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A new approach to the transformations of the matrices of the fractional linear systems with desired eigenvalues is proposed. Conditions for the existence of the solution to the transformation problem of the linear system to its asymptotically stable controllable and observable canonical forms with desired eigenvalues are given and illustrated by numerical examples of fractional linear systems. [ABSTRACT FROM AUTHOR]

We consider interval systems of linear equations and inequalities with a restriction to some a priori given set. We focus on a characterization of strong solvability, that is, solvability for each realization of interval values, and we compare this with an existence of a strong solution defined analogously. The motivation comes from the area of interval-valued optimization problems, where strong solvability means guaranteed feasibility of any realization of the problem. Strong solvability with strict inequalities implies the robust Slater condition, which ensures that standard optimality conditions can be used. We apply the issues particularly in two optimization classes, convex quadratic programming with quadratic constraints and posynomial geometric programming. For the former, we also utilize the presented result to improve a characterization of the worst case optimal value. Eventually, we state several open problems that emerged while deriving the results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the uniform exponential stability of the system dx(t)dt=Ax(t)−ρBx(t),(ρ>0),$$ \frac{dx(t)}{dt}= Ax(t)-\rho Bx(t),\kern3.0235pt \left(\rho >0\right), $$ where the unbounded operator A$$ A $$ is the infinitesimal generator of a linear C0−$$ {C}_0- $$semigroup of contractions S(t)$$ S(t) $$ in a Hilbert space X$$ X $$ and B$$ B $$ is a Desch–Schappacher operator. Then we give sufficient conditions for exponential stability of the above system. The obtained stability result is then applied to prove the uniform exponential stabilization of some bilinear partial differential equations. [ABSTRACT FROM AUTHOR]

Summary This paper studies the problem of sensor fault estimation for linear systems in the presence of unknown inputs in both input and output channels. Relying on matrix equation theory, the unknown inputs are first removed from the output channel. Second, an augmented descriptor system is constructed by treating both system state and sensor fault as (pseudo)‐state variables. Third, the augmented descriptor system is transformed into a regular one, followed by the elimination of the unknown input from the state equation. Finally, a reduced‐order prescribed‐time sensor fault estimator is obtained by using periodic delayed observers. This matrix equation based approach is then complemented by an alternative one, which relies on successive transformations of state, input and outputs. This approach sheds light on the adopted solvability conditions throughout the paper and on the role of different directions in the output space in handling unknown inputs, sensor faults and the core state estimation problem. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed methodologies. [ABSTRACT FROM AUTHOR]

GMRES-based iterative refinement in three precisions (GMRES-IR3), proposed by Carson and Higham in 2018, uses a low precision LU factorization to accelerate the solution of a linear system without compromising numerical stability or robustness. GMRES-IR3 solves the update equation of iterative refinement using GMRES preconditioned by the LU factors, where all operations within GMRES are carried out in the working precision u, except for the matrix--vector products and the application of the preconditioner, which require the use of extra precision u2. The use of extra precision can be expensive, and is especially unattractive if it is not available in hardware; for this reason, existing implementations have not used extra precision, despite the absence of an error analysis for this approach. In this article, we propose to relax the requirements on the precisions used within GMRES, allowing the use of arbitrary precisions up for applying the preconditioned matrix--vector product and ug for the rest of the operations. We obtain the fiveprecision GMRES-based iterative refinement (GMRES-IR5) algorithm which has the potential to solve relatively badly conditioned problems in less time and memory than GMRES-IR3. We develop a rounding error analysis that generalizes that of GMRES-IR3, obtaining conditions under which the forward and backward errors converge to their limiting values. Our analysis makes use of a new result on the backward stability of MGS-GMRES in two precisions. On hardware where three or more arithmetics are available, which is becoming very common, the number of possible combinations of precisions in GMRES-IR5 is extremely large. We provide an analysis of our theoretical results that identifies a relatively small subset of relevant combinations. By choosing from within this subset one can achieve different levels of tradeoff between cost and robustness, which allows for a finer choice of precisions depending on the problem difficulty and the available hardware. We carry out numerical experiments on random dense and SuiteSparse matrices to validate our theoretical analysis and discuss the complexity of GMRES-IR5. [ABSTRACT FROM AUTHOR]

We introduce a mixed precision algorithm for computing sparse matrix-vector products and use it to accelerate the solution of sparse linear systems by iterative methods. Our approach is based on the idea of adapting the precision of each matrix element to their magnitude: we split the elements into buckets and use progressively lower precisions for the buckets of progressively smaller elements. We carry out a rounding error analysis of this algorithm that provides us with an explicit rule to decide which element goes into which bucket and allows us to rigorously control the accuracy of the algorithm. We implement the algorithm on a multicore computer and obtain significant speedups (up to a factor 7\times) with respect to uniform precision algorithms, without loss of accuracy, on a range of sparse matrices from real-life applications. We showcase the effectiveness of our algorithm by plugging it into various Krylov solvers for sparse linear systems and observe that the convergence of the solution is essentially unaffected by the use of adaptive precision. [ABSTRACT FROM AUTHOR]

In response to the influence of motor interference, damping, friction, and other uncertain factors on the operation of electric power steering systems under extreme working conditions, this study proposes a control strategy for electric power steering systems based on an active disturbance rejection algorithm. In ADRC, the fastest tracking differentiator is used to arrange the transition process for the target signal, and the extended state observer compensates for the total disturbance in the system. Phase compensation has been performed on the monitoring torque by using the torque differentiation method. The Simulink/Carsim simulation results show that ADRC has significantly improved anti-disturbance performance compared to PID and fuzzy PID. When using ADRC, the tracking accuracy of the assisted current is enhanced by 45.8%–75.8%, and the current adjustment time is reduced by 35.6%–61.7%. After phase compensation, the monitoring torque overshoot is reduced by 83.3%. Therefore, the proposed control strategy improves EPS's robustness and steering feel. [ABSTRACT FROM AUTHOR]

A new approach to the transformations of the matrices of linear continuous-time systems to their canonical forms with desired eigenvalues is proposed. Conditions for the existence of solutions to the problems were given and illustrated by simple numerical examples. [ABSTRACT FROM AUTHOR]

The eigenvalues assignment problems for descriptor linear systems with state and its derivative feedbacks are considered herein. Necessary and sufficient conditions for the existence of solutions to the problems are established. The Euler and Tustin approximations of the continuous-time systems are analyzed. Procedures for computation of the feedbacks are given and illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

The conjugate gradient (CG) method is a classic Krylov subspace method for solving symmetric positive definite linear systems. We analyze an analogous semi-conjugate gradient (SCG) method, a special case of the existing semi-conjugate direction (SCD) methods, for unsymmetric positive definite linear systems. Unlike CG, SCG requires the solution of a lower triangular linear system to produce each semi-conjugate direction. We prove that SCG is theoretically equivalent to the full orthogonalization method (FOM), which is based on the Arnoldi process and converges in a finite number of steps. Because SCG's triangular system increases in size each iteration, Dai and Yuan [Study on semi-conjugate direction methods for non-symmetric systems, Int. J. Numer. Meth. Eng. 60(8) (2004), pp. 1383–1399] proposed a sliding window implementation (SWI) to improve efficiency. We show that the directions produced are still locally semi-conjugate. A counter-example illustrates that SWI is different from the direct incomplete orthogonalization method (DIOM), which is FOM with a sliding window. Numerical experiments from the convection-diffusion equation and other applications show that SCG is robust and that the sliding window implementation SWI allows SCG to solve large systems efficiently. [ABSTRACT FROM AUTHOR]

A Kalman reduced form is obtained for linear systems over Hermite rings. This reduced form gives information of the set of assignable polynomials to a given linear system. [ABSTRACT FROM AUTHOR]

A novel approach is proposed to suppress bounded exogenous disturbances in linear control systems using a PI controller. The approach is based on reducing the original problem to a nonconvex matrix optimization problem. A gradient method for finding the controller's parameters is derived and its justification is provided. The corresponding recurrence procedure is rather effective and yields quite satisfactory controllers in terms of engineering performance criteria. This paper continues a series of the author's research works devoted to the design of feedback control laws from an optimization point of view. [ABSTRACT FROM AUTHOR]

We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems. For the dynamic responses of continuous medium structures, the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation. Thus, a symplectic finite element method with energy conservation is constructed in this paper. A linear elastic system can be discretized into multiple elements, and a Hamiltonian system of each element can be constructed. The single element is discretized by the Galerkin method, and then the Hamiltonian system is constructed into the Birkhoffian system. Finally, all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme. Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate, it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm. The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic systems. [ABSTRACT FROM AUTHOR]

A problem on how to generate chaos from two 3D linear systems via switching control is investigated. Each linear system has the simplest algebraic structure with three parameters. Two basic conditions of all parameters are given. One of two linear systems is stable. The other is unstable. Switching signals of different quadratic surfaces are designed respectively to generate chaotic dynamical behaviors. The constructed quadratic surfaces can be bounded or unbounded. Numerical examples and corresponding simulations verify the feasibility and effectiveness of the designed switching signals of quadratic surfaces for generating chaos. [ABSTRACT FROM AUTHOR]

In this paper, a new preconditioning method is proposed for the linear system arising from the elliptic optimal control problem. It is based on row permutations of the linear system and approximations of the corresponding Schur complement inspired by the matching strategy. The eigenvalue bounds of the preconditioned matrices are shown to be independent of mesh size and regularization parameter. Numerical results illustrate the efficiency of the proposed preconditioning methods. [ABSTRACT FROM AUTHOR]

This study deals with the numerical solution of a class of linear systems of second-order boundary value problems (BVPs) using a new symmetric cubic B-spline method (NCBM). This is a typical cubic B-spline collocation method powered by new approximations for second-order derivatives. The flexibility and high order precision of B-spline functions allow them to approximate the answers. These functions have a symmetrical property. The new second-order approximation plays an important role in producing more accurate results up to a fifth-order accuracy. To verify the proposed method's accuracy, it is tested on three linear systems of ordinary differential equations with multiple step sizes. The numerical findings by the present method are quite similar to the exact solutions available in the literature. We discovered that when the step size decreased, the computational errors decreased, resulting in better precision. In addition, details of maximum errors are investigated. Moreover, simple implementation and straightforward computations are the main advantages of the offered method. This method yields improved results, even if it does not require using free parameters. Thus, it can be concluded that the offered scheme is reliable and efficient. [ABSTRACT FROM AUTHOR]

In this paper we propose a new probability distribution for the randomized Kaczmarz (RK) algorithm where each row of the coefficient matrix is selected in the current iteration with the probability proportional to the square of the sine of the angle between it and the chosen row in the previous iteration. This probability distribution is helpful to accelerate the convergence of the RK algorithm. We obtain the linear convergence rate estimation in expectation for the RK algorithm with the new probability distribution (RKn), which is different from the existing results. In order to avoid the calculation and storage of probability matrix when solving the large-scale linear systems, one practical probability distribution is also proposed. Furthermore, an acceleration for the RKn algorithm and its simplified version are introduced respectively, whose core is to replace the projection onto one hyperplane with that onto the intersection of two hyperplanes. The accelerated schemes are proved to have faster convergence rates. Finally, numerical experiments are provided to illustrate the performance of the proposed algorithms by comparing with the RK algorithm and Motzkin's algorithm. [ABSTRACT FROM AUTHOR]

Semiring is an algebraic structure of (S, +, ×). Similar to a ring, but without the condition that each element must have an inverse to the adding operation. The forms (S, +) and (S, ×) are semigroups that satisfy the distributive law of multiplication and addition. In matrix theory, there is a term known as the Kronecker product. This operation transforms two matrices into a larger matrix containing all possible products of the entries in the two matrices. This Kronecker product has several properties often used to solve the complex problems of linear algebra and its applications. The Kronecker product is related to the Lyapunov equation of a linear system. Based on previous research in the Lyapunov equation in conventional linear algebra, this paper will describe the characteristics of the Lyapunov equation in a semiring linear system in terms of the Kronecker product. [ABSTRACT FROM AUTHOR]

This paper introduces the Nyström preconditioned conjugate gradient (PCG) algorithm for solving a symmetric positive-definite linear system. The algorithm applies the randomized Nyström method to form a low-rank approximation of the matrix, which leads to an efficient preconditioner that can be deployed with the conjugate gradient algorithm. Theoretical analysis shows that the preconditioned system has constant condition number as soon as the rank of the approximation is comparable with the number of effective degrees of freedom in the matrix. The paper also develops adaptive methods that provably achieve similar performance without knowledge of the effective dimension. Numerical tests show that Nyström PCG can rapidly solve large linear systems that arise in data analysis problems, and it surpasses several competing methods from the literature. [ABSTRACT FROM AUTHOR]

We propose a new approach to filtering under arbitrary bounded exogenous disturbances based on reducing this problem to an optimization problem. The approach has a low computational complexity since only Lyapunov equations are solved at each iteration. At the same time, it possesses advantages essential from an engineering-practical point of view, namely, the possibilities to limit the filter matrix and to construct optimal filter matrices separately for each coordinate of the system's state vector. A gradient method for finding the filter matrix is presented. According to the examples, the proposed recurrence procedure is rather effective and yields quite satisfactory results. This paper continues the series of research works devoted to feedback control design from an optimization perspective. [ABSTRACT FROM AUTHOR]

The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed-precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so, we first develop a new error analysis for LU- and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that mixed-precision iterative refinement combined with approximate sparse factorization can lead to considerable reductions of both the time and memory consumption. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study three different types of algebraic symbolic 2-plithogenic equations. The symbolic 2-plithogenic linear Diophantine equations, symbolic 2-plithogenic quadratic equations, and linear system of symbolic 2-plithgenic equations will be discussed and handled, where algorithms to solve the previous types will be presented and proved by transforming them to classical algebraic systems of equations. [ABSTRACT FROM AUTHOR]

In this paper, the fractional centered difference formula is employed to discretize the Riesz derivative, while the Crank-Nicolson scheme is approximated the time derivative, and the explicit linearized technique is used to deal with nonlinear term, a second-order finite difference method is obtained for the nonlinear Riesz space-fractional diffusion equations, and the resulting system is symmetric positive definite ill-conditioned Toeplitz matrix and then the fast sine transform can be used to reduce the computational cost of the matrix-vector multiplication. The preconditioned conjugate gradient method with a preconditioner based on sine transform is proposed to solve the linear system. Theoretically, the spectrum of the preconditioned matrix falling in an open interval (1/2,3/2) is proved, which can guarantee the linear convergence rate of the proposed methods. By the similar technique, the two-dimension case is also studied. Finally, numerical experiments are carried out to demonstrate that the proposed preconditioner works well. [ABSTRACT FROM AUTHOR]

The bilateral filter is a non-linear edge-preserving filter that can be adopted in a variety of tasks in computer photography. However, the naive bilateral filter is computationally expensive. Existing researches on the acceleration of bilateral filter mostly concentrate on range approximation. Nevertheless, the range kernel has more impact on the bilateral filter than the spatial kernel. Range approximation would have more side effects. In this paper, we propose a novel approximation of the bilateral filter with spatial subsampling, where the affinity matrix is estimated from a subset of it. We show that the main computational burden of our approximation is a large linear system, for which we propose an efficient iterative algorithm to solve. We have carried out both quantitative and qualitative experiments to evaluate our fast bilateral filter. Experimental results suggest that the proposed filter outperforms the state-of-the-art methods in approximation accuracy. The proposed filter is highly efficient; under a moderate sampling rate, i.e., (1 / 5) × (1 / 5) , it needs 0.29s to process a color image with 1 megapixel on an Intel i7-9700 CPU. [ABSTRACT FROM AUTHOR]

This article studies the simultaneous fault‐tolerant control and disturbance rejection for systems with fault in polynomial form. The unknown external disturbance is considered to be caused by the external source system, and the fault can be expressed as a polynomial function of time. Combining the fault observer and the disturbance observer, a controller is designed, which can not only diagnose the fault, but also improve the anti‐disturbance performance. Finally, a numerical example is given to verify the validity of the results. [ABSTRACT FROM AUTHOR]

On‐board training of artificial neural network (ANN) is important in instances where real time data are required for model training. Provision of on‐board intelligence enables the developed systems to self‐recalibrate and enhances their efficiencies. In this work, investigations have been performed to determine optimized parameters of ANN model for linear systems. The performance parameters that is, model parameters, memory requirements, accuracy and processing time are chosen by considering the model to be installed on commercially available microcontrollers that have very limited on‐board memory. Minimum data requirements for training ANN models of linear systems are also explored for better performance. All dataset ranges are normalized in order to exclude the effects of range differences. It is shown that for linear systems, 1–3–1 architecture produces best results against ≤100 data points when Bayesian Regularization (BR) training function is used along with Log Sigmoid Activation function. Simulations for 1–3–1 architecture are then performed for datasets having 10, 25, 50 and 100 data points. The results show that training with 25 data points produces over‐all better performance than other datasets. A large dataset utilizes more training time and memory whereas a smaller dataset produces relatively lesser accuracy. The effects of clustered data and uniformly distributed data are also explored. It is found that total epochs in case of clustered data are significantly higher than uniformly distributed data. The combination of these optimized parameters that is, 1–3–1 architecture, with BR and Log function, for ≤100 data points can be used for the development and implementation of linear components or systems in resource‐constrained embedded systems. [ABSTRACT FROM AUTHOR]

We define degree of mutual dependence of three events in a probability space by using Boltzmann-Shannon entropy function of an appropriate variable distribution produced by these events and depending on four parameters varying, in general, within of a polytope. It turns out that the entropy function attains its absolute maximum exactly when the three events are mutually independent and its absolute minimum at some vertices of the polytope where the events are "maximally" dependent. By composing the entropy function with an appropriate linear function we obtain a continuous "degree of mutual dependence" function with the same domain and the interval [0, 1] as a target. It attains value 0 when the events are mutually independent (the entropy is maximal) and value 1 when they are "maximally" dependent (the entropy is minimal). A link is available for downloading a Java code which evaluates the degree of mutual dependence of three events in the classical case of a sample space with equally likely outcomes. [ABSTRACT FROM AUTHOR]

We propose a new approach to the problem of tuning and optimizing the parameters of a PID controller based on reducing the problem to an optimization problem. In this case, the performance of the controller is evaluated by a quadratic criterion of the system output: the PID controller is tuned against the uncertainty in the initial conditions so that the system output is uniformly small; in this case, a given degree of stability of the closed-loop system is additionally guaranteed. A gradient method for finding the PID controller parameters is given. Numerous examples show that the recurrent procedure proposed is very efficient and leads to PID controllers that are quite satisfactory in terms of engineering performance indices. The article continues a series of papers by the present authors devoted to synthesizing feedback in control problems from the standpoint of optimization. [ABSTRACT FROM AUTHOR]

This article describes the consensus control issue of networked multi-agent system with controller area network bus–induced delay in the frequency domain. Initially, causes of the controller area network bus–induced delay in distributed networked control systems and features of controller area network buses are analyzed. Then, networked multi-agent systems composed of distributed networked control systems are modeled in the frequency domain. An analytical controller based on the parameterization technique is proposed to improve the H 2 performance criterion of distributed networked control systems. The performance tracking and robustness of the controller is analyzed by its sensitivity functions. After the performance and robustness analysis, the delay margin criteria and consensus conditions of networked multi-agent systems are derived in the existence of controller area network bus–induced delay. We verify the efficiency of the proposed method by simulating examples. [ABSTRACT FROM AUTHOR]

Recently, it was demonstrated that a modification of the Kalman-filtering model with a pointwise delay of the signal noise could improve communication with considerably distanced spacecraft. However, a complete and correct derivation of the equations of the Kalman-type filter for this case has not yet been provided. In this paper, we close this gap. The method of derivation is based on a passage from the distributed delay to pointwise by means of a delta function. The derived equations constitute a system of first-order partial differential equations with the initial and boundary conditions. [ABSTRACT FROM AUTHOR]

In order to solve the problem of trajectory shift in iterative learning control, an adaptive iterative learning control strategy was proposed. For linear time-varying systems with unknown parameters, an adaptive parameter updating algorithm was constructed by solving a quadratic programming problem. The estimated parameter information and trajectory information with shift were used to design an adaptive iterative learning control strategy. The results show that the estimation error of parameters is bounded, and the tracking error of the system is bounded when the trajectory shift is bounded. The effectiveness and practicability of the proposed adaptive iterative learning control strategy are verified by numerical simulation. [ABSTRACT FROM AUTHOR]

G. Pareschi and M. Popa [J. Amer.Math. Soc. 16 (2003), pp. 285–302] give criterions for global generation and surjectivity of multiplication maps of global sections of coherent sheaves on abelian varieties in the theory of M-regularity. In this paper, we refine some of their criterions via the M-regularity of \mathbb {Q}-twisted sheaves introduced by Z. Jiang and G. Pareschi [Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), pp. 815–846]. As an application, we show that the M-regularity of a suitable \mathbb {Q}-twisted sheaf implies property (N_p) and jet-ampleness for ample line bundles on abelian varieties. [ABSTRACT FROM AUTHOR]

Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M-matrix in the Gauss–Seidel double SOR method to obtain a comparison result. [ABSTRACT FROM AUTHOR]

We present an inherently parallel method using a Taylor polynomial as an estimator for several numerical examples of strictly diagonally dominant and banded triangular Toeplitz systems. We demonstrate the effectiveness of the method using multiple precisions for each of four test cases, and compare the results versus classical Jacobi and Newton methods. We also show that the residual norms can be driven down to any arbitrary precision with less hardware requirements than existing methods. Finally, we use truncated Taylor series as initial estimates for the Newton, and in each case, we obtain equal or better accuracy than the Newton method alone, while using less vector computations. [ABSTRACT FROM AUTHOR]

Recently, Wang (2017) has introduced the K-nonnegative double splitting using the notion of matrices that leave a cone K ⊆ ℝn invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for K-weak regular and K-nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory for these sub-classes of matrices. We then obtain the comparison results for two double splittings of a K-monotone matrix. Most of these results are completely new even for K = ℝ + n . The convergence behavior is discussed by performing numerical experiments for different matrices derived from the discretized Poisson equation. [ABSTRACT FROM AUTHOR]

In this paper, we present a new iterative method for solving a linear system, whose coefficient matrix is an M-matrix. This method includes four parameters that are obtained by the accelerated overrelaxation (AOR) splitting and using the Taylor approximation. First, under some standard assumptions, we establish the convergence properties of the new method. Then, by minimizing the Frobenius norm of the iteration matrix, we find the optimal parameters. Meanwhile, numerical results on test examples show the efficiency of the new proposed method in contrast with the Hermitian and skew-Hermitian splitting (HSS), AOR methods and a modified version of the AOR (QAOR) iteration. [ABSTRACT FROM AUTHOR]

For the recently developed quasi-reversible BCH codes with long lengths and high error-correcting capability, this paper is aimed at proposing a new and faster decoding procedure. It consists of four steps: 1) compute the consecutive syndromes; 2) calculate the syndrome functions by the forward and backward recursions; 3) solve a linear subsystem together with one matrix multiplication in order to find an error-locator polynomial; 4) determine the errors from the obtained polynomial by using the root-finding algorithm. This procedure, especially in Steps 2 and 3, differs greatly from the conventional procedures, which determine an error-locator polynomial directly from solving a linear system with the aid of the consecutive syndromes. The key idea behind this decoding technique is that the computational complexity of such a small subsystem instead of an originally large linear system can be significantly reduced, although there are additional forward and backward syndrome calculations with low complexity increasing. Finally, the illustrative examples and numerical simulations can be helpful to demonstrate the accuracy and efficacy of the presented decoding technique at different error-correcting capabilities. [ABSTRACT FROM AUTHOR]

The paper further refines the clan composition technique that is considered a way of matrix partitioning into a union of block-diagonal and block-column matrices. This enables solving the individual systems for each horizontal block on a separate computing node, followed by solving the composition system. The size of minimal clans, obtained as a result of matrix decomposition, varies considerably. For load balancing, early versions of ParAd software were using dynamic scheduling of jobs. The present paper studies a task of static balancing the clan size. Rather good results are obtained using a fast bin packing algorithm with the first fit on a sorted array which are considerably improved applying a multi-objective graph partitioning with software package METIS. Aggregation of clans allows us to obtain up to three times extra speed-up, including systems over fields of real numbers, on matrices from Model Checking Contest and Matrix Market. [ABSTRACT FROM AUTHOR]

A new approach is proposed for solving the problem of suppressing nonrandom bounded exogenous disturbances in linear control systems using dynamic output feedback. The approach is based on reducing the problem to a matrix optimization problem with the feedback matrix and the observer matrix as the variables. A gradient method for finding dynamic output feedback is written out and justified. A number of examples are considered. [ABSTRACT FROM AUTHOR]

The problem of minimizing an integral quadratic functional on the trajectories of a linear singularly perturbed system with linear terminal constraints is considered. Asymptotic approximations are constructed in the form of a program and feedback to the optimal control in this problem. The main advantage of the proposed computational procedures is that, in their implementation, the original problem is decomposed into two unperturbed optimal control problems of lower dimension. [ABSTRACT FROM AUTHOR]

Efficient sampling from a high-dimensional Gaussian distribution is an old but high-stakes issue. Vanilla Cholesky samplers imply a computational cost and memory requirements that can rapidly become prohibitive in high dimensions. To tackle these issues, multiple methods have been proposed from different communities ranging from iterative numerical linear algebra to Markov chain Monte Carlo (MCMC) approaches. Surprisingly, no complete review and comparison of these methods has been conducted. This paper aims to review all these approaches by pointing out their differences, close relations, benefits, and limitations. In addition to reviewing the state of the art, this paper proposes a unifying Gaussian simulation framework by deriving a stochastic counterpart of the celebrated proximal point algorithm in optimization. This framework offers a novel and unifying revisiting of most of the existing MCMC approaches while also extending them. Guidelines to choosing the appropriate Gaussian simulation method for a given sampling problem in high dimensions are proposed and illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

Parameter uncertainties and measurement outliers unavoidably exist in a real linear system. Such uncertainties and outliers make the true joint state-measurement distributions (induced by the true system model) deviate from the nominal ones (induced by the nominal system model) so that the performance of the optimal state estimator designed for the nominal model becomes unsatisfactory or even unacceptable in practice. The challenges are to quantitatively describe the uncertainties in the model and the outliers in the measurements, and then robustify the estimator in a right way. This article studies a distributionally robust state estimation framework for linear systems subject to parameter uncertainties and measurement outliers. It utilizes a family of distributions near the nominal one to implicitly describe the uncertainties and outliers, and the robust state estimation in the worst case is made over the least-favorable distribution. The advantages of the presented framework include: 1) it only uses a few scalars to parameterize the method and does not require the structural information of uncertainties; 2) it generalizes several classical filters (e.g., the fading Kalman filter, risk-sensitive Kalman filter, relative-entropy Kalman filter, outlier-insensitive Kalman filters) into a unified framework. We show that the distributionally robust state estimation problem can be reformulated into a linear semi-definite program and in some special cases it can be analytically solved. Comprehensive comparisons with existing state estimation frameworks that are insensitive to parameter uncertainties and measurement outliers are also conducted. [ABSTRACT FROM AUTHOR]

In this study, we introduce the topic for exponential stability criterion of delayed linear system via intermittent feedback control. Utilizing the concept of fundamental for intermittent control is designed as continuous-time controller to generate open-loop predictions of states, for foreseeing the behavior of linear system in the future over the time-varying delay defined in (2.2). The open loop trajectory would declare the set of control I (t) into a closed-loop feedback control. For new sufficient condition of this system obtained in forms of linear matrix inequalities (LMIs), we apply the modified Lyapunov-Krasovskii functionals (LKF) merged with Jensen's inequality and Leibniz-Newton's technique. Numerical example is demonstrated the usefulness of the obtained intermittent feedback control (IFC). [ABSTRACT FROM AUTHOR]

It is shown that in an uncontrollable linear system ˙ x = Ax+Bu it is possible to assign arbitrarily the eigenvalues of the closedloop system with state feedbacks u = Kx, K €Rnxn if rank [A B] = n. The design procedure consists of two steps. In step 1, a nonsingular matrix M €nn is chosen so that the pair (MA, MB) is controllable. In step 2, the feedback matrix K is chosen so that the closed-loop matrix Ac = A-BK has the desired eigenvalues. The procedure is illustrated by a simple example. [ABSTRACT FROM AUTHOR]

We work over an algebraically closed field of characteristic zero. For a nondegenerate projective variety X ⊆ P N , the locus of points from which X is projected nonbirationally onto its image, is called the Segre locus of X. The purpose here is to give an upper bound of the number of the irreducible components of the Segre locus of a projective variety in terms of its invariants. [ABSTRACT FROM AUTHOR]

In this paper, a novel event-triggered observer is proposed to decrease the information exchange in continuous-time linear networked systems. The structure of the proposed observer consists of two main parts. The first one, called the event-triggered mechanism (ETM), is placed by the system and is used to determine when the measured output of the system must be sent through the network. The ETM consists of a traditional Luenberger observer, an event-triggered local observer, and an event-detector. The second part is a copy of the event-triggered local observer which is used to estimate the system state using the transmitted data from the sensors. Under the observability assumption, it is proved that the error between the system state and its estimation is asymptotically limited to a predetermined bound. Moreover, it is guaranteed that the Zeno behaviour is avoided. Numerical simulations are provided to demonstrate the design procedure of the proposed event-triggered observer. [ABSTRACT FROM AUTHOR]

It is well known that a finite-dimensional linear system cannot be chaotic. In this article, by introducing a weak topology into a two-dimensional Euclidean space, it shows that Li–Yorke chaos can be generated by a linear map, where the weak topology is induced by a linear functional. Some examples of linear systems are presented, some are chaotic while some others regular. Consequently, several open problems are posted. [ABSTRACT FROM AUTHOR]

A topoplane is a mild deformation of a linear hyperplane contained in a given smooth manifold that is homeomorphic to a Euclidean space. We consider solidly transsective topoplane arrangements. These collections generalize pseudohyperplane arrangements. Even though the topoplane arrangements locally look like hyperplane arrangements, the global coning procedure is absent here. The main aim of the paper is to introduce the Varchenko matrix in this context and show that the determinant has a similar factorization as in the case of hyperplane arrangements. We achieve this by suitably generalizing the strategy of Aguiar and Mahajan. We also study a system of linear equations introduced by them and describe its solution space in the context of topoplane arrangements. [ABSTRACT FROM AUTHOR]