Generalized Inverse Of Matrix Research Articles

Bivariate matrix-valued Newton-Thiele-type mixed rational interpolants

By means of Newton’s difference quotient and Thiele type continued fraction combined with the generalized inverse matrix, we present a kind of bivariate matrix-valued blending difference quotient. On this basis, we construct the bivariate Newton-Thiele type rational interpolants and give its iterative algorithm.

Study on Equivalent Distributed Stress Determination Technique based on Generalized Matrix Inversion (Part 1) －Analysis of Two-dimensional Edge Cracks based on the Center Cracked Plate’s Weight Function－

Study on Equivalent Distributed Stress Determination Technique based on Generalized Matrix Inversion (Part 1) －Analysis of Two-dimensional Edge Cracks based on the Center Cracked Plate’s Weight Function－

GMDS-ZNN Model 3 and its Ten-Instant Discrete Algorithm for Time-Variant Matrix Inversion Compared With Other Multiple-Instant Ones

The online time-variant matrix inversion problem has attracted extensive attention and study, because of its considerable appearance and application in scientific research and industrial production. For various control optimization problems, the demand for the high-precision and rapid-convergence of matrix inversion algorithm is increasing. It remains an ongoing challenge due to the rigorous requirements of precision and convergence of the algorithm. In this paper, on the basis of our previous works, by using Zhang neural network (ZNN) method, a continuous time-variant matrix inversion model, which is also a Getz-Marsden dynamic system (i.e., GMDS-ZNN model 3), is proposed. Besides, a general ten-instant Zhang et al discretization (ZeaD) formula is presented and investigated, with corresponding theoretical results being provided. Next, by applying this general formula to discretize the continuous time-variant matrix inversion model, a general ten-instant discrete time-variant matrix inversion (DTVMI) algorithm with the sixth-order precision is proposed. For comparison, four other DTVMI algorithms, with the second-, third-, fourth-, and fifth-order precisions, are also proposed and presented, respectively, by using other ZeaD formulas to discretize the continuous time-variant matrix inversion model. Besides, for the situation of coefficient matrix derivative being unknown, we provided the formula of estimating it with the fifth-order precision. With the help of the proposed matrix derivative estimation formula, the actual application field of GMDS-ZNN model 3 is expanded evidently. Finally, theoretical analyses and simulation experiment results highlight the effectiveness and accuracy of the proposed GMDS-ZNN model 3 and DTVMI algorithms.

LU Decomposition method to solve bipolar fuzzy linear systems

The aim of this work is to solve the linear system of equations using LU decomposition method in bipolar fuzzy environment. We assume a special case when the coefficient matrix of the system is symmetric positive definite. We discuss this point in detail by giving some numerical examples. Moreover, we investigate m × n inconsistent bipolar fuzzy matrix equation and find the least square solution of the inconsistent bipolar fuzzy matrix using the generalized inverse matrix theory. The existence of the strong bipolar fuzzy least square solution of the inconsistent bipolar fuzzy matrix is discussed. In the end, a numerical example is presented to illustrate our proposed method.

The specificities of the research in the economics’ development of Sevastopol

A comprehensive study of regional processes implies a qualitative analysis of indicators over time, which is necessary not only to identify current trends, but also to make forecasts that are used in the development of regional development strategies and programs. In order to study the development of the city of Sevastopol on the basis of statistical data on the state of the economy in the Ukrainian and Russian periods, as well as determine the possibility of their use for making forecasts, it is necessary to solve the issue of homogeneity of the series of economic indicators. The existing criteria for verifying the homogeneity of data are not applicable to the solution of the issue of compatibility of multidimensional series belonging to different time intervals. The article proposes the use of exploratory factor analysis to solve this problem. However, the lack of statistical data leads to a degeneration of the matrix of pairwise correlations of economic indicators. To obtain estimates of the parameters of the factor model, a generalized inverse matrix is used, which is obtained as a result of a matrix iterative procedure. Exploratory factor models for the Ukrainian and Russian periods of Sevastopol have fundamental differences, and the corresponding multidimensional series cannot be combined for a holistic study of economic processes in the region.

Extension of the Measurable Wavelength Range for a Near-Infrared Spectrometer Using a Plasmonic Au Grating on a Si Substrate.

In this paper, we proposed near-infrared spectroscopy based on a Si photodetector equipped with a gold grating and extended the measurable wavelength range to cover 1200–1600 nm by improving a spectrum derivation procedure. In the spectrum derivation, photocurrent data during alteration of the incidence angle of the measured light were converted using a responsivity matrix R, which determines the spectroscopic characteristics of the photodetector device. A generalized inverse matrix of R was used to obtain the spectrum and to fit a situation where multiple surface plasmon resonance (SPR) peaks appeared in the scanning range. When light composed of two wavelengths, 1250 nm and 1450 nm, was irradiated, the two wavelengths were distinctively discriminated using the improved method.

Applying a Unit-Consistent Generalized Matrix Inverse for Stable Control of Robotic Systems

Abstract It is well understood that the robustness of mechanical and robotic control systems depends critically on minimizing sensitivity to arbitrary application-specific details whenever possible. For example, if a system is defined and performs well in one particular Euclidean coordinate frame then it should be expected to perform identically if that coordinate frame is arbitrarily rotated or scaled. Similarly, the performance of the system should not be affected if its key parameters are all consistently defined in metric units or in imperial units. In this paper we show that a recently introduced generalized matrix inverse permits performance consistency to be rigorously guaranteed in control systems that require solutions to underdetermined and/or overdetermined systems of equations. We analyze and empirically demonstrate how these theoretical guarantees can be directly obtained in a practical robotic arm system.

Stochastic response determination of structural systems modeled via dependent coordinates: a frequency domain treatment based on generalized modal analysis

Generalized independent coordinates are typically utilized within an analytical dynamics framework to model the motion of structural and mechanical engineering systems. Nevertheless, for complex systems, such as multi-body structures, an explicit formulation of the equations of motion by utilizing generalized, independent, coordinates can be a daunting task. In this regard, employing a set of redundant coordinates can facilitate the formulation of the governing dynamics equations. In this setting, however, standard response analysis techniques cannot be applied in a straightforward manner. For instance, defining and determining a transfer function within a frequency domain response analysis framework is challenging due to the presence of singular matrices, and thus, the machinery of generalized matrix inverses needs to be employed. An efficient frequency domain response analysis methodology for structural dynamical systems modeled via dependent coordinates is developed herein. This is done by resorting to the Moore–Penrose generalized matrix inverse in conjunction with a recently proposed extended modal analysis treatment. It is shown that not only the formulation is efficient in drastically reducing the computational cost when compared to a straightforward numerical evaluation of the involved generalized inverses, but also facilitates the derivation and implementation of the celebrated random vibration input–output frequency domain relationship between the excitation and the response power spectrum matrices. The validity of the methodology is demonstrated by considering a multi-degree-of-freedom shear type structure and a multi-body structural system as numerical examples.

On the Relative Gain Array (RGA) with singular and rectangular matrices

This paper identifies a significant deficiency in the literature on the application of the Relative Gain Array (RGA) formalism in the case of singular matrices. Specifically, it is shown that the conventional use of the Moore–Penrose pseudoinverse is inappropriate because it fails to preserve critical properties that can be assumed in the nonsingular case. It is then shown that such properties can be rigorously preserved using an alternative generalized matrix inverse.

A Rank-Preserving Generalized Matrix Inverse for Consistency With Respect to Similarity

There has recently been renewed recognition of the need to understand the consistency properties that must be preserved when a generalized matrix inverse is required. The most widely known generalized inverse, the Moore-Penrose pseudoinverse, provides consistency with respect to orthonormal transformations (e.g., rotations of a coordinate frame), and a recently derived inverse provides consistency with respect to diagonal transformations (e.g., a change of units on state variables). Another well-known and theoretically important generalized inverse is the Drazin inverse, which preserves consistency with respect to similarity transformations. In this paper we note a limitation of the Drazin inverse is that it does not generally preserve the rank of the linear system of interest. We then introduce an alternative generalized inverse that both preserves rank and provides consistency with respect to similarity transformations. Lastly we provide an example and discuss experiments which suggest the need for algorithms with improved numerical stability.

Concentration of the Frobenius Norm of Generalized Matrix Inverses

In many applications it is useful to replace the Moore-Penrose pseudoinverse (MPP) by a different generalized inverse with more favorable properties. We may want, for example, to have many zero entries, but without giving up too much of the stability of the MPP. One way to quantify stability is by how much the Frobenius norm of a generalized inverse exceeds that of the MPP. In this paper we derive finite-size concentration bounds for the Frobenius norm of $\ell^p$-minimal general inverses of iid Gaussian matrices, with $1 \leq p \leq 2$. For $p = 1$ we prove exponential concentration of the Frobenius norm of the sparse pseudoinverse; for $p = 2$, we get a similar concentration bound for the MPP. Our proof is based on the convex Gaussian min-max theorem, but unlike previous applications which give asymptotic results, we derive finite-size bounds.

A structural morphogenesis method based on a linkage mechanism system

This paper presents an elements-grouped morphogenesis method for the design of grid shells based on finding the linkage mechanism system shape that corresponds to the state of minimum potential energy. This method assigns the elements to several groups according to architectural design intentions and requirements to optimize a structural shape. A shape transformation equation is derived to constrain the total length of elements in each element group in the morphogenesis process and the generalized inverse matrix theory is then used to resolve the equation. The positions of nodes are, thus, progressively updated until the system reaches the minimum potential energy state. This method is characterized by the settings of element groups, temporary elements, and temporary forces. Finally, several numerical examples illustrate the characteristics and effectiveness of the proposed method.

Reduction of residual noise based on eigencomponent filtering for speech enhancement

In this paper, residual noise of corrupted speech observations is further restrained based on eigencomponent (an eigenvalue and its corresponding eigenvector) filtering. Three relevant algorithms are proposed to obtain the core eigencomponents that deeply affect enhancement quality of speech fragments by joint diagonalization of clean speech and noise covariance matrix. In addition, the generalized inverse matrix transform is introduced to the recovery of enhanced speech signal for the issue of matrix irreversibility after eigencomponents are filtered. Experiment results show that the proposed methods work better than many other methods under various conditions on both noise reduction and speech distortion.

Exploration of Deep Magnetite Deposit Under Thick and Conductive Overburden with Ex Component of SOTEM: A Case Study in China

In northern China, thick loess strata with low resistivity are widely present and can shield the deep induced electromagnetic field signal and influence the accuracy of deep targets while electromagnetic prospecting is being performed. To achieve deep target exploration with high resolution, a new technique known as the short offset transient electromagnetic method (SOTEM) is applied in this research area. This method can be used to perform near-source configuration surveys, but shows no obvious difference between the deep magnetite deposit and low-resistivity layer in this survey area. Therefore, it cannot explore the magnetite deposit directly using the traditional magnetic component (Hz) of SOTEM. The Ex component, which can more effectively distinguish the high-resistivity surrounding rock and overlying low-resistivity layer than the Hz component, is used for exploration. First, 2D forward modeling and sensitivity analysis of the high-resistivity body are carried out to determine the different responses of the Ex and Hz components, respectively. Moreover, to better determine the interface between the surrounding rock and low-resistivity layer, 1D Occam inversion and the generalized inverse matrix inversion method are used. Furthermore, aiming at the static effect of the Ex component in field exploration, joint inversion of the Ex and Hz components is carried out for comprehensive interpretation. Finally, a field test is conducted, and the drilling results are used for verification. The research results can provide an effective geophysical model for the exploration of contact metasomatic magnetite deposits.

Generalized inverse matrix-exterior penalty function (GIM-EPF) algorithm for data processing of multi-wavelength pyrometer (MWP).

The unknown emissivity of materials is a huge obstacle in multi-wavelength pyrometry (MWP). It leads to a set of ill-posed equations that cannot be directly inverted to obtain the true temperature from a set of multi-wavelength measurements. Constraint optimization algorithms such as the gradient projection (GP) and internal penalty function (IPF) algorithms provide solutions without any emissivity model assumptions but require a narrow fixed emissivity range and an appropriate initial emissivity input value, otherwise, accuracy and computational efficiency are greatly affected. Here, we propose a generalized inverse matrix-exterior penalty function (GIM-EPF) algorithm to realize an efficient and accurate inversion without limiting the emissivity range in advance. First, a set of emissivities is obtained by the generalized inverse matrix method; these emissivities are used as initial values in the exterior penalty function iteration algorithm, from which temperature and spectral emissivity are obtained. Simulation results show that the GIM-EPF algorithm provides results superior to IPF, especially in computational efficiency. The proposed GIM-EPF method is 8 times faster than the IPF method with a 0.56% relative error at the 1800 K true temperature. The GIM-EPF method also allows near real-time diagnosis of rocket exhaust flame temperature. Rocket nozzle temperature experiment results show that the temperatures derived by the GIM-EPF algorithm agree well with the theoretical design temperature and the IPF inversion temperature.

The Moore–Penrose Dual Generalized Inverse Matrix With Application to Kinematic Synthesis of Spatial Linkages

The paper initially reports about the properties of an expression of dual generalized inverse matrix currently available in the literature. It is demonstrated that such a matrix does not fulfill all the Penrose conditions. Hence, novel and computationally efficient algorithms/formulas for the computation of the Moore–Penrose dual generalized inverse (MPDGI) are herein proposed. The paper also contains a new algorithm for the singular value decomposition (SVD) of a dual matrix. The availability of these formulas allows the simultaneous solution of overdetermined systems of dual linear equations without requiring the traditional separation in primal and dual parts. This should prove useful for the solution of many kinematic problems. The algorithms/formulas herein deduced have been also tested on the kinematic synthesis of the constant transmission ratio RCCC spatial linkage.

A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals

Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula.

An improved EMD method for modal identification and a combined static-dynamic method for damage detection

Empirical mode decomposition (EMD) is a highly adaptable signal processing method. However, the EMD approach has certain drawbacks, including distortions from end effects and mode mixing. In the present study, these two problems are addressed using an end extension method based on the support vector regression machine (SVRM) and a modal decomposition method based on the characteristics of the Hilbert transform. The algorithm includes two steps: using the SVRM, the time series data are extended at both endpoints to reduce the end effects, and then, a modified EMD method using the characteristics of the Hilbert transform is performed on the resulting signal to reduce mode mixing. A new combined static-dynamic method for identifying structural damage is presented. This method combines the static and dynamic information in an equilibrium equation that can be solved using the Moore-Penrose generalized matrix inverse. The combination method uses the differences in displacements of the structure with and without damage and variations in the modal force vector. Tests on a four-story, steel-frame structure were conducted to obtain static and dynamic responses of the structure. The modal parameters are identified using data from the dynamic tests and improved EMD method. The new method is shown to be more accurate and effective than the traditional EMD method. Through tests with a shear-type test frame, the higher performance of the proposed static-dynamic damage detection approach, which can detect both single and multiple damage locations and the degree of the damage, is demonstrated. For structures with multiple damage, the combined approach is more effective than either the static or dynamic method. The proposed EMD method and static-dynamic damage detection method offer improved modal identification and damage detection, respectively, in structures.

A Generalized Matrix Inverse That Is Consistent with Respect to Diagonal Transformations

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general solution to a longstanding open problem relevant to a wide variety of applications in robotics, tracking, and control systems. The new inverse complements the Drazin inverse (which is consistent with respect to similarity transformations) and the Moore--Penrose inverse (which is consistent with respect to unitary/orthogonal transformations) to complete a trilogy of generalized matrix inverses that exhausts the standard family of analytically important linear system transformations. Results are generalized to obtain unit-consistent and unit-invariant matrix decompositions, and examples of their use are described.

Flowrate Determination for Arbitrary Multipath Arrangement Based on Generalized Inverse of Matrix

As a widely used method in flow measurement, Gaussian quadrature is traditionally applied to determine the flowrate with multiple acoustic paths. However, it is only available to parallel path arrangement and the path position cannot be customized by users because of the nature of numerical integration principle. Accuracy of flowrate determination is greatly restricted meanwhile. In this paper, a new method based on generalized inverse of matrix (GIM) is proposed to work out the flowrate in arbitrary multipath arrangement. With an integrable model of flow distribution established, the GIM method will find the optimal weights and achieve highly accurate flowrate. Three perfect profile assumptions, including Jacobi, optimal weighted integration for circular sections, and typical power-law profile, are, respectively, adopted in this paper to explore the performance of GIM. Its effects are at first experimentally validated in parallel path arrangement of Gauss–Jacobi on a commercially available ultrasonic flowmeter. Then, four representative types that Gaussian quadrature is not feasible to are studied via numerical simulation, including a parallel and three crossed path arrangements. Performance of the proposed GIM method is verified under straight pipe, single bend, and double bend with an expander as well as on eight installation angles. Results reveal that GIM is applicable to accurate flowrate determination for arbitrary multipath arrangement and crossed paths could provide high determination accuracy and good robustness even under complicated flow field.