convergence behavior of generalized parameterized uzawa method for singular saddle point problems in this paper we will seek the least squares solution for singular saddle point problems the parameterized uzawa pu method is further studied and a generalized pu gpu proper splitting is proposed the convergence behavior of the corresponding gpu iteration is studied it is proved that the gpu iteration method can converge to the best least squares solutions of the singular saddle point problems in addition we prove that the gpu preconditioned gmres for singular saddle point problems will also determine the least squares solution at breakdown the eigenvalue distributions of the gpu preconditioned matrix are derived numerical experiments are presented which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the gpu iteration has better convergence behavior than the pu iteration both as a solver and a preconditioner of gmres

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

approximate linearization by state feedback and coordinate change a nonlinear system can always be approximated to first order by linear systems it has been shown by jakubczyk and respondek 4 and hunt and su 3 that certain nonlinear systems are the exact transforms of linear systems under nonlinear state coordinate change and nonlinear state feedback in this paper we give necessary and sufficient conditions for a nonlinear system to be approximated to higher order by the transform of a linear system the use of this technique in the design of nonlinear compensators has been suggested recently by the author 6

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

a note on preconditioning nonsymmetric matrices the preconditioners for indefinite matrices of kkt form in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 are extended to general nonsymmetric matrices

analysis of preconditioners for saddle point problems mixed finite element formulations give rise to large sparse block linear systems of equations the solution of which is often sought via a preconditioned iterative technique in this work we present a general analysis of block preconditioners based on the stability conditions inherited from the formulation of the finite element method the babuska brezzi or inf sup conditions the analysis is motivated by the notions of norm equivalence and field of values equivalence of matrices in particular we give sufficient conditions for diagonal and triangular block preconditioners to be norm and field of values equivalent to the system matrix

a note on the positive stable block triangular preconditioner for generalized saddle point problems abstract in this paper we give a sharp eigenvalue bound for the positive stable block triangular preconditioned matrix presented in a recent paper by cao z h cao positive stable block triangular preconditioners for symmetric saddle point problems appl numer math 57 2007 899 910 the intervals containing these eigenvalues of the preconditioned matrix remove the origin which is benefit for further study numerical experiments of a model stokes problem are presented to show the estimate and the effectiveness of the positive stable block triangular preconditioners

restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems the restrictively preconditioned conjugate gradient rpcg method for solving large sparse system of linear equations of a symmetric positive definite and block two by two coefficient matrix is further studied in fact this rpcg method is essentially the classical preconditioned conjugate gradient pcg method with a specially structured preconditioner within this setting we present algorithmic descriptions of two restrictive preconditioners that respectively employ the block jacobi and the block symmetric gauss seidel matrix splitting matrices as approximations to certain matrices involved in them and give convergence analyses of the correspondingly induced two pcg methods numerical results show that these restrictive preconditioners can lead to practical and effective pcg methods for solving large sparse systems of linear equations of symmetric positive definite and block two by two coefficient matrices

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

global convergence of nonlinear cascade flows with morse bott zero dynamics abstract we derive a sufficient condition that a flow captures the dynamics on an invariant submanifold this leads to a refinement of the lasalle invariance principle as a consequence we generalize a well known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set this includes the case where a globally asymptotically stable system is coupled to a morse bott flow

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

an alternating preconditioner for saddle point problems based on matrix splittings a new alternating preconditioner with two parameters is proposed for solving saddle point problems some theoretical analyses for the eigenvalues of the associated preconditioned matrix are given the choice of the parameters is considered and the quasi optimal parameters are obtained the new preconditioner with these quasi optimal parameters significantly improves the convergence rate of the generalized minimal residual gmres iteration numerical experiments from the linearized navier stokes equations demonstrate the efficiency of the new preconditioner especially on the larger viscosity parameter n further extensions of the preconditioner to generalized saddle point matrices are also checked

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

global stabilization of nonlinear cascade systems abstract we present sufficient conditions for the global stabilizability of two cascade connected nonlinear systems these are based on general results concerning global asymptotic stability of triangular systems which are proved in the last section for polynomial systems in particular the stabilizing feedback is given explicitly

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

on generalized parameterized inexact uzawa method for a block two by two linear system recently chen and jiang f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 presented a parameterized inexact uzawa piu algorithm for solving symmetric saddle point problems where the 1 2 and the 2 1 blocks are the transpose of each other in this paper we extend the piu method to the block two by two linear system by allowing the 1 2 block to be not equal to the transpose of the 2 1 block and the 2 2 block may not be zero we prove that the iteration method is convergent under certain conditions with different choices of the parameter matrices we obtain several new algorithms for solving the block two by two linear system numerical experiments confirm our theoretical results and show that our method is feasible and effective

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

on parameterized block triangular preconditioners for generalized saddle point problems in this paper we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems these preconditioners generalize the common block diagonal and triangular preconditioners we will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues numerical experiments of a model stokes problem are presented

a unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations abstract a unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods suitable to the simd and mimd multiprocessor systems respectively is presented and its convergence theory is established under rather weak conditions these afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting in addition how the known parallel matrix multisplitting iterative methods can be classified into this new framework and what novel ones can be generated by it are shown in detail

on the convergence of nonstationary multisplitting two stage iteration methods for hermitian positive definite linear systems convergence properties of the nonstationary multisplitting two stage iteration methods for solving large sparse system of linear equations are further studied when the coefficient matrices are hermitian positive definite matrices

stabilization of a class of nonlinear systems by a smooth feedback control we consider the stabilizability problem for systems of the form x f x ub in the neighborhood of an equilibrium point of f x first by means of center manifold theory a lower order system is introduced if this system is stabilizable then so is the original system results on the stabilization of the low order system are presented no resort is taken to liapunov theory the relation between stabilizability and controllability is discussed

optimization of the parameterized uzawa preconditioners for saddle point matrices the parameterized uzawa preconditioners for saddle point problems are studied in this paper the eigenvalues of the preconditioned matrix are located in 0 2 by choosing the suitable parameters furthermore we give two strategies to optimize the rate of convergence by finding the suitable values of parameters numerical computations show that the parameterized uzawa preconditioners can lead to practical and effective preconditioned gmres methods for solving the saddle point problems

persistence in dynamical systems soit un espace metrique localement compact de metrique d et soit e un sous ensemble clos de a frontiere e et interieur e on suppose que l on a un flot continu 7b f defini sur e tel que e est invariant par 7b f on considere la notion de persistance et on donne un critere de persistance

determinantal representations for the j transformation the iterative j transformation homeier h h h 1993 some applications of nonlinear convergence accelerators int j quantum chem 45 545 562 is of similar generality as the well known e algorithm brezinski c 1980 a general extrapolation algorithm numer math 35 175 180 h vie t 1979 generalized neville type extrapolation schemes bit 19 204 213 the properties of the j transformation were studied recently in two companion papers homeier h h h 1994a a hierarchically consistent iterative sequence transformation numer algo 8 47 81 homeier h h h 1994b analytical and numerical studies of the convergence behavior of the j transformation j comput appl math to appear in the present contribution explicit determinantal representations for this sequence transformation are derived the relation to the brezinski walz theory brezinski c walz g 1991 sequences of transformations and triangular recursion schemes with applications in numerical analysis j comput appl math 34 361 383 is discussed overholt s process overholt k j 1965 extended aitken acceleration bit 5 122 132 is shown to be a special case of the j transformation consequently explicit determinantal representations of overholt s process are derived which do not depend on lower order transforms also families of sequences are given for which overholt s process is exact as a numerical example the euler series is summed using the j transformation the results indicate that the j transformation is a very powerful numerical tool