on modified factorizations for large scale linearly constrained optimization we consider the algebraic issues concerning the solution of general large scale linearly constrained nonlinear optimization problems particular attention is given to suitable methods for solving the linear systems that occur at each iteration of such methods the main issue addressed is how to ensure that a quadratic model of the objective function is positive definite in the null space of the constraints while neither adversely affecting the convergence of newton s method nor incurring a significant computational overhead numerical evidence to support the theoretical developments is provided

approximate factorization constraint preconditioners for saddle point matrices we consider the application of the conjugate gradient method to the solution of large symmetric indefinite linear systems special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given numerical experiments validate these conclusions

variants of the deteriorated pss preconditioner for saddle point problems two new preconditioners which can be viewed as variants of the deteriorated positive definite and skew hermitian splitting preconditioner are proposed for solving saddle point problems the corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks the choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated cayley transform which result in faster convergence rate and more clustered spectrum compared with some preconditioners of similar structures the new preconditioners have better convergence properties and spectrum distributions in addition more practical preconditioning variants of the new preconditioners are considered numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate gmres c 2018 elsevier ltd all rights reserved

a parameterized deteriorated pss preconditioner and its optimization for nonsymmetric saddle point problems abstract based on the deteriorated positive definite and skew hermitian splitting dpss preconditioner we develop a parameterized dpss pdpss preconditioner for the nonsymmetric saddle point linear systems the spectral properties of the pdpss preconditioned coefficient matrix are analyzed moreover an upper bound of the degree of the minimal polynomial of the pdpss preconditioned matrix is also obtained inasmuch as the efficiency of the pdpss preconditioner depends on the values of its parameters we further derive a fast and effective method to compute the optimal parameter values of the pdpss preconditioner finally numerical examples are employed to illustrate the feasibility and the efficiency of the pdpss preconditioner

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

an augmented lagrangian approach to linearized problems in hydrodynamic stability the solution of linear systems arising from the linear stability analysis of solutions of the navier stokes equations is considered due to indefiniteness of the submatrix corresponding to the velocities these systems pose a serious challenge for iterative solution methods in this paper the augmented lagrangian based block triangular preconditioner introduced by the authors in siam j sci comput 28 2006 pp 2095 2113 is extended to this class of problems we prove eigenvalue estimates for the velocity submatrix and deduce several representations of the schur complement operator which are relevant to numerical properties of the augmented system numerical experiments on several model problems demonstrate the effectiveness and robustness of the preconditioner over a wide range of problem parameters

a generalized variant of simplified hss preconditioner for generalized saddle point problems abstract based on the simplified hermitian and skew hermitian splitting shss preconditioner developed by cao ren and shi bit numer math 2016 a generalized variant of the shss gshss preconditioner is presented for solving generalized saddle point problems the convergence of the gshss iteration method and the spectral properties of the preconditioned matrix are studied in details eigenvalue bounds and the degree of minimal polynomials of the preconditioned matrix are obtained in particular a practical and feasible choice strategy of iteration parameter is given which also provides a feasible way to select the accelerated parameters for some other preconditioners numerical experiments are presented which show the effectiveness of the proposed preconditioner and the feasibility of the practical choice strategy of accelerated parameter

a block product preconditioner for saddle point problems abstract in this paper a block product bp preconditioner is established for saddle point problems spectral properties of the bp preconditioned matrix are investigated a strategy for practical choice of quasi optimal parameter is given numerical results on saddle point linear systems arising from stokes problems and weighted least square problems show that the proposed bp preconditioner is more economic to implement within krylov subspace acceleration than some extensively studied preconditioners

an approach to robust stability of matrix polytopes through copositive homogeneous polynomials an approach to guaranteeing the stability of a polytope of matrices is proposed previous results in the literature have made the obvious connection between various robust stability problems and a test for positivity of a multivariable polynomial such results are extended in order to demonstrate that all matrices within a polytope are stable is and only if an associated homogeneous polynomial is strictly copositive the additional structure obtained by exploiting homogeneity of this multivariable polynomial leads to several computationally tractable sufficiency tests for establishing either robust stability or instability of a polytope of matrices

on signed incomplete cholesky factorization preconditioners for saddle point systems limited memory incomplete cholesky factorizations can provide robust preconditioners for sparse symmetric positive definite linear systems in this paper the focus is on extending the approach to sparse symmetric indefinite systems in saddle point form a limited memory signed incomplete cholesky factorization of the form ldl t is proposed where the diagonal matrix d has entries pm 1 the main advantage of this approach is its simplicity as it avoids the use of numerical pivoting instead a global shift strategy involving two shifts one for the 1 1 block and one for the 2 2 block of the saddle point matrix is used to prevent breakdown and to improve performance the matrix is optionally prescaled and preordered using a standard sparse matrix ordering scheme that is then postprocessed to give a constrained ordering that reduces the likelihood of breakdown and need for shifts the use of intermediate memory memory used in the construction of the incomplete factorization but subsequently d

a necessary and sufficient criterion for the stability of a convex set of matrices a hurwitz stability criterion for the convex hull of two matrices due to c r johnson 1976 is generalized to the case of any compact convex set of matrices it is shown that the same criterion is not available in the case of schur stability

parameterized approximate block lu preconditioners for generalized saddle point problems abstract in this paper we are concerned with the iteration solution of generalized saddle point problems based on the exact block lu factorization of the coefficient matrix we construct a class of parameterized approximate block lu factorization preconditioners which rely on suitable approximations of the schur complement of the 1 1 block of the coefficient matrix convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced algebraic characteristics of the related preconditioned matrices are discussed including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial the established results extend those of the approximate factorization and variants of the hermitian and skew hermitian splitting and positive and skew hermitian splitting preconditioners for saddle point problems numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

an augmented lagrangian based approach to the oseen problem we describe an effective solver for the discrete oseen problem based on an augmented lagrangian formulation of the corresponding saddle point system the proposed method is a block triangular preconditioner used with a krylov subspace iteration like bicgstab the crucial ingredient is a novel multigrid approach for the 1 1 block which extends a technique introduced by schoberl for elasticity problems to nonsymmetric problems our analysis indicates that this approach results in fast convergence independent of the mesh size and largely insensitive to the viscosity we present experimental evidence for both isop2 p0 and isop2 p1 finite elements in support of our conclusions we also show results of a comparison with two state of the art preconditioners showing the competitiveness of our approach

maximal unidirectional perturbation bounds for stability of polynomials and matrices abstract given a stable polynomial or matrix we consider a class of unidirectional perturbations in this note we provide a closed form for the maximal perturbation bounds under which stability is preserved the results are then used to derive a closed form for the maximal stability box around the coefficients of a nominally stable polynomial

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

a relaxed dimensional factorization preconditioner for the incompressible navier stokes equations in this paper we introduce a relaxed dimensional factorization rdf preconditioner for saddle point problems properties of the preconditioned matrix are analyzed and compared with those of the closely related dimensional splitting ds preconditioner recently introduced by benzi and guo 7 numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the rdf preconditioner with respect to both mesh size and viscosity

field of values convergence analysis of augmented lagrangian preconditioners for the linearized navier stokes problem we study a block triangular preconditioner for finite element approximations of the linearized navier stokes equations the preconditioner is based on the augmented lagrangian formulation of the problem and was introduced by the authors in siam j sci comput 28 2006 pp 2095 2113 in this paper we prove field of values type estimates for the preconditioned system which lead to optimal convergence bounds for the gmres algorithm applied to solve the system two variants of the preconditioner are considered an ideal one based on exact solves for the velocity submatrix and a more practical variant based on block triangular approximations of the velocity submatrix

the finite primitive permutation groups containing an abelian regular subgroup a complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup this solves a long standing open problem in permutation group theory initiated by w burnside in 1900

the minimal dimension of stable faces required to guarantee stability of a matrix polytope considers the problem of determining whether each point in a polytope n n matrices is stable the approach is to check stability of certain faces of the polytope for n or 3 the authors show that stability of each point in every 2n 4 dimensional face guarantees stability of the entire polytope furthermore they prove that for any k

stability of a polytope of matrices counterexamples while there have been significant breakthroughs for the stability of a polytope of polynomials since v l kharitonov s 1978 seminal result on interval polynomials for a polytope of matrices the stability problem is considered far from completely resolved counterexamples are provided for three conjectures that are directly motivated by the results in the polynomial case these counterexamples illustrate the fundamental differences between polynomial stability and matrix stability problems and indicate that some obvious lines of attack on the matrix polytope stability problem will fail

new local generalized shift splitting preconditioners for saddle point problems based on a local generalized shift splitting of the saddle point matrix with symmetric positive definite 1 1 block and symmetric positive semidefinite 2 2 block a new local generalized shift splitting preconditioner with two shift parameters for solving saddle point problems is proposed the preconditioner is extracted from a new local generalized shift splitting iteration and can lead to the unconditional convergence of the iteration in addition we consider solving the saddle point systems by preconditioned krylov subspace methods and discuss some properties of the preconditioned saddle point matrix with a deteriorated preconditioner such as eigenvalues eigenvectors and degree of the minimal polynomial numerical experiments arising from a finite element discretization model of the stokes problem are given to validate the effectiveness of the proposed preconditioner

solution of indefinite linear systems using an lq decomposition for the linear constraints in this paper indefinite linear systems with linear constraints are considered we present a special decomposition that makes use of the lq decomposition and retains the constraints in the factors the resulting decomposition is of a structure similar to that obtained using the bunch kaufman parlett algorithm the decomposition can be used in a direct solution algorithm for indefinite systems but it can also be used to construct effective preconditioners combinations of the latter with conjugate gradient type methods have been demonstrated to be very useful

sparse block factorization of saddle point matrices abstract the factorization method presented in this paper takes advantage of the special structures and properties of saddle point matrices a variant of gaussian elimination equivalent to the cholesky s factorization is suggested and implemented for factorizing the saddle point matrices block wise with small blocks of orders 1 and 2 the gaussian elimination applied to these small blocks on block level also induces a block 3 3 structured factorization of which the blocks have special properties we compare the new block factorization with the schilders factorization in terms of sparsity and computational complexity the factorization can be used as a direct method and also anticipate for preconditioning techniques

on augmentation block triangular preconditioners for regularized saddle point problems two augmentation block triangular preconditioners were introduced by shen et al 2012 for the regularized saddle point problem however the spectral analysis of the preconditioner based on the augmentation of the 2 2 block was not throughly derived there in this short paper we give a detailed spectral analysis on this preconditioner a numerical example is employed to validate the efficiency of the presented theoretical results

finite permutation groups with a regular dihedral subgroup and edge transitive dihedrants abstract we study finite permutation groups that contain a regular dihedral subgroup called d groups in particular a classification is given of bi quasiprimitive d groups it is then applied to classify edge transitive dihedrants of order 2 p m and valency coprime to p where p is an odd prime and m is a positive integer

transitive characteristically simple subgroups of finite quasiprimitive permutation groups abstract the first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product this allows us to characterize nonabelian transitive characteristically simple subgroups h of finite quasiprimitive permutation groups g if the socle of g denoted by soc g is nonabelian then either h is intransitive or soc g is an alternating group in its natural action an explicit description is given for the possibilities of h under the condition that h does not contain a nontrivial normal subgroup of soc g

a pair of matrices sharing common lyapunov solutions a closer look abstract let a b be a pair of matrices with regular inertia if ha a h and hb b h are both positive definite for some hermitian matrix h then all matrices in conv a a 1 b b 1 have identical regular inertia this in turn implies that both conv a b and conv a b 1 consist of non singular matrices in general neither of the converse implications holds in this paper we seek situations where they do hold in particular when a and b are real 2 2 matrices several aspects of the above statements for n n matrices are discussed a connection to the characterization of the convex hull of matrices with regular inertia is introduced differences between the real and the complex case are indicated

on the np hardness of checking matrix polytope stability and continuous time switching stability motivated by questions in robust control and switched linear dynamical systems we consider the problem checking whether all convex combinations of k matrices in rntimesn are stable in particular we are interested whether there exist algorithms which can solve this problem in time polynomial in n and k we show that if k nd for any fixed real d 0 then the problem is np hard meaning that no polynomial time algorithm in n exists provided that p ne np a widely believed conjecture in computer science on the other hand when k is a constant independent of n then it is known that the problem may be solved in polynomial time in n using these results and the method of measurable switching rules we prove our main statement verifying the absolute asymptotic stability of a continuous time switched linear system with more than nd matrices ai isin rntimesn satisfying 0 ges ai ai t is np hard

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

numerical contour integral methods for unsteady stokes equations abstract the unsteady stokes equations are semi discretized in space to obtain a system of linear time invariant differential algebraic equations daes i e the unsteady discrete stokes equations the solution to unsteady discrete stokes equations is represented as an integral along a smooth curve in the complex plane with singularities of the integrand located on the left of and not too close to the curve truncated quadrature rules based on the sinc function are then employed to evaluate the solution this results in a number of complex linear systems to solve leading to major expense in practical implementation constraint preconditioners are proposed to work with the krylov subspace methods for solving those complex linear systems numerical examples illustrate that the numerical contour integral methods are more effective than the time stepping methods in addition the constraint preconditioners significantly improve the behavior of krylov subspace methods for solving the involved complex linear systems

on quasi shift splitting iteration method for a class of saddle point problems abstract motivated by a technique of scale splitting for matrix we put forward a quasi shift splitting named after qss iteration method to address a class of large scale sparse saddle point problems this novel method also naturally leads to the corresponding qss preconditioner in addition an extrapolated variant of qss named after eqss iteration method is constructed further some useful convergence properties of the qss and eqss iteration methods and spectral properties of qss preconditioner are studied finally a saddle point equations stem from model navier stokes problem is supplied to illustrate the effectiveness of our new two methods and qss preconditioner

a new relaxed pss preconditioner for nonsymmetric saddle point problems a new relaxed pss like iteration scheme for the nonsymmetric saddle point problem is proposed as a stationary iterative method the new variant is proved to converge unconditionally when used for preconditioning the preconditioner differs from the coefficient matrix only in the upper right components the theoretical analysis shows that the preconditioned matrix has a well clustered eigenvalues around 1 0 with a reasonable choice of the relaxation parameter this sound property is desirable in that the related krylov subspace method can converge much faster which is validated by numerical examples

a relaxed positive definite and skew hermitian splitting preconditioner for saddle point problems for saddle point linear system whose coefficient matrix is non hermitian of positive definite hermitian part pan et al 2006 proposed a positive definite and skew hermitian splitting pss preconditioner in this paper based on the pss preconditioner we present a relaxed positive definite and skew hermitian splitting rpss preconditioner we analyze the spectral properties of the new preconditioned matrix and conclude that the rpss preconditioner has much better properties than pss numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the new preconditioner

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

analysis of augmented lagrangian based preconditioners for the steady incompressible navier stokes equations we analyze a class of modified augmented lagrangian based preconditioners for both stable and stabilized finite element discretizations of the steady incompressible navier stokes equations we study the eigenvalues of the preconditioned matrices obtained from picard linearization and we devise a simple and effective method for the choice of the augmentation parameter gamma based on fourier analysis numerical experiments on a wide range of model problems demonstrate the robustness of these preconditioners yielding fast convergence independent of mesh size and only mildly dependent on viscosity on both uniform and stretched grids good results are also obtained on linear systems arising from newton linearization we also show that performing inexact preconditioner solves with an algebraic multigrid algorithm results in excellent scalability comparisons of the modified augmented lagrangian preconditioners with other state of the art techniques show the competitiveness of our approach

two modified block triangular splitting preconditioners for generalized saddle point problems abstract for the generalized saddle point problems firstly we introduce a modified generalized relaxed splitting mgrs preconditioner to accelerate the convergence rate of the krylov subspace methods based on a block triangular splitting of the saddle point matrix secondly we propose a modified block triangular splitting mbts this new preconditioner is easily implemented since it has simple block structure the spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed respectively moreover we apply the mgrs and the mbts preconditioners to three dimensional linearized navier stokes equations then we derive the quasi optimal parameters of the mgrs and the mbts preconditioners for two and three dimensional navier stokes equations respectively finally numerical experiments are illustrated to show the preconditioning effects of the two new preconditioners

threshold incomplete factorization constraint preconditioners for saddle point matrices abstract this paper presents a drop threshold incomplete ld 1lt factorization constraint preconditioner for saddle point systems using a threshold parameter a transformed saddle point matrix is partitioned into a block structure with blocks of order 1 and 2 constituting a priori pivots based on these pivots an incomplete ld 1lt factorization constraint preconditioner is computed that approaches an exact form as approaches zero we prove that both the exact and incomplete factorizations exist such that the entries of the constraint block remain unaltered in the triangular factors numerical results are presented for validation

on the spectrum of convex sets of matrices let k be an arbitrary compact convex set of square matrices this paper presents necessary and sufficient conditions for the spectrum of k to have no eigenvalues in a prespecified closed convex subset of the complex plane the obtained result implies different criteria for analysis of the spectral set of k in particular we have formulated criteria for nonsingularity inertia and hurwitz stability of k which can be used in the robustness analysis of linear control systems with uncertain parameters

preconditioning iterative methods for the optimal control of the stokes equations solving problems regarding the optimal control of partial differential equations pdes also known as pde constrained optimization is a frontier area of numerical analysis of particular interest is the problem of flow control where one would like to effect some desired flow by exerting for example an external force the bottleneck in many current algorithms is the solution of the optimality system a system of equations in saddle point form that is usually very large and ill conditioned in this paper we describe two preconditioners a block diagonal preconditioner for the minimal residual method and a block lower triangular preconditioner for a nonstandard conjugate gradient method which can be effective when applied to such problems where the pdes are the stokes equations we consider only distributed control here although we believe other problems could be treated in the same way we give numerical results and we compare these with those obtained by solving the equivalent forward problem using similar techniques

preconditioning linear least squares problems by identifying a basis matrix we study the solution of the linear least squares problem min x b a x 2 2 where the matrix a in bbb r m times n m ge n has rank n and is large and sparse we assume that a is available as a matrix not an operator the preconditioning of this problem is difficult because the matrix a does not have the properties of differential problems that make standard preconditioners effective incomplete cholesky techniques applied to the normal equations do not produce a well conditioned problem we attempt to bypass the ill conditioning by finding an n times n nonsingular submatrix b of a that reduces the euclidean norm of ab 1 we use b to precondition a symmetric quasi definite linear system whose condition number is then independent of the condition number of a and has the same solution as the original least squares problem we illustrate the performance of our approach on some standard test problems and show it is competitive with other approaches

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

analysis of the relaxed deteriorated pss preconditioner for singular saddle point linear systems abstract the relaxed deteriorated pss rdpss preconditioner proposed by cao et al 13 is further generalized to solve singular saddle point linear systems properties of the rdpss splitting and convergence of the corresponding iteration method are studied by using the moore penrose inverse the rdpss iteration method and the rdpss preconditioned gmres are both proved to converge to the least squares solution of the singular saddle point linear system the rdpss preconditioned matrix is also analyzed results about eigenvalue distributions are derived moreover the rdpss preconditioner is generalized to a class of more general preconditioners the corresponding convergence and eigenvalue distribution results are analyzed numerical experiments are presented to illustrate the effectiveness of the rdpss preconditioner and its variants to accelerate gmres for solving singular saddle point linear systems

two new variants of the hss preconditioner for regularized saddle point problems two new preconditioners which can be viewed as improved variants of the hermitian and skew hermitian splitting hss preconditioner are presented for regularized saddle point problems the unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed moreover eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained compared with the hss preconditioner they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions numerical experiments from the stokes problem are presented which illustrates the advantages of the two preconditioners over the hss preconditioner and some other preconditioners to accelerate the convergence rate of gmres

a preconditioner for linear systems arising from interior point optimization methods we explore a preconditioning technique applied to the problem of solving linear systems arising from primal dual interior point algorithms in linear and quadratic programming the preconditioner has the attractive property of improved eigenvalue clustering with increased ill conditioning of the 1 1 block of the saddle point matrix it fits well into the optimization framework since the interior point iterates yield increasingly ill conditioned linear systems as the solution is approached we analyze the spectral characteristics of the preconditioner utilizing projections onto the null space of the constraint matrix and demonstrate performance on problems from the netlib and cuter test suites the numerical experiments include results based on inexact inner iterations

on the stability of a convex set of matrices in this paper we give an alternative proof of the constant inertia theorem for convex compact sets of complex matrices it is shown that the companion matrix whose non trivial column is negative satisfies the directional lyapunov condition inclusion for real multiplier vectors an example of a real matrix polytope that satisfies the directional lyapunov condition for real multiplier vectors and which has nonconstant inertia is given a new stability criterion for convex compact sets of real z matrices is given this criterion uses only real vectors and positive definite diagonal matrices

necessary and sufficient conditions for the hurwitz and schur stability of interval matrices establishes a set of new sufficient conditions for the hurwitz and schur stability of interval matrices the authors use these results to establish necessary and sufficient conditions for the hurwitz and schur stability of interval matrices the authors relate the above results to the existence of quadratic lyapunov functions for linear time invariant systems with interval valued coefficient matrices using the above results the authors develop an algorithm to determine the hurwitz and the schur stability properties of interval matrices the authors demonstrate the applicability of their results by means of two specific examples

convex invertible sets and matrix sign function it is proved that in two dimensional unital real algebras a step by step construction of minimal convex invertible sets terminates in two steps an analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps and the complex field is the only finite dimensional complex algebra with this property connections are made with the matrix sign function the matrix sign function of a matrix belongs to the convex invertible set generated by the matrix an example is given to show that the matrix sign function cannot always be obtained in two steps by following the step by step construction

on finite permutation groups with a transitive cyclic subgroup abstract in this paper we study the structure of finite permutation groups with a transitive cyclic subgroup in particular we extend the classification due to w feit and g a jones of the primitive groups containing such subgroups to permutation groups that are i quasiprimitive ii almost simple and iii innately transitive we also analyse the actions of normal subgroups of arbitrary groups with transitive cyclic subgroups we give an application to circulant homogeneous factorisations of complete graphs obtaining new structural information and giving a new proof of their possible parameters

new variant of the hss iteration method for weighted toeplitz regularized least squares problems from image restoration abstract for fast solving weighted toeplitz least squares problems from image restoration ng and pan 2014 studied a new hermitian and skew hermitian splitting nhss preconditioner in this paper a generalization of the nhss preconditioner and the corresponding iterative method are presented convergence for the new iteration method is studied and optimal choice of the parameters is discussed bounds on the eigenvalues and the corresponding eigenvector distributions are proposed the degree of the minimal polynomial of the preconditioned matrix is obtained numerical experiments arising from image restoration are provided which show the proposed iteration method is effective and confirm our theoretical results are correct

on nonsingularity of a polytope of matrices abstract the nonsingularity problem of a polytope of real matrices and its relation to the robust stability problem is considered this problem is investigated by using the bernstein expansion of the determinant function here we adapt the known bernstein algorithm for checking the positivity of a multivariate polynomial on a box to the nonsingularity problem it is shown that for a family of z matrices the positive stability problem is equivalent to the nonsingularity if this family has a stable member it is established that the stability of the convex hull of real matrices a 1 a 2 a k is equivalent to the nonsingularity of the convex hull of matrices a 1 a 2 a k j i if a 1 is stable

primitive permutation groups with a regular subgroup abstract this paper starts the classification of the primitive permutation groups g such that g contains a regular subgroup x we determine all the triples g x with soc g an alternating or a sporadic or an exceptional group of lie type further we construct all the examples g x with g a classical group which are known to us our particular interest is in the 8 dimensional orthogonal groups of witt index 4 we determine all the triples g x with soc g p 8 q in order to obtain all these triples we also study the almost simple groups g with g p 2 n 1 q the case g u n q is started in this paper and finished in b baumeister primitive permutation groups of unitary type with a regular subgroup bull belg math soc 112 5 2006 657 673 a group x is called a burnside group or short a b group if each primitive permutation group which contains a regular subgroup isomorphic to x is necessarily 2 transitive in the end of the paper we discuss b groups

crout versions of ilu for general sparse matrices this paper presents an efficient implementation of the incomplete lu ilu factorization derived from the crout version of gaussian elimination at step k of the elimination the kth row of u and the kth column of l are computed using previously computed rows of u and columns of l the data structure and implementation borrow from already known techniques used in developing both sparse direct solution codes and incomplete cholesky factorizations this version of ilu can be computed much faster than standard threshold based ilu factorizations computed rowwise or columnwise in addition the data structure allows efficient implementations of more rigorous and effective dropping strategies

finite edge transitive cayley graphs and rotary cayley maps this paper aims to develop a theory for studying cayley graphs especially for those with a high degree of symmetry the theory consists of analysing several types of basic cayley graphs normal bi normal and core free and analysing several operations of cayley graphs core quotient normal quotient and imprimitive quotient it provides methods for constructing and characterising various combinatorial objects such as half transitive graphs orientable and non orientable regular cayley maps vertex transitive non cayley graphs and permutation groups containing certain regular subgroups in particular a characterisation is given of locally primitive holomorph cayley graphs and a classification is given of rotary cayley maps of simple groups also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup

a modified product preconditioner for indefinite and asymmetric generalized saddle point matrices in this paper we present a new modified product preconditioner mp for a class of large sparse linear systems with indefinite and asymmetric matrices the eigenvalue distribution and form of the eigenvectors of the presented new preconditioned matrix and its minimal polynomial are investigated some numerical experiments illustrate that the proposed new preconditioner performs better than block diagonal preconditioner block triangular preconditioner constraint preconditioner and product preconditioner