exponential stability and solution bounds for systems with bounded nonlinearities estimation of lyapunov exponents of systems with bounded nonlinearities plays an essential part in their robust control known results in this field are based on the gronwall inequality yielding relatively conservative bounds for lyapunov exponents in this note we obtained more accurate upper bounds for the general lyapunov exponent for systems consisting of a known linear time varying part and an unknown nonlinear component with a bounded lipschitz constant at zero consequently a sufficient condition for exponential stability of this system is formulated the systems are indicated for which the obtained bound is precise i e cannot be improved without additional information on the nonlinear term in the presence of a persisting perturbation an upper bound for the solution norm is derived and expressed in the norm of the solution of the corresponding linear system using the obtained results a condition for exponential stability of a linear time varying control system with a nonlinear feedback is derived numerical results are obtained for a second order time varying system and for the lienard equation in the latter case they are favorably compared with stability conditions previously obtained using the lyapunov function method

brief paper robust control with a terminal state constraint this paper introduces a problem of robust control with a terminal state constraint this problem involves finding an output feedback controller to steer the initial state of a given system into an ellipsoidal bounding set in a specified time the underlying system in this problem is a time varying uncertain linear system in which the uncertainty satisfies an integral quadratic constraint necessary and sufficient conditions are given for the solution to this problem in terms of a pair of parameter dependent riccati differential equations

how to prove that a preconditioner cannot be superlinear in the general case of multilevel toeplitz matrices we recently proved that any multilevel circulant preconditioner is not superlinear a cluster it may provide cannot be proper the proof was based on the concept of quasi equimodular matrices although this concept does not apply for example to the sine transform matrices in this paper with a new concept of partially equimodular matrices we cover all trigonometric matrix algebras widely used in the literature we propose a technique for proving the nonsuperlinearity of certain frequently used preconditioners for some representative sample multilevel matrices at the same time we show that these preconditioners are in a certain sense the best among the sublinear preconditioners with only a general cluster for multilevel toeplitz matrices

singular values and eigenvalues of non hermitian block toeplitz matrices in this paper we are concerned with the analysis of the distribution and localization of the singular values of toeplitz matrices ftn f g generated by a p variate lebesgue integrable matrix valued function f q p c hk q we prove that the union of the essential ranges of the singular values of f is a proper weak cluster for the whole set of the singular values of ftn f g by showing that the number of outliers is strongly depending on the regularity features of the underlying function f in particular if f is continuous or from the krein algebra and p 1 then the cluster is proper other results concerning the extreme spectral behavior of ftn f g second order ergodic formulas and localization of eigenvalues of preconditioned matrices ft 1 n g tn f g are presented some examples of applications to the preconditioning of these results are also discussed c 1999 elsevier science b v all rights reserved msc 15a12 15a18 65f10

spectral and computational properties of band symmetric toeplitz matrices abstract we are investigating spectral properties of band symmetric toeplitz matrices bst matrices by giving a suitable representation of a bst matrix we achieve separation results and multiplicity conditions for the eigenvalues of a 7 r 5 diagonal bst matrix and also structural properties of the eigenvectors we give eigenvalue bounds for a 2k 1 diagonal bst matrix and also necessary and sufficient conditions for positive definiteness which are easy to check the same conditions apply in the case of block bst matrices either with full blocks or with bst blocks we exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a bst matrix either for sequential or for parallel computations two algorithms based on the bisection technique and newton s method are shown to be very fast for computing the eigenvalues of a 7 or 5 diagonal bst matrix

gain scheduling of minimax optimal state feedback controllers for uncertain lpv systems a new robust control gain scheduling scheme for uncertain linear parameter varying lpv systems is proposed the gain scheduled controller consists of a set of minimax optimal robust controllers and incorporates a new interpolation rule to achieve continuity of the controller gain over a range of operating conditions for every fixed system parameter the proposed controller guarantees a certain bound on the worst case performance of the corresponding uncertain closed loop system furthermore a bound on the rate of parameter variations is obtained under which the closed loop lpv system is robustly stable

functions of noncommuting self adjoint operators under perturbation and estimates of triple operator integrals abstract we define functions of noncommuting self adjoint operators with the help of double operator integrals we are studying the problem to find conditions on a function f on r 2 for which the map a b f a b is lipschitz in the operator norm and in schatten von neumann norms s p it turns out that for functions f in the besov class b 1 1 r 2 the above map is lipschitz in the s p norm for p 1 2 however it is not lipschitz in the operator norm nor in the s p norm for p 2 the main tool is triple operator integrals to obtain the results we introduce new haagerup like tensor products of l spaces and obtain schatten von neumann norm estimates of triple operator integrals we also obtain similar results for functions of noncommuting unitary operators

robustness of trajectories with finite time extent the problem of estimating perturbation bounds of finite trajectories is considered the trajectory is assumed to be generated by a linear system with uncertainty characterized in terms of integral quadratic constraints it is shown that such perturbation bounds can be obtained as the solution to a nonconvex quadratic optimization problem which can be addressed using lagrange relaxation the result can be used in robustness analysis of hybrid systems and switched dynamical systems

brief paper robust output feedback stabilizability via controller switching the paper considers the output feedback robust stabilizability problem for uncertain dynamical systems the uncertain system under consideration is a composite of a continuous time plant and a switched controller a necessary and sufficient condition of absolute stabilizability is given in terms of the existense of suitable solutions to a dynamic programming equation and a riccati algebraic equation of the h filtering type a real time implementable method for robust stabilization is also presented

the rate of convergence of toeplitz based pcg methods for second order nonlinear boundary value problems in previous works 21 23 we proposed the use of tau 5 and band toeplitz based preconditioners for the solution of 1d and 2d boundary value problems bvp by means of the preconditioned conjugate gradient pcg methods as tau and band toeplitz linear systems can be solved 4 by using fast sine transforms 8 these methods become especially attractive in a parallel environment of computation in this paper we extend this technique to the nonlinear nonsymmetric case and in addition we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear therefore these methods work much finer than those based on separable preconditioners 18 45 on incomplete lu factorizations 36 13 27 and on circulant preconditioners 9 30 35 since the latter two techniques do not assure a linear rate of convergence on the other hand the proposed technique has a wider range of application since it can be naturally used for nonlinear nonsymmetric problems and for bvp in which the coefficients of the differential operator are not strictly positive and only piecewise smooth finally the several numerical experiments performed here and in 22 23 confirm the effectiveness of the theoretical analysis

c g preconditioning for toeplitz matrices abstract we consider the problem of solving a toeplitz system of equations by conjugate gradient method when a sequence of nested toeplitz matrices is associated to a function the spectral behaviour of the matrices involved is closely related to the analytical properties of the generating function thus it is possible to devise efficient preconditioning techniques by using various functional approximation strategies this approach leads to attractive results in the case of ill conditioned matrices for which a wide class of preconditioners are proposed

essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices

superlinear convergence for pcg using band plus algebra preconditioners for toeplitz systems the paper studies fast and efficient solution algorithms for nxn symmetric ill conditioned toeplitz systems t n f x b where the generating function f is known a priori real valued nonnegative and has isolated roots of even order the preconditioner that we propose is a product of a band toeplitz matrix and matrices that belong to a certain trigonometric algebra the basic idea behind the proposed scheme is to combine the advantages of all components of the product that are well known when every component is used as a stand alone preconditioner as a result we obtain a flexible preconditioner which can be applied to the system t n f x b infusing superlinear convergence to the pcg method the important feature of the proposed technique is that it can be extended to cover the 2d case i e ill conditioned block toeplitz matrices with toeplitz blocks we perform many numerical experiments whose results confirm the theoretical analysis and effectiveness of the proposed strategy

absolute stability criteria for a generalized lur e problem with delay in the feedback a nonautonomous linear system controlled by a nonlinear sector restricted feedback with a time varying delay is considered delay independent sufficient conditions for absolute stability and instability expressed in the transfer function of the linear part and the sector bounds are established for a system with an exponentially stable linear part an upper bound for the lyapunov exponent is found it is shown that if the transfer function is sign constant asymptotic stability of the system with the margin linear feedback guarantees absolute stability of the considered system thus such systems satisfy the known aizerman conjecture they include in particular a closed loop system consisting of any number of time varying first order links and feedback with arbitrary delay under some additional condition which is certainly true for a time invariant linear block the obtained stability criterion is precise the approach employed in the proofs is based on a direct analysis of the corresponding volterra equation which contains only the transfer function of the linear block and therefore embraces a wide range of control systems as an example a second order system is considered it is shown that here the obtained stability bound is reached for a linear feedback with a periodic delay function

estimates of operator moduli of continuity abstract in aleksandrov and peller 2010 2 we obtained general estimates of the operator moduli of continuity of functions on the real line in this paper we improve the estimates obtained in aleksandrov and peller 2010 2 for certain special classes of functions in particular we improve estimates of kato 1973 18 and show that s t c s t log 2 log s t s t for all bounded operators s and t on hilbert space here s def s s 1 2 moreover we show that this inequality is sharp we prove in this paper that if f is a nondecreasing continuous function on r that vanishes on 0 and is concave on 0 then its operator modulus of continuity f admits the estimate f const e f t d t t 2 log t 0 we also study the problem of sharpness of estimates obtained in aleksandrov and peller 2010 2 3 we construct a c function f on r such that f l 1 f lip 1 and f const log 2 0 1 in the last section of the paper we obtain sharp estimates of f a f b in the case when the spectrum of a has n points moreover we obtain a more general result in terms of the e entropy of the spectrum that also improves the estimate of the operator moduli of continuity of lipschitz functions on finite intervals which was obtained in aleksandrov and peller 2010 2

two level toeplitz preconditioning approximation results for matrices and functions large 2 level toeplitz systems arise in a variety of applications see e g r h chan and m ng siam rev 38 1996 pp 427 482 for which efficient numerical methods for their solution are required some successful numerical techniques need the explicit knowledge of the generating function f of the considered system t n f rm x b an assumption that usually is not fulfilled in real applications in this paper we analyze and complete the procedure proposed in d noutsas s serra capizzano and p vassalos numer linear algebra appl 12 2005 pp 231 239 for the 2 level case in such a way from the knowledge of the coefficients of t n f we determine optimal preconditioning strategies for the solution of our systems finally some numerical experiments are performed and discussed in connection with our theoretical analysis

operator holder zygmund functions abstract it is well known that a lipschitz function on the real line does not have to be operator lipschitz we show that the situation changes dramatically if we pass to holder classes namely we prove that if f belongs to the holder class r with 0 1 then f a f b const a b for arbitrary self adjoint operators a and b we prove a similar result for functions f in the zygmund class 1 r for arbitrary self adjoint operators a and k we have f a k 2 f a f a k const k we also obtain analogs of this result for all holder zygmund classes r 0 then we find a sharp estimate for f a f b for functions f of class def f f const for an arbitrary modulus of continuity in particular we study moduli of continuity for which f a f b const a b for self adjoint a and b and for an arbitrary function f in we obtain similar estimates for commutators f a q q f a and quasicommutators f a q q f b finally we estimate the norms of finite differences j 0 m 1 m j m j f a j k for f in the class m that is defined in terms of finite differences and a modulus continuity of order m we also obtain similar results for unitary operators and for contractions

preconditioners for ill conditioned toeplitz systems constructed from positive kernels in this paper we are interested in the iterative solution of ill conditioned toeplitz systems generated by continuous nonnegative real valued functions f with a finite number of zeros we construct new w circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f kn with a suitable positive reproducing kernel kn by the restriction to positive kernels we obtain positive definite preconditioners moreover if f has only zeros of even order le 2s then we can prove that the property int pi pi t 2k k n t mbox d t le c n 2k k 0 hspace 1 5pt ldots s of the kernel is necessary and sufficient to ensure the convergence of the pcg method in a number of iteration steps independent of the dimension n of the system our theoretical results were confirmed by numerical tests

spectral behavior of preconditioned non hermitian multilevel block toeplitz matrices with matrix valued symbol this note is devoted to preconditioning strategies for non hermitian multilevel block toeplitz linear systems associated with a multivariate lebesgue integrable matrix valued symbol in particular we consider special preconditioned matrices where the preconditioner has a band multilevel block toeplitz structure and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices in this respect our main result is as follows let i k k let m s be the linear space of complex s i s matrices and let f g i k m s be functions whose components f ij g ij i k c i j 1 s belong to l consider the matrices t n 1 g t n f where n n 1 n k varies in n k and t n f t n g are the multilevel block toeplitz matrices of size n 1 n k s generated by f g then t n 1 g t n f n n k g 1 f i e the family of matrices t n 1 g t n f n n k has a global asymptotic spectral distribution described by the function g 1 f provided g possesses certain properties which ensure in particular the invertibility of t n g for all n and the following topological conditions are met the essential range of g 1 f defined as the union of the essential ranges of the eigenvalue functions j g 1 f j 1 s does not disconnect the complex plane and has empty interior this result generalizes the one obtained by donatelli neytcheva serra capizzano in a previous work concerning the non preconditioned case g 1 the last part of this note is devoted to numerical experiments which confirm the theoretical analysis and suggest the choice of optimal gmres preconditioning techniques to be used for the considered linear systems

a note on algebraic multigrid methods for the discrete weighted laplacian in recent contributions algebraic multigrid methods have been designed and studied from the viewpoint of spectral complementarity in this note we focus our efforts on specific applications and more precisely on large linear systems arising from the approximation of the weighted laplacian with various boundary conditions we adapt the multigrid idea to this specific setting and we present and critically discuss a wide set of numerical experiments showing the potentiality of the considered approach

multigrid methods for symmetric positive definite block toeplitz matrices with nonnegative generating functions in this paper we introduce a generalized multigrid method for solving linear systems bf t n m bf x bf b where bf t n m in re nm times nm is a symmetric block toeplitz matrix with symmetric toeplitz blocks we use a special choice of the projection operator whose coefficients simply depend on the generating functions associated with the proposed class of matrices this choice leads to iterative methods with convergence rates independent of the euclidean condition number kappa 2 bf t n m and of the dimension of the involved matrices the total arithmetic cost is o nm log nm for dense matrices and o nm for band matrices with band blocks in the pram model only o log nm parallel steps are required this algorithm is therefore competitive with the preconditioned conjugate gradient methods proposed by chan and jin siam j sci statist comput 13 1992 pp 1218 1235 ku and kuo siam j sci statist comput 13 1992 pp 948 966 di benedetto siam j sci comput 17 1996 and serra bit 34 1994 pp 579 594 for dense matrices and improves those results for band block toeplitz matrices with band toeplitz blocks

functions of operators under perturbations of class sp this is a continuation of our paper 2 we prove that for functions f in the holder class r and 1 p the operator f a f b belongs to sp whenever a and b are self adjoint operators with a b sp we also obtain sharp estimates for the schatten von neumann norms f a f b sp in terms of a b sp and establish similar results for other operator ideals we also estimate schatten von neumann norms of higher order differences j 0m 1 m j mj f a jk we prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions then we find necessary conditions on f for f a f b to belong to sq under the assumption that a b sp we also obtain schatten von neumann estimates for quasicommutators f a r rf b and introduce a spectral shift function and find a trace formula for operators of the form f a k 2f a f a k

a connection between h control and the absolute stabilizability of discrete time uncertain linear systems we consider the problem of stabilizing a class of uncertain discrete time systems in which the uncertainty satisfies a certain sum constraint we consider a notion of absolute stabilizability for discrete time systems and show that a necessary and sufficient condition for a discrete time uncertain system to be absolutely stabilizable is existence of a solution to a corresponding discrete time h control problem we also show that if it is possible to stabilize an uncertain system via nonlinear output feedback control then it is possible to absolutely stabilize it via linear output feedback control

stability of the notion of approximating class of sequences and applications given an approximating class of sequences b n m n m for a n n we prove that b n m n m x being the pseudo inverse of moore penrose is an approximating class of sequences for a n n where a n n is a sparsely vanishing sequence of matrices a n of size d n with d k d q for k q k email protected n as a consequence we extend distributional spectral results on the algebra generated by toeplitz sequences by including the pseudo inversion operation in the case where the sequences that are pseudo inverted are distributed as sparsely vanishing symbols applications to preconditioning and a potential use in image signal restoration problems are presented

dichotomy and absolute stability of nonlinear systems with periodically nonstationary linear part abstract systems with periodically nonstationary linear part are considered inputs and outputs of nonlinearities satisfy a certain integral quadratic constraint the conditions necessary and sufficient of dichotomy and absolute stability are obtained

band preconditioners for block toeplitz toeplitz block systems abstract preconditioned conjugate gradient methods are employed to solve symmetric positive definite m by m block toeplitz with n by n toeplitz block systems am nx b where am n are generated by 2 periodic nonnegative functions with zeros serra has proposed using band block toeplitz with band toeplitz block matrices bm n with their external and internal bandwidths independent of m and n as preconditioners serra showed that if the hessians of the generating function at the zeros are positive definite then the condition number of b 1m nam n is uniformly bounded by a constant independent of m and n whereas the condition number of am n tends to infinity as m and n tend to infinity in this paper we provide a method for deriving band preconditioners for block toeplitz toeplitz block matrices numerical examples are given to illustrate the performance of the method

symbol based multigrid methods for galerkin b spline isogeometric analysis we consider the stiffness matrices arising from the galerkin b spline isogeometric analysis discretization of classical elliptic problems by exploiting their specific spectral properties compactly described by a symbol we design an efficient multigrid method for the fast solution of the related linear systems the convergence rate of general purpose multigrid methods based on classical stationary smoothers is optimal i e bounded independently of the matrix size but it also worsens exponentially with respect to the spline degree the symbol allows us to give a detailed theoretical explanation of this exponential worsening in the case of the two grid scheme in addition thanks to a specific factorization of the symbol we are able to design an ad hoc multigrid method with an effective preconditioned cg or gmres smoother at the finest level in the spirit of the multi iterative idea the convergence rate of this multi iterative multigrid method is not only optimal but also robust i e bounded su

generalized locally toeplitz sequences spectral analysis and applications to discretized partial differential equations abstract starting from the finite difference discretization of an elliptic second order pde as i j 1 d x i a i j x x j u x b x over a bounded domain we introduce the notion of generalized locally toeplitz sequence of matrices the singular value distribution the eigenvalue distribution in the hermitian case is studied and characterized for generalized locally toeplitz sequences in terms of weighted multidimensional szeg formulas this extends preceding results attributed to tilli which concern the unilevel case the application of this theoretic analysis to the numerical solution of pdes is finally discussed

a new approach to the lur e problem in the theory of absolute stability the classical lur e problem consists of finding conditions for absolute stability of a linear system with a nonlinear feedback contained within a prescribed sector most of the results obtained on this problem are based on the frequency domain or lyapunov functions methods which are applied to systems with a time invariant or periodic linear block this paper develops a new approach providing a sufficient stability criterion for systems with time variable coefficients which is expressed in the transfer function of the linear block and the sector margins of the nonlinear block the systems for which this criterion is precise are found it is shown that stability of a system with a sign constant transfer function is guaranteed by stability of the system with a limit linear feedback so that for such systems the famous aizerman conjecture is true this in particular is the case for systems with a linear block consisting of an arbitrary number of first order time dependent links as an example the stability criterion is applied to a second order system for which the obtained results are contrasted with ones delivered by the popov criterion

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

weak robust controllability and observability of uncertain linear systems this paper introduces notions of weak robust observability full information weak robust controllability and partial information weak robust controllability for a class of uncertain linear systems in which the uncertainty satisfies an integral quadratic constraint the paper gives characterizations of these properties in terms of the existence of suitable solutions to riccati differential equations of the type which occur in h sup spl infin control furthermore it is shown that the conditions of weak robust observability and full information weak robust controllability are necessary but not sufficient for partial information weak robust controllability

nonlinear versus linear control in the absolute stabilizability of uncertain systems with structured uncertainty this note considers a stabilization problem for a class of uncertain linear systems containing structured uncertainty described by a certain integral quadratic constraint the notion of stabilizability considered is that of absolute stabilizability the main result gives a necessary and sufficient condition for the absolute stabilizability of this class of uncertain systems in terms of the existence of a solution to a corresponding diagonally scaled h sup spl infin control problem it follows from this result that absolute stabilizability via nonlinear control implies absolute stabilizability via linear control

analysis of preconditioning techniques for ill conditioned toeplitz matrices several preconditioning techniques for solving toeplitz systems are known in literature but their convergence features are completely understood only in the well conditioned case we study the application of tau circulant and hartley preconditioners to ill conditioned toeplitz matrices by proving that only the first class realizes a rate of convergence not depending on the dimension of the system

optimal quasi optimal and superlinear band toeplitz preconditioners for asymptotically ill conditioned positive definite toeplitz systems in this paper we are concerned with the solution of n x n hermitian toeplitz systems with nonnegative generating functions f the preconditioned conjugate gradient pcg method with the well known circulant preconditioners fails in the case where f has zeros in this paper we consider as preconditioners band toeplitz matrices generated by trigonometric polynomials g of fixed degree l we use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n for each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy we obtain different estimates of l for which the total cost of the proposed pcg methods is optimal and the related rates of convergence are superlinear finally for the most economical strategy we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems

bttb preconditioners for bttb least squares problems abstract in this paper we consider solving the least squares problem min x b t x 2 by using preconditioned conjugate gradient pcg methods where t is a large rectangular matrix which consists of several square block toeplitz toeplitz block bttb matrices and b is a column vector we propose a bttb preconditioner to speed up the pcg method and prove that the bttb preconditioner is a good preconditioner we then discuss the construction of the bttb preconditioner numerical examples including image restoration problems are given to illustrate the efficiency of our bttb preconditioner numerical results show that our bttb preconditioner is more efficient than the well known level 1 and level 2 circulant preconditioners

a cutting plane algorithm for robustness analysis of periodically time varying systems an algorithm for robustness analysis of periodic systems is derived the system under consideration consists of a linear periodically time varying plant in feedback interconnection with a structured uncertainty conditions for robust stability and robust performance can be formulated in terms of periodic integral quadratic constraints iqcs in this way the robustness analysis becomes a problem of optimizing the parameters of the iqc a cutting plane algorithm is suggested for solving this infinite dimensional optimization problem

extreme singular values and eigenvalues of non hermitian block toeplitz matrices abstract in this paper we are concerned with the analysis of the distribution and localization of the singular values of toeplitz matrices tn f generated by a p variate lebesgue integrable matrix valued function f q p c h k q we prove that the union of the essential ranges of the singular values of f is a proper weak cluster for the whole set of the singular values of tn f by showing that the number of outliers is strongly depending on the regularity features of the underlying function f in particular if f is continuous or from the krein algebra and p 1 then the cluster is proper other results concerning the extreme spectral behavior of tn f second order ergodic formulas and localization of eigenvalues of preconditioned matrices tn 1 g tn f are presented some examples of applications to the preconditioning of these results are also discussed

matrix algebra preconditioners for multilevel toeplitz matrices are not superlinear let f be a d variate 2 periodic continuous function and let tn f n n n1 n d be the multiindexed sequence of multilevel n n toeplitz matrices n n n i ni generated by f let a an n be a sequence of matrix algebras simultaneously diagonalized by unitary transforms we show that there exist infinitely many linearly independent trigonometric polynomials and continuous nonpolynomial functions f such that rank tn f pn o n n d 1 n 1 i for any matrix sequence p pn a this implies that no superlinear matrix algebra preconditioner exists in the multilevel toeplitz case the above mentioned result improves the analysis of the author and e tyrtyshnikov siam j matrix anal appl 21 2 1999 431 where the same was proved under the assumption that the involved algebras are of circulant type 2002 elsevier science inc all rights reserved

functions of perturbed n tuples of commuting self adjoint operators let a1 an and b1 bn be n tuples of commuting self adjoint operators on hilbert space for functions f on rn satisfying certain conditions we obtain sharp estimates of the operator norms or norms in operator ideals of f a1 an f b1 bn in terms of the corresponding norms of aj bj 1 j n we obtain analogs of earlier results on estimates for functions of perturbed self adjoint and normal operators it turns out that for n 3 the methods that were used for self adjoint and normal operators do not work we propose a new method that works for arbitrary n we also get sharp estimates for quasicommutators f a1 an r rf b1 bn in terms of norms of ajr rbj 1 j n for a bounded linear operator r

superoptimal approximation for unbounded symbols abstract the superoptimal frobenius approximation of toeplitz matrices is considered in connection with the case of unbounded symbols in particular we use the superoptimal approximation as preconditioner for the cg method when a fisher hartwig singularity is present in the symbol with special regard to systems coming from times series and financial applications a theoretical discussion concerning classical circulant preconditioners and a numerical comparison with the strang and with the optimal approximations are presented particularly with reference to the presence of noise

structured dissipativity and absolute stability of nonlinear systems the paper considers a problem of absolute stability for a nonlinear uncertain system with structured uncertainty an s procedure for nonlinear systems is established and using this the paper shows that the absolute stability problem is equivalent to an absolute stability problem for a nonlinear uncertain system with unstructured uncertainty the paper also introduces a new notion of structured dissipativeness for nonlinear systems this notion extends the standard notion of dissipativeness in that it allows for multiple supply functions the paper shows that a nonlinear uncertain system containing structured uncertainties will be absolutely stable if and only if a corresponding structured dissipativeness condition is satisfied it is also shown that structured dissipativeness is equivalent to the standard notion of dissipativeness with a parameter dependent supply function

convergence analysis of two grid methods for elliptic toeplitz and pdes matrix sequences summary the solution of large toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers 13 14 22 the technique was modified in 6 36 and a rigorous proof of convergence of the tgm two grid method was given in the special case where the generating function has only a zero at x 0 0 of order at most two here by extending the latter approach we perform a complete analysis of convergence of the tgm under the sole assumption that f is nonnegative and with a zero at x 0 0 of finite order an extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic pdes with nonconstant coefficients and of any order is then discussed

band toeplitz preconditioners for non symmetric real toeplitz systems by preconditioned gmres method abstract in this paper we study n n real non symmetric ill conditioned toeplitz systems of the form t n f x b the corresponding generating function is a complex valued one of the form f f 1 i f 2 where i 1 is known a priori and has roots we note that f 1 is a 2 periodic even function while f 2 is a 2 periodic odd one in order to solve the above system efficiently we use the preconditioned generalized minimal residual pgmres method and the preconditioned conjugate gradient method applied to the normal equations pcgn we present a specific preconditioning technique that combines elimination of the roots of f and best uniform approximation or interpolation by trigonometric polynomials the proposed preconditioner is a band toeplitz matrix t n p generated by the trigonometric polynomial p g q the trigonometric polynomial g is an appropriate complex polynomial having the same zeros as f while q is the trigonometric polynomial derived by the uniform approximation or interpolation of the function f g using t n p as a preconditioner we achieve a good clustering of the singular values of the preconditioned matrix in a small interval around 1 as well as a good clustering of the eigenvalues in a small domain in the right half plane of the complex plane far from zero moreover we describe on how the presented preconditioning technique can be extended to the two level toeplitz systems finally we show by various numerical experiments that the proposed preconditioning technique can solve a toeplitz system of the above form in a small number of iterations

multigrid methods for anisotropic bttb systems multigrid methods are highly efficient solution techniques for large sparse multilevel toeplitz systems which are positive definite and ill conditioned in this paper we develop multigrid methods which are especially designed for anisotropic two level toeplitz bttb matrices first a method is described for systems with anisotropy along coordinate axes as a suitable combination of semicoarsening and full coarsening steps although the basic idea is known from the solution of partial differential equations we present it here in a more formal way using generating functions and their level curves this enables us not only to prove the optimal convergence of the two grid method but also to carry over the results to systems with anisotropy in other directions we introduce new coordinates in order to describe these more complicated systems in terms of generating functions this enables us to solve them with the same efficiency for the two level method we present a convergence proof in this more general case

minimax optimal control of stochastic uncertain systems with relative entropy constraints this paper considers a new class of discrete time stochastic uncertain systems in which the uncertainty is described by a constraint on the relative entropy between a nominal noise distribution and the perturbed noise distribution this uncertainty description is a natural extension to the case of stochastic uncertain systems of the sum quadratic constraint uncertainty description this paper solves problems of worst case robust performance analysis and output feedback minimax optimal controller synthesis in a general nonlinear setting specializing these results to the linear case leads to a minimax linear quadratic gaussian lqg optimal controller this controller is defined by riccati difference equations and a kalman filter like state equation the paper also shows that the minimax lqg problem will have a solution if and only if a corresponding h sup spl infin control problem has a solution a linear example is presented to illustrate the minimax lqg methodology

a connection between h control and the absolute stabilizability of uncertain systems abstract this paper considers a stabilization problem for a class of uncertain systems in which the uncertainty satisfies a certain integral quadratic constraint such uncertain systems have been previously studied in the theory of absolute stability with most publications in this area being in the russian literature the paper introduces a notion of absolute stabilizability and shows that an uncertain system is absolutely stabilizable if and only if there exists a solution to a corresponding h control problem the paper also shows that if an uncertain system can be absolutely stabilized via nonlinear output feedback control then it can be absolutely stabilized via linear output feedback control

matrix algebra preconditioners for multilevel toeplitz systems do not insure optimal convergence rate in the last decades several matrix algebra optimal and superlinear preconditioners those assuring a strong clustering at the unity have been proposed for the solution of polynomially ill conditioned toeplitz linear systems the corresponding generalizations for multilevel structures are neither optimal nor superlinear see e g contemp math 281 2001 193 concerning the notion of superlinearity it has been recently shown that the proper clustering cannot be obtained in general see linear algebra appl 343 344 2002 303 siam j matrix anal appl 22 1 1999 431 math comput 72 2003 1305 in this paper by exploiting a proof technique previously proposed by the authors see contemp math 323 2003 313 we prove that the spectral equivalence and the essential spectral equivalence up to a constant number of diverging eigenvalues are impossible too in conclusion optimal matrix algebra preconditioners in the multilevel setting simply do not exist in general and therefore the search for optimal iterative solvers should be oriented to different directions with special attention to multilevel multigrid techniques

functions of normal operators under perturbations abstract in peller 1980 27 peller 1985 28 aleksandrov and peller 2009 2 aleksandrov and peller 2010 3 and aleksandrov and peller 2010 4 sharp estimates for f a f b were obtained for self adjoint operators a and b and for various classes of functions f on the real line r in this paper we extend those results to the case of functions of normal operators we show that if a function f belongs to the holder class r 2 0 1 of functions of two variables and n 1 and n 2 are normal operators then f n 1 f n 2 const f n 1 n 2 we obtain a more general result for functions in the space r 2 f f 1 f 2 const 1 2 for an arbitrary modulus of continuity we prove that if f belongs to the besov class b 1 1 r 2 then it is operator lipschitz i e f n 1 f n 2 const f b 1 1 n 1 n 2 we also study properties of f n 1 f n 2 in the case when f r 2 and n 1 n 2 belongs to the schatten von neumann class s p

block band toeplitz preconditioners derived from generating function approximations analysis and applications we are concerned with the study and the design of optimal preconditioners for ill conditioned toeplitz systems that arise from a priori known real valued nonnegative generating functions f x y having roots of even multiplicities our preconditioned matrix is constructed by using a trigonometric polynomial x y obtained from fourier kernel approximations or from the use of a proper interpolation scheme both of the above techniques produce a trigonometric polynomial x y which approximates the generating function f x y and hence the preconditioned matrix is forced to have clustered spectrum as x y is chosen to be a trigonometric polynomial the preconditioner is a block band toeplitz matrix with toeplitz blocks and therefore its inversion does not increase the total complexity of the pcg method preconditioning by block toeplitz matrices has been treated in the literature in several papers we compare our method with their results and we show the efficiency of our proposal through various numerical experiments