Search

In this paper for linear discrete time switched systems, the problem of existence of a common solution to Stein inequalities is considered. A sufficient condition for robust Schur stability of a matrix polyope by using Schur complement lemma and a necessary and sufficient condition for the existence of a common solution of Stein equation are given. As in the case of continuous time systems, the problem of existence of a common solution is reduced to a convex optimization one. An efficient solution algorithm which requires solving a linear minimax problem at each step is suggested. The algorithm is supported with a number of examples from the literature and observed that the method desired results fastly. [ABSTRACT FROM AUTHOR]

For Lipschitz test functions we propose a new bound of the solution of the Stein equation related to the generalized inverse Gaussian (resp. the Kummer) distribution. This bound is derived using the general approach established in [4] for distributions satisfying a certain differential equation, and thus is optimal for Lipschitz test functions. Themain contribution of this paper is to establish an explicit expression of the bound as a function of the parameters of the distribution in terms of the modified Bessel function of the third kind (resp. the confluent hypergeometric function of the second kind). Under a restriction on the parameters we also obtain an optimal bound for the first derivative of the solution. A recurrence formula is established using the iterative technique developed in [4, 5] in order to bound derivatives of any order, for test functions smooth enough. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of large scale singular (continuous‐time) Lyapunov equations of the form AX + XA⊤ + BB⊤ = 0, where A is semistable, that is, its spectrum is contained in the left half plane, with the exception of a few semisimple eigenvalues at zero. We also consider the case of a few semisimple eigenvalues on the imaginary axis. We assume that we know these few eigenvalues (zero or imaginary), and that we have or can compute the corresponding invariant subspaces. We use this information to build an appropriate newly proposed subspace on which to project the Lyapunov equations, and then compute a low‐rank approximation to the least squares solution. Selected illustrative numerical examples are provided. [ABSTRACT FROM AUTHOR]

A classical result in random matrix theory reveals that the limiting spectral distribution of a Wigner matrix whose entries have a common variance and satisfy other regular assumptions almost surely converges to the semicircular law. In the paper, we will relax the assumption of uniform covariance of each entry, when the average of the normalized sums of the variances in each row of the data matrix converges to a constant, we prove that the same limiting spectral distribution holds. A similar result on a sample covariance matrix is also established. The proofs mainly depend on the Stein equation and the generalized Stein equation of independent random variables. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a c-stable linear equation in the tensor product space [ABSTRACT FROM AUTHOR]

Given the square matrices A,B,D,E and the matrix C of conforming dimensions, we consider the linear matrix equation AXE + DXB = C in the unknown matrix X. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing. [ABSTRACT FROM AUTHOR]

We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and *-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of 2 × 2 block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations. [ABSTRACT FROM AUTHOR]

SUMMARY A squared Smith type algorithm for solving large-scale discrete-time Stein equations is developed. The algorithm uses restarted Krylov spaces to compute approximations of the squared Smith iterations in low-rank factored form. Fast convergence results when very few iterations of the alternating direction implicit method are applied to the Stein equation beforehand. The convergence of the algorithm is discussed and its performance is demonstrated by several test examples. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this survey paper, a summary of results which are to be found in a series of papers, is presented. The subject of interest is focused on matrix algebraic properties of the Fisher information matrix (FIM) of stationary processes. The FIM is an ingredient of the Cram'er-Rao inequality, and belongs to the basics of asymptotic estimation theory in mathematical statistics. The FIM is interconnected with the Sylvester, Bezout and tensor Sylvester matrices. Through these interconnections it is shown that the FIM of scalar and multiple stationary processes fulfill the resultant matrix property. A statistical distance measure involving entries of the FIM is presented. In quantum information, a different statistical distance measure is set forth. It is related to the Fisher information but where the information about one parameter in a particular measurement procedure is considered. The FIM of scalar stationary processes is also interconnected to the solutions of appropriate Stein equations, conditions for the FIM to verify certain Stein equations are formulated. The presence of Vandermonde matrices is also emphasized. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the factored alternating directions implicit (ADI) iteration for large and sparse Sylvester equations. A novel low-rank expression for the associated Sylvester residual is established which enables cheap computations of the residual norm along the iteration, and which yields a reformulated factored ADI iteration. The application to generalized Sylvester equations is considered as well. We also discuss the efficient handling of complex shift parameters and reveal interconnections between the ADI iterates w.r.t. those complex shifts. This yields a further modification of the factored ADI iteration which employs only an absolutely necessary amount of complex arithmetic operations and storage, and which produces low-rank solution factors consisting of entirely real data. Certain linear matrix equations, such as, e.g., cross Gramian Sylvester and Stein equations, are in fact special cases of generalized Sylvester equations and we show how specially tailored low-rank ADI iterations can be deduced from the generalized factored ADI iteration. [Copyright &y& Elsevier]

In this survey paper, a summary of results which are to be found in a series of papers, is presented. The subject of interest is focused on matrix algebraic properties of the Fisher information matrix (FIM) of stationary processes. The FIM is an ingredient of the Cram'er-Rao inequality, and belongs to the basics of asymptotic estimation theory in mathematical statistics. The FIM is interconnected with the Sylvester, Bezout and tensor Sylvester matrices. Through these interconnections it is shown that the FIM of scalar and multiple stationary processes fulfill the resultant matrix property. A statistical distance measure involving entries of the FIM is presented. In quantum information, a different statistical distance measure is set forth. It is related to the Fisher information but where the information about one parameter in a particular measurement procedure is considered. The FIM of scalar stationary processes is also interconnected to the solutions of appropriate Stein equations, conditions for the FIM to verify certain Stein equations are formulated. The presence of Vandermonde matrices is also emphasized. [ABSTRACT FROM AUTHOR]

We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with $A_k = A^{2^k}$ not explicitly computed but in the recursive form $A_k = A_{k-1}^{2}$, and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension n, the resulting algorithms are of an efficient O( n) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results. [ABSTRACT FROM AUTHOR]

Efficient numerical algorithms for the solution of large and sparse matrix Riccati and Lyapunov equations based on the low rank alternating directions implicit (ADI) iteration have become available around the year 2000. Over the decade that passed since then, additional methods based on extended and rational Krylov subspace projection have entered the field and proved to be competitive alternatives. In this survey we sketch both types of methods and discuss their advantages and drawbacks. We focus on the continuous time case here, but corresponding results for discrete time problems can for most results be found in the available literature. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Abstract: We consider the problem of testing the hypothesis on marginal distribution of ergodic diffusion with Fisher–Snedecor invariant distribution, to be called Fisher–Snedecor diffusion. We propose a GMM approach to testing this statistical hypothesis where the moment condition is based on eigenfunctions of the diffusion infinitesimal generator—Fisher–Snedecor polynomials. Statistical test is observed in two different settings: (1) for known values of parameters of the process; (2) for consistent moment based estimators of parameters. Results are illustrated in a short simulation study. [Copyright &y& Elsevier]

One of the main uses of radiation aerial monitoring is in the mapping of an area contaminated by radioactive isotopes. In this work, we will introduce two independent approaches, redundancy and complementary methods, to estimate the radiation activity level of a contaminated area, using air monitoring information. Due to mathematical limitations, aerial monitoring of a radioactive field can not provide a 3d distribution of the contamination within the radioactive cloud. However, a reasonably good estimation of the 2d contamination distribution on the ground can be obtained by measuring the radioactive field from a constant height of 100 to 500 meters. Two complementary independent approaches have to be used in order to solve this problem: the elementary matrix solution after a conditional simulation and the application based on Stein equation. [ABSTRACT FROM AUTHOR]

We consider the problem of parameter estimation for an ergodic diffusion with the symmetric scaled Student invariant distribution, where the spectral representation of the transition density is given in terms of the finite number of polynomial eigenfunctions (Routh-Romanovski polynomials) and absolutely continuous spectrum of the negative infinitesimal generator of observed diffusion. We prove the consistency and asymptotic normality of the proposed estimators and, based on the Stein equation for Student diffusion, consider the statistical test for the Student distributional assumptions. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, we propose preconditioned Krylov methods for solving large Lyapunov matrix equations . Such problems appear in control theory, model reduction, circuit simulation and others. Using the Alternating Direction Implicit (ADI) iteration method, we transform the original Lyapunov equation to an equivalent symmetric Stein equation depending on some ADI parameters. We then define the Smith and the low rank ADI preconditioners. To solve the obtained Stein matrix equation, we apply the global Arnoldi method and get low rank approximate solutions. We give some theoretical results and report numerical tests to show the effectiveness of the proposed approaches. [Copyright &y& Elsevier]

Abstract: We consider the problem of parameter estimation for an ergodic diffusion with reciprocal gamma invariant distribution. Spectral decomposition of the transition density of such a Markov process is presented in terms of a finite number of discrete eigenfunctions (Bessel polynomials) and eigenfunctions related to a continuous part of the spectrum of the negative infinitesimal generator of reciprocal gamma diffusion. Consistency and asymptotical normality of proposed estimators are presented. Based on the Stein equation for reciprocal gamma diffusion and Bessel polynomials, the hypothesis testing procedure is considered. [Copyright &y& Elsevier]

Abstract: We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control of Markovian jump linear systems. Two iterations for computing a symmetric (maximal) solution of this system are investigated. We construct sequences of the solutions of the decoupled Stein equations and show that these sequences converge to a solution of the considered system. Numerical experiments are given. [Copyright &y& Elsevier]

Abstract: We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control. Two iterations for computing a symmetric solution of this system are investigated. New iterations are based on the properties of a Stein equation. It is necessary to solve a Stein equation at each step of proposed algorithms. We adapt the conditions for convergence several previous iterations presented in the literature for solving rational Riccati equations arising in stochastic control. [Copyright &y& Elsevier]

Abstract: Let and let be Hermitian matrices. Some already known results, including the general inertia theorem, give partial answers to the following problem: find a complete set of relations between the similarity class of L and the congruence classes of H and K, when the Lyapunov equation is satisfied. In this paper, we solve this problem when L is nonderogatory, H is nonsingular and K has at least one eigenvalue with positive real part and one eigenvalue with negative real part. Our result generalizes a previous paper by L. M. DeAlba. The corresponding problem with the Stein equation follows easily using a Cayley transform. [Copyright &y& Elsevier]

Abstract: We consider different iterative methods for computing a Hermitian or maximal Hermitian solution of two types of rational Riccati equations arising in stochastic control. The classical Newton procedure and its modification applied to equations are very expensive. New less expensive iterations for these equations are introduced and some convergence properties of new iterations are proved. [Copyright &y& Elsevier]

Abstract: Let and let be Hermitian matrices. The general inertia theorem gives a complete set of relations between the similarity class of L and the congruence class of H, when the Lyapunov equation LH + HL ∗ = K is satisfied and K >0. In this paper, we give some relations between the similarity class of L and the congruence class of K, when the Lyapunov equation is satisfied and H >0. We also consider the corresponding problem with the Stein equation. [Copyright &y& Elsevier]

A new stability preserving model reduction algorithm for discrete linear SISO-systems based on a least squares approach is proposed. Similar to the Pade´ approximation, an equation system for the Markov parameters involving a high dimensional Hankel matrix is considered. It is proved that approximate solutions, computed via the Moore–Penrose pseudo-inverse, give rise to a stability preserving reduction scheme. Furthermore, the proposed algorithm is compared to the balanced truncation method, showing comparable performance of the reduced systems. [Copyright &y& Elsevier]