a generalized shift splitting preconditioner for complex symmetric linear systems abstract in this paper the generalized shift splitting gss preconditioner is implemented for solving a class of generalized saddle point problems which stem from the solution of complex symmetric linear systems the gss preconditioner is induced by the generalized shift splitting iterative method theoretical analysis shows that the generalized shift splitting iterative method is unconditionally convergent some numerical experiments are provided to show the effectiveness of the proposed preconditioner

uzawa type algorithms for nonsymmetric saddle point problems in this paper we consider iterative algorithms of uzawa type for solving linear nonsymmetric saddle point problems specifically we consider systems written as usual in block form where the upper left block is an invertible linear operator with positive definite symmetric part such saddle point problems arise for example in certain finite element and finite difference discretizations of navier stokes equations oseen equations and mixed finite element discretization of second order convection diffusion problems we consider two algorithms each of which utilizes a preconditioner for the operator in the upper left block convergence results for the algorithms are established in appropriate norms the convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator the other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block applications to the solution of steady state navier stokes equations are discussed and finally the results of numerical experiments involving the algorithms are presented

a new nonlinear uzawa algorithm for generalized saddle point problems abstract in j h bramble i e pasciak a t vassilev analysis of the inexact uzawa algorithm for saddle point problems siam j numer anal 34 1997 1072 1092 a nonlinear uzawa algorithm for solving generalized saddle point problems iteratively was considered in z cao fast uzawa algorithm for generalized saddle point problems appl numer math 46 2003 157 171 cao gave another nonlinear uzawa algorithm in order to accelerate convergence in this paper a new nonlinear uzawa method which is defined by two nonlinear approximate inverses is proposed and its convergence result is deduced at the same time we analyze convergence results of three nonlinear uzawa methods the results of numerical experiments are presented when we apply them to stokes equations discretized by mixed finite element

aor uzawa iterative method for a class of complex symmetric linear system of equations abstract in this paper based on accelerated overrelaxation aor method and uzawa method we present aor uzawa iterative method for solving a broad class of complex symmetric linear systems we investigate its convergence properties in addition the resulting aor uzawa preconditioner is used to precondition krylov subspace methods such as the restarted generalized minimal residual gmres method for solving the real equivalent formulation of the complex symmetric system finally two numerical examples are given to show the effectiveness of the proposed method

on semi convergence of the generalized shift splitting iteration method for singular nonsymmetric saddle point problems recently a class of generalized shift splitting iteration methods were proposed for solving nonsingular nonsymmetric saddle point problems cao et al 2015 in this paper the generalized shift splitting iteration method is extended to solve singular nonsymmetric saddle point problems semi convergence of the generalized shift splitting iteration method is carefully analyzed which shows that the iterative sequence generated by the generalized shift splitting iteration method converges to a solution of the singular saddle point problem unconditionally numerical results verify the robustness and efficiency of the generalized shift splitting iteration method

schur complements on hilbert spaces and saddle point systems for any continuous bilinear form defined on a pair of hilbert spaces satisfying the compatibility ladyshenskaya babusca brezzi condition symmetric schur complement operators can be defined on each of the two hilbert spaces in this paper we find bounds for the spectrum of the schur operators only in terms of the compatibility and continuity constants in light of the new spectral results for the schur complements we review the classical babusca brezzi theory find sharp stability estimates and improve a convergence result for the inexact uzawa algorithm we prove that for any symmetric saddle point problem the inexact uzawa algorithm converges provided that the inexact process for inverting the residual at each step has the relative error smaller than 1 3 as a consequence we provide a new type of algorithm for discretizing saddle point problems which combines the inexact uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions

on semi convergence of parameterized uzawa methods for singular saddle point problems abstract bai et al recently proposed an efficient parameterized uzawa method for solving the nonsingular saddle point problems z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 in this paper we further prove the semi convergence of this method when it is applied to solve the singular saddle point problems under suitable restrictions on the involved iteration parameters the optimal iteration parameters and the corresponding optimal semi convergence factor for the parameterized uzawa method are determined in addition numerical experiments are used to show the feasibility and effectiveness of the parameterized uzawa method for solving singular saddle point problems

on preconditioned generalized shift splitting iteration methods for saddle point problems abstract we study a preconditioned generalized shift splitting iteration method for solving saddle point problems the unconditional convergence theory of the preconditioned generalized shift splitting iteration method is established when the splitting matrix is used as a preconditioner we analyze eigenvalue distribution of the preconditioned saddle point matrix it is proved that complex eigenvalues having nonzero imaginary parts of the preconditioned matrix are located in an intersection of two circles and the real parts of all eigenvalues of the preconditioned matrix are located in a positive interval numerical experiments are used to verify our theoretical results and illustrate effectiveness of the proposed iteration method and the corresponding splitting preconditioner

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

moment of cepstrum and its applications it is shown that the moments of a signal sequence and of its corresponding cepstral sequence are related in a way which obviates the need for the direct calculation of the cepstral coefficients hence an ideal calculation for the moments of the cepstrum is possible even if the duration of the cepstrum is infinite it is also possible to describe all the properties of a signal sequence from its moments the effectiveness of this method is demonstrated in the problems of homomorphic deconvolution and echo removal it is also shown that in linear filtering problems the moments of the minimum phase sequence are calculated from the corresponding linear phase sequence without actually calculating the minimum phase sequence for a discrete time real signal sequence with finite duration complete equivalence is demonstrated between the signal sequence and its moments this permits a new description and a new set of design specifications based on moments for such systems

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

algorithms for linear groups of finite rank let g be a finitely generated solvable by finite linear group we present an algorithm to compute the torsion free rank of g and a bound on the prufer rank of g this yields in turn an algorithm to decide whether a finitely generated subgroup of g has finite index the algorithms are implemented in magma for groups over algebraic number fields

on an iterative method for saddle point problems convergence behavior of a nested iterative scheme presented in a paper by bank welfert and yserentant is studied it is shown that this scheme converges under conditions weaker than that stated in their paper

inexact uzawa algorithms for variational inequalities of the second kind in this paper we discuss inexact uzawa algorithms and inexact non linear uzawa algorithms to solve discretized variational inequalities of the second kind we prove convergence results for the algorithms numerical examples are included to show the effectiveness of the algorithms

direct nonrecursive relations between cepstrum and predictor coefficients direct i e nonrecursive relations are derived for the cepstrum in terms of the predictor coefficients and vice versa connections with algebraic roots symmetric functions statistical moments and cumulants are pointed out some implications for pitch detection are also discussed

a unified approach for uzawa algorithms we present a unified approach in analyzing uzawa iterative algorithms for saddle point problems we study the classical uzawa method the augmented lagrangian method and two versions of inexact uzawa algorithms the target application is the stokes system but other saddle point systems e g arising from mortar methods or lagrange multipliers methods can benefit from our study we prove convergence of uzawa algorithms and find optimal rates of convergence in an abstract setting on finite or infinite dimensional hilbert spaces the results can be used to design multilevel or adaptive algorithms for solving saddle point problems the discrete spaces do not have to satisfy the lbb stability condition

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

calculating the complex cepstrum without phase unwrapping or integration the complex cepstrum as developed by oppenheim 4 and oppenheim schafer and stockham 5 originated from the pioneering paper of bogert healy and tukey 3 it is a nonlinear transformation from one signal space into another which converts two superposed signals into their sum the cepstrum is usually computed by inverse fourier transformation of the frequency domain complex logarithm of the original signal the phase term or imaginary part of this complex logarithm presents a computational difficulty which is sometimes approached by integration of the derivative of the phase spectrum an excellent approach to phase computation is the adaptive integration methodology as proposed by tribolet 10 unfortunately this technique can be computationally intensive this note describes a simple technique for efficient computation of the complex cepstrum which requires no phase unwrapping or integration the procedure is given in both one and two dimensions

on the iterative algorithm for large sparse saddle point problems in this paper we discuss some iterated method for solving the saddle point problem we propose some new schemes and prove its convergence the method has weaker convergence condition than the classic uzawa method the analysis is supported by numerical experiments

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

on the semi convergence of regularized hss iteration methods for singular saddle point problems abstract recently bai and benzi proposed a class of regularized hermitian and skew hermitian splitting rhss iteration methods for solving the nonsingular saddle point problem in this paper we apply this method to solve the singular saddle point problem in the process of the semi convergence analysis we get that the rhss method and the hss method are unconditionally semi convergent which has improved the previous results then some spectral properties of the corresponding preconditioned matrices and a class of improved preconditioned matrices are analyzed finally some numerical experiments on linear systems arising from the discretization of the stokes equation are presented to illustrate the feasibility and effectiveness of this method and the corresponding preconditioners

generalized shift splitting preconditioners for nonsingular and singular generalized saddle point problems based on the shift splitting technique a class of generalized shift splitting preconditioners are proposed for both nonsingular and singular generalized saddle point problems the generalized shift splitting preconditioner is induced by a generalized shift splitting of the generalized saddle point matrix resulting in a generalized shift splitting fixed point iteration theoretical analyses show that the generalized shift splitting iteration method is convergent and semi convergent unconditionally for solving the nonsingular and the singular generalized saddle point problems respectively numerical experiments of a model navier stokes problem are implemented to demonstrate the feasibility and effectiveness of the proposed preconditioners

residual reduction algorithms for nonsymmetric saddle point problems in this paper we introduce and analyze uzawa algorithms for non symmetric saddle point systems convergence for the algorithms is established based on new spectral results about schur complements a new uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state navier stokes equations

on semi convergence of the uzawa hss method for singular saddle point problems based on the hermitian and skew hermitian splitting hss iteration scheme an efficient uzawa hss iteration method has been proposed to solve the nonsingular saddle point problems in this paper we discuss the feasibility of the uzawa hss method used for solving singular saddle point problems the semi convergence properties of the uzawa hss iteration method are carefully analyzed which show that the iterative sequence generated by the uzawa hss method converges to a solution of the singular saddle point problem if the iteration parameters satisfy suitable restrictions numerical results verify the robustness and efficiency of the uzawa hss method

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

scalar lpc quantization based on format jnd s efficient scalar quantization tables for lpc k parameters were developed using a distortion measure based on just noticeable differences jnd s in formant parameters of the speech spectrum envelope forty percent fewer bits were required than the 41 frame used in conventional approaches an empirical technique was developed for relating perturbations in k parameters and formant parameters new estimates were obtained for the values of the formant jnd s they are about four times the steady state values reported by flanagan 6 and increase sharply above approximately 1 5 khz

on the nonlinear inexact uzawa algorithm for saddle point problems this paper discusses convergence behavior of the nonlinear inexact uzawa algorithm for solving saddle point problems presented in a recent paper of bramble pasciak and vassilev siam j numer anal 34 1997 pp 1072 1092 it is shown that this algorithm converges under a condition weaker than that stated in their paper

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

a class of efficient parameterized shift splitting preconditioners for block two by two linear systems abstract based on the effective constructing an equivalent block two by two linear systems we establish a class of efficient parameterized shift splitting eps preconditioners for solving block two by two linear systems theoretical analysis shows that the eps iteration method is unconditionally convergent and then we study the spectral properties of the corresponding preconditioned matrix by removing one of the shift term of the eps preconditioners two classes of local eps preconditioners are presented and the eigenvalues distribution of the preconditioned matrices are also studied numerical experiments are provided to demonstrate the feasibility and effectiveness of the proposed preconditioners

cohomological finiteness conditions for elementary amenable groups it is proved that every elementary amenable group of type fp admits a cocompact classifying space for proper actions

the generalized modified shift splitting preconditioners for nonsymmetric saddle point problems for a nonsymmetric saddle point problem the modified shift splitting mss preconditioner has been proposed by zhou et al by replacing the parameter in 2 2 block in the mss preconditioner by another parameter a generalized mss gmss preconditioner is established in this paper which results in a fixed point iteration called the gmss iteration method we provide the convergent and semi convergent analysis of the gmss iteration method which show that this method is convergence and semi convergence if the related parameters satisfy suitable restrictions meanwhile the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail finally numerical examples show that the gmss method is more feasibility and robustness than the mss uzawa hss and pu sts methods as a solver and the gmss preconditioner outperforms the gsor uzawa hss mss and lmss preconditioners for the gmres method for solving both the nonsingular and the singular saddle point problems with nonsymmetric positive definite and symmetric dominant 1 1 parts

time domain cepstral transformations the realization of homomorphic systems for convolution is addressed motivated by the limitations of the fourier transform based method commonly used in homomorphic complex cepstral filtering the time domain cepstral transformation tdct method which is entirely based on time domain calculations thus avoiding or minimizing the problems associated with the ft method is introduced explicit transformations of an ordinary mixed phase time sequence belonging to convolution space into its complex cepstrum time sequence belonging to additive cepstrum vector space and vice versa are derived the method does not require unwrapped phase calculations and no specific windows are used to precondition the signal in order to produce a more accurate representation of the complex cepstrum it trades reduced computational efficiency for improved performance examples comparing fourier based and td cepstrum transformations are presented

convergence analysis of modified pgss methods for singular saddle point problems abstract recently variants of shift splitting iteration method have been proposed for solving singular saddle point problems however these methods can only be proved to converge to one of the solutions of the consistent singular linear system not knowing any further information about this solution in this work we consider a modified preconditioned generalized shift splitting mpgss iteration method for solving both consistent and inconsistent singular saddle point problems this method is proved to converge to the best least squares solution moreover based on the iteration form a preconditioner is obtained to accelerate krylov subspace methods theoretical analysis shows that the preconditioned gmres method also converges to the best least squares solution of the consistent singular saddle point problem in addition numerical results are presented to show the effectiveness and robustness of the proposed iteration method and preconditioner

a modification of the generalized shift splitting method for singular saddle point problems abstract a modification of the generalized shift splitting gss method is presented for solving singular saddle point problems in this kind of modification the diagonal shift matrix is replaced by a block diagonal matrix which is symmetric positive definite semi convergence of the proposed method is investigated the induced preconditioner is applied to the saddle point problem and the preconditioned system is solved by the restarted generalized minimal residual method eigenvalue distribution of the preconditioned matrix is also discussed finally some numerical experiments are given to show the effectiveness and robustness of the new preconditioner numerical results show that the modified gss method is superior to the classical gss method

parameterized generalized shift splitting preconditioners for nonsymmetric saddle point problems abstract to solve nonsymmetric saddle point problems the parameterized generalized shift splitting pgss preconditioner is presented and analyzed the corresponding pgss iteration method can be applied not only to the nonsingular saddle point problems but also to the singular ones the convergence and semi convergence of the pgss iteration method are discussed carefully meanwhile the spectral properties of the preconditioned matrix and the strategy of the choices of the parameters are given numerical experiments further demonstrate that the pgss iteration method and the pgss preconditioner are efficient and have better performance than some existing iteration methods and newly proposed preconditioners respectively for solving both the nonsingular and singular nonsymmetric saddle point problems

two new variants of nonlinear inexact uzawa algorithms for saddle point problems in this paper we consider some nonlinear inexact uzawa methods for iteratively solving linear saddle point problems by means of a new technique we first give an essential improvement on the convergence results of bramble paschiak vassilev for a known nonlinear inexact uzawa algorithm then we propose two new algorithms which can be viewed as a combination of the known nonlinear inexact uzawa method with the classical steepest descent method and conjugate gradient method respectively the two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved including the preconditioned schur complement numerical results of the algorithms applied for the stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms

dimension of elementary amenable groups this paper has three parts it is conjectured that for every elementary amenable group g and every non zero commutative ring k the homological dimension hdk g is equal to the hirsch length h g whenever g has no k torsion in part i this conjecture is proved for several classes including the abelian by polycyclic groups in part ii it is shown that the elementary amenable groups of homological dimension one are colimits of systems of groups of cohomological dimension one in part iii the deep problem of calculating the cohomological dimension of elementary amenable groups is tackled with particular emphasis on the nilpotent by polycyclic case where a complete answer is obtained over q for countable groups

the nonlinear inexact uzawa hybrid algorithms based on one step newton method for solving nonlinear saddle point problems this paper mainly consider the construction of efficient uzawa type algorithms for solving the nonlinear saddle point problem nspp two nonlinear inexact uzawa hybrid algorithms based on one step newton method are proposed for solving the nspps by using the so called energy norm the convergence of the proposed algorithms is verified under some practical conditions in addition some numerical results are reported to illustrate the behavior of the considered algorithm