on robust hurwitz polynomials in this note kharitonov s theorem on robust hurwitz polynomials is simplified for low order polynomials specifically for n 3 4 and 5 the number of polynomials required to check robust stability is one two and three respectively instead of four furthermore it is shown that for n geq 6 the number of polynomials for robust stability checking is necessarily four thus further simplification is not possible the same simplifications arise in robust schur polynomials by using the bilinear transformation applications of these simplifications to two dimensional polynomials as well as to robustness for single parameters are indicated

optimal preconditioners for systems defined by functions of toeplitz matrices abstract we propose several circulant preconditioners for systems defined by some functions g of toeplitz matrices a n in this paper we are interested in solving g a n x b by the preconditioned conjugate method or the preconditioned minimal residual method namely in the cases when g z are the functions e z sin z and cos z numerical results are given to show the effectiveness of the proposed preconditioners

circulant preconditioners for complex toeplitz matrices the solution of n by n complex toeplitz systems a n x b by the preconditioned conjugate gradient method is studied the preconditioner c n is the circulant matrix that minimizes b n a n f over all circulant matrices b n the authors prove that if the generating function of a n is a 2 mu periodic continuous complex valued function without any zeros then the spectrum of the normalized preconditioned matrix c n 1 a n c n 1 a n will be clustered around one hence they show that if the condition number of a n is of o n alpha the conjugate gradient method when applied to solving the normalized preconditioned system converges in at most o alpha log n 1 steps thus the total complexity of the algorithm is o alpha n log 2 n 1 log n

some parseval goldstein type identities involving the fs 2 transform the fc 2 transform and the p4 transform and their applications abstract in the present paper the authors consider several new integral transforms including the f s 2 transform the f c 2 transform and the p 4 transform as generalizations of the classical fourier sine transform the classical fourier cosine transform and the classical stieltjes transform respectively many identities involving these transforms are given by making use of these identities a number of new parseval goldstein type identities are obtained for these and many other well known integral transforms the identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions some examples are also given as illustrations of the results presented here

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

shift invert arnoldi approximation to the toeplitz matrix exponential the shift invert arnoldi method is employed to generate an orthonormal basis from the krylov subspace corresponding to a real toeplitz matrix and an initial vector the vectors and recurrence coefficients produced by this method are exploited to approximate the toeplitz matrix exponential toeplitz matrix inversion formula and rapid toeplitz matrix vector multiplications are utilized to lower the computational costs for convergence analysis a sufficient condition is established to guarantee that the error bound is independent of the norm of the matrix numerical results are given to demonstrate the efficiency of the method

approximate inverse circulant plus diagonal preconditioners for toeplitz plus diagonal matrices we consider the solutions of hermitian positive definite toeplitz plus diagonal systems t d x b where t is a toeplitz matrix and d is diagonal and positive however unlike the case of toeplitz systems no fast direct solvers have been developed for solving them in this paper we employ the preconditioned conjugate gradient method with approximate inverse circulant plus diagonal preconditioners to solving such systems the proposed preconditioner can be constructed and implemented efficiently using fast fourier transforms we show that if the entries of t decay away exponentially from the main diagonals the preconditioned conjugate gradient method applied to the preconditioned system converges very quickly numerical examples including spatial regularization for image deconvolution application are given to illustrate the effectiveness of the proposed preconditioner

a new method of solving bessel s differential equation using the l 2 transform in the present paper a new method of solving bessel s differential equation is given using the l 2 transform

fast computation of the multivariable stability margin for real interrelated uncertain parameters an alternative implementation of an algorithm for the computation of the multivariable stability margin to check the robust stability of feedback systems with real parametric uncertainty is proposed this method eliminates the need for the frequency search by reducing it to the testing of a finite number of conditions these conditions have a special structure that allows a significant improvement in the speed of computation

identities for the e2 1 transform and their applications in the present paper the authors introduce the e 2 1 transform with kernel the exponential integral function it is shown that the third iterate of the l 2 transform is the exponential integral transform and some identities involving the new transform the l 2 transform and the widder potential transform are given using the identities parseval goldstein type results involving these transforms are proved some illustrative examples are also given

characterization and efficient computation of the structured singular value the concept of structured singular value was recently introduced by doyle 1 as a tool for the analysis and synthesis of feedback systems with structured uncertainties it is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the h infty approach to control system design in this paper it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems properties of these optimization problems are exhibited leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3 and often does for block structures of larger size

identities and parseval type relations for the l2 transform abstract in the present paper the authors introduce identities for the l 2 transform which yield a parseval type relationship for the l 2 transform the parseval type relationship proven in this paper give rise to useful corollaries for evaluating indefinite integrals of special functions some examples are also given as illustrations of the results presented here

identities for the hankel transform and their applications abstract in the present paper the authors show that iterations of the hankel transform with k transform is a constant multiple of the widder transform using these iteration identities several parseval goldstein type theorems for these transforms are given by making use of these results a number of new goldstein type exchange identities are obtained for these and the laplace transform the identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions some examples are also given as illustration of the results presented here

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

superlinear pcg methods for symmetric toeplitz systems in this paper we deal with the solution by means of preconditioned conjugate gradient pcg methods of n x n symmetric toeplitz systems a n f x b with nonnegative generating function f here the function f is assumed to be continuous and strictly positive or is assumed to have isolated zeros of even order in the first case we use as preconditioner the natural and the optimal approximation of a n f proposed by bini and di benedetto and we prove that the related pcg method has a superlinear rate of convergence and a total arithmetic cost of o n log n ops under the second hypothesis we cannot guarantee that the natural matrix is positive definite while for the optimal we show that in the ill conditioned case this can be really a bad choice consequently we define a new matrix for preconditioning the given system then by applying the sherman morrison woodbury inversion formula to the preconditioned system we introduce a small constant number of subsidiary systems which can be solved again by means of the previous pcg method finally we perform some numerical experiments that show the effectiveness of the devised technique and the adherence with the theoretical analysis

a note on the spectral distribution of symmetrized toeplitz sequences abstract the singular value and spectral distribution of toeplitz matrix sequences with lebesgue integrable generating functions is well studied early results were provided in the classical szeg theorem and the avram parter theorem in which the singular value symbol coincides with the generating function more general versions of the theorem were later proved by zamarashkin and tyrtyshnikov and tilli considering real nonsymmetric toeplitz matrix sequences we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically by using the notion of approximating class of sequences in particular under the assumption that the symbol is sparsely vanishing we show that roughly half of the eigenvalues of the symmetrized toeplitz matrix i e a hankel matrix are negative positive for sufficiently large dimension i e the matrix sequence is symmetric asymptotically indefinite

robust stability with structured real parameter perturbations this paper considers the problem of robust stabilization of a linear time invariant system subject to variations of a real parameter vector for a given controller the radius of the largest stability hypersphere in this parameter space is calculated this radius is a measure of the stability margin of the closed loop system the results developed are applicable to all systems where the closed loop characteristic polynomial coefficients are linear functions of the parameters of interest in particular this always occurs for single input multioutput or single output multiinput systems where the transfer function coefficients are linear or affine functions of the parameters many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters the largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined based on these calculations a design procedure is proposed to robustify a given stabilizing controller this algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller io stabilize a plant subject to given ranges of parameter excursions these results are illustrated by an example

circulant preconditioners for functions of hermitian toeplitz matrices abstract circulant preconditioners for function of matrices have been recently of interest in particular several authors proposed the use of the optimal circulant preconditioners as well as the superoptimal circulant preconditioners in this context and numerically illustrated that such preconditioners are effective for certain functions of toeplitz matrices motivated by their results we propose in this work the absolute value superoptimal circulant preconditioners and provide several theorems that analytically show the effectiveness of such circulant preconditioners for systems defined by functions of toeplitz matrices namely we show that the eigenvalues of the preconditioned matrices are clustered around 1 and rapid convergence of krylov subspace methods can therefore be expected moreover we show that our results can be extended to functions of block toeplitz matrices with toeplitz blocks provided that the optimal block circulant matrices with circulant blocks are used as preconditioners numerical examples are given to support our theoretical results

optimal preconditioners for functions of matrices abstract the optimal preconditioner c u a of a given matrix a was proposed in 1988 by t chan 6 since then it has been proved to be efficient for solving a large class of structured systems in this paper we construct the optimal preconditioners for different functions of matrices more precisely let f be a function of matrices from c n n to c n n given a c n n there are two possible optimal preconditioners for f a c u f a and f c u a in the paper we study properties of both c u f a and f c u a for different functions of matrices numerical experiments are given to illustrate the efficiency of the optimal preconditioners when they are used to solve f a x b

optimal trigonometric preconditioners for nonsymmetric toeplitz systems abstract this paper is concerned with the solution of systems of linear equations tnxn bn where tn n n denotes a sequence of nonsingular nonsymmetric toeplitz matrices arising from a generating function of the wiener class we present a technique for the fast construction of optimal trigonometric preconditioners mn mn t ntn of the corresponding normal equation which can be extended to toeplitz least squares problems in a straightforward way moreover we prove that the spectrum of the preconditioned matrix mn1t ntn is clustered at 1 such that the pcg method applied to the normal equation converges superlinearly numerical tests confirm the theoretical expectations

computing the exponential of large block triangular block toeplitz matrices encountered in fluid queues the erlangian approximation of markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block triangular block toeplitz structure to this end we propose some algorithms which exploit the toeplitz structure and the properties of generators such algorithms allow us to compute the exponential of very large matrices which would otherwise be untreatable with standard methods we also prove interesting decay properties of the exponential of a generator having a block triangular block toeplitz structure