Jacobi Method Research Articles

Field Programmable Gate Array (FPGA) Implementation of Parallel Jacobi for Eigen-Decomposition in Direction of Arrival (DOA) Estimation Algorithm

The eigen-decomposition of a covariance matrix is a key step in the Direction of Arrival (DOA) estimation algorithms such as subspace classes. Eigen-decomposition using the parallel Jacobi algorithm implemented on FPGA offers excellent parallelism and real-time performance. Addressing the high complexity and resource consumption of the traditional parallel Jacobi algorithm implemented on FPGA, this study proposes an improved FPGA-based parallel Jacobi algorithm for eigen-decomposition. By analyzing the relationship between angle calculation and rotation during the Jacobi algorithm decomposition process, leveraging parallelism in the data processing, and based on the concepts of time-division multiplexing and parallel partition processing, this approach effectively reduces FPGA resource consumption. The improved parallel Jacobi algorithm is then applied to the classic DOA estimation algorithm, the MUSIC algorithm, and implemented on Xilinx’s Zynq FPGA. Experimental results demonstrate that this parallel approach can reduce resource consumption by approximately 75% compared to the traditional method but introduces little additional time consumption. The proposed method in this paper will solve the problem of great hardware consumption of eigen-decomposition based on FPGA in DOA applications.

Adaptive hyperbolic-cross-space mapped Jacobi method on unbounded domains with applications to solving multidimensional spatiotemporal integrodifferential equations

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

Two highly accurate and efficient numerical methods for solving the fractional Liénard’s equation arising in oscillating circuits

In this paper, we investigate the fractional model of the Liénard and Duffing equations with the Liouville-Caputo fractional derivative. These equations grow with the evolution of radio and vacuum tube technology, which describe oscillating circuits and generalize the spring–mass device equation. We compare two numerical approaches, namely Jacobi and Haar wavelet collocation methods. The given approaches are used to discretize and transform the equation into a system of algebraic equations, and the Broyden-Quasi Newton algorithm is applied to solve the resulting nonlinear system of equations. A complete error analysis and convergence rates for different grid sizes are derived for both methods, which are used to compare the accuracy and efficiency of the two approaches. While both approaches produce correct solutions, according to the numerical findings, the Jacobi collocation method is more efficient and accurate than the Haar wavelet collocation method.

Performance evaluation of symbol detection algorithms in massive MIMO communication systems

Abstract Massive MIMO (mMIMO) is the essential technique for attaining the exponential increase in the data rate needed for future communication systems. These advancements have been significantly supported by progress in VLSI technology, which enables the integration of an enormous number of antennas and the complex signal processing essential for massive MIMO systems on a single chip. This work conducts a comprehensive analysis of the intricacy of seven matrix decomposition techniques for symbol detection in future mMIMO communication systems: ADMM-based infinity norm (ADMIN), Neumann series (NS), Newton iteration (NI), Jacobi iteration (Ja), improved Gauss-Seidel (IGS), conjugate gradient (CG), and QR decomposition (QR), against linear and near-optimal minimum mean square error (MMSE). QR, GS, Ja, and CG belong to linear algebraic methods based on matrix decomposition. MMSE and ADMIN are nonlinear optimization methods; NS and NI are iterative methods. The analysis examines into the complexity of these detection algorithms, considering the symbol error rate, the convergence rate, the initial solution vector, and the correlation factor. Performance evaluations are conducted on 8 and 16-user mMIMO systems with 64 and 128 base station antennas, modulation schemes (32-QAM and 64-QAM), iteration counts (p = 1, 2, 3, 4), correlation factors (α = 0.2, 0.4, and 0.6), and initial solution vectors (zero vector, D −1 and W 2 −1 yMF ). The result clearly shows that if the initial solution and number of iterations are chosen properly, the IGS-based linear detector achieves near-optimal performance with a lower computational complexity of O(K 2) compared to the nonlinear MMSE-based detector with a computational complexity of O(K 3).

Higher-dimensional MOG dark compact object: shadow behaviour in the light of EHT observations

Consideration of extra spatial dimensions is motivated by the unification of gravity with other interactions, the achievement of the ultimate framework of quantum gravity, and fundamental problems in particle physics and cosmology. Much attention has been focused on the effect of these extra dimensions on the modified theories of gravity. Analytically examining astrophysical phenomena like black hole shadows is one approach to understand how extra dimensions would affect the modified gravitational theories. The purpose of this study is to derive a higher-dimensional metric for a dark compact object in STVG theory and then examine the behaviour of the shadow shapes for this solution in STVG theory in higher dimensions. We apply the Carter method to formulate the geodesic equations and the Hamilton–Jacobi method to find photon orbits around this higher-dimensional MOG dark compact object. We investigate the effects of extra dimensions and the STVG parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} on the black hole’s shadow size. Next, we compare the shadow radius of this higher-dimensional MOG dark compact object to the shadow size of the supermassive black hole M87*, which has been realized by the Event Horizon Telescope (EHT) collaborations, in order to restrict these parameters. We find that extra dimensions in the STVG theory typically lead to a reduction in the shadow size of the higher-dimensional MOG dark compact object, whereas the effect of parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} on this black hole’s shadow is suppressible. Remarkably, given the constraints from EHT observations, we find that the shadow size of the four-dimensional MOG dark compact object lies in the confidence levels of the EHT data. Finally, we investigate the issue of acceleration bounds in higher-dimensional MOG dark compact object in confrontation with EHT data of M87*.

A compartmental modal for type2 diabetes on the effect of plasma glucose -insulin regulatory system by Jacobes model

A compartmental model for diabetes is developed. The model describes the dynamics of the spread of Type 2 diabetes. A theoretical analysis in the non-adherence to drugs is investigated. A system of differential equations is analyzed by stability analysis; the non-trivial critical point obtained is locally asymptotically firm under the given conditions. In consideration of Mathematical model for glucose tolerance test (GTT) is considered, actual glucose data values are fixed using MATLAB least squares curve fitting technique. Two methods are used to numerically work out the distributions of steady states of diabetic sub-populations. The Gauss-Seidel method is more accurate than the Jacobi method. The result show that more than 50% of clinical diagnosis attempt needs to be applied to have more diagnosed population than undiagnosed. The GTT model shows that if severe diet and medication is followed diabetes can be controlled.

On derivation of Vlasov–Maxwell–Einstein equations from the principle of least action, Hamilton–Jacobi method and Milne–McCree Model

On derivation of Vlasov–Maxwell–Einstein equations from the principle of least action, Hamilton–Jacobi method and Milne–McCree Model

Real-time explicit co-simulation of wire-rope systems for industrial mobile harbor cranes

This paper presents an investigation into the real-time explicit co-simulation of mobile harbor cranes under hoisting operation. The system is divided into two subsystems, with Subsystem 1 representing the payload and Subsystem 2 encompassing the wire-rope system. To capture the real-time behavior of the wire ropes accurately, the ALEM (Arbitrary Lagrangian–Eulerian Modal) method is employed in this study. Using this formulation, the intricate behavior of the wire ropes, encompassing elasticity, bending effects, and dynamic influences, is rigorously considered. The dynamic equations governing the payload are solved using the Runge–Kutta4 method, while the Generalized Alpha method is utilized to solve the wire-rope system. Both the Gauss–Seidel and Jacobi methods are investigated as two coupling techniques to connect the subsystems. Additionally, the study delves into the impact of the extrapolation method and macro time step on results accuracy and efficiency. The findings demonstrate that co-simulation employing the Gauss–Seidel method and FOH (First-Order Hold) extrapolation yields optimum and accurate simulations. Moreover, the study successfully achieves real-time simulation by optimizing the wire-rope system simulation, accounting for all degrees of freedom inherent in a 3D system. Remarkably, the maximum error observed in the axial force amounts to a mere 0.6% when employing real-time simulation. These findings can hold practical value for employing the current wire-rope subsystem in the interface modeling of real-world industrial cranes.

Jacobi method for dual quaternion Hermitian eigenvalue problems and applications

Jacobi method for dual quaternion Hermitian eigenvalue problems and applications

Observational signatures of charged rotating traversable wormhole: shadows and light rings with different accretions

In this work, we investigate the optical appearance of qualitatively new observational features of accretion disk images around the charged rotating traversable wormhole (TWH) space-time for different spin, throat, and charge values. To accomplish this, we first consider the Hamilton–Jacobi method to derive the geodesic equations for the motion of photons and study the effects of parameters on the photon orbit in the observer’s sky. We found that each parameter affects the size and shape of the wormhole (WH) shadow and flatness is observed in the shadow because of spin and other parameters. To produce shadow images of sufficient visual quality but within manageable computational times, we adopt the ray-trace procedure and characterize the significant features of light trajectories on the observer’s screen, depending on the interaction between the space-time structure and the accretion disk. In addition, we consider the static spherically symmetric accretion flow model to observe the specific intensity around the traversable WH space-time geometry. It is found that the intensity and positions of the photon ring vary with respect to the involved parameters. In future observation, this type of study may provide a fertile playground to test the nature of compact objects, specifically the WH in the strong-field regime.

Convergence analysis of Jacobi spectral tau-collocation method in solving a system of weakly singular Volterra integral equations

The main purpose of this paper is to solve a system of weakly singular Volterra integral equations using the Jacobi spectral tau-collocation method from two perspectives. Since the solutions of the main system exhibit discontinuity at the origin, classical Jacobi methods may yield less accuracy. Therefore, in the first approach, we transform the proposed system through a suitable transformation into an alternative type whose solutions are as smooth as desired. Subsequently, we derive a matrix formulation of the method and analyze its convergence properties in both L2 and L∞-norms.In the second approach, instead of employing a smoothing transformation, we select fractional Jacobi polynomials as basis functions for the approximation space. This choice is motivated by their similar behavior to the exact solutions. We then derive a matrix formulation of the method and perform an error analysis analogous to the first approach. Finally, we present several illustrative examples to demonstrate the accuracy of our method.

A double‐layer Jacobi method for partial differential equation‐constrained nonlinear model predictive control

AbstractThis paper presents a real‐time optimization method for nonlinear model predictive control (NMPC) of systems governed by partial differential equations (PDEs). The NMPC problem to be solved is formulated by discretizing the PDE system in space and time by using the finite difference method. The proposed method is called the double‐layer Jacobi method, which exploits both the spatial and temporal sparsities of the PDE‐constrained NMPC problem. In the upper layer, the NMPC problem is solved by ignoring the temporal couplings of either the state or costate (Lagrange multiplier corresponding to the state equation) equations so that the spatial sparsity is preserved. The lower‐layer Jacobi method is a linear solver dedicated to PDE‐constrained NMPC problems by exploiting the spatial sparsity. Convergence analysis indicates that the convergence of the proposed method is related to the prediction horizon. Results of a numerical experiment of controlling a heat transfer process show that the proposed method can be two orders of magnitude faster than the conventional Newton's method exploiting the banded structure of NMPC problems.

A Unitary Transformation Extension of PolSAR Four-Component Target Decomposition

As an improvement on the traditional model-based Yamaguchi four-component decomposition method, in recent years, to fully utilize the polarization information in the coherency matrix, four-component target decomposition methods Y4R and S4R have been proposed, which are based on the rotation of the coherency matrix and the expansion of the volume model, respectively. At the same time, there is also an improved G4U method proposed based on Y4R and S4R. Although these methods have achieved certain decomposition results, there are still problems with overestimation of volume scattering and insufficient utilization of polarization information. In this paper, a unitary transformation extension to the four-component target decomposition method of PolSAR based on the properties of the Jacobi method is proposed. By analyzing the terms in the basic scattering models, such as volume scattering, in the existing four-component decomposition methods, it is clear that the reason for the existence of the residual matrix in the existing decomposition methods is that the off-diagonal term T13 and the real part of T23 of the coherency matrix T do not participate in the four-component decomposition. On this basis, a matrix transformation method is proposed to decouple terms T13 and ReT23, and the residual matrix decomposed based on this method is derived. The performance of the proposed method was validated and evaluated using two datasets. The experimental results indicate that, compared with model-based methods such as Y4R, S4R and G4U, the proposed method can enhance the contribution of double-bounce scattering and odd-bounce scattering power in urban areas in both sets of data. The computational time of the proposed method is equivalent to Y4R, S4R, etc.

Modeling H2S solubility in aqueous MDEA, MEA and DEA solutions by the electrolyte SRK-CPA EOS

In this study, an electrolyte cubic plus association equation of state (e-CPA EOS) has been proposed to predict the vapor–liquid equilibrium of (H2S) in aqueous solutions of monoethanolamine (MEA), diethanolamine (DEA), and methyldiethanolamine (MDEA). This EOS is composed of repulsive forces, short-range interactions, short-range ionic interactions, association term, long-range ionic interactions, and Born term. The Soave– Redlich– Kwong EOS is used as a non-electrolyte part. A new iterative procedure based on the Jacobi method is introduced to obtain the compositions of all species in the liquid phase. The genetic algorithm (GA) is used to find the optimal values of binary interaction coefficients between molecules and ions. Compared with experimental data, the e-CPA EOS model successfully describes the vapor–liquid equilibrium of H2S in the aqueous solutions of MEA, DEA, and MDEA. The average absolute deviations for 313 data of MEA-H2S-H2O, 156 data of DEA-H2S-H2O, and 145 data of MDEA-H2S-H2O systems were obtained as 8.20 %, 8.04 %, and 9.41 %, respectively. In addition, for some similar cases, the capability of the e-CPA EOS was compared with the Clegg–Pitzer, N-Wilson-NRF, e-NRTL, and e-CTS models. The results indicated that the e-CPA EOS has less error than the four other models.

Geometrical and Numerical Analysis of Predator–Prey System Based on the Allee Effect in Predator

This study explores the complex dynamics of the predator–prey interactions, with a specific emphasis on the influence of the Allee effect on the predator population. We examined the fundamental mathematical characteristics of the model under consideration, such as the positivity of the system and the boundedness of the solutions. We investigated the equilibrium points and analyzed their stability using the Jacobi and Lyapunov methods. A comprehensive examination was carried out on the geometric properties of the dynamical system to compute the five invariants of the KCC theory. In particular, the deviation curvature tensor and its eigenvalues are investigated to demonstrate the behavior of the system stability. We have also obtained the necessary and sufficient conditions for the given set of parameters of the system in order to have the Jacobi stability (instability) near the equilibrium point. To visualize the dynamical behavior of the predator–prey model with the Allee effect in the predator density, numerical simulations were conducted. The investigation encompasses an examination of the system’s behavior from both geometric and numerical standpoints, with the objective of attaining a thorough comprehension using few examples.

Massive Charged Fermion Emissions from Kerr – Newman – Vaidya Black Holes

In this article, we consider charged fermion particle radiation around Kerr – Newman – Vaidya black holes. Using the Hamilton – Jacobi method we derived the emission probability and the temperature of the Hawking radiations. We obtained that the temperature is equal to that due to scalar particles times a factor that contains some characteristics of fermions.

The Intersection of Knowledge Management, the Jacobi Method, and Operational Research: A Paradigmatic Example of Serendipity

The Intersection of Knowledge Management, the Jacobi Method, and Operational Research: A Paradigmatic Example of Serendipity

Tunneling of rotational black string with nonlinear electromagnetic fields

In the background of anti-de Sitter spaces, a novel solution of a rotating black string (RBS) attributed to a nonlinear electromagnetic field has been derived. The authors have deciphered two metric functions which are called RBS1 and RBS2. The two real roots, the two metrics (RBS1 and RBS2), have been found based on Newton–Raphson method. Furthermore, the Hawking radiation of scalar particles has been studied by employing the Hamilton–Jacobi (HJ) method to calculate the Hawking temperature of RBS1 and RBS2.

Singular value decomposition for complex matrices using two-sided Jacobi method

Singular value decomposition for complex matrices using two-sided Jacobi method

On Derivation of Vlasov–Maxwell–Einstein Equations from the Principle of Least Action, the Hamilton–Jacobi Method, and the Milne–McCrea Model

On Derivation of Vlasov–Maxwell–Einstein Equations from the Principle of Least Action, the Hamilton–Jacobi Method, and the Milne–McCrea Model