Matrix Diagonalization Technique Research Articles

A Global Optimization Algorithm for Solving Linearly Constrained Quadratic Fractional Problems

This paper first proposes a new and enhanced second order cone programming relaxation using the simultaneous matrix diagonalization technique for the linearly constrained quadratic fractional programming problem. The problem has wide applications in statics, economics and signal processing. Thus, fast and effective algorithm is required. The enhanced second order cone programming relaxation improves the relaxation effect and computational efficiency compared to the classical second order cone programming relaxation. Moreover, although the bound quality of the enhanced second order cone programming relaxation is worse than that of the copositive relaxation, the computational efficiency is significantly enhanced. Then we present a global algorithm based on the branch and bound framework. Extensive numerical experiments show that the enhanced second order cone programming relaxation-based branch and bound algorithm globally solves the problem in less computing time than the copositive relaxation approach.

Numerical simulation of quantum anharmonic oscillator, embedded within an infinite square well potential, by matrix methods using Gnumeric spreadsheet

The quantum anharmonic oscillator has been solved numerically using matrix diagonalization technique. The interaction potential consisting of quadratic () and quartic (λx4) terms is embedded within an infinite square well potential of appropriate width, ‘a’ and its sine eigen functions are used as basis functions ‘N’ for the employed matrix method. The energy eigen values for the resultant Hamiltonian are solved in a free open source software (FOSS), Gnumeric, a simple worksheet environment. The numerical parameters ‘a’ and ‘N’ are optimized to converge to the expected energies for harmonic oscillator and those for anharmonic oscillator from perturbation theory for small values of physical parameter, ‘λ’. The pure quartic oscillator is studied for both small and large values of λ and validated with results obtained from other numerical techniques. The breakdown of perturbation approximation for large values of λ is also shown.

A New Spatial Branch and Bound Algorithm for Quadratic Program with One Quadratic Constraint and Linear Constraints

This paper proposes a novel second-order cone programming (SOCP) relaxation for a quadratic program with one quadratic constraint and several linear constraints (QCQP) that arises in various real-life fields. This new SOCP relaxation fully exploits the simultaneous matrix diagonalization technique which has become an attractive tool in the area of quadratic programming in the literature. We first demonstrate that the new SOCP relaxation is as tight as the semidefinite programming (SDP) relaxation for the QCQP when the objective matrix and constraint matrix are simultaneously diagonalizable. We further derive a spatial branch-and-bound algorithm based on the new SOCP relaxation in order to obtain the global optimal solution. Extensive numerical experiments are conducted between the new SOCP relaxation-based branch-and-bound algorithm and the SDP relaxation-based branch-and-bound algorithm. The computational results illustrate that the new SOCP relaxation achieves a good balance between the bound quality and computational efficiency and thus leads to a high-efficiency global algorithm.

Split-step spectral Galerkin method for the two-dimensional nonlinear space-fractional Schrödinger equation

In this paper, we propose a highly accurate and conservative split-step spectral Galerkin (SSSG) scheme, which combines the standard second-order Strang split-step method for handling the nonlinear and the potential terms with the Legendre spectral Galerkin method for approximating the Riesz space-fractional derivatives, for the two-dimensional nonlinear space-fractional Schrödinger equation. The mass conservation property of the numerical solution is proved and the optimal error estimate with respect to spatial discretization is established by introducing an orthogonal projection operator. In addition, a matrix diagonalization technique is introduced to resolve the multi-dimensional difficulty and reduce the computational complexity in the implementation. Numerical experiments are presented to illustrate the accuracy and robustness of the SSSG method.

A splitting Chebyshev collocation method for Schrödinger–Poisson system

We develop a splitting Chebyshev collocation (SCC) method for the time-dependent Schrodinger–Poisson (SP) system arising from theoretical analysis of quantum plasmas. By means of splitting technique in time, the time-dependant SP system is first reduced to uncoupled Schrodinger and Poisson equations at every time step. The space variables in Schrodinger and Poisson equations are next represented by high-order Chebyshev polynomials, and the resulting system are discretized by the spectral collocation method. Finally, matrix diagonalization technique is applied to solve the fully discretized system in one dimension, two dimensions and three dimensions, respectively. The newly proposed method not only achieves spectral accuracy in space but also reduces the computer-memory requirements and the computational time in comparison with conventional solver. Numerical results confirm the spectral accuracy and efficiency of this method, and indicate that the SCC method could be an efficient alternative method for simulating the dynamics of quantum plasmas.

Modal Analysis of Fluid Flows: An Overview

Modal Analysis of Fluid Flows: An Overview

A Multitarget Active Backscattering 2-D Positioning System With Superresolution Time Series Post-Processing Technique

An active backscattering 2-D positioning system is studied and demonstrated in this paper. The hardware is based on an investigated principle of switched injection phase locking for wireless ranging. A new post-processing technique and a system modeling for multitarget time-of-arrival (TOA) and direction-of-arrival (DOA) estimations are proposed and studied. The technique features a joint singular value decomposition-based ESPRIT algorithm and a matrix diagonalization technique using Moore–Penrose pseudoinverse for model order estimation as well as the TOA/DOA extraction. They make use of the shift invariant property of a backscattered signal in the time-domain. The accuracy and resolution are studied and are shown to be superior to those of the conventional frequency-domain technique using a fast Fourier transform (FFT). Moreover, the proposed algorithms take full advantage of the chirp bandwidth and allow all the targets to operate at the same modulation frequency and arbitrary carrier frequency. These merits avoid a commonly exercised compromise of the FFT method among the resolution, target quantity, signal-to-noise ratio, and other critical specifications. The system is simulated, and the demonstrator is constructed for experimental verifications in both indoor and outdoor environments, with a bandwidth of 600 MHz centered at 5.6 GHz.

Examining the eigenvalues effect to the computational cost in mobile robot simultaneous localization and mapping

One of the biggest factors that contributes to the computational cost of extended Kalman filter-based simultaneous localization and mapping is the computation of the covariance update. This results from the multiplications of the covariance matrix with other parameters along with the increment of its dimension, which is twice the number of landmarks. This study attempts to look for an optimal solution to decrease the computational complexity of the covariance matrix without compromising the accuracy of the state estimation through eigenvalue approach. This paper presents a study on the matrix-diagonalization technique, which is applied to the covariance matrix in extended Kalman filter-based simultaneous localization and mapping to simplify the multiplication process. The behavior of estimation and covariance were observed based on four case studies to analyze the performance of the proposed technique.

An iteration free backward semi-Lagrangian scheme for solving incompressible Navier–Stokes equations

A backward semi-Lagrangian method based on the error correction method is designed to solve incompressible Navier–Stokes equations. The time derivative of the Stokes equation is discretized with the second order backward differentiation formula. For the induced steady Stokes equation, a projection method is used to split it into velocity and pressure. Fourth-order finite differences for partial derivatives are used to the boundary value problems for the velocity and the pressure. Also, finite linear systems for Poisson equations and Helmholtz equations are solved with a matrix-diagonalization technique. For characteristic curves satisfying highly nonlinear self-consistent initial value problems, the departure points are solved with an error correction strategy having a temporal convergence of order two. The constructed algorithm turns out to be completely iteration free. In particular, the suggested algorithm possesses a good behavior of the total energy conservation compared to existing methods. To assess the effectiveness of the method, two-dimensional lid-driven cavity problems with large different Reynolds numbers are solved. The doubly periodic shear layer flows are also used to assess the efficiency of the algorithm.

Optical rectification and third harmonic generation of spherical quantum dots: Controlling via external factors

In this paper simultaneous effects of pressure, temperature, external electric field and laser radiation on the optical rectification and third harmonic generation of a spherical quantum dot with parabolic confinement and dressed impurity are studied. By means of matrix diagonalization technique, energy eigenvalues and functions are evaluated and used to find the optical rectification coefficient and third harmonic generation of the system via density operator method. It is shown that these nonlinear optical quantities strongly depend on pressure, temperature, electric field, confinement frequency and dressing laser intensity. Obvious effects of these external factors propose new facilities with different effects to control nonlinear optical properties of such systems.

Optical properties of parabolic quantum dots with dressed impurity: Combined effects of pressure, temperature and laser intensity

In this paper simultaneous effects of pressure, temperature and laser radiation on the optical absorption coefficient and refractive index of a spherical quantum dot with parabolic confinement and dressed impurity are studied. By means of matrix diagonalization technique, energy eigenvalues and functions are evaluated and used to find the optical properties of the system via density operator method. It is shown that linear and nonlinear optical properties strongly depend on pressure, temperature and dressing laser intensity. The interesting point is that the influence of laser radiation depends on pressure and temperature.

A GPU parallelized spectral method for elliptic equations in rectangular domains

We design and implement a polynomial-based spectral method on graphic processing units (GPUs). The key to success lies in the seamless integration of the matrix diagonalization technique and the new generation CUDA tools. The method is applicable to elliptic equations in rectangular domains with general boundary condition. We show remarkable speedups of up to 15 times in the 2-D case and more than 35 times in the 3-D case.

Fast two-dimensional weighted least squares techniques for the design of two-dimensional finite impulse response filters

This paper focuses on the design of two-dimensional (2D) quadrantally symmetric finite impulse response (FIR) filters, and presents three very efficient algorithms for the weighted least squares (WLS) design with a weight matrix that assigns four different weights to four different frequency bands. The first algorithm seeks for iterative solutions to the matrix equation describing the optimality condition of the design problem. The second algorithm aims at the limit solution of the solution sequence to the first algorithm, analytically obtained by using matrix diagonalization techniques. The third algorithm belongs to the category of iterative reweighting techniques. It uses the second algorithm as its iteration core, and aims at reducing the maximum magnitude error of the filter by iteratively adjusting the four entry values of the weight matrix. Design examples are provided to demonstrate the performance of the proposed algorithms.

Linear and Nonlinear Optical Properties of Spherical Quantum Dots: Effects of Hydrogenic Impurity and Conduction Band Non-Parabolicity

Simultaneous effects of an on-center hydrogenic impurity and band edge non-parabolicity on intersubband optical absorption coefficients and refractive index changes of a typical GaAs/AlxGa1−x As spherical quantum dot are theoretically investigated, using the Luttinger—Kohn effective mass equation. So, electronic structure and optical properties of the system are studied by means of the matrix diagonalization technique and compact density matrix approach, respectively. Finally, effects of an impurity, band edge non-parabolicity, incident light intensity and the dot size on the linear, the third-order nonlinear and the total optical absorption coefficients and refractive index changes are investigated. Our results indicate that, the magnitudes of these optical quantities increase and their peaks shift to higher energies as the influences of the impurity and the band edge non-parabolicity are considered. Moreover, incident light intensity and the dot size have considerable effects on the optical absorption coefficients and refractive index changes.

Electronic structure and nonlinear optical properties of a two-dimensional hexagonal quantum dot

In this work electronic structure, the linear and the third-order nonlinear refractive index changes as well as optical absorption coefficients of a two-dimensional hexagonal quantum dot are investigated. Energy eigenvalues and eigenfunctions of the system are calculated by the matrix diagonalization technique, and optical properties are also obtained using the compact density matrix approach. As our results indicate, both the dot size and the confinement potential have a great influence on the intersubband energy intervals, the transition probability and consequently, the linear and the third-order nonlinear refractive index changes and optical absorption coefficients.

Double‐CO32− Centered [CoII5] Wheel and Modeling of Its Magnetic Properties

A high-spin Co(II) cluster with a rare pentagonal molecular structure and formula [Co(5)(CO(3))(2)(bpp)(5)]ClO(4) (1; Hbpp is 2,6-bis(phenyliminomethyl)-4-methylphenolate) has been synthesized and characterized by single-crystal X-ray diffraction. This topology arises from fusing five [Co(2)(bpp)] moieties in a cyclic manner around two CO(3)(2-) central ligands, resulting in propeller-like configuration. The irregular coordination of the carbonate ions to the metal centers results in a combination of coordination numbers (CNs) of the Co(II) ions of five and six. The bulk magnetization of this complicated magnetically exchanged system has been modeled successfully by employing a matrix diagonalization technique. For this, the combination of S=3/2 ions (CN=5) with ions exhibiting strong spin-orbit coupling (CN=6) has been considered and a perturbative approach to handle the data in the whole studied range of temperatures (2-300 K) yielding parameters of g and D (for the five-coordinate Co(II) ions), of A, κ, λ, and Δ (for the metals with spin-orbit coupling) and of the exchange constants J. The agreement with results from DFT calculations, also presented here, is remarkable.

Third-order nonlinear optical properties of a one- and two-electron spherical quantum dot with and without a hydrogenic impurity

In this study, a detailed investigation of the nonlinear optical properties of a spherical quantum dot (QD) containing one and two electrons has been carried out. First, intersublevel linear, third-order, and total absorptions of both single- and double-electron QDs have been investigated for cases with (i.e., D0 and D− donor) and without impurity. Second, linear, third-order, and total refractive indices have been examined in the same structure. In the calculation of optical properties, for the density of electrons in the QD considered, some symbolic values chosen are not similar to those in previous studies. The full numeric matrix diagonalization technique has been employed in determining sublevel energy eigenvalues and their wave functions. The Poisson–Schrödinger equations have been solved self-consistently in the Hartree approximation. In addition, the quantum mechanical many-body effects have been taken into account in local density approximation. The results are presented as a function of QD radii, photon intensities, and photon energies. It is found out that although magnitudes of the linear absorption coefficients are the same for cases with and without impurity, the third-order ones are larger, and so the total absorption coefficients are smaller in the case without impurity. It is also observed that the size of the QD and the intensity of the illumination have drastic effects on nonlinear optical properties.

Self-consistent computation of electronic and optical properties of a single exciton in a spherical quantum dot via matrix diagonalization method

In this study, we develop and demonstrate an efficient self-consistent calculation schema that computes the electronic structure and optical properties of a single exciton in a spherical quantum dot (QD) with an interacting pair of electron and hole wave functions. To observe modifications on bands, wave functions, and energies due to the attractive Coulomb potential, the full numeric matrix diagonalization technique is employed to determine sublevel energy eigenvalues and their wave functions in effective mass approximation. This treatment allows to observe that the conduction and valance band edges bend, that the electron and hole wave functions strongly localize in the QD, and that the excitonic energy level exhibits redshift. In our approach for the Coulomb term between electron and hole, the Poisson–Schrödinger equations are solved self-consistently in the Hartree approximation. Subsequently, exciton binding energies and associated optical properties are computed. The results are presented as a function of QD radii and photon energies. We conclude that all of these numerical results are in agreement with the experimental studies.

Size Effect of a Negatively Charged Exciton in a Two-Dimensional Quantum Dot

In this paper we study a negatively charged exciton (NCE), which is trapped by a two-dimensional (2D) parabolic potential. By using matrix diagonalization techniques, the correlation energies of the low-lying states with L = 0, 1, and 2 are calculated as a function of confinement strength. We find that the size effects of different states are different. This phenomenon can be explained as a hidden symmetry, which is originated purely from symmetry. Based on symmetry, the features of the low-lying states are discussed in the influence of the 2D parabolic potential well. It is found that the confinement may cause accidental degeneracies between levels with different low-excited states. It is shown that the effect of quantum confinement on the binding energy of the heavy hole is stronger than that of a light hole.

Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods

A robust methodology is presented for efficiently solving partial differential equations using Chebyshev spectral techniques. It is well known that differential equations in one dimension can be solved efficiently with Chebyshev discretizations, O( N) operations for N unknowns, however this efficiency is lost in higher dimensions due to the coupling between modes. This paper presents the “quasi-inverse“ technique (QIT), which combines optimizations of one-dimensional spectral differentiation matrices with Kronecker matrix products to build efficient multi-dimensional operators. This strategy results in O( N 2 D−1 ) operations for N D unknowns, independent of the form of the differential operators. QIT is compared to the matrix diagonalization technique (MDT) of Haidvogel and Zang [D.B. Haidvogel, T. Zang, The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials, J. Comput. Phys. 30 (1979) 167–180] and Shen [J. Shen, Efficient spectral-Galerkin method. II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comp. 16 (1) (1995) 74–87]. While the cost for MDT and QIT are the same in two dimensions, there are significant differences. MDT utilizes an eigenvalue/eigenvector decomposition and can only be used for relatively simple differential equations. QIT is based upon intrinsic properties of the Chebyshev polynomials and is adaptable to linear PDEs with constant coefficients in simple domains. We present results for a standard suite of test problems, and discuss of the adaptability of QIT to more complicated problems.