preconditioning by fast direct methods for nonself adjoint nonseparable elliptic equations we consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself adjoint elliptic boundary value problems we derive bounds on convergence rates that are independent of discretization mesh size for two dimensional problems on rectangular domains discretized on an n times n grid these bounds lead to asymptotic operation counts of o n 2 log n log varepsilon 1 to achieve relative error varepsilon and o n 2 log n 2 to reach truncation error

finite element approximations of a ladyzhenskaya model for stationary incompressible viscous flow some finite element approximation procedures are presented for a model proposed by ladyzhenskaya for stationary incompressible viscous flow the approximate problems are proved to be well posed and stable under standard assumptions on the finite element families the solutions of the approximate problems converge to the solution of the original problem under minimum regularity assumptions some error estimates are derived the optimal order of accuracy is assured with or even without using exact integration rules in the approximation procedure iterative methods for solving the discrete nonlinear problems and comments on some computational experiments are provided special attention is also paid to the common properties as well as differences between the approximation procedure presented here and the approximation for the stationary navier stokes equations

anisotropic branching random walks on homogeneous trees symmetric branching random walk on a homogeneous tree exhibits a weak survival phase for parameter values in a certain interval the population survives forever with positive probability but with probability one eventually vacates every finite subset of the tree in this phase particle trails must converge to the geometric boundary of the tree the random subset of the boundary consisting of all ends of the tree in which the population survives called the limit set of the process is shown to have hausdorff dimension no larger than one half the hausdorff dimension of the entire geometric boundary moreover there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic it is further shown that in all cases there is a distinguished probability measure supported by such that the hausdorff dimension of where is the set of generic points of converges to one half the hausdorff dimension of at the phase separation point exact formulas are obtained for the hausdorff dimensions of and and it is shown that the log hausdorff dimension of has critical exponent 1 2 at the phase separation point

branching random walks on free products of groups we study certain phase transitions of branching random walks brw on cayley graphs of free products the aim of this paper is to compare the size and structural properties of the trace that is the subgraph that consists of all edges and vertices that were visited by some particle with those of the original cayley graph we investigate the phase when the growth parameter is small enough such that the process survives but the trace is not the original graph a first result is that the box counting dimension of the boundary of the trace exists is almost surely constant and equals the hausdorff dimension which we denote by the main result states that the function has only one point of discontinuity which is at c r where r is the radius of convergence of the green function of the underlying random walk furthermore r is bounded by one half the hausdorff dimension of the boundary of the original cayley graph and the behaviour of r as r is classified in the case of free products of infinite groups the end boundary can be decomposed into words of finite and words of infinite length we prove the existence of a phase transition such that if c the end boundary of the trace consists only of infinite words and if c it also contains finite words in the last case the hausdorff dimension of the set of ends of the trace and the original graph induced by finite words is strictly smaller than the one of the ends induced by infinite words

converting bicr method for linear equations with complex symmetric matrices abstract the bi conjugate gradient bicg method for symmetric complex matrices scbicg which can be derived from bicg has been proposed for solving linear equations with complex symmetric matrices however an alternative method derived from the bi conjugate residual bicr method for complex symmetric matrices has not previously been proposed we therefore design bicr for symmetric complex matrices scbicr by using the same analogy as that discussed in scbicg coefficients ci with real number defined in scbicg need to be set by users before starting the iteration and we have had the numerical results with several combinations when the coefficients ci are real that the residual norms of scbicg do not converge we therefore design an alternative implementation such that the coefficients ci can be complex and are appropriately determined at each step of the algorithm we give the preconditioned algorithms moreover the factor in the loss of convergence speed is analyzed to clarify the difference of convergence between scbicg and our proposed scbicr numerical experiments demonstrate that the residual norms of our proposed variant with the complex coefficients ci converge fairly faster than those of the conjugate orthogonal conjugate gradient cocg method and several implementations of scbicg

preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems abstract in this paper by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method agsor proposed by edalatpour et al 2015 we establish the preconditioned agsor pagsor iteration method for solving a class of complex symmetric linear systems the convergence conditions optimal iteration parameters and corresponding optimal convergence factor of the pagsor iteration method are determined besides we prove that the spectral radius of the pagsor iteration method is smaller than that of the agsor one under proper restrictions and its optimal convergence factor is smaller than that of the preconditioned symmetric block triangular splitting psbts one put forward by zhang et al 2018 recently the spectral properties of the preconditioned pagsor matrix are also proposed numerical experiments illustrate the correctness of the theories and the effectiveness of the proposed iteration method and the preconditioner for the generalized minimal residual gmres method

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

multifractality of entangled random walks and non uniform hyperbolic spaces multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a non symmetric lattice of obstacles are investigated using the equivalence of the model to random walks on a locally non symmetric tree statistical properties of topological invariants such as drift and return probabilities have been studied by means of a renormalization group rg technique the comparison of the analytical rg results with numerical simulations as well as with the rigorous results of gerl and woess demonstrates clearly the validity of our approach it is shown explicitly by direct counting for the discrete version of the model and by conformal methods for the continuous version that multifractality occurs when local uniformity of the phase space which has an exponentially large number of states has been broken

a class of accelerated parameterized inexact uzawa algorithms for complex symmetric linear systems abstract we establish a class of accelerated parameterized inexact uzawa apiu algorithms for solving the complex symmetric linear systems our main contribution is accelerating the convergence of the piu algorithm by making use of the extrapolation technique which is based on the eigenvalues of the iterative matrix these accelerated parameterized inexact uzawa algorithms involve two iteration parameters whose special choices can recover the parameterized inexact uzawa algorithm and some other methods first the accelerated model for the piu algorithm is established and the accelerated piu algorithm is presented then we study the convergence of the corrected piu algorithm moreover we present the optimal iteration parameter and the corresponding optimal convergence factor for the piu method we also consider acceleration of the piu iteraton by krylov subspace methods numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method

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

anti gaussian quadrature formulas an anti gaussian quadrature formula is an n 1 point formula of degree 2n 1 which integrates polynomials of degree up to 2n 1 with an error equal in magnitude but of opposite sign to that of the n point gaussian formula its intended application is to estimate the error incurred in gaussian integration by halving the difference between the results obtained from the two formulas we show that an anti gaussian formula has positive weights and that its nodes are in the integration interval and are interlaced by those of the corresponding gaussian formula similar results for gaussian formulas with respect to a positive weight are given except that for some weight functions at most two of the nodes may be outside the integration interval the anti gaussian formula has only interior nodes in many cases when the kronrod extension does not and is as easy to compute as the n 1 point gaussian formula

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

on convergence and semi convergence of ssor like methods for augmented linear systems abstract in this paper we analyze the convergence and semi convergence of a class of ssor like methods with four real functions for augmented systems the class takes most existed ssor like methods as its special cases for nonsingular systems we obtain the minimum of convergence factors of all the ssor like methods in the class and study when it can be reached by the convergence factors of methods in the class by considering the equivalence of methods we show that most of the existed ssor like methods have the same minimum of convergence factors

on semi convergence of the parameterized generalized mhss method for singular complex linear systems for solving a class of complex symmetric singular linear systems we propose a parameterized generalized mhss pgmhss iteration method and investigate the semi convergence conditions by analyzing the spectrum of the iteration matrix then we analyze the optimal iteration parameters that minimize the upper bound of the semi convergence factor numerical experiments are used to test the feasibility and the effectiveness of the pgmhss iteration method

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

modified hss iteration methods for a class of non hermitian positive definite linear systems abstract we consider the numerical solution of a class of non hermitian positive definite linear systems by the modified hermitian and skew hermitian splitting mhss iteration method we show that the mhss iteration method converges unconditionally even when the real and the imaginary parts of the coefficient matrix are nonsymmetric and positive semidefinite and at least one of them is positive definite at each step the mhss iteration method requires to solve two linear sub systems with real nonsymmetric positive definite coefficient matrices we propose to use inner iteration methods to compute approximate solutions of these linear sub systems we illustrate the performance of the mhss method and its inexact variant by two numerical examples

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

the full martin boundary of the bi tree are interested in the martin compactification of the bi tree with respect to each eigenvalue t p of the transition operator acting on nonnegative functions the minimal martin boundary i e set of minimal t harmonic functions is well understood see 20 and also 22 for cartesian products of markov chains in general on the other hand the description of the full martin compactification has remained an open problem some evidence has been obtained by picardello and sjogren 21 who studied boundary behaviour of harmonic functions on the bi tree the situation is similar for the hyperbolic bi disk equipped with the laplace beltrami operator in this case the minimal martin boundary is known via the work of karpelevic 13 and guivarc h 10 see also the general result of freire 7 and taylor 26 concerning cartesian products of manifolds the problem of determining the full martin boundary was proposed by guivarc h and taylor 11 who gave the answer only for the critical eigenvalue where the behaviour is different in the present paper we describe the full martin compactification of the bitree with respect to all eigenvalues t p this is possible thanks to recent work of lalley 14 who has given a local limit theorem for aperiodic nearestneighbour random walks on homogeneous trees that is an asymptotic estimate of the n step transition probabilities which is uniform both in time and space a considerable improvement of previous work of sawyer 24 gerl and woess

an iteration for indefinite systems and its application to the navier stokes equations for large sparse systems of linear equations iterative solution techniques are attractive in this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner we apply our technique to the linearized navier stokes equations the oseen equations

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

the poisson transform and representations of a free group abstract let g be a free group with r generators 1 r g which plays the same role of the laplace beltrami operator on semisimple lie groups are characterized furthermore an analytic family of representations z of g on functions on the boundary is considered defined by z x p z x x 1 where p x is the poisson kernel relative to the action of g on it is proved that for 0 s re z z is uniformly bounded on an appropriate hilbert space h s finally the uniform boundedness of other special representations of g obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of gl 2 q p is proved

on stratified extensions of gauss laguerre and gauss hermite quadrature formulas for the practical estimation of the error of gauss laguerre and gauss hermite quadrature formulas it is well known that real positive kronrod formulas do not exist among the alternatives which are available in the literature the stratified rules introduced by laurie are of particular interest in this note we determine the degree optimal stratified extensions for the gauss laguerre and gauss hermite formulas and we investigate their properties

local limits and harmonic functions for nonisotropic random walks on free groups nearest neighbour random walks on the homogeneous tree representing a free group withs generators 2 s are investigated by use of generating functions and their analytic properties a local limit theorem is derived a study of the harmonic functions corresponding to the random walk leads to properties that characterize ther harmonic function connected with the local limits

a generalization of preconditioned mhss iteration method for complex symmetric indefinite linear systems we present a generalization of the preconditioned modified hss pmhss iteration method for solving two classes of complex symmetric indefinite linear systems and show that under suitable conditions the new iteration method is unconditionally convergent each iteration of the new method requires the solution of two linear systems with real symmetric positive definite coefficient matrices numerical experiments are given to show the effectiveness of the new method

two variants of the pmhss iteration method for a class of complex symmetric indefinite linear systems for a class of complex symmetric indefinite linear systems we present two variants of the preconditioned modified hermitian and skew hermitian splitting pmhss iteration method one is a generalization of the pmhss iteration method and the other is a generalization of the lopsided pmhss iteration method we study the convergence properties of these two variants by analyzing the upper bounds of spectral radiuses of the corresponding iteration matrices and compare these upper bounds numerical examples are illustrated to show the effectiveness of the proposed iteration methods

on semi convergence of parameterized shss method for a class of singular complex symmetric linear systems abstract in this paper we use the parameterized single step hss p shss iterative method to solve a broad class of singular complex symmetric linear systems the semi convergence properties of the p shss method are derived under suitable conditions moreover some properties of the preconditioned matrix and the optimal parameters are analyzed in detail numerical experiments are given to support our theoretical results and show the effectiveness of the p shss method either as a solver or as a preconditioner

gauss quadrature rules for a generalized hermite weight function abstract in this paper we give an algorithm for finding the coefficients in the three term recurrence relation for the polynomials orthogonal with respect to the weight function w x x b 2 exp x c 2 on r where b c r 2 1 the algorithm is based on the discretization technique proposed by gautschi with the lanczos algorithm using the coefficients obtained by this algorithm we construct the gauss quadrature rules relative to the weight function w x to show the performance of the new algorithm we give some numerical examples

on semi convergence of a class of uzawa methods for singular saddle point problems in this paper a class of uzawa methods are presented for singular saddle point problems these methods contain the recently proposed uzawa aor and uzawa saor methods as special cases the 1 1 block of the corresponding uzawa preconditioner is positive definite both nonsingular and singular preconditioning matrices are considered the semi convergence of these methods is analyzed by using the techniques of singular value decomposition and moore penrose inverse numerical results show that they need less workload per iteration step comparing with the gsor and piu methods in some situations so they are feasible and effective for singular saddle point problems in some situations

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

the resolvent for simple random walks on the free product of two discrete groups on montre que si g est le produit libre de 2 groupes discrets g 1 g 2 et si est une mesure sur g qui est une combinaison convexe de 2 mesures de probabilite 1 et 2 sur g 1 et g 2 alors la resolvante zse 1 est proportionnelle au produit libre des resolvantes i se i 1 ou e est l element identite et i depend analytiquement de z

the rate of escape for anisotropic random walks in a tree let g be the group generated by l free involutions whose cayley graph t is the infinite homogeneous tree with l edges at every node a general central limit theorem and law of the iterated logarithm is proven for left invariant random walks z n on g or t which applies to the distance of z n from a fixed point as well as to the distribution of the last r letters in z n for nearest neighbor random walks we also derive a generating function identity that yields formulas for the asymptotic mean and variance of the distance from a fixed point a generalization for z n with a finitely supported step distribution is derived and discussed

on adi like iteration method for fractional diffusion equations abstract the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of the identity matrix with two diagonal times toeplitz matrices in this paper the alternating direction implicit adi like method based on the classical adi iteration is proposed to solve the discretized linear system theoretical analyses show that the adi like iteration method is convergent each iteration of this method requires the solutions of two linear systems with hessenberg coefficient matrices these two systems can be solved effectively by hessenberg lu factorization numerical examples are presented to illustrate the effectiveness of the adi like iteration method

structured least squares problems and inverse eigenvalue problems for p q reflexive matrices abstract a new meaningful structured matrix p q reflexive matrix is defined without the common assumption that p or q is unitary a general solution is derived for its structured least squares problem as a necessary and sufficient condition being presented for the solvability of its structured inverse eigenvalue problem structured constrains are firstly given to guarantee the existence of the solution the optimal approximation problem is also considered under spectral constrains

iterative system solvers for the frequency analysis of linear mechanical systems abstract the paper deals with the numerical treatment of the direct frequency domain dfd analysis of linear mechanical systems attention is mainly focused on the solution of the complex system of equations needed for each analyzed frequency strategies for transforming the system into the shifted form t i z d are proposed and discussed where is related to the frequency for two formulations of the shifted system the performance of some krylov sub space iterative solvers is tested and compared to that of a multi frontal direct method advantage is taken of the shifted form in solving simultaneously a large number of systems resulting from different values of the shift frequency numerical experiments on some prototype structural dynamics problems are reported the results shown demonstrate how the devised strategies for the iterative solution can outperform in many cases the direct solver

a variant of the hss preconditioner for complex symmetric indefinite linear systems abstract using the equivalent block two by two real linear systems we establish a new variant of the hermitian and skew hermitian splitting hss preconditioner for a class of complex symmetric indefinite linear systems the new preconditioner is not only a better approximation to the block two by two real coefficient matrix than the well known hss preconditioner but also resulting in an unconditional convergent fixed point iteration the quasi optimal parameter which minimizes an upper bound of the spectral radius of the iteration matrix is analyzed eigen properties and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed finally two numerical examples are provided to show the efficiency of the new preconditioner

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

the existence and construction of rational gauss type quadrature rules abstract consider a hermitian positive definite linear functional f and assume we have m distinct nodes fixed in advance anywhere on the real line in this paper we then study the existence and construction of n th rational gauss radau m 1 and gauss lobatto m 2 quadrature formulas that approximate f f these are quadrature formulas with n positive weights and with the n m remaining nodes real and distinct so that the quadrature is exact in a 2 n m dimensional space of rational functions further we also consider the case in which the functional is defined by a positive bounded borel measure on an interval for which it is required in addition that the nodes are all in the support of the measure

splitting based block preconditioning methods for block two by two matrices of real square blocks recently bai proposed rotated block preconditioners for block two by two matrices of real square blocks these rotated block preconditioners have the product form of a scaled orthogonal matrix and a block two by two triangular matrix theoretical and numerical results have shown the superiority of these rotated block preconditioners see bai 2013 10 in this paper inspired by the efficiency and the special structure of the rotated block preconditioners we establish a new equivalent linear system to the original linear system in which an orthogonal matrix arises we construct block jacobi and block gauss seidel splitting iteration methods based on the coefficient matrix of the new linear system the convergence of these splitting iterations is also demonstrated then by utilizing the proposed block jacobi and block gauss seidel splittings we put forward block preconditioners which are of the product form of a scaled orthogonal matrix and a block two by two diagonal or a block two by two triangular matrix spectral distributions of these preconditioned matrices and numerical experiments show that the proposed splitting based block preconditioners can be quite competitive with the rotated block preconditioners when they are used to accelerate krylov subspace iteration methods such as gmres for solving the block two by two liner systems

modified unsymmetric sor method for saddle point problems in this paper we propose a modified unsymmetric sor musor method for solving singular or nonsingular saddle point problem which is an extension of the ssor iteration method for the musor method we prove its convergence for solving nonsingular saddle point problem and semi convergence for solving singular saddle point problem under suitable conditions moreover we study the choices of the optimal parameters numerical experiments are presented which support our theoretical results and reveal that the new method is effective and feasible for solving both singular and nonsingular saddle point problems

on numerical computation of integrals with integrands of the form f x sin w xr on 0 1 with existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form 0 1f x sin wx r dxor 0 1f x cos wx r dx r 0 where f is a sufficiently smooth function on 0 1 gautschi has developed software as scripts in matlab for computing these integrals for the special case r w 1 in this paper an algorithm as a mathematica program is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range numerical examples are given of testing the performance of the algorithm program

random walks on three strand braids and on related hyperbolic groups we investigate the statistical properties of random walks on the simplest nontrivial braid group b3 and on related hyperbolic groups we provide a method using cayley graphs of groups allowing us to compute explicitly the probability distribution of the basic statistical characteristics of random trajectories the drift and the return probability the action of the groups under consideration in the hyperbolic plane is investigated and the distribution of a geometric invariant the hyperbolic distance is analysed it is shown that a random walk on b3 can be viewed as a magnetic random walk on the group psl 2

on inexact hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems we study theoretical properties of two inexact hermitian skew hermitian splitting ihss iteration methods for the large sparse non hermitian positive definite system of linear equations in the inner iteration processes we employ the conjugate gradient cg method to solve the linear systems associated with the hermitian part and the lanczos or conjugate gradient for normal equations cgne method to solve the linear systems associated with the skew hermitian part respectively resulting in ihss cg lanczos and ihss cg cgne iteration methods correspondingly theoretical analyses show that both ihss cg lanczos and ihss cg cgne converge unconditionally to the exact solution of the non hermitian positive definite linear system moreover their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the hermitian part but are less dependent on the spectrum of the skew hermitian part and are independent of the eigenvectors of the matrices involved optimal choices of the inner iteration steps in the ihss cg lanczos and ihss cg cgne iterations are discussed in detail by considering both global convergence speed and overall computation workload and computational efficiencies of both inexact iterations are analyzed and compared deliberately

semi convergence analysis of uzawa methods for singular saddle point problems recently zheng bai and yang studied the parameterized uzawa method for solving singular saddle point problems b zheng z z bai x yang on semi convergence of parameterized uzawa methods for singular saddle point problems linear algebra appl 431 2009 808 817 in this paper we discuss the inexact uzawa method which covers the uzawa method the preconditioned uzawa method and the parameterized uzawa method to solve the singular saddle point problems we prove the semi convergence result under restrictions by verifying two necessary and sufficient conditions that is all elementary divisors associated with the eigenvalue 1 of its iterative matrix are linear and the pseudo spectral radius of the iterative matrix is less than 1 sufficient conditions for the semi convergence of several uzawa type methods are also provided in addition numerical examples are given to demonstrate the semi convergence of uzawa type methods

a fast algorithm to solve systems of nonlinear equations abstract a new hss based algorithm for solving systems of nonlinear equations is presented and its semilocal convergence is proved spectral properties of the new method are investigated performance profile for the new scheme is computed and compared with hss algorithm besides by a numerical example in which a two dimensional nonlinear convection diffusion equation is solved we compare the new method and the newton hss method numerical results show that the new scheme solves the problem faster than the newton hss scheme in terms of cpu time and number of iterations moreover the application of the new method is found to be fast reliable flexible accurate and has small cpu time

a new generalized parameterized inexact uzawa method for solving saddle point problems recently bai parlett and wang presented a class of parameterized inexact uzawa piu methods for solving saddle point problems bai et al 2005 in this paper we develop a new generalized piu method for solving both nonsingular and singular saddle point problems the necessary and sufficient conditions of the convergence semi convergence for solving nonsingular singular saddle point problems are derived meanwhile the characteristic of eigenvalues of the iteration matrix corresponding to the above iteration method is discussed we further show that the generalized piu type method proposed in this paper has a wider convergence semi convergence region than some classical uzawa methods such as the inexact uzawa method the sor like method the gsor method and so on finally numerical examples are given to illustrate the feasibility and efficiency of this method

a generalization of parameterized inexact uzawa method for singular saddle point problems in this paper we further study the generalized parameterized inexact uzawa method for solving singular saddle point problems obtaining the generalized parameterized inexact uzawa gpiu method theoretical analysis shows that the semi convergence of this new method can be guaranteed by suitable choices of the iteration parameters numerical experiments are used to demonstrate the feasibility and effectiveness of the generalized parameterized inexact uzawa method for solving singular saddle point problems

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

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

stratified nested and related quadrature rules the stratified nested quadrature procedure due to laurie is discussed together with an alternative computational procedure which leads to the concept of hybrid gkp rules in the context of the approximation of stratified nested sequences the work of krogh and van snyder on the representation of the gkp rules is considered and a generalisation of this employing hybrid rules of special form is discussed

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

a systematization of the unscented kalman filter theory in this paper we propose a systematization of the discrete time unscented kalman filter ukf theory we gather all available ukf variants in the literature present corrections to theoretical inconsistencies and provide a tool for the construction of new ukf s in a consistent way this systematization is done mainly by revisiting the concepts of sigma representation unscented transformation ut scaled unscented transformation sut ukf and square root unscented kalman filter srukf inconsistencies are related to 1 matching the order of the transformed covariance and cross covariance matrices of both the ut and the sut 2 multiple ukf definitions 3 issue with some reduced sets of sigma points described in the literature 4 the conservativeness of the sut 5 the scaling effect of the sut on both its transformed covariance and cross covariance matrices and 6 possibly ill conditioned results in srukf s with the proposed systematization the symmetric sets of sigma points in the literature are formally justified and we are able to provide new consistent variations for ukf s such as the scaled srukf s and the ukf s composed by the minimum number of sigma points furthermore our proposed srukf has improved computational properties when compared to state of the art methods

on the computation of spectra in free probability abstract we use free probability techniques to compute borders of spectra of non hermitian operators in finite von neumann algebras which arise as free sums of simple operators to this end the resolvent is analyzed with the aid of the haagerup inequality concrete examples coming from reduced c algebras of free product groups and leading to systems of polynomial equations illustrate the approach

sufficient conditions for the convergent splittings of non hermitian positive definite matrices abstract we present sufficient conditions for the convergent splitting of a non hermitian positive definite matrix these results are applicable to identify the convergence of iterative methods for solving large sparse system of linear equations

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

the relaxed nonlinear phss like iteration method for absolute value equations finding the solution of the absolute value equation ave a x x b has attracted much attention in recent years in this paper we propose a relaxed nonlinear phss like iterative method which is more efficient than the picard hss iterative method for the ave and is a generalization of the nonlinear hss like iteration method for the ave by using the theory of nonsmooth analysis we prove the convergence of the relaxed nonlinear phss like iterative method for the ave numerical experiments are given to demonstrate the feasibility robustness and effectiveness of the relaxed nonlinear hss like method

on hss and ahss iteration methods for nonsymmetric positive definite toeplitz systems two iteration methods are proposed to solve real nonsymmetric positive definite toeplitz systems of linear equations these methods are based on hermitian and skew hermitian splitting hss and accelerated hermitian and skew hermitian splitting ahss by constructing an orthogonal matrix and using a similarity transformation the real toeplitz linear system is transformed into a generalized saddle point problem then the structured hss and the structured ahss iteration methods are established by applying the hss and the ahss iteration methods to the generalized saddle point problem we discuss efficient implementations and demonstrate that the structured hss and the structured ahss iteration methods have better behavior than the hss iteration method in terms of both computational complexity and convergence speed moreover the structured ahss iteration method outperforms the hss and the structured hss iteration methods the structured ahss iteration method also converges unconditionally to the unique solution of the toeplitz linear system in addition an upper bound for the contraction factor of the structured ahss iteration method is derived numerical experiments are used to illustrate the effectiveness of the structured ahss iteration method

fast iterative solution of stabilised stokes systems part i using simple diagonal 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 the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem