Nonsingular Systems Research Articles

Design of a Residue Number System Based Linear System Solver in Hardware

This paper is focused on error-free solution of dense linear systems using residual arithmetic in hardware. The designed Modular System uses hardware identical Residual Processors (RP)s for solving independent systems of linear congruences and combines their solutions into the solution of the given linear system. This approach uses the residue number system which is based on the Chinese remainder theorem. In order to efficiently exploit parallel processing and cooperation of the individual components, a hardware architecture of the Modular System with several RPs is designed. In order to verify the proposed architecture, a Xilinx FPGA with a MicroBlaze processor was used. Experimental results are obtained for an evaluation FPGA board with Virtex 6. Results from implementation serve for subsequent theoretical analysis of the system performance for various linear system sizes and further improvement of the system. The proposed system can be useful as a special hardware peripheral or a part of an embedded system for solving large nonsingular systems of linear equations with integer, rational or floating-point coefficients with arbitrary precision.

The general solutions of singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense

In this paper, we generalize the time-varying descriptor systems to the case of fractional order in matrix forms. Moreover, we present the general exact solutions of the linear singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense by using a new attractive method. Finally, two illustrated examples are also given to show our new approach.

A QMR-Type Algorithm for Drazin-Inverse Solution of Singular Nonsymmetric Linear Systems

In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.

Isochronous Liénard-type nonlinear oscillators of arbitrary dimensions

In this paper, we briefly present an overview of the recent developments made in identifying/generating systems of Lienard-type nonlinear oscillators exhibiting isochronous properties, including linear, quadratic and mixed cases and their higher-order generalizations. There exists several procedures/methods in the literature to identify/generate isochronous systems. The application of local as well as nonlocal transformations and Ω-modified Hamiltonian method in identifying and generating systems exhibiting isochronous properties of arbitrary dimensions is also discussed in detail. The identified oscillators include singular and nonsingular Hamiltonian systems and PT-symmetric systems.

Limited frequency interval Gramian‐based model reduction for generalised non‐singular discrete time systems

A limited frequency interval Gramians-based model reduction technique for generalised non-singular discrete time systems is presented. The technique generalises the results of existing limited frequency interval Gramians-based model reduction (of discrete time systems) schemes to general non-singular discrete time systems. Numerical examples are also presented to illustrate the proposed technique.

On Li-Yorke measurable sensitivity

The notion of Li-Yorke sensitivity has been studied extensively in the case of topological dynamical systems. We introduce a measurable version of Li-Yorke sensitivity, for nonsingular (and measure-preserving) dynamical systems, and compare it with various mixing notions. It is known that in the case of nonsingular dynamical systems, ergodic Cartesian square implies double ergodicity, which in turn implies weak mixing, but the converses do not hold in general, though they are all equivalent in the finite measure-preserving case. We show that for nonsingular systems, ergodic Cartesian square implies Li-Yorke measurable sensitivity, which in turn implies weak mixing. As a consequence we obtain that, in the finite measure-preserving case, Li-Yorke measurable sensitivity is equivalent to weak mixing. We also show that with respect to totally bounded metrics, double ergodicity implies Li-Yorke measurable sensitivity.

On Nonsingular Saddle-Point Systems with a Maximally Rank Deficient Leading Block

We consider nonsingular saddle-point matrices whose leading block is maximally rank deficient, and show that the inverse in this case has unique mathematical properties. We then develop a class of indefinite block preconditioners that rely on approximating the null space of the leading block. The preconditioned matrix is a product of two indefinite matrices, but under certain conditions the conjugate gradient method can be applied and is rapidly convergent. Spectral properties of the preconditioners are observed and validated by numerical experiments.

Nonsingular direct neural control of air-breathing hypersonic vehicle via back-stepping

This paper investigates the design of nonsingular direct neural control systems for the longitudinal dynamics of a generic air-breathing hypersonic vehicle (AHV). The control objective is to steer the velocity and altitude to track the given commands in the presence of unmatched disturbances. For the velocity subsystem, a neural network (NN) is employed to approximate the developed control law directly where a direct neural controller is designed. For the altitude subsystem that is transformed into strict feedback form, two NNs are used to estimate the virtual and actual control laws derived from back-stepping design. Hence the problem of possible control singularity and “explosion of terms” are avoided. More specially, a nonlinear tracking differentiator is introduced to reconstruct the flight-path angle and angle of attack that are difficult and costly to measure in practice. Numerical simulations are presented to verify the feasibility of the proposed control approach.

Model reduction of generalized non-singular systems using limited frequency interval Gramians

Limited frequency interval Gramians-based model reduction techniques for generalized non-singular systems are presented. These techniques generalize the results of existing limited frequency interval Gramians schemes for general non-singular systems. Numerical examples are also presented to illustrate the effectiveness of proposed techniques.

Transformations of matrix structures work again

Matrices with the structures of Toeplitz, Hankel, Vandermonde and Cauchy types are omnipresent in modern computations in Sciences, Engineering, and Signal and Image Processing. These four matrix classes have distinct features, but in our 1990 paper in Mathematics of Computation we showed that Vandermonde and Hankel multipliers can be applied to transform each class into the others, and then we demonstrated how by using these transforms we can readily extend any successful matrix inversion algorithm from one of these classes to all the others. The power of this approach was widely recognized later, when novel numerically stable algorithms solved nonsingular Toeplitz linear systems of equations in quadratic (versus classical cubic) arithmetic time based on transforming Toeplitz into Cauchy matrix structures. More recent papers combined the same transformation with a link of the Cauchy matrices to the Hierarchical Semiseparable matrix structure, which is a specialization of matrix representations employed by the Fast Multipole Method. This produced numerically stable algorithms that approximated the solution of a nonsingular Toeplitz linear system of equations in nearly linear arithmetic time. We first revisit the successful method of structure transformation, covering it comprehensively. Then we analyze the latter approximation algorithms for Toeplitz linear systems and extend them to approximate multiplication of Vandermonde and Cauchy matrices by a vector and to approximate solution of Vandermonde and Cauchy linear systems of equations provided that they are nonsingular and well-conditioned. We decrease the arithmetic cost of the known numerical approximation algorithms for these tasks from quadratic to nearly linear, and similarly for the computations with the matrices having structures that generalize the structures of Vandermonde and Cauchy matrices and for polynomial and rational evaluation and interpolation. We also accelerate a little further the known numerical approximation algorithms for a nonsingular Toeplitz or Toeplitz-like linear system by employing distinct transformations of matrix structures, and we briefly comment on some natural research challenges, particularly some promising applications of our techniques to high precision computations.

On semi-convergence of generalized skew-Hermitian triangular splitting iteration methods for singular saddle-point problems

Recently, Krukier et al. (2014) [13] proposed an efficient generalized skew-Hermitian triangular splitting (GSTS) iteration method for nonsingular saddle-point linear systems with strong skew-Hermitian parts. In this work, we further use the GSTS method to solve singular saddle-point problems. The semi-convergence properties of GSTS method are analyzed by using singular value decomposition and Moore–Penrose inverse, under suitable restrictions on the involved iteration parameters. Numerical results are presented to demonstrate the feasibility and efficiency of the GSTS iteration methods, both used as solvers and preconditioners for GMRES method.

Comparison Results between Jacobi and USSOR Iterative Methods

In this paper, some comparison results between Jacobi and USSOR iteration for solving nonsingular linear systems are presented. It is showed that spectral radius of Jacobi iteration matrix B is less than that of USSOR iterative matrix under some conditions. A numerical example is also given to illustrate our results.

A topological classification of the Chaplygin systems in the dynamics of a rigid body in a fluid

The paper is concerned with the topological analysis of the Chaplygin integrable case in the dynamics of a rigid body in a fluid. A full list of the topological types of Chaplygin systems in their dependence on the energy level is compiled on the basis of the Fomenko-Zieschang theory. An effective description of the topology of the Liouville foliation in terms of natural coordinate variables is also presented, which opens a direct way to calculating topological invariants. It turns out that on all nonsingular energy levels Chaplygin systems are Liouville equivalent to the well-known Euler case in the dynamics of a rigid body with fixed point.Bibliography: 23 titles.

Convergence of P-regular splitting iterative methods for non-Hermitian positive semidefinite linear systems

In this paper, we study the convergence of P-regular splitting iterative methods for singular and non-singular non-Hermitian positive semidefinite linear systems, which generalize the known results. Some numerical experiments are performed to illustrate the convergence results.

A Framework for Deflated and Augmented Krylov Subspace Methods

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately. We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MINRES method. Numerical experiments illustrate properties of different variants of deflated MINRES analyzed in this paper.

Classical Time Crystals

We consider the possibility that classical dynamical systems display motion in their lowest-energy state, forming a time analogue of crystalline spatial order. Challenges facing that idea are identified and overcome. We display arbitrary orbits of an angular variable as lowest-energy trajectories for nonsingular Lagrangian systems. Dynamics within orbits of broken symmetry provide a natural arena for formation of time crystals. We exhibit models of that kind, including a model with traveling density waves.

Convergence of parallel block SSOR multisplitting method for block H-matrix

In this paper we investigate block SSOR multisplittings. When the coefficient matrix is a block H-matrix or a (generalized) block strictly diagonally dominant matrix, the convergence of the parallel block SSOR multisplitting method for solving nonsingular linear systems is proved. Two numerical examples are given to illustrate the theoretical results.

Identification and frequency domain quasi-maximum likelihood estimation of linearized dynamic stochastic general equilibrium models

This paper considers issues related to identification, inference, and computation in linearized dynamic stochastic general equilibrium (DSGE) models. We first provide a necessary and sufficient condition for the local identification of the structural parameters based on the (first and) second order properties of the process. The condition allows for arbitrary relations between the number of observed endogenous variables and structural shocks, and is simple to verify. The extensions, including identification through a subset of frequencies, partial identification, conditional identification, and identification under general nonlinear constraints, are also studied. When lack of identification is detected, the method can be further used to trace out nonidentification curves. For estimation, restricting our attention to nonsingular systems, we consider a frequency domain quasi-maximum likelihood estimator and present its asymptotic properties. The limiting distribution of the estimator can be different from results in the related literature due to the structure of the DSGE model. Finally, we discuss a quasi-Bayesian procedure for estimation and inference. The procedure can be used to incorporate relevant prior distributions and is computationally attractive.

Simultaneous semi-global [formula omitted]-stabilization and asymptotical stabilization for singular systems subject to input saturation

The simultaneous semi-global Lp-stabilization and asymptotic stabilization in both semi-global and global cases are considered for continuous-time linear singular systems subject to input saturation. Based on the asymptotical property of the parametric generalized algebraic Riccati equations, a low-and-high gain state feedback control law and a scheduling technique are proposed. When applied to the singular system subject to input saturation and external disturbance, the semi-global finite-gain Lp-stabilization and semi-global/global asymptotic stabilization are obtained simultaneously. The obtained results reduce to the existing ones for non-singular linear systems when the linear singular system reduces to a non-singular system. Furthermore, our results also improve the semi-global stabilization results for singular systems with saturation in the absence of the external disturbance.

Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation

We establish a partial regularity result for weak solutions of nonsingular parabolic systems with subquadratic growth of the type \partial_t u - \mathrm{div} a(x,t,u,Du) = B(x,t,u,Du), where the structure function a satisfies ellipticity and growth conditions with growth rate \frac{2n}{n+2} < p < 2 . We prove Hölder continuity of the spatial gradient of solutions away from a negligible set. The proof is based on a variant of a harmonic type approximation lemma adapted to parabolic systems with subquadratic growth.