear matrix equations and in inverting nonsingular square matrices. To speed up the convergence, a restarted technique with frequency m is proposed, namely, DOIA(m); it outperforms the DOIA. The pseudoinverse of a rectangular matrix can be sought using the DOIA and DOIA(m). The Moore–Penrose iterative algorithm (MPIA) and MPIA(m) based on the polynomial-type matrix pencil and the optimized hyperpower iterative algorithm OHPIA(m) are developed. They are efficient and accurate iterative methods for finding the pseudoinverse, especially the MPIA(m) and OHPIA(m).

Dual rectangular fuzzy complex matrix equations: extended solution, algebraic solution, solution and their calculating1

In this paper, we focus on generalized fuzzy complex numbers and propose a straightforward matrix method to solve the dual rectangular fuzzy complex matrix equations C · Z ˜ + L ˜ = R · Z ˜ + W ˜ , in which C and R are crisp complex matrices and Z ˜ , L ˜ and M ˜ are fuzzy complex number matrices. The existing methods for solving fuzzy complex matrix equations involve separately calculating the extended solution and the corresponding parameters of the real and imaginary parts, whereby we obtain the algebraic solution of the equations. By means of the interval arithmetic and embedding approach, the n × n dual rectangular fuzzy complex linear systems could be converted into 2n × 2n fuzzy linear systems, which are also equivalent to the 4n × 4n real linear systems. By directly solving the 4n × 4n real linear systems, the algebraic solutions can be obtained. The general dual rectangular fuzzy complex matrix equations and dual rectangular fuzzy complex linear systems are investigated by the generalized inverses of matrices. Finally, some examples are given to illustrate the effectiveness of method.

On the periodicity of singular vectors and the holomorphic block-circulant SVD on the unit circumference

We investigate the singular value decomposition of a rectangular matrix that is analytic on the complex unit circumference, which occurs, e.g., with the matrix of transfer functions representing a broadband multiple-input multiple-output channel. Our analysis is based on the Puiseux series expansion of the eigenvalue decomposition of analytic para-Hermitian matrices on the complex unit circumference. We study the case in which the rectangular matrix does not admit a full analytic singular value factorization, either due to partly multiplexed systems or to sign ambiguity. We show how to find an SVD factorization in the ring of Puiseux series where each singular value and the associated singular vectors present the same period and multiplexing structure, and we prove that it is always possible to find an analytic pseudo-circulant factorization, meaning that any arbitrary arrangements of multiplexed systems can be converted into a parallel form. In particular, one can show that the sign ambiguity can be overcome by allowing non-real holomorphic singular values.

The solvability of inhomogeneous boundary-value problems in Sobolev spaces

The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be overdetermined or underdetermined. They may contain derivatives, of the unknown vector-valued function, whose integer or fractional orders exceed the order of the differential equation. Similar problems arise naturally in various applications. The theory introduces the notion of a rectangular number characteristic matrix of the problem. The index and Fredholm numbers of this matrix coincide, respectively, with the index and Fredholm numbers of the inhomogeneous boundary-value problem. Unlike the index, the Fredholm numbers (i.e., the dimensions of the problem kernel and co-kernel) are unstable even with respect to small (in the norm) finite-dimensional perturbations. We give examples in which the characteristic matrix can be explicitly found. We also prove a limit theorem for a sequence of characteristic matrices. Specifically, it follows from this theorem that the Fredholm numbers of the problems under investigation are semicontinuous in the strong operator topology. Such a property ceases to be valid in the general case.

Randomized Block Kaczmarz Methods for Inner Inverses of a Matrix

In this paper, two randomized block Kaczmarz methods to compute inner inverses of any rectangular matrix A are presented. These are iterative methods without matrix multiplications and their convergence is proved. The numerical results show that the proposed methods are more efficient than iterative methods involving matrix multiplications for the high-dimensional matrix.

Fiedler Linearizations of Rectangular Rational Matrix Functions

Fiedler Linearizations of Rectangular Rational Matrix Functions

CONSTRUCTION OF A SET OF DIFFERENTIAL EQUATIONS SYSTEMSAND STABILITY IN THE VICINITY OF A PROGRAM MANIFOLD

The problem of constructing a entire set of differential equation systems for a given program manifold is addressed. Necessary and sufficient conditions have been compiled to ensure that the program manifold is integral to the systems being developed. These conditions form a rectangular linear algebraic system in relation to the required functions. Using the property of a rectangular matrix, the set of differential equation systems was constructed. Additionally, the problem of designing indirect automatic control systems with rigid feedback is explored. Since the given program is not always executed perfectly due to initial or ongoing disturbances, it is reasonable to require stability of the program manifold with respect to a certain function. This leads to the analysis of the stability of the system of equations in relation to the given program manifold. Through the construction of Lyapunov functions for the system in canonical form, sufficient conditions for the absolute stability of the program manifold are derived. These results can be applied to the design of stable automatic indirect control systems.

Stability and Ultimate Boundedness of Solutions of Certain Third Order Nonlinear Rectangular Matrix Differential Equations

We present in this paper the qualitative study of solutions of certain third order nonlinear matrix differential equations where the unknown function X is matrix-valued. The properties of solutions were investigated using Lyapunov’s direct method by employing the use of suitable Lyapunov functionals obtained from the differential equations describing the system satisfying certain requirements for establishing the stability and boundedness of solutions of the system considered. An example is given to demonstrate the significance of the results obtained as well as analysis through geometric graphs describing the dynamics of the system’s solutions. The results obtained are novel and will significantly enhance and extend the results of those mentioned in the literature.

Kalman Filter for Linear Discrete-Time Rectangular Singular Systems Considering Causality

This paper proposes a Kalman filter for linear rectangular singular discrete-time systems, where the singular matrix in the system is a rectangular matrix without full column rank. By using two different restricted equivalent transformation methods and adding the measurement equation to the state equation, the system is transformed into a square singular system satisfying regularity and observability. During this process, the causality of the system is taken into account, and multiple matrix transformations are applied accordingly. Based on these modifications, state estimation results are obtained using the Kalman filter. Finally, a numerical example is employed to demonstrate the effectiveness of our approach.

Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models

AbstractStandard multiparameter eigenvalue problems (MEPs) are systems of linear ‐parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time‐invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.

Risk Analysis of Potential Hazards by FMEA & Boston Rectangular Matrix in FF-Steel Mills

Risk Analysis of Potential Hazards by FMEA & Boston Rectangular Matrix in FF-Steel Mills

Determination of solutions for chorus excitation mechanism characteristic equation by means of Van Allen probe data analysis

In this paper, we focus on studying the quantitative characteristics of the excitation mechanism of chorus emissions closely related to the problem of relativistic electron flux formation in the radiation belts. The study is based on the processing of information accumulated by the Van Allen Probe spacecraft through the analysis of high-resolution data. We have chosen two typical examples of chorus emissions with spectral forms predominantly in the upper-frequency band (above half the electron cyclotron frequency) in the region of the local minimum of the magnetic field outside the plasmapause in the middle magnetosphere. We have developed and implemented a calculation algorithm that enables us to represent the results of wave field measurements in a high-resolution data channel in the form of a rectangular event matrix, each row of which corresponds to a cycle of the wave process. In the event matrix, we select the rows corresponding to fragments of chorus emissions that best characterize the natural source of short electromagnetic pulses. This method made it possible to determine the complex eigenvalues of the characteristic equation of the chorus emissions source. The proposed method for processing experimental information makes it possible to determine the main characteristics of the linear mechanism of chorus excitation. The results obtained in this way from observational data are in good agreement with relevant theory of chorus emissions excitation by amplifying noise electromagnetic pulses in a duct with a depleted density of a cold plasma.

Cache Optimization and Performance Modeling of Batched, Small, and Rectangular Matrix Multiplication on Intel, AMD, and Fujitsu Processors

Factorization and multiplication of dense matrices and tensors are critical, yet extremely expensive pieces of the scientific toolbox. Careful use of low rank approximation can drastically reduce the computation and memory requirements of these operations. In addition to a lower arithmetic complexity, such methods can, by their structure, be designed to efficiently exploit modern hardware architectures. The majority of existing work relies on batched BLAS libraries to handle the computation of many small dense matrices. We show that through careful analysis of the cache utilization, register accumulation using SIMD registers and a redesign of the implementation, one can achieve significantly higher throughput for these types of batched low-rank matrices across a large range of block and batch sizes. We test our algorithm on three CPUs using diverse ISAs – the Fujitsu A64FX using ARM SVE, the Intel Xeon 6148 using AVX-512, and AMD EPYC 7502 using AVX-2, and show that our new batching methodology is able to obtain more than twice the throughput of vendor optimized libraries for all CPU architectures and problem sizes.

Asymptotics of rectangular spherical integrals

In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as m,n go to infinity while m/n converges to a positive constant. We also improve and give a new proof of the asymptotics for the Harish-Chandra-Itzykson-Zuber integral derived in [37,38].

On computing modified moments for half-range Hermite weights

AbstractIn this paper, we consider the computation of the modified moments for the system of Laguerre polynomials on the real semiaxis with the Hermite weight. These moments can be used for the computation of integrals with the Hermite weight on the real semiaxis via product rules. We propose a new computational method based on the construction of the null-space of a rectangular matrix derived from the three-term recurrence relation of the system of orthonormal Laguerre polynomials. It is shown that the proposed algorithm computes the modified moments with high relative accuracy and linear complexity. Numerical examples illustrate the effectiveness of the proposed method.

Receiving a Response Image of a Solar Cell Efficiency by the Photo Scanning Method

The device for quality analysis of solar cells with different sensitive surface areas using radiation from a non-mechanical planar system is described. Current solutions for quality analysis of solar cells have disadvantages: work with analogy data, large dimensions of mechanical focusing system, interaction with small areas active zone of solar cells without possibility of identifying point pattern efficiency. Accordingly, a method is proposed for measuring the efficiency of solar cells at the production stage and during the operational period, which makes it possible to identify places of its insensitivity with the possibility of obtaining quantitative data on efficiency for the purpose of their mathematical processing. The developed photo scanning control device can be suitable for solar cells diagnostics with a sensitive zone area from 9 mm2 to A5 sheet size. The method makes it possible to measure local photocurrent and to find places of solar cell shunting. Process of diagnostics is controlled by one personal computer, which performs functions of controlling illumination of a monitor, setting diagnostic modes, obtaining data on the efficiency of the solar cells, storing and processing them. The device can be based on a laptop and is considered as mobile. Special software is used to manage, visualize and process the output data in a form of a rectangular matrix. An experiment was carried out on artificial darkening of the working areas of solar cells of different sensitivity. This confirmed effect of data changes and a visual display of obscuring zone on the output results. Tests were carried out on solar cells based on polycrystalline silicon. The results showed a high level of visibility and repeatability of the darkened area, and the quality of the data obtained made it possible to identify a small measurement error associated with external “noise”.

Data‐driven linear complexity low‐rank approximation of general kernel matrices: A geometric approach

SummaryA general, rectangular kernel matrix may be defined as where is a kernel function and where and are two sets of points. In this paper, we seek a low‐rank approximation to a kernel matrix where the sets of points and are large and are arbitrarily distributed, such as away from each other, “intermingled”, identical, and so forth. Such rectangular kernel matrices may arise, for example, in Gaussian process regression where corresponds to the training data and corresponds to the test data. In this case, the points are often high‐dimensional. Since the point sets are large, we must exploit the fact that the matrix arises from a kernel function, and avoid forming the matrix, and thus ruling out most algebraic techniques. In particular, we seek methods that can scale linearly or nearly linearly with respect to the size of data for a fixed approximation rank. The main idea in this paper is to geometrically select appropriate subsets of points to construct a low rank approximation. An analysis in this paper guides how this selection should be performed.