Matrix Iterative Method Research Articles

A fast convergent numerical method for matrix sign function with application in SDEs

In this article, a new matrix iterative method for computing the sign of a matrix is offered. Convergence along with asymptotical stability of the presented method is studied. An accelerated form of the iteration will further be constructed. Finally, the application of the given approach in numerical solution of stochastic differential equations and in solving algebraic Riccati equations is pointed out.

The inexact fixed matrix iteration for solving large linear inequalities in a least squares sense

A fixed matrix iteration algorithm was proposed by A. Dax (Numer. Algor. 50, 97---114 2009) for solving linear inequalities in a least squares sense. However, a great deal of computation for this algorithm is required, especially for large-scale problems, because a least squares subproblem should be solved accurately at each iteration. We present a modified method, the inexact fixed iteration method, which is a generalization of the fixed matrix iteration method. In this inexact iteration process, the classical LSQR method is implemented to determine an approximate solution of each least squares subproblem with less computational effort. The convergence of this algorithm is analyzed and several numerical examples are presented to illustrate the efficiency of the inexact fixed matrix iteration algorithm for solving large linear inequalities.

Generalized Schultz iterative methods for the computation of outer inverses

We consider a general matrix iterative method of the type Xk+1=Xkp(AXk) for computing an outer inverse AR(G),N(G)(2), for given matrices A∈Cm×n and G∈Cn×m such that AR(G)⊕N(G)=Cm. Here p(x) is an arbitrary polynomial of degree d. The convergence of the method is proven under certain necessary conditions and the characterization of all methods having order r is given. The obtained results provide a direct generalization of all known iterative methods of the same type. Moreover, we introduce one new method and show a procedure how to improve the convergence order of existing methods. This procedure is demonstrated on one concrete method and the improvement is confirmed by numerical examples.

Approximating the Matrix Sign Function Using a Novel Iterative Method

This study presents a matrix iterative method for finding the sign of a square complex matrix. It is shown that the sequence of iterates converges to the sign and has asymptotical stability, provided that the initial matrix is appropriately chosen. Some illustrations are presented to support the theory.

Case Retrieval Algorithm Based on Structure Matrix

Case retrieval is the focal stage of Case-Based Reasoning systems whose quality is determined by the speed and accuracy of retrieva1. In this work, we aim at developing a better Case retrieval algorithm by using vector model and propose its new case retrieval Algorithm based on Structure Matrix, wich is derived to learn the kernel matrix for capturing the relations between the case structure units based on matrix iterative analysis. The experimental comparison of similarity shows that using of the structural information of the case, the accuracy of their experimental results has a general increase, when opposed to some of the existing similarity measure. The same learning algorithm based on matrix iteration method compared to other methods, have higher accuracy (5% to 8%), and required less training documents, the computational cost is smaller

Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations

The problem of solving matrix equations has many applications in control and system theory. This paper is concerned with the iterative solutions of the Sylvester-transpose matrix equation∑i=1k(AiXBi+CiXTDi)=E,and the periodic Sylvester matrix equationA^jX^jB^j+C^jX^j+1D^j=E^jforj=1,2,…,λ.The basic idea is to develop the conjugate gradients squared (CGS) and bi-conjugate gradient stabilized (Bi-CGSTAB) methods for obtaining matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. Numerical test results are given to compare matrix iterative methods with other well-known methods.

The generalized QMRCGSTAB algorithm for solving Sylvester-transpose matrix equations

In this work we generalize the Quasi-Minimal Residual Variant of the Bi-CGSTAB algorithm (QMRCGSTAB) to obtain a matrix iterative method for solving the Sylvester-transpose matrix equation ∑ i = 1 l ( A i X B i + C i X T D i ) = E . In order to compare the generalized algorithm with some existing methods, we present two numerical examples.

New matrix iterative methods for constraint solutions of the matrix equation [formula omitted

In this paper, two new matrix iterative methods are presented to solve the matrix equation A X B = C , the minimum residual problem min X ∈ S ‖ A X B − C ‖ and the matrix nearness problem min X ∈ S E ‖ X − X ∗ ‖ , where S is the set of constraint matrices, such as symmetric, symmetric R -symmetric and ( R , S ) -symmetric, and S E is the solution set of above matrix equation or minimum residual problem. These matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in Deng et al. (2006) [13], Huang et al. (2008) [15], Peng (2005) [16] and Lei and Liao (2007) [17]. Paige’s algorithms are used as the frame method for deriving these matrix iterative methods. Numerical examples are used to illustrate the efficiency of these new methods.

A matrix LSQR iterative method to solve matrix equation AXB=C

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈𝒮, where 𝒮 is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X 0∈𝒮, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution , which is nearest to a given matrix ˜X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C , J. Comput. Appl. Math. 212 (2008), pp. 231–244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499–513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C , Appl. Math. Comput. 170 (2005), pp. 711–723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C , Appl. Math. Comput. 160 (2005), pp. 763–777].

Green evaluation of microwave-assisted leaching process of high titanium slag on life cycle assessment

A greenness evaluation index and system of microwave-assisted leaching method were established. The effects of the life cycle assessment variables, such as the resource consumption, environment impact, cost, time and quality, were investigated, and the concept of green degree was applied in the production of synthetic rutile. An analytic hierarchy process was utilized to assess matrix of greenness evaluation. The Gauss-Seidel iterative matrix method was employed to solve the assessment matrix and obtain the weights and membership functions of all evaluation indexes. A fuzzy decision-making method was applied to build the greenness evaluation model, and then the scores of green degree in microwave-assisted leaching process was obtained. The greenness evaluation model was applied to the life cycle assessment of the microwave-assisted leaching process. The results show that the microwave-assisted leaching process has advantages over the conventional ones, with respect to energy-consumption, processing time and environmental protection.

Ionization of atomic hydrogen in strong infrared laser fields

We have used the matrix iteration method of Nurhuda and Faisal [Phys. Rev. A 60, 3125 (1999)] to treat ionization of atomic hydrogen by a strong laser pulse. After testing our predictions against a variety of previous calculations, we present ejected-electron spectra as well as angular distributions for few-cycle infrared laser pulses with peak intensities of up to ${10}^{15}$ W/cm${}^{2}$. It is shown that the convergence of the results with the number of partial waves is a serious issue, which can be managed in a satisfactory way by using the velocity form of the electric dipole operator in connection with an efficient time-propagation scheme.

Effect of direct coupling of microrings on the gain bandwidth of cascade microring Raman amplifier

A combination of coupled microrings that are coupled to a bus waveguide is proposed as a wide-band microring Raman amplifier. This structure is called one channel coupled microrings (OCCM) Raman amplifier. It is shown analytically that the number of poles in the OCCM transfer function is greater than the square of microrings number ( N). The gain bandwidth and gain ripple of the proposed amplifier are increased and decreased, respectively due to increasing the number of poles and zeros in the transfer function. An iterative matrix method is introduced to solve Raman equations in the amplifier with arbitrary number of microrings and Genetic algorithm is employed to maximize the gain flatness.

INTERSUBBAND OPTICAL ABSORPTION COEFFICIENTS AND REFRACTIVE INDEX CHANGES OF AN ELLIPSOIDAL FINITE POTENTIAL QUANTUM DOT

Intersubband optical absorption coefficient and refractive index changes of a weakly prolate ellipsoidal quantum dot, using the compact-density matrix formalism and iterative method, are investigated. In this regard, the linear and nonlinear intersubband optical absorption coefficient and refractive index changes of a GaAs / Al x Ga 1-x As ellipsoidal quantum dot, as functions of the dot radius, ellipticity constant, stoichiometric ratio and incident light intensity are calculated. The results indicate that absorption coefficient and refractive index changes strongly depend on the light intensity, size and geometry of the dot and structure parameters such as aluminium concentration in GaAs / Al x Ga 1-x As structures.

A regularized pairwise multi-classification knowledge-based machine and applications

This paper presents a novel knowledge-based linear classification model for multi-category discrimination of sets or objects with prior knowledge. The prior knowledge is in the form of multiple polyhedral sets belonging to one or more categories or classes and it is introduced as additional constraints into the formulation of the Tikhonov linear least squares multi-class support vector machine model. The resulting formulation leads to a least squares problem that can be solved using matrix methods or iterative methods. Investigations include the development of a linear knowledge-based classification model extended to the case of multi-categorical discrimination and expressed as a single unconstrained optimization problem. Advantages of this formulation include explicit expressions for the classification weights of the classifier(s) and its ability to incorporate and handle prior knowledge directly to the classifiers. In addition it can provide fast solutions to the optimal classification weights for multi-categorical separation without the use of specialized solver-software. To evaluate the model, data and prior knowledge from the Wisconsin breast cancer prognosis and two-phase flow regimes in pipes were used to train and test the proposed formulation.

FULL-WAVE MODELING OF MULTIPLE VIAS USING DIFFERENTIAL SIGNALING AND SHARED ANTIPAD IN MULTILAYERED HIGH SPEED VERTICAL INTERCONNECTS

A 3D full-wave approach, based on the Foldy-Lax multiple scattering equations, is successfully extended to model massively- coupled multiple vias using difierential signaling and shared antipad in high speed vertical interconnects. For the flrst time, this method has been used and tested on via-pair with shared antipad in multilayered structure. The magnetic frill current source on the port is calculated by using the flnite difierence method. Banded matrix iterative method is applied to accelerate the flnite difierence calculation. Numerical example of 15 signal via-pairs and 20 ground shielding vias in 6- layer board demonstrates that this approach is able to model the via-pair with shared antipad and to include all the coupling efiects among multiple vias. The electrical performances of difierent signal driving schemes are provided and discussed. The coupling crosstalk on various via-pairs is compared. The improvement of signal integrity is shown by using difierential signaling and shared antipad for via-pair in multilayered structure. The results are compared with HFSS and SIwave in accuracy and CPU. The CPU using Foldy-Lax approach is three orders of magnitude faster than using HFSS, and two orders of magnitude faster than using SIwave. The accuracy of Foldy-Lax is within 2% difierence from HFSS up to 20GHz, and outperforms SIwave in accuracy.

Methods for calculating the main-axis Lyapunov exponents of a type of chaotic systems with delay

In this paper，we propose a new method for calculating the main-axis Lyapunov exponents of a type of chaotic systems with delays，which we call Matrix iteration method. We discuss this new method theoretically by detailed analysis. Also，we introduce another method called Taylor expansion method. The method invented by Wolf is also introduced and we discuss the advantages and disadvantages of these three methods. Finally，we calculate the main-axis Lyapunov exponents of a type of chaotic systems with delay to test these method.

A nonlinear multi-classification knowledge-based kernel machine

This paper presents a knowledge-based nonlinear kernel classification model for multi-category discrimination of sets or objects with prior knowledge. A kernel function is employed to find a nonlinear classifier capable of discriminating future points into an appropriate class. The prior knowledge is in the form of multiple polyhedral sets belonging to one or more categories or classes, and it is introduced as additional constraints into the formulation of the regularized nonlinear kernel least squares multi-class support vector machine model. The resulting formulation leads to a linear system of equations that can be solved using matrix methods or iterative methods. This work extends previous work (Oladunni et al. in ICCS 2006, Lecture notes in Computer Science, Part I, LNCS, vol 3991. Springer, Berlin, pp 188–195, 2006) that incorporated similar prior knowledge into a regularized linear least squares multi-class model. To evaluate the model, data and prior knowledge from the two-phase flow regimes in pipes were used to train and test the proposed formulation.

Expansion of weighted pseudoinverse matrices with singular weights into matrix power products and iteration methods

We obtain expansions of weighted pseudoinverse matrices with singular weights into matrix power products with negative exponents and arbitrary positive parameters. We show that the rate of convergence of these expansions depends on a parameter. On the basis of the proposed expansions, we construct and investigate iteration methods with quadratic rate of convergence for the calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions. Iteration methods for the calculation of weighted normal pseudosolutions are adapted to the solution of least-squares problems with constraints.

An iterative matrix solution of the Boltzmann equation for the calculation of diffusion and noise in semiconductors

We extend the iterative matrix method for the solution of the Boltzmann transport equation in multi-band semiconductors to the calculation of second order parameters such as correlation functions and the noise temperature. The method allows also the extraction of the diffusion coefficient in each valley. The results for the case of GaAs with two valleys show that the diffusion in this material is dominated mainly by interactions in the Γ valley. The calculated noise temperature is in good agreement with Monte Carlo results.

An efficient solver for the high-order accurate time-discontinuous Galerkin (TDG) method for second-order hyperbolic systems

An efficient predictor/multi-corrector algorithm for the high-order accurate time-discontinuous Galerkin (TDG) method is presented. The algorithm overcomes the demanding storage and computational effort of the direct solution of the TDG matrix equations. The success of the block matrix iterative solution method is a result of a specific form of time approximation. An analysis shows that the algorithm needs only a few iteration passes with a single matrix factorization of size equal to the number of spatial degrees-of-freedom to reach the 2 p + 1 order of accuracy of the parent TDG solution obtained from a direct solve. For linear ( p = 1 ) and quadratic ( p = 2 ) approximations in time, each iteration pass retains the unconditional and asymptotic stability of the TDG parent solution. Numerical results demonstrate the efficiency and accuracy of the time-integration algorithm over second-order accurate single-step/single-solve (SS/SS) methods.