on computing the distance to stability for matrices using linear dissipative hamiltonian systems abstract in this paper we consider the problem of computing the nearest stable matrix to an unstable one we propose new algorithms to solve this problem based on a reformulation using linear dissipative hamiltonian systems we show that a matrix a is stable if and only if it can be written as a j r q where j j t r 0 and q 0 that is r is positive semidefinite and q is positive definite this reformulation results in an equivalent optimization problem with a simple convex feasible set we propose three strategies to solve the problem in variables j r q i a block coordinate descent method ii a projected gradient descent method and iii a fast gradient method inspired from smooth convex optimization these methods require o n 3 operations per iteration where n is the size of a we show the effectiveness of the fast gradient method compared to the other approaches and to several state of the art algorithms

numerical solution of the discrete time convergent non negative definite lyapunov equation abstract an algorithm for the numerical solution of the real discrete time non negative definite lyapunov equation is discussed an improved rank 2 updating formula is proposed for the cholesky factor that arises during the numerical solution

the projective method for solving linear matrix inequalities numerous problems in control and systems theory can be formulated in terms of linear matrix inequalities lmi since solving an lmi amounts to a convex optimization problem such formulations are known to be numerically tractable however the interest in lmi based design techniques has really surged with the introduction of efficient interior point methods for solving lmis with a polynomial time complexity this paper describes one particular method called the projective method simple geometrical arguments are used to clarify the strategy and convergence mechanism of the projective algorithm a complexity analysis is provided and applications to two generic lmi problems feasibility and linear objective minimization are discussed

properties of the entire set of hurwitz polynomials and stability analysis of polynomial families it is proved in this paper that all hurwitz polynomials of order not less than n form two simply connected borel cones in the polynomial parameter space based on this result edge theorems for hurwitz stability of general polyhedrons of polynomials and boundary theorems for hurwitz stability of compact sets of polynomials are obtained both cases of families of polynomials with dependent and independent coefficients are considered different from the previous ones our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomial families and do not require the convexity or the connectivity of the set of polynomials moreover our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine

on the inverse and determinant of general bordered tridiagonal matrices in the current paper we present a symbolic algorithm that never suffers from breakdown for computing the inverse of general bordered tridiagonal matrices meanwhile an explicit formula of evaluating the determinant of a bordered tridiagonal matrix is also derived the implementation of the proposed algorithm using computer algebra systems cass is straightforward some numerical examples are given in order to demonstrate the performance and validity of the presented results

analytical inversion of general periodic tridiagonal matrices in this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed toeplitz for both periodic and non periodic tridiagonal matrices an efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given it is suited for implementation using computer algebra systems such as maple matlab macsyma and mathematica an example is also given to illustrate the algorithm

computing a hurwitz factorization of a polynomial a polynomial is called a hurwitz polynomial sometimes when the coefficients are real a stable polynomial if all its roots have real part strictly less than zero in this paper we present a numerical method for computing the coefficients of the hurwitz factor f z of a polynomial p z it is based on a polynomial description of the classical lr algorithm for solving the matrix eigenvalue problem similarly with the matrix iteration it turns out that the proposed scheme has a global linear convergence and moreover the convergence rate can be improved by considering the technique of shifting our numerical experiments performed with several test polynomials indicate that the algorithm has good stability properties since the computed approximation errors are generally in accordance with the estimated condition numbers of the desired factors

matrix stabilization using differential equations we consider the problem of stabilizing a matrix by a correction of minimal norm given a square matrix that has some eigenvalues with positive real part find the nearest matrix having no eigenvalu

brief paper a hierarchy of lmi inner approximations of the set of stable polynomials exploiting spectral properties of symmetric banded toeplitz matrices we describe simple sufficient conditions for the positivity of a trigonometric polynomial formulated as linear matrix inequalities lmis in the coefficients as an application of these results we derive a hierarchy of convex lmi inner approximations affine sections of the cone of positive definite matrices of size m of the nonconvex set of schur stable polynomials of given degree n

approximating the nearest stable discrete time system abstract in this paper we consider the problem of stabilizing discrete time linear systems by computing a nearby stable matrix to an unstable one to do so we provide a new characterization for the set of stable matrices we show that a matrix a is stable if and only if it can be written as a s 1 u b s where s is positive definite u is orthogonal and b is a positive semidefinite contraction that is the singular values of b are less or equal to 1 this characterization results in an equivalent non convex optimization problem with a feasible set on which it is easy to project we propose a very efficient fast projected gradient method to tackle the problem in variables s u b and generate locally optimal solutions we show the effectiveness of the proposed method compared to other approaches

oscillation of solutions of neutral partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of neutral partial functional differential equations of the form t p t t u x t i 1 l i t u x t i a t u x t k 1 s a k t u x t k t q x t u x t j 1 m q j x t f j u x t j x t 0 g where is a bounded domain inrnwith a piecewise smooth boundary and is the laplacian in the euclideann spacern

the use of the sherman morrison woodbury formula to solve cyclic block tri diagonal and cyclic block penta diagonal linear systems of equations the article presents a new theoretical viewpoint of batista s algorithms for solving cyclic tri diagonal and penta diagonal linear systems the theory is based on the sherman morrison woodbury formula

stabilization via nonsmooth nonconvex optimization nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixed order controller design we support this claim by applying nonsmooth analysis and methods to a challenging belgian chocolate stabilization problem posed in 1994 find a stable minimum phase rational controller that stabilizes a specified second order plant although easily stated this particular problem remained unsolved until 2002 when a solution was found using an eleventh order controller our computational methods find a stabilizing third order controller without difficulty suggesting explicit formulas for the controller and for the closed loop system which has only one pole with multiplicity 5 furthermore our analytical techniques prove that this controller is locally optimal in the sense that there is no nearby controller with the same order for which the closed loop system has all its poles further left in the complex plane although the focus of the paper is stabilization once a stabilizing controller is obtained the same computational techniques can be used to optimize various measures of the closed loop system including its complex stability radius or hinfin performance

subspace identification with guaranteed stability using constrained optimization in system identification the true system is often known to be stable however due to finite sample constraints modeling errors plant disturbances and measurement noise the identified model may be unstable we present a constrained optimization method to ensure asymptotic stability of the identified model in the context of subspace identification methods in subspace identification we first obtain an estimate of the state sequence or extended observability matrix and then solve a least squares optimization problem to estimate the system parameters to ensure asymptotic stability of the identified model we write the least squares optimization problem as a convex linear programming problem with mixed equality quadratic and positive semidefinite constraints suitable for existing convex optimization codes such as sedumi we present examples to illustrate the method and compare to existing approaches

brief paper smooth trivial vector bundle structure of the space of hurwitz polynomials due to the importance of the robustness of stable systems the properties of the set of hurwitz real polynomials are interesting to study among these properties is that the set is an open and not a convex set and recently it was proved that the set of hurwitz polynomials of degree n with positive coefficients h n is contractible now we will prove that the set h n is a smooth trivial vector bundle over h n k of rank k

on the variation of the spectrum of a normal matrix abstract let a and a be two n n normal matrices with spectra and j then by the hoffman wielandt theorem there is a permutation of 1 2 n such that n j 1 j j 2 a a f where f denotes the frobenius norm however if a is normal but a nonnormal it may be asked how to relate the eigenvalues of a to those of a an answer is given in this paper there is a permutation of 1 2 n such that n j 1 j j 2 n a a f and the factor n is best possible as a corollary we have j j n a a 2 for the spectral norm 2 thus the known upper bound 2n 1 a a 2 is reduced by a factor of about two

optimality conditions for maximizing a function over a polyhedron we present new first and second order optimality conditions for maximizing a function over a polyhedron these conditions are expressed in terms of the first and second order directional derivatives along the edges of the polyhedron and an edge description of the polyhedron if the objective function is quadratic and edge convex and the constraint polyhedron includes a box constraint then local optimality can be checked in polynomial time the theory is applied to continuous formulations of the vertex and edge separator problems it is seen that the necessary and sufficient optimality conditions for these problems are related to the existence of edges at specific locations in the graph

inversion of k tridiagonal matrices with toeplitz structure in this paper we consider an inverse problem with the k tridiagonal toeplitz matrices a theoretical result is obtained that under certain assumptions the explicit inverse of a k tridiagonal toeplitz matrix can be derived immediately two numerical examples are given to demonstrate the validity of our results

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

determining the closest stable polynomial to an unstable one the problem of being given a polynomial whose zeros do not all lie on or inside the unit circle and finding the closest polynomial whose zeros are all on or inside the unit circle is considered the measure of closeness used is the weighted euclidean distance in coefficient space the algorithm can be extended to other measures of closeness as well because the direct minimization on the coefficient space is difficult the problem is approached in schur coefficient space in this way the stability condition is easily guaranteed a very efficient algorithm for obtaining the optimum solution is developed

numerical algorithms for solving comrade linear systems based on tridiagonal solvers recently two efficient algorithms for solving comrade linear systems have been proposed by karawia a a karawia two algorithms for solving comrade linear systems appl math comput 189 2007 291 297 the two algorithms are based on the lu decomposition of the comrade matrix in this paper two algorithms are presented for solving the comrade linear systems based on the use of conventional fast tridiagonal solvers and an efficient way of evaluating the determinant of the comrade matrix is discussed

on nonsingularity verification of uncertain matrices over a quadratically constrained set nonsingularity of a matrix is investigated in this technical note that depends affinely on several uncertainty matrices whose real and imaginary parts are constrained by some quadratic forms based on some equivalent descriptions for these uncertainty matrices a sufficient condition is derived this condition is given in a linear matrix inequality form and can in principle be efficiently validated the obtained result includes some well known results in systems and control theory as a special case such as the lyapunov equation based stability criterion for linear time invariant systems the d g scaling based upper bound for the mixed structured singular value a sufficient condition for the stability of spatially interconnected systems etc a computationally verifiable condition is also provided that indicates situations under which the aforementioned condition becomes necessary

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

continuous quadratic programming formulations of optimization problems on graphs four np hard optimization problems on graphs are studied the vertex separator problem the edge separator problem the maximum clique problem and the maximum independent set problem we show that the vertex separator problem is equivalent to a continuous bilinear quadratic program this continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem the maximum clique problem and the maximum independent set problem all of these formulations when expressed as maximization problems are shown to follow from the convexity properties of the objective function along the edges of the feasible set an algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators computational results are given

a new recursive algorithm for inverting general tridiagonal and anti tridiagonal matrices in the current article the authors present a new recursive symbolic computational algorithm for inverting general tridiagonal and anti tridiagonal matrices an illustrative example is given

directional quasi convexity asymmetric schur convexity and optimality of consecutive partitions the current paper has the following distinct goals 1 to generalize standard quasi convexity to obtain a weaker property of functions that suffices for optimality of extreme points specifically it is required that intersections of the level sets of the function with line segments having direction in a prescribed set are convex 2 to characterize schur convexity for symmetric functions through the above generalization of quasi convexity thereby obtaining a previously unknown relationship of schur convexity and standard convexity the new characterization is used to extend the definition of schur convexity to functions which are not symmetric 3 to obtain new sufficient conditions for the optimality of consecutive partitions by using the new definition of schur convexity for functions that are not necessarily symmetric the conclusion that schur convexity is an instance of a convexity property that implies the optimality of extreme points unifies two approaches that were used in the literature to prove optimality of subsets of the domain of real valued functions the two tools quasi convexity and schur convexity were previously considered as distinct techniques

nearest stable system using successive convex approximations stability is a crucial property in the study of dynamical systems we focus on the problem of enforcing the stability of a system a posteriori the system can be a matrix or a polynomial either in continuous time or in discrete time we present an algorithm that constructs a sequence of successive stable iterates that tend to a nearby stable approximation x of a given system a the stable iterates are obtained by projecting a onto the convex approximations of the set of stable systems some possible applications for this method are correcting the error arising from some noise in system identification and a possible solver for bilinear matrix inequalities based on convex approximations in the case of polynomials a fair complexity is achieved by finding a closed form solution to first order optimality conditions

new algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems abstract recently an efficient computational algorithm for solving periodic pentadiagonal linear systems has been proposed by karawia a a karawia a computational algorithm for solving periodic pentadiagonal linear systems appl math comput 174 2006 613 618 the algorithm is based on the lu factorization of the periodic pentadiagonal matrix in this paper new algorithms are presented for solving periodic pentadiagonal linear systems based on the use of any pentadiagonal linear solver in addition an efficient way of evaluating the determinant of a periodic pentadiagonal matrix is discussed the corresponding results in this paper can be readily obtained for solving periodic tridiagonal linear systems