a new method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation abstract we present a fast method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation the method is close to the symplectic method for finding all the eigenvalues of a hamiltonian matrix and is based on a g orthogonal transformation which preserves structure and has desirable numerical properties the algorithm requires about one fourth the number of floating point operations and one half the storage of the qz algorithm

on computing the stabilizing solution of the discrete time riccati equation abstract a necessary and sufficient condition is given for the existence of the stabilizing solution of the discrete time riccati equation without any invertibility or positivity assumptions the result involves a special associated matrix pencil

on discrete algebraic riccati equations a rank characterization of solutions abstract we give a rank characterization of the solution set of discrete algebraic riccati equations dares involving real matrices our characterization involves the use of invariant subspaces of the system matrix we observe that a dare can be considered as a specific type of a non symmetric continuous time algebraic riccati equation we show how to extract the low rank solutions from the full rank solution of a homogeneous dare we also study the eigen structure of various feedback closed loop matrices associated with solutions of a dare we show as a consequence of our observations that a solution x of a dare for controllable systems and certain specific types of uncontrollable systems satisfies the inequality x m i n x x m a x we characterize conditions under which the solution set of dare is bounded unbounded we use the method of invariant subspaces to identify some invariant sets of a difference riccati equation

structured maximal perturbations for hamiltonian eigenvalue problems we discuss the effect of structure preserving perturbations on complex or real hamiltonian eigenproblems and characterize the structured worst case effect perturbations we derive explicit expressions for the maximal hamiltonian perturbations

balanced parametrization of classes of linear systems canonical forms and parametrizations are presented for several sets of minimal systems of given dimension asymptotically stable systems allpass systems bounded real systems positive real systems minimum phase systems and the class of all minimal systems the approach is based on balancing techniques for these classes of systems applications of these results to hankel operators and model reduction are discussed

regularization of descriptor systems by output feedback conditions are given under which a descriptor or generalized state space system can be regularized by output feedback it is shown that under these conditions proportional and derivative output feedback controls can be constructed such that the closed loop system is regular and has index at most one this property ensures the solvability of the resulting system of dynamic algebraic equations a reduced form is given that allows the system properties as well as the feedback to be determined the construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way

hermitian solutions of the discrete algebraic riccati equation hermitian solutions of the discrete algebraic riccati equation play an important role in the least squares optimal control problem for discrete linear systems in this paper we describe the set of hermitian solutions in various ways in terms of factorizations of rational matrix functions which take hermitian values on the unit circle in terms of certain invariant subspaces of a matrix which is unitary in an indefinite scalar product and in terms of all invariant subspaces of a certain matrix these results are inspired by known results for the algebraic riccati equation arising in the least squares optimal control problem for continuous linear systems

some remarks on the complex j symmetric eigenproblem abstract the eigenproblem for complex j symmetric matrices h c a c d a t a c c t d d t c n n is considered a proof of the existence of a transformation to the complex j symmetric schur form proposed in 21 is given the complex symplectic unitary qr decomposition and the complex symplectic sr decomposition are discussed it is shown that a qr like method based on the complex symplectic unitary qr decomposition is not feasible here a complex symplectic sr algorithm is presented which can be implemented such that one step of the sr algorithm can be carried out in o n arithmetic operations based on this a complex symplectic lanczos method can be derived moreover it is discussed how the 2 n 2 n complex j symmetric matrix h c can be embedded in a 4 n 4 n real hamiltonian matrix

the smoothed spectral abscissa for robust stability optimization this paper concerns the stability optimization of parameterized matrices a x a problem typically arising in the design of fixed order or fixed structured feedback controllers it is well known that the minimization of the spectral abscissa function alpha a gives rise to very difficult optimization problems since alpha a is not everywhere differentiable and even not everywhere lipschitz we therefore propose a new stability measure namely the smoothed spectral abscissa tilde alpha epsilon a which is based on the inversion of a relaxed h 2 type cost function the regularization parameter epsilon allows tuning the degree of smoothness for epsilon approaching zero the smoothed spectral abscissa converges towards the nonsmooth spectral abscissa from above so that tilde alpha epsilon a leq0 guarantees asymptotic stability evaluation of the smoothed spectral abscissa and its derivatives w r t matrix parameters x can be performed at the cost of solving a primal dual lyapunov equation pair allowing for an efficient integration into a derivative based optimization framework two optimization problems are considered on the one hand the minimization of the smoothed spectral abscissa tilde alpha epsilon a x as a function of the matrix parameters for a fixed value of epsilon and on the other hand the maximization of epsilon such that the stability requirement tilde alpha epsilon a x leq0 is still satisfied the latter problem can be interpreted as an h 2 norm minimization problem and its solution additionally implies an upper bound on the corresponding h infty norm or a lower bound on the distance to instability in both cases additional equality and inequality constraints on the variables can be naturally taken into account in the optimization problem

riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices until recently it was believed that a necessary and sufficient condition for convergence of the riccati difference equation of optimal filtering was that the system be both delectable and stabilizable recently it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular the present paper generalizes these results in several directions first properties of the algebraic riccati equation are established for the case of singular state transition matrix second several assumptions previously imposed in establishing convergence of the riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability weakening the conditions on the initial covariance and allowing the state transition matrix to be singular third results on the convergence and properties of the riccati equations are expressed as both necessary and sufficient conditions whereas previous results were only sufficient these extensions mean that the results have wider applicability including fixed lag smoothing problems and filtering for systems with time delays the implications of the results in the dual problem of optimal control are also studied

an updating algorithm for subspace tracking in certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components the usual tool for doing this is the singular value decomposition however the singular value decomposition has the drawback that it requires o p sup 3 operations to recompute when a new sample arrives it is shown that a different decomposition called the urv decomposition is equally effective in exhibiting the null space and can be updated in o p sup 2 time the updating technique can be run on a linear array of p processors in o p time

a morley finite element method for the displacement obstacle problem of clamped kirchhoff plates abstract we study a morley finite element method for the displacement obstacle problem of clamped kirchhoff plates on polygonal domains error estimates are derived in the energy norm and the l norm the performance of the method is illustrated by numerical experiments

double set parameter finite element method for two sided displacement obstacle problem of clamped plate abstract double set parameter nonconforming finite element method fem for the two sided displacement obstacle problem of clamped plate is considered firstly a double set parameter nonconforming triangular element named veubeke zienkiewicz element is introduced which has the same global dimension as zienkiewicz element and can pass the generalized patch test for any mesh secondly we skillfully construct an enriching operator for connecting the veubeke zienkiewicz and the c 1 conforming bell element spaces with the help of this operator and an auxiliary obstacle problem built as a bridge between the continuous and discrete obstacle problems we present the optimal order error estimates in the broken energy norm and l norm respectively finally we establish a general framework of double set parameter nonconforming elements approximating for the two sided displacement obstacle problem of clamped plate

a robust numerical method for the γ iteration in h control abstract we present a numerical method for the solution of the optimal h control problem based on the iteration and a novel extended matrix pencil formulation of the state space solution to the sub optimal h control problem in particular instead of algebraic riccati equations or unstructured matrix pencils our approach is solely based on solving even generalized eigenproblems the enhanced numerical robustness of the method is derived from the fact that using the structure of the problem spectral symmetries are preserved moreover these methods are also applicable even if the pencil has eigenvalues on the imaginary axis we compare the new method with conventional methods and present several examples

balancing sparse hamiltonian eigenproblems balancing a matrix by a simple and accurate similarity transformation can improve the speed and accuracy of numerical methods for computing eigenvalues we describe balancing strategies for a large and sparse hamiltonian matrix h it is first shown how to permute h to irreducible form while retaining its structure this form can be used to decompose the hamiltonian eigenproblem into smaller sized problems next we discuss the computation of a symplectic scaling matrix d so that the norm of d 1 h d is reduced the considered scaling algorithm is solely based on matrix vector products and thus particularly suitable if the elements of h are not explicitly given the merits of balancing for eigenvalue computations are illustrated by several practically relevant examples c 2004 elsevier inc all rights reserved

a step toward a unified treatment of continuous and discrete time control problems in this paper we introduce a new approach for a unified theory for continuous and discrete time optimal control problems based on the generalized cayley transformation we also relate the associated discrete and continuous generalized algebraic riccati equations we demonstrate the potential of this new approach by proving a new result for discrete algebraic riccati equations but we also discuss where this new approach as well as all other approaches still is nonsatisfactory we explain a discrepancy observed between the discrete and continuous case and show that this discrepancy is partly due to the consideration of the wrong analogues we also present an idea for an implication scheme that relates general theorems for discrete and continuous control problems

computing the density of states for optical spectra of molecules by low rank and qtt tensor approximation abstract in this paper we introduce a new interpolation scheme to approximate the density of states dos for a class of rank structured matrices with application to the tamm dancoff approximation tda of the bethe salpeter equation bse the presented approach for approximating the dos is based on two main techniques first we propose an economical method for calculating the traces of parametric matrix resolvents at interpolation points by taking advantage of the block diagonal plus low rank matrix structure described in 6 3 for the bse tda problem this allows us to overcome the computational difficulties of the traditional schemes since we avoid the construction of the matrix inverse and hence the need of stochastic sampling second we show that a regularized or smoothed dos discretized on a fine grid of size n can be accurately represented in a low rank quantized tensor train qtt format that can be determined through a least squares fitting procedure the qtt tensor provides good approximation properties for strictly oscillating dos functions with multiple gaps in contrast to interpolation by problem independent functions like polynomials trigonometric functions etc moreover the qtt approximant requires asymptotically much fewer e g o log n functional calls compared with the full grid size n numerical tests indicate that the qtt approach yields accurate recovery of dos associated with problems that contain relatively large spectral gaps the qtt tensor rank only weakly depends on the size of a molecular system that paves the way for treatment of large scale spectral problems

a new real structure preserving quaternion qr algorithm abstract new real structure preserving decompositions are introduced to develop fast and robust algorithms for the right eigenproblem of general quaternion matrices under the orthogonally j r s symplectic transformations the francis j r s qr step and the j r s qr algorithm are firstly proposed for j r s symmetric matrices and then applied to calculate the schur form of quaternion matrices a novel quaternion givens matrix is defined and utilized to compute the qr factorization of quaternion hessenberg matrices an implicit double shift quaternion qr algorithm is presented with a technique for automatically choosing shifts and within real operations numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms

a quadratic c 0 interior penalty method for the displacement obstacle problem of clamped kirchhoff plates we study a quadratic c 0 interior penalty method for the displacement obstacle problem of kirchhoff plates with general dirichlet boundary conditions on general polygonal domains under the conditions that the obstacles are sufficiently smooth and separated from each other and the boundary displacement we prove that the magnitudes of the errors in the energy norm and the l infty norm are o h alpha where h is the mesh size and alpha frac12 is determined by the interior angles of the polygonal domain we also address the approximations of the coincidence set and the free boundary the performance of the method is illustrated by numerical results

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

an implicitly restarted krylov subspace method for real symmetric skew symmetric eigenproblems abstract a new implicitly restarted krylov subspace method for real symmetric skew symmetric generalized eigenvalue problems is presented the new method improves and generalizes the shira method of mehrmann and watkins 2001 37 to the case where the skew symmetric matrix is singular it computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point several applications from control theory are presented and the properties of the new method are illustrated by benchmark examples

a numerical method for quadratic eigenvalue problems of gyroscopic systems abstract we consider the quadratic eigenvalues problem qep of gyroscopic systems 2 m g k x 0 where m m g g and k k r n n with m being positive definite guo numerical solution of a quadratic eigenvalue problem linear algebra and its applications 385 2004 391 406 showed that all eigenvalues of the qep can be found by solving the maximal solution of a nonlinear matrix equation z a z 1 a q with quadratic convergence when the qep has no eigenvalues on the imaginary axis the convergence becomes linear or more slower guo 2004 when the qep allows purely imaginary eigenvalues having even partial multiplicities in this paper we consider the general case when the qep has eigenvalues on the imaginary axis we propose an eigenvalue shifting technique to transform the original gyroscopic system to a new gyroscopic system which shifts purely imaginary eigenvalues to eigenvalues with nonzero real parts while keeps other eigenpairs unchanged this transformation ensures that the new method for computing the maximal solution of the nonlinear matrix equation converges quadratically numerical examples illustrate the efficiency of our method

a new error analysis of nonconforming bergan s energy orthogonal element for the extended fisher kolmogorov equation abstract a nonconforming finite element method fem for short is proposed and analyzed for the fourth order extended fisher kolmogorov efk for short equation by employing the bergan s energy orthogonal plate element because the shape function and its first derivatives of this element are discontinuous at the element s vertices which is quite different from the conventional finite elements used in the existing literature a series of novel approaches including some a priori bounds and interpolation function splitting are developed to present a new error analysis for deriving optimal estimates of order o h in energy norm for both the semi discrete and backward euler fully discrete schemes at last numerical experiments are also provided to verify the theoretical analysis

a classification of biconservative hypersurfaces in a pseudo euclidean space abstract in this paper we study biconservative hypersurfaces of index 2 in e 2 5 we give the complete classification of biconservative hypersurfaces with diagonalizable shape operator with exactly three distinct principal curvatures we also give an explicit example of biconservative hypersurfaces with four distinct principal curvatures

on structured componentwise condition numbers for hamiltonian eigenvalue problems abstract in this paper we introduce the structured componentwise perturbation analysis to hamiltonian eigenvalue problems the explicit expressions for the relative structured componentwise condition number for hamiltonian eigenvalue problems are derived also we will compare the relative structured componentwise condition number with the relative unstructured componentwise condition number and the relative structured normwise condition number respectively we show the superiority of the relative structured componentwise condition number through numerical examples

numerical solution of the discrete time periodic riccati equation in this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic riccati equation this method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the euler lagrange difference equations in doing so it is possible to extract a basis for the stable deflating subspace of the extended pencil from which the riccati solution is obtained this algorithm is an extension of the standard qz algorithm and retains its attractive features such as quadratic convergence and small relative backward error a method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with

bilinear transformation of infinite dimensional state space systems and balanced realizations of nonrational transfer functions the bilinear transform maps the open right half plane to the open unit disk and is therefore a suitable tool for carrying over results for continuous time systems to discrete time systems and vice versa corresponding state space formulae are widely used and well understood for the case of finite dimensional systems in this paper infinite dimensional generalizations of these formulae are studied for a general class of infinite dimensional state space systems in particular it is shown that reachability and observability are carried over and that the reachability and observability gramians are preserved under this transformation young showed that a wide class of nonrational discrete time transfer functions admit a balanced state space representation it is shown that this result carries over to the continuous time situation via the bilinear transformation

hermitian solutions of the discrete time algebraic riccati equation this paper deals with the existence and uniqueness of unmixed solutions of the discrete time algebraic riccati equation dare sets of solutions of the dare are parametrized by invariant subspaces of the state transition matrix

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

on a class of generalized eigenvalue problems and equivalent eigenvalue problems that arise in systems and control theory systems and control theory has long been a rich source of problems for the numerical linear algebra community in many problems conditions on analytic functions of a complex variable are usually evaluated by solving a special generalized eigenvalue problem in this paper we develop a general framework for studying such problems we show that for these problems solutions can be obtained by either solving a generalized eigenvalue problem or by solving an equivalent eigenvalue problem a consequence of this observation is that these problems can always be solved by finding the eigenvalues of a hamiltonian or discrete time counterpart matrix even in cases where an associated hamiltonian matrix cannot normally be defined we also derive a number of new compact tests for determining whether or not a transfer function matrix is strictly positive real these tests which are of independent interest due to the fact that many problems can be recast as spr problems are defined even in the case when the matrix d d is singular and can be formulated without requiring inversion of the system matrix a

finite element methods for the displacement obstacle problem of clamped plates we study finite element methods for the displacement obstacle problem of clamped kirchhoff plates a unified convergence analysis is provided for c1 finite element methods classical nonconforming finite element methods and c0 interior penalty methods under the condition that the obstacles are sufficiently smooth and that they are separated from each other and the zero displacement boundary constraint we prove that the convergence in the energy norm is o h for convex domains

on the existence of a negative semidefinite antistabilizing solution to the discrete time algebraic riccati equation in the problem of infimizing a not necessarily positive semidefinite quadratic cost subject to a linear dynamical constraint it is usually expected that the existence of a lower bound to the cost is equivalent to the existence of a negative semidefinite antistabilizing solution to the algebraic riccati equation by a counterexample it is shown that this equivalence breaks down in the discrete time case this phenomenon as well as the whole question of the existence of the appropriate solution to the algebraic riccati equation are investigated in detail

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

hamilton and jacobi come full circle jacobi algorithms for structured hamiltonian eigenproblems we develop jacobi algorithms for solving the complete eigenproblem for hamiltonian and skew hamiltonian matrices that are also symmetric or skew symmetric based on the direct solution of 4 4 and in one case 8 8 subproblems these structure preserving algorithms produce symplectic orthogonal bases for the invariant subspaces associated with a matrix in any one of the four classes under consideration the key step in the construction of the algorithms is a quaternion characterization of the 4 4 symplectic orthogonal group and the subspaces of 4 4 hamiltonian skew hamiltonian symmetric and skew symmetric matrices in addition to preserving structure these algorithms are inherently parallelizable numerically stable and show asymptotic quadratic convergence

on computing the eigenvalues of a symplectic pencil this paper presents an algorithm for computing the eigenvalues of a symplectic pencil that arises in one of the commonly used approaches for solving the discrete time algebraic riccati equation the algorithm is numerically efficient and reliable in that it employs only orthogonal transformations and makes use of the structure of the symplectic pencil it requires about one fourth the number of floating point operations that the qz algorithm uses to compute the eigenvalues of the pencil directly the proposed method can be regarded as being analogous for the case of symplectic pencils to the method developed by van loan for computing the eigenvalues of hamiltonian matrices

continuous and discrete time riccati theory a popov function approach abstract necessary and sufficient conditions for the existence of the stabilizing solution of the algebraic riccati equation are derived for both the continuous and the discrete time case under the weakest possible assumptions imposed on the initial data both frequency domain conditions involving an associated popov function and time domain conditions involving an associated matrix pencil are investigated

a new error analysis of bergan s energy orthogonal element for two sided displacement obstacle problem of clamped plate abstract the main aim of this paper is to present a new error analysis of the bergan s energy orthogonal plate element for the two sided displacement obstacle problem of clamped plate due to the exact solution does not belong to h l o c 4 and the shape function and its first derivatives are discontinuous at the element s vertices while all previous theoretical analysis always required the continuity of the shape function at element s vertices for the conventional finite elements a series of novel approaches including the definition of approximation subset the constructions of a one to one transferring operator and an enriching operator are developed to derive error estimates of order o h both in energy norm and l norm it seems that these results have never been seen in the existing literature

on biconservative surfaces in 4 dimensional euclidean space abstract in this paper we study non cmc biconservative surfaces with flat normal bundle in the euclidean 4 space e 4 first we determine locally all parallel normalized mean curvature vector pnmcv biconservative surfaces in e 4 showing that they are certain rotational surfaces then we consider meridian surfaces lying on a rotational hypersurface of e 4 and construct a family of biconservative surfaces with flat normal bundle and non parallel normalized mean curvature vector

a new block method for computing the hamiltonian schur form abstract a generalization of the method of chu et al d chu x liu v mehrmann a numerical method for computing the hamiltonian schur form numer math 105 2007 375 412 for the computation of the hamiltonian real schur form is presented the new method avoids some of the difficulties that may arise when a hamiltonian matrix has tightly clustered groups of eigenvalues a detailed analysis of the method is presented and several numerical examples demonstrate the superior behavior of the method

fast iterative solution of the bethe salpeter eigenvalue problem using low rank and qtt tensor approximation abstract in this paper we propose and study two approaches to approximate the solution of the bethe salpeter equation bse by using structured iterative eigenvalue solvers both approaches are based on the reduced basis method and low rank factorizations of the generating matrices we also propose to represent the static screen interaction part in the bse matrix by a small active sub block with a size balancing the storage for rank structured representations of other matrix blocks we demonstrate by various numerical tests that the combination of the diagonal plus low rank plus reduced block approximation exhibits higher precision with low numerical cost providing as well a distinct two sided error estimate for the smallest eigenvalues of the bethe salpeter operator the complexity is reduced to o n b 2 in the size of the atomic orbitals basis set n b instead of the practically intractable o n b 6 scaling for the direct diagonalization in the second approach we apply the quantized tt qtt tensor representation to both the long eigenvectors and the column vectors in the rank structured bse matrix blocks and combine this with the als type iteration in block qtt format the qtt rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems n o n b hence the overall asymptotic complexity for solving the bse problem by the qtt approximation is estimated by o log n o n o 2 we confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chain type molecules while supporting sufficient accuracy

numerical methods in control we study classical control problems like pole assignment stabilization linear quadratic control and h control from a numerical analysis point of view we present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework we also discuss some new algorithmic approaches

numerical solution of differential algebraic riccati equations abstract we consider riccati matrix differential algebraic equations arising from singular or descriptor control problems we discuss the solvability of such equations under different conditions in order to apply numerical methods for differential algebraic systems one has to transform the equation unfortunately these equations then have a linear part which is described by a singular pencil and thus the usual integration methods do not apply under some conditions which we discuss these singularities can be removed by a preprocessing algorithm and the equation can then be solved by well known methods for differential algebraic systems like dassl of l petzold or limex of deuflhard hairer and zugck we discuss the numerical procedures and give some numerical examples

cmc biconservative surfaces in sn r and hn r abstract we classify non minimal biconservative surfaces with parallel mean curvature vector field in s n r and h n r when these surfaces do not lie in s n or h n and they are not vertical cylinders we find their explicit local equation we also prove a result on the compactness of biconservative surfaces with constant mean curvature in hadamard manifolds

first order structured perturbation theory for multiple zero eigenvalues of skew adjoint matrices abstract given a matrix belonging to some class of structured matrices we consider the question of comparing the sensitivity of its eigenvalues under two different kinds of perturbations either unstructured i e arbitrary or structured i e those belonging to the same class of matrices as the unperturbed one in a previous paper kressner et al 2009 13 the authors compared the structured and unstructured condition numbers of possibly multiple eigenvalues for several different matrix and pencil structures only one case was left out of the analysis namely the one where the asymptotic order of perturbed eigenvalues under structured perturbations is different from the asymptotic order under unstructured ones this is precisely the case we consider in the present paper given a matrix which is skew adjoint with respect to a symmetric scalar product and has a zero eigenvalue with a certain jordan structure first order expansions are obtained for the perturbed eigenvalues under structured perturbation as well as bounds on the structured condition number similar results are obtained for structured perturbations of symmetric skew symmetric and palindromic matrix pencils

uniformly superconvergent analysis of an efficient two grid method for nonlinear bi wave singular perturbation problem abstract the main aim of this paper is to present a two grid method for the fourth order nonlinear bi wave singular perturbation problem with low order nonconforming finite element based on the ciarlet raviart scheme the existence and uniqueness of the approximation solution are demonstrated through the brouwer fixed point theorem and the uniform superconvergent estimates in the broken h 1 norm and l 2 norm are obtained which are independent of the perturbation parameter some numerical results indicate that the proposed method is indeed an efficient algorithm

complete biconservative surfaces in the hyperbolic space h3 abstract we construct simply connected complete non c m c biconservative surfaces in the 3 dimensional hyperbolic space h 3 in an intrinsic and extrinsic way we obtain three families of such surfaces and for each surface the set of points where the gradient of the mean curvature function does not vanish is dense and has two connected components in the intrinsic approach we first construct a simply connected complete abstract surface and then prove that it admits a unique biconservative immersion in h 3 working extrinsically we use the images of the explicit parametric equations and a gluing process to obtain our surfaces they are made up of circles or hyperbolas or parabolas respectively which lie in 2 affine parallel planes and touch a certain curve lying in a totally geodesic hyperbolic surface h 2 in h 3

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

an svd like matrix decomposition and its applications this author is supported by nsf under grant no eps 9874732 and matching support from the state of kansas

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

a new method for computing the stable invariant subspace of a real hamiltonian matrix abstract a new backward stable structure preserving method of complexity o n3 is presented for computing the stable invariant subspace of a real hamiltonian matrix and the stabilizing solution of the continuous time algebraic riccati equation the new method is based on the relationship between the invariant subspaces of the hamiltonian matrix h and the extended matrix 0 h h 0 and makes use of the symplectic urv like decomposition that was recently introduced by the authors

discontinuous galerkin finite element method for the fourth order obstacle problem in this paper we consider the discontinuous galerkin finite element method for the fourth order obstacle problem which is the variational inequality problem of the first kind by defining the finite element approximation solution we obtain the optimal error estimate

on equivalence of pencils from discrete time and continuous time control we study two matrix pencils that arise respectively in discrete time and continuous time optimal and robust control we introduce a one to one transformation between these two pencils we show that for the pencils under the transformation their regularity is preserved and their eigenvalues and deflating subspaces are equivalently related the eigen structures of the pencils under consideration have strong connections with the associated control problems our result may be applied to connect the discrete time and continuous time control problems and eventually lead to a unified treatment of these two types of control problems