new estimates for solutions of lyapunov equations results for estimating the solution of differential and algebraic lyapunov matrix equations are obtained and some of the well known results are generalized

fixed points koebe uniformization and circle packings a domain in the riemann sphere hat mathbb c is called a circle domain if every connected component of its boundary is either a circle or a point in 1908 p koebe ko1 posed the following conjecture known as koebe s kreisnormierungsproblem any plane domain is conformally homeomorphic to a circle domain in hat mathbb c when the domain is simply connected this is the content of the riemann mapping theorem

fractional hadamard powers of positive semidefinite matrices we consider the class sn of all real positive semidefinite n n matrices and the subclass sn of all a sn with non negative entries for a positive non integer number and some a sn when will the fractional hadamard power a again belong to sn it is known that for a specific this holds for all a sn if and only if n 2 now let a sn be of the form a t v where t sn has rank 1 and v sn has rank p 1 if the hadamard quotient of t and v is hadamard independent in general position and v has sufficently small entries then a complete answer is given depending on n p and special attention is given to the case that p 1

new iterative algorithm for algebraic riccati equation related to h sub spl infin control problem of singularly perturbed systems we present the solution to the algebraic riccati equation are with indefinite sign quadratic term related to the h sub spl infin control problem for singularly perturbed systems by means of a kleinman type algorithm the resulting algorithm is very efficient from the numerical point of view because the are is solvable even if the quadratic term has an indefinite sign moreover the resulting iterative algorithm is quadratically convergent we also present an algorithm for solving the generalized algebraic lyapunov equation on the basis of the fixed point algorithm

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf 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

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

influence of boundary stress singularities on the vibration of clamped and simply supported sectorial plates with arbitrary radial edge conditions abstract this paper extends previous studies made for sectorial plates having re entrant i e interior corners causing stress singularities to provide accurate frequencies when the circular edge is either clamped or simply supported an extensive review of the literature is also given herein spanning nearly the past two decades explaining the free vibration characteristics of sectorial plates in this work the classical ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements these sets include 1 mathematically complete algebraic trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained and 2 corner functions which account for the bending moment singularities at the re entrant vertex corner of the radial edges having arbitrary edge conditions extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of non dimensional frequencies for sectorial plates having either a clamped or hinged circumferential edge and various combinations of clamped hinged and free conditions on the radial edges accurate to at least four significant figure frequencies and normalized contours of the transverse vibratory displacement are presented for the spectra of sector angles 90 180 semi circular 270 300 330 350 355 360 complete circular causing a re entrant vertex corner of the radial edges for sector angles of 360 a clamped clamped clamped hinged clamped free hinged free or free free radial crack ensues one general observation is the substantial reduction in the first six frequencies as the sector angle increases for all plates except in the first two modes of plates having free free radial edges

a generalized adi iterative method the adi iterative method for the solution of sylvester s equationax xb c proceeds by strictly alternating between the solution of the two equations begin gathered left a delta k 1 i right x 2k 1 x 2k left b delta k 1 i right c hfill x 2k 2 left b tau k 1 i right left a tau k 1 i right x 2k 1 c hfill end gathered fork 0 1 2 herex o is a given initial approximate solution and the k and k are real or complex parameters chosen so that the computed approximate solutionsx k converge rapidly to the solutionx of the sylvester equation ask increases this paper discusses the possibility of solving one of the equations in the adi iterative method more often than the other one i e relaxing the strict alternation requirement in order to achieve a higher rate of convergence our analysis based on potential theory shows that this generalization of the adi iterative method can give faster convergence than when strict alternation is required

on two classes of mixed type lyapunov equations abstract the hermitian positive definite solutions of the mixed type lyapunov equations x axb bxa q and ax xa bxb q 0 are studied in this paper based on the bhaskar and lakshmikantham s fixed point theorem new sufficient conditions for the existence of hermitian positive definite solutions are derived iterative methods are proposed to compute the hermitian positive definite solutions numerical examples are used to illustrate the convergence of the new methods

preconditioned hss iteration method and its non alternating variant for continuous sylvester equations abstract bai 2010 proposed an efficient hermitian and skew hermitian splitting hss iteration method for solving a broad class of large sparse continuous sylvester equations to further improve the efficiency of the hss method in this paper we present a preconditioned hss phss iteration method and its non alternating variant nphss for this matrix equation the convergence properties of the phss and nphss methods are studied in depth and the quasi optimal values of the iteration parameters for the two methods are also derived moreover to reduce the computational cost we establish the inexact variants of the two iteration methods numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants

deterministic continuation of stochastic metastable equilibria via lyapunov equations and ellipsoids numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary partial and delay differential equations here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations we stress that we do not develop a new technical tool but that we combine results and methods from probability theory dynamical systems numerical analysis optimization and control theory into an algorithm that augments classical equilibrium continuation methods in particular we use ellipsoids defining regions of high concentration of sample paths it is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numer

removability rigidity of circle domains and koebe s conjecture abstract a circle domain in the riemann sphere is conformally rigid if every conformal map of onto another circle domain is the restriction of a mobius transformation we show that two rigidity conjectures of he and schramm are in fact equivalent at least for a large family of circle domains the proof follows from a result on the removability of countable unions of certain conformally removable sets we also introduce trans quasiconformal deformation of schottky groups to prove that a circle domain is conformally rigid if and only if it is quasiconformally rigid thereby providing new evidence for the aforementioned conjectures

global fom and gmres algorithms for a class of complex matrix equations abstract in this paper a generalized global arnoldi process is given to produce an f orthonormal basis of complex matrix space then we propose the global full orthogonalization method gl fom and global generalized minimum residual gl gmres method for solving a class of complex matrix equations the new methods can be implemented by the original coefficient matrices by using schur complement we give the expressions of the approximate solutions and the corresponding residuals some convergence results of the proposed methods are also derived finally numerical examples are given to illustrate the effectiveness with comparison to other existing methods

fast algorithms for the sylvester equation ax xb t c for given matrices a fm m b fn n and c fm n over an arbitrary field f the matrix equation ax xbt c has a unique solution x fm n if and only if a and b have disjoint spectra we describe an algorithm that computes the solution x for m n n with o n logn arithmetic operations in f where 2 is such that m m matrices can be multiplied with o m arithmetic operations e g 2 376 it seems that before no better bound than o m3 n3 arithmetic operations was known the state of the art in numerical analysis is o n3 m3 flops but these algorithms due to bartels stewart and golub nash van loan involve schur decompositions i e they compute the eigenvalues of at least one of a and b and can hence not be transferred for general f

exact analytical solutions for free vibrations of thick sectorial plates with simply supported radial edges abstract the first known exact analytical solutions are derived for the free vibrations of thick mindlin sectorial plates having simply supported radial edges and arbitrary conditions along the circular edge the general solutions to the mindlin differential equations of motion contain noninteger order ordinary and modified bessel functions of the first and second kinds and six arbitrary constants of integration by exercising a careful limiting process three regularity conditions at the vertex of the radial edges are invoked to yield three equations of constraint among the six constants for sector angles exceeding 180 re entrant corners three additional linearly independent equations among the six constants are obtained by satisfying the three boundary conditions along the circular edge frequency determinant equations are derived for mindlin sectorial plates with circular boundaries which are clamped simply supported or free nondimensional frequency parameters are presented for over a wide range of salient and re entrant sector angles 30 360 and thickness to radius ratios of 0 1 0 2 and 0 4 frequency results obtained for mindlin sectorial plates are compared to those determined for classically thin sectorial plates and the results are found to be considerably different than those derived from thin plate theory particularly for the fundamental frequencies of plates having sector angles slightly greater than 180 when the circular boundary is free the frequencies for 360 sectorial plates i e circular plates having a hinged crack are compared with those for complete circular ones

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

vibration of annular sector mindlin plates with internal radial line and circumferential arc supports an energy approach is presented for the free vibration analysis of thick annular sector plates with internal line arc supports this has been a topic of practical interest but there appears to have been no previous work reported based on the stationary energy principle and the recently proposed pb 2 rayleigh ritz method the governing eigenvalue equation is derived for internally supported thick annular sector plates of different radii relative thickness ratios sector angles and with various end conditions along the radial and circumferential edges the proposed solution algorithm involves 1 sets of mathematically complete two dimensional polynomials p 2 used as the admissible displacement and rotational functions and 2 a basic function b formed from the piecewise expressions for the boundary and internal line arc supports these functions automatically satisfy all the kinematic boundary conditions of the four edges and internal supports the numerical procedure has been demonstrated by several example plate problems convergence and comparison studies for simple examples are used to verify the accuracy of the present method

explicit solutions of the discrete time lyapunov matrix equation and kalman yakubovich equations a general solution for the nonsquare nonsymmetric lyapunov matrix equation in a canonical form is presented the solution is shown to be a toeplitz matrix which may be calculated using the backwards levinson algorithm this solution is then applied to the kalman yakubovich equations to derive a method for generating strictly positive real functions via the positive real lemma this latter result has an application in system identification

parallel solutions of very large sparse lyapunov equations by balanced bbd decompositions in this paper a method for the parallel solution of large sparse matrix equations is presented the method is based on the balanced bordered block diagonal bbd decomposition which is applied in conjunction with the successive over relaxation sor iterative method to solve large lyapunov equations in the kronecker sum representation a variety of experimental results will be presented including solutions of lyapunov equations with matrices as large as 1993 spl times 1993

simultaneous eigenvalue lower bounds for the lyapunov matrix equation simultaneous eigenvalue lower bounds for the solution of the lyapunov equation are presented the n summation and the n product bounds generalize existing bounds such as those for the trace and the determinant these bounds are stronger than the majority of the relevant bounds shown in the literature

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

extending bicg and bicr methods to solve the stein tensor equation abstract the biconjugate gradients bicg and biconjugate residual bicr methods are attractive ways for solving nonsymmetric linear system a x b in this paper the bicg and bicr methods are extended to solve the high order stein tensor equation the convergent properties of the developed iterative algorithms are studied numerical examples are provided to confirm the theoretical results which demonstrate that the proposed algorithms are effective and feasible for solving the stein tensor equation

upper summation and product bounds for solution eigenvalues of the lyapunov matrix equation upper bounds for summations including the trace and for products including the determinant of the eigenvalues of the solution of the continuous algebraic lyapunov matrix equation are presented the majority of the bounds are tighter than those in the literature and some are new

gradient based iterative algorithms for generalized coupled sylvester conjugate matrix equations abstract by applying the hierarchical identification principle the gradient based iterative algorithm is suggested to solve a class of complex matrix equations with the real representation of a complex matrix as a tool the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices also we solve the problem which is proposed by wu et al 2010 finally some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

bcr method for solving generalized coupled sylvester equations over centrosymmetric or anti centrosymmetric matrix abstract this paper introduces another version of biconjugate residual method bcr for solving the generalized coupled sylvester matrix equations over centrosymmetric or anti centrosymmetric matrix we prove this version of bcr algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round off errors furthermore a method is provided for choosing the initial matrices to obtain the least norm solution of the problem at last some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed

optimal alternating direction implicit parameters for nonsymmetric systems of linear equations the determination of optimal parameters for the well known alternating direction implicit adi method leads to a minimization problem for rational functions of given degree l on compact sets e and f that contain the eigenvalues of the matrices involved if e and f are real intervals the solution of this problem was given by zolotarev more than 100 years ago and by wachspress in the 1960s however in the complex case e g if the linear system arises from the discretization of non self adjoint elliptic differential equations little is known about this problem in this paper two approaches to the zolotarev problem in the complex plane are considered first generalized leja points are considered as an example for rational functions which are asymptotically minimal in addition for rectangular eigenvalue domains e and f the exact minimal solutions for l 2 are constructed and it is shown by numerical examples that the results obtained using the optimal parameters of degree 2 are comparable to tho

iterative solutions to matrix equations of the form aixbi fi this paper is concerned with the numerical solutions to the linear matrix equations a 1xb 1 f 1 and a 2xb 2 f 2 two iterative algorithms are presented to obtain the solutions for any initial value it is proved that the iterative solutions obtained by the proposed algorithms converge to their true values finally simulation examples are given to verify the proposed convergence theorems

an implicit deflation method for ill conditioned sylvester and lyapunov equations this paper describes an effective algorithm for solving ill conditioned sylvester or lyapunov equations these equations can be solved by conventional methods for ill conditioned linear systems however such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data for these ill conditioned matrix equations an implicit deflation algorithm is proposed to implement a certain svd based minimum norm least squares approximate solution the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical details are also discussed

hierarchical least squares based iterative estimation algorithm for multivariable box jenkins like systems using the auxiliary model abstract this paper presents a hierarchical least squares iterative algorithm to estimate the parameters of multivariable box jenkins like systems by combining the hierarchical identification principle and the auxiliary model identification idea the key is to decompose a multivariable systems into two subsystems by using the hierarchical identification principle as there exist the unmeasurable noise free outputs and noise terms in the information vector the solution is using the auxiliary model identification idea to replace the unmeasurable variables with the outputs of the auxiliary model and the estimated residuals a numerical example is given to show the performance of the proposed algorithm

unified type non stationary lyapunov matrix equation simultaneous eigenvalue bounds abstract simultaneous eigenvalue bounds for the solution of the unified non stationary lyapunov matrix equation are presented when the solution becomes stationary the results reduce to bounds of the unified type algebraic lyapunov equation in the limiting cases the results reduce to bounds for the solution of the differential and difference lyapunov equations the bounds given in this paper are a generalization of some existing bounds obtained separately for the continuous and discrete type stationary and non stationary lyapunov equations

upper and lower matrix bounds of the solution for the discrete lyapunov equation new matrix bounds of the solution for the discrete algebraic matrix lyapunov equation are established the upper and lower eigenvalue bounds such as each eigenvalue including the extreme ones the trace and the determinant are also determined by these matrix bounds for the same solution the present schemes are tighter as compared with the majority of existing results

multirate sampled data systems all h sub spl infin suboptimal controllers and the minimum entropy controller for a general multirate sampled data sd system the authors characterize explicitly the set of all causal stabilizing controllers that achieve a certain h sub spl infin norm bound moreover they give explicitly a particular controller that further minimizes an entropy function for the sd system the characterization lays the groundwork for synthesizing multirate control systems with multiple mixed control specifications

iterative algorithms for the minimum norm solution and the least squares solution of the linear matrix equations abstract in this paper two iterative algorithms are proposed to solve the linear matrix equations a 1 xb 1 c 1 x t d 1 m 1 a 2 xb 2 c 2 x t d 2 m 2 when the matrix equations are consistent by the first algorithm a solution x can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore the minimum norm solution can be got by choosing a special kind of initial matrix additionally the unique optimal approximation solution to a given matrix x 0 can be derived by finding the minimum norm solution of a new matrix equations a 1 x b 1 c 1 x t d 1 m 1 a 2 x b 2 c 2 x t d 2 m 2 when the matrix equations are inconsistent we present the second algorithm to find the least squares solution with the minimum norm finally two numerical examples are tested by matlab the results show that these iterative algorithms are efficient

a brief summary of the bounds on the solution of the algebraic matrix equations in control theory abstract in recent years several bounds have been reported for different measures of the extent or size of the solution of the algebraic matrix equations arising in control theory such as the riccati equation and the lyapunov equation this paper collects the bounds that have been presented up to now and summarizes them in an unified form this will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

the solution of the matrix equations axb cxd e and ya dz yc bz e f the conditions for the existence of a unique solution of the matrix equation axb cxd e are proved to be that i the pencils a c and d b are regular and ii the spectra of the pencils have an empty intersection a numerical algorithm for solving the equation is proposed the possibility of a least squares type solution is briefly discussed the set of equations ya dz yc bz e f is proved to be equivalent to the aforementioned equation and its solution is also investigated a numerical algorithm is proposed

planar relative schottky sets and quasisymmetric maps a relative schottky set in a planar domain is a subset of obtained by removing from open geometric discs whose closures are in and are pairwise disjoint in this paper we study quasisymmetric and related maps between relative schottky sets of measure zero we prove in particular that quasisymmetric maps between such sets in jordan domains are conformal locally bi lipschitz and that their first derivatives are locally lipschitz we also provide a locally bi lipschitz uniformization result for relative schottky sets in jordan domains and establish rigidity with respect to local quasisymmetric maps for relative schottky sets in the unit disc

the relationship between hadamard and conventional multiplication for positive definite matrices abstract it is known that if a is positive definite hermitian then a a 1 i in the positive semidefinite ordering our principal new result is a converse to this inequality under certain weak regularity assumptions about a function f on the positive definite matrices a f a af a for all positive definite a if and only if f a is a positive multiple of a 1 in addition to the inequality a a 1 i it is known that a a 1 t i and stronger that min a b min ab t for a b positive definite hermitian we also show that min a b min ab and note that min ab and min ab t can be quite different for a b positive definite hermitian we utilize a simple technique for dealing with the hadamard product which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under hadamard multiplication and of the inequalities mentioned

weighted least squares solutions to general coupled sylvester matrix equations this paper is concerned with weighted least squares solutions to general coupled sylvester matrix equations gradient based iterative algorithms are proposed to solve this problem this type of iterative algorithm includes a wide class of iterative algorithms and two special cases of them are studied in detail in this paper necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented sufficient conditions that are easy to compute are also given the optimal step sizes such that the convergence rates of the algorithms which are properly defined in this paper are maximized and established several special cases of the weighted least squares problem such as a least squares solution to the coupled sylvester matrix equations problem solutions to the general coupled sylvester matrix equations problem and a weighted least squares solution to the linear matrix equation problem are simultaneously solved several numerical examples are given to illustrate the effectiveness of the proposed algorithms

convergence and comparison theorems for a generalized alternating iterative method using as a principal tool the convergence results of standard iterative process for the solution of linear systems alternating iterative methods are studied we extend the convergence theorem for the stationary alternating iterative method of benzi and szyld numererische mathematik 76 1997 309 for weak nonnegative splittings of the first type of a monotone matrix to weak nonnegative splitting of the second type on the other hand we introduce a more general method the nonstationary alternating iterative method establishing convergence results for weak nonnegative splittings of a monotone matrix and for p regular splittings of a symmetric positive definite matrix

rigidity of circle domains whose boundary hasσ finite linear measure let be a circle domain in the riemann sphere whose boundary has finite linear measure we show that is rigid in the sense that any conformal homeomorphism of onto any other circle domain is equal to the restriction of a mobius transformation previously kaufman and bishop have independently found examples of non rigid circle domains whose boundary is a cantor set of hausdorff dimension one

on the adi method for sylvester equations this paper is concerned with the numerical solution of large scale sylvester equations ax xb c lyapunov equations as a special case in particular included with c having very small rank for stable lyapunov equations penzl 2000 22 and li and white 2002 20 demonstrated that the so called cholesky factor adi method with decent shift parameters can be very effective in this paper we present a generalization of the cholesky factor adi method for sylvester equations an easily implementable extension of penz s shift strategy for the lyapunov equation is presented for the current case it is demonstrated that galerkin projection via adi subspaces often produces much more accurate solutions than adi solutions

bounds for the eigenvalues of the solution of the discrete riccati and lyapunov equations and the continuous lyapunov equation we present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete riccati and lyapunov matrix equations and the continuous lyapunov matrix equation nearly all of our bounds for the discrete riccati equation are new the bounds for the discrete and continuous lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation

gradient based iterative solutions for general linear matrix equations in this paper we present a gradient based iterative algorithm for solving general linear matrix equations by extending the jacobi iteration and by applying the hierarchical identification principle convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices two numerical examples are provided to show that the proposed algorithm is effective

numerical solution of the lyapunov equation by approximate power iteration abstract we present the approximate power iteration api algorithm for the computation of the dominant invariant subspace of the solution x of large order lyapunov equations ax xa t q 0 without first computing the matrix x itself the api algorithm is an iterative procedure that uses krylov subspace bases in computing estimates of matrix vector products x v in a power iteration sequence application of the api algorithm requires that a a t x admits a good low rank solution then api provides an orthogonal basis of a subspace that closely approximates the dominant x invariant subspace of corresponding dimension analytical convergence results are also presented

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

a note on the iterative solutions of general coupled matrix equation abstract recently ding and chen f ding t chen on iterative solutions of general coupled matrix equations siam j control optim 44 2006 2269 2284 developed a gradient based iterative method for solving a class of coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified so that the iterative solutions are obtained by applying hierarchical identification principle in this note by considering the coupled sylvester matrix equation as a linear operator equation we give a natural way to derive this algorithm we also propose some faster algorithms and present some numerical results

an analysis of block successive overrelaxation for a class of matrices with complex spectra optimal block successive overrelaxation bsor parameters are derived for a special class of matrices having complex spectra these matrices arise from the central difference approximation to singularly perturbed convection diffusion equations using a coarse grid and thus have large cell reynolds number a simple problem is examined in detail and it is shown that bsor is an excellent method for cell reynolds numbers near 1 and that the rate of convergence is invariant to the cell reynolds number for large cell reynolds number

the minkowski lyapunov equation for linear dynamics theoretical foundations we consider the lyapunov equation for the linear dynamics which arises naturally when one seeks for a lyapunov function with a uniform exact decrease in this setting a solution to the lyapunov equation has been characterized only for quadratic lyapunov functions we demonstrate that the lyapunov equation is a well posed equation for strictly stable dynamics and a much more general class of lyapunov functions specified via minkowski functions of proper c sets which include euclidean and weighted euclidean vector norms polytopic and weighted polytopic 1 vector norms as well as vector semi norms induced by the minkowski functions of proper c sets furthermore we establish that the lyapunov equation admits a basic solution i e the unique solution within the class of minkowski functions associated with proper c sets finally we provide a characterization of the lower and upper approximations of the basic solution that converge pointwise and compactly to it while in addition the upper approximations satisfy the classical lyapunov inequality

new wavelet based approach for time domain simulations a new numerical method for the time domain solution of maxwell s equations in linear media is proposed in this paper the field quantities derived from the spatial discretization of maxwell s equations are expanded in the time domain by wavelets on the interval this choice yields a new arrangement of the unknowns into a matrix instead of the usual vector and transforms the differential equations in time in a sylvester 1983 matrix equation the memory requirements are proportional to the number of spatial unknowns and the time evolution of the space quantities is obtained with better accuracy than in conventional marching on time techniques

a numerical algorithm for lyapunov equations in this paper we give a new numerical iterative method for lyapunov matrix equations which is called kronecker products iterative method kpim by means of the kronecker products we get an iterative convergent sequence from the lyapunov matrix equations at the same time we give its algorithms and corresponding improved algorithm in order to reduce the computation cost finally we discuss how to choose the optimal parameter of kpim for the continuous lyapunov matrix equations in the case of symmetry

solution of underdetermined sylvester equations in sensor array signal processing abstract we propose efficient solution of an underdetermined matrix equation the matrix equations studied here appear in the study of subspace based high resolution multiparameter direction of arrival estimation the proposed gradient based technique is at least an order of magnitude computational improvement over the existing approaches simulation results are provided to illustrated the proposed technique

an implicit sequential algorithm for solving coupled lyapunov equations of continuous time markovian jump systems in this paper an implicit sequential algorithm is presented for solving coupled lyapunov matrix equations of continuous time markovian jump linear systems first some existing iterative algorithms which can be utilized to solve the coupled lyapunov matrix equations are reviewed and discussed next based on the existing parallel iterative algorithm an implicit sequential algorithm is proposed by using the latest updated information the proposed algorithm fills the current gap of implicit algorithms for solving continuous coupled lyapunov matrix equations it is shown that the proposed algorithm with zero initial conditions can monotonically converge to the unique positive definite solutions of the coupled lyapunov matrix equations if the associated markovian jump system is stochastically stable moreover a necessary and sufficient condition is established for the proposed algorithm to be convergent the algorithm presented in this paper has much better convergence performance than other existing iterative algorithms and requires less storage capacity finally a numerical example is given to show the effectiveness of the proposed algorithm

index of hadamard multiplication by positive matrices ii for each n n positive semidefinite matrix a we define the minimal index i a max 0 a b b for all b 0 and for each norm n the n index in a min n a b b 0 and n b 1 where a b aijbij is the hadamard or schur product of a aij and b bij and b 0 means that b is a positive semidefinite matrix a comparison between these indexes is done for different choices of the norm n as an application we find for each bounded invertible selfadjoint operator s on a hilbert space the best constant m s such that sts s 1ts 1 m s t for all t 0

computation of approximate null vectors of sylvester and lyapunov operators this paper describes an effective algorithm for computing approximate null vectors of certain matrix operators associated with sylvester or lyapunov equations the singular value decomposition and rank revealing qr methods are two widely used stable algorithms for numerical determination of the rank and nullity of a matrix a these methods however are not readily applicable to sylvester and lyapunov operators since they require on the order of n sup 6 arithmetic operations on order n sup 2 data for these problems a variant of inverse power iteration is employed to compute orthonormal bases for singular subspaces associated with the small singular values the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical aspects are considered and a direct refinement technique is proposed to enhance the convergence of the algorithm

parameterized preconditioned hermitian and skew hermitian splitting iteration method for a class of linear matrix equations recently wang et al 2013 proposed an efficient hermitian and skew hermitian splitting hss iteration method to approximate the solution of linear matrix equation to further improve the efficiency of the hss method we in this paper establish a parameterized preconditioned hss pphss iteration method by introducing two hermitian positive definite preconditioners and two more different iteration parameters for the hss method the convergence properties of the pphss method used for solving linear matrix equation a x b c are strictly analyzed and the quasi optimal values of the iteration parameters for the pphss method are also derived numerical results demonstrate the feasibility and the efficiency of the pphss iteration method

rigidity of schottky sets we call the complement of a union of at least three disjoint round open balls in the unit sphere bbb s n a schottky set we prove that every quasisymmetric homeomorphism of a schottky set of spherical measure zero to another schottky set is the restriction of a m obius transformation on bbb s n in the other direction we show that every schottky set in bbb s 2 of positive measure admits nontrivial quasisymmetric maps to other schottky sets these results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries

the least squares anti bisymmetric solution and the optimal approximation solution for sylvester equation in this paper a modified conjugate gradient iterative method for solving sylvester equation is presented by using this iterative method the least squares anti bisymmetric solution and the optimal approximation solution can be obtained here we present the derivation and theoretical analysis of our iterative method numerical results illustrate the feasibility and effectiveness of the proposed iterative method

an analytical solution for vibrations of a polarly orthotropic mindlin sectorial plate with simply supported radial edges abstract this paper presents the first known analytical solution for vibrations of a polarly orthotropic mindlin sectorial plate with simply supported radial edges the solution is a series solution constructed using the frobenius method and exactly satisfies not only the boundary conditions along the radial and circular edges but also the regularity conditions at the vertex of the radial edges the moment and shear force singularities at the vertex are exactly considered in the solution the correctness of the proposed solution is confirmed by comparing non dimensional frequencies of isotropic plates obtained from the present solution with published data obtained from a closed form solution this paper also investigates the effects of elastic and shear moduli on the vibration frequencies of the sectorial plates with free or fix boundary conditions along the circumferential edge a study is also carried out about the influence of elastic and shear moduli on the moment and shear force singularities at the plate origin r 0 for different vertex angles

technical communique upper bounds for the solution of the discrete algebraic lyapunov equation new upper bounds for the solution of the discrete algebraic lyapunov equation dale p apa t q are presented the only restriction on their applicability is that a be stable there are no restrictions on the singular values of a nor on the diagonalizability of a the new bounds relate the size of p to the radius of stability of a the upper bounds are computable when the large dimension of a make direct solution of the dale impossible the new bounds are shown to reflect the dependence of p on a better than previously known upper bounds

bounds for the solution of the lyapunov matrix equation a unified approach abstract in recent years several bounds have been reported for the solution of the continuous and the discrete lyapunov equations using the unified lyapunov equation we give in this paper bounds for the solution of this equation in the limiting cases the bounds reduce to existing bounds for both the continuous and discrete lyapunov equations

rigidity theorems for circle domains a circle domain omega in the riemann sphere is conformally rigid if every conformal map from omega onto another circle domain is the restriction of a mobius transformation we show that circle domains satisfying a certain quasihyperbolic condition which was considered by jones and smirnov ark mat 38 263 279 2000 are conformally rigid in particular holder circle domains and john circle domains are all conformally rigid this provides new evidence for a conjecture of he and schramm relating rigidity and conformal removability

a matrix crs iterative method for solving a class of coupled sylvester transpose matrix equations abstract in this paper we apply kronecker product and vectorization operator to extend the conjugate residual squared crs method for solving a class of coupled sylvester transpose matrix equations some numerical examples are given to compare the accuracy and efficiency of the new matrix iterative method with other methods presented in the literature numerical results validate that the proposed method can be much more efficient than some existing methods

lower bounds for the solution of the discrete algebraic lyapunov equation lower bounds for the determinant and for the trace of the matrix solution of the discrete algebraic lyapunov equation are presented the determinant is tighter than bounds in the literature and the trace supplements them

on an inequality for the hadamard product of an m matrix or an h matrix and its inverse abstract let a be an n n matrix q a min a and a denote the spectrum of a from fiedler and markham linear algebra appl 101 1988 1 song linear algebra appl 305 2000 99 and yong linear algebra appl 320 2000 167 for the hadamard products of n n m matrices and their inverses the infimum of q a a 1 is 2 n in this paper the following results are presented if q ak ak 1 tends to the infimum 2 n for n n n 2 m matrices ak k 1 2 then the spectral radius jk of the jacobi iterative matrix of ak tends to 1 that is if q a a 1 is close to 2 n then j is close to 1 and another lower bound is given for a being an n n m matrix q a a 1 max 1 j 2 1 j 1 n 2 1 n 1 j 1 n 2 where j is the spectral radius of the jacobi iterative matrix of a furthermore if a is an h matrix then q a a 1 1 jm a 2 1 jm a 2 where jm a is the spectral radius of the jacobi iterative matrix of the comparison matrix m a

on modified hss iteration methods for continuous sylvester equations we introduce and analyze a modification of the hermitian and skew hermitian splitting iteration method for solving large sparse continuous sylvester equations with non hermitian and complex symmetric positive definite semidefinite matrices it is found that the modified hermitian and skew hermitian splitting mhss iteration method is unconditionally convergent each iteration in this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices these two systems can be solved inexactly numerical results show that the mhss iteration method and its inexact variant are efficient and robust solvers for this class of continuous sylvester equations

free vibration analysis of mindlin sector plates numerical solutions by differential quadrature method abstract in this paper the free vibration analysis of moderately thick sector plates based on mindlin s first order shear deformation theory is examined the solutions to this problem are determined using the differential quadrature dq method for numerical evaluation the dq formulations for free vibration of the mindlin sector plates in two dimensional polar co ordinate system are derived to demonstrate the accuracy of the dq method convergence and comparison studies are carried out in parametric study the first eight frequency parameters are computed for sector plates with different boundary conditions relative thickness ratios and sector angles 30 360 the effects of sector angle and relative thickness ratio on the frequency parameters are discussed

sor for ax xb c we consider a new approach to the block sor method applied to linear systems of equations which can be written as a matrix equation ax xb c such systems arise for example from finite difference discretizations of seperable elliptic boundary value problems on rectangular domains on one hand this gives us an iterative method for the solution of such matrix equations e g lyapunov s matrix equation where b at and on the other hand the problem of choosing appropriate parameters for the block sor method can be written in a more compact form which may be helpful especially for non self adjoint problems i e if a and b are nonsymmetric using this technique we determine under more general assumptions than those of chin and manteuffel the optimal sor parameters for the model problem of a convection diffusion equation

on solution bounds for three types of lyapunov matrix equations continuous discrete and unified equations an approach is proposed to obtain a series of solution bounds for the unified algebraic lyapunov matrix equation by making use of its feature i e a mediator between the continuous time and discrete time lyapunov equations due to the approach every existing solution bound for both the continuous and discrete equations becomes intact in the sense that it yields a corresponding bound for the unified equation the approach is endorsed by this simple fact the three types of lyapunov equations share the solution provided certain relationships are maintained among corresponding coefficient matrices of the equations some examples are presented to show that the proposed method actually produces new bounds and builds bridges over existing bounds

minimal residual methods augmented with eigenvectors for solving sylvester equations and generalized sylvester equations in this paper we propose minimal residual methods augmented with eigenvectors for solving large sylvester matrix equations ax xb c and generalized sylvester matrix equations axb x c the subspace from which the approximate solution is extracted is the kronecker product subspace of two augmented krylov subspaces numerical experiments report the effectiveness of these methods

krylov subspace methods for the sylvester equation abstract we describe galerkin and minimal residual algorithms for the solution of sylvester s equation ax xb c the algorithms use krylov subspaces forwhich orthogonal bases are generated by the arnoldi process for certain choices of krylov subspaces the computation of the solution splits into the solution of many independent subproblems this makes the algorithms suitable for implementation on parallel computers

parallel computation of the solutions of coupled algebraic lyapunov equations a parallel algorithm for solving systems of coupled lyapunov equations associated with linear jump parameter systems is introduced the recursive scheme is based on solving independent reduced order lyapunov equations monotonicity of convergence is established

unitary dilation approach to contractive matrix completion in this paper a new method to construct all contractive matrix completion to a block triangular matrix is given this new method uses only elementary matrix operations and has the advantages of better numerical reliability and ease in implementation on computers

conjugate gradient least squares algorithm for solving the generalized coupled sylvester matrix equations abstract this paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled sylvester matrix equations j 1 q a i j x j b i j f i i 1 2 p we prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum norm solution

iterative methods for cyclically reduced non self adjoint linear systems we study iterative methods for solving linear systems of the type arising from two cyclic discretizations of non self adjoint two dimensional ellip tic partial differential equations a prototype is the convection diffusion equa tion the methods consist of applying one step of cyclic reduction resulting in a reduced system of half the order of the original discrete problem com bined with a reordering and a block iterative technique for solving the reduced system for constant coefficient problems we present analytic bounds on the spectral radii of the iteration matrices in terms of cell reynolds numbers that show the methods to be rapidly convergent in addition we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the unreduced system

lsqr iterative common symmetric solutions to matrix equations axb e and cxd f abstract a new matrix based iterative method is presented to compute common symmetric solution or common symmetric least squares solution of the pair of matrix equations axb e and cxd f by this iterative method for any initial matrix x0 a solution x can be obtained within finite iteration steps if exact arithmetic was used and the solution x with the minimum frobenius norm can be obtained by choosing a special kind of initial matrix in addition the unique nearest common symmetric solution or common symmetric least squares solution x to given matrix x in frobenius norm can be obtained by first finding the minimum frobenius norm common symmetric solution or common symmetric least squares solution of the new pair of matrix equations the given numerical examples show that the matrix based iterative method proposed in this paper has faster convergence than the iterative methods proposed in 1 2 to solve the same problems