a quantization of conjugacy classes of matrices abstract we construct a generator system of the annihilator of the generalized verma module of g l n c induced from any character of any parabolic subalgebra as an analogue of minors and elementary divisors the generator system has a quantization parameter e and it generates the defining ideal of the conjugacy class of square matrices at the classical limit e 0

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

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

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

block diagonally preconditioned piu methods of saddle point problem for large sparse saddle point problems we firstly introduce the block diagonally preconditioned gauss seidl method pbgs which reduces to the gsor method z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 and piu method z z bai z q wang on parameterized inexact uzawa methods for generalized saddle point problems linear algebra appl 428 2008 2900 2932 when the preconditioners equal to different matrices respectively then we generalize the pbgs method to the ppiu method and discuss the sufficient conditions such that the spectral radius of the ppiu method is much less than one furthermore some rules are considered for choices of the preconditioners including the splitting method of the 1 1 block matrix in the piu method and numerical examples are given to show the superiority of the new method to the piu method

linear differential equations on the riemann sphere and representations of quivers our interest in this article is a generalization of the additive deligne simpson problem which was originally defined for fuchsian differential equations on the riemann sphere we extend this problem to differential equations having an arbitrary number of unramified irregular singular points and we determine the existence of solutions of the generalized additive deligne simpson problems moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the riemann sphere namely open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces furthermore we detail the connectedness of the moduli spaces

a block alternating splitting iteration method for a class of block two by two complex linear systems optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering in this paper we focus on solving a class of block two by two complex linear systems arising from the distributed optimal control with time periodic parabolic equations a new block alternating splitting bas iteration method is presented for solving the class of complex linear systems the convergence theory and the spectral properties of the bas iteration method are discussed numerical experiments are presented to illustrate the efficiency of the bas iteration as a solver as well as a preconditioner for krylov subspace methods

linear differential equations on p1 and root systems abstract in this paper we study the euler transform on linear ordinary differential operators on p 1 the spectral type is the tuple of integers which count the multiplicities of local formal solutions with the same leading terms we compute the changes of spectral types under the action of the euler transform and show that the changes of spectral types generate a transformation group of a z lattice which is isomorphic to a quotient lattice of a kac moody root lattice with the weyl group as the transformation group

a class of accelerated uzawa algorithms for saddle point problems in this paper we establish a class of accelerated uzawa au algorithms for solving the large sparse nonsingular saddle point problems by making use of the extrapolation technique this extrapolation technique is based on the eigenvalues of the iterative matrix these au algorithms involve two iteration parameters whose special choices can cover the known classical uzawa method as well as yield new ones firstly the accelerated model for the uzawa algorithm is established and the detail algorithm description of au method is presented then the convergence analyse of the au method is given moreover theoretical analyses show that the au algorithm converges faster than some uzawa type methods the uzawa method is also included in when the eigenvalues of the iterative matrix and the parameter satisfy some conditions numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the au method

a note on gpiu method for generalized saddle point problems abstract in this note the generalized parameterized inexact uzawa method abbreviated as the gpiu method for solving the generalized saddle point problems with symmetric positive semi definite 2 2 block is studied the convergence of the method is established which is an extension of the results obtained in a recent paper by zhou and zhang 2009 22

further note on constraint preconditioning for nonsymmetric indefinite matrices in this note we further study the constraint preconditioner for indefinite linear systems which extends the result by cao siam j matrix anal appl 24 2002 121

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

dual isomonodromic deformations and moment maps to loop algebras the hamiltonian structure of the monodromy preserving deformation equations of jimboet al jmms is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras the nonautonomous hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps this construction is shown to lead to dual pairs of matrix differential operators whose monodromy is preserved under the same family of deformations as illustrative examples involving discrete and continuous reductions a higher rank generalization of the hamiltonian equations governing the correlation functions for an impenetrable bose gas is obtained as well as dual pairs of isomonodromy representations for the equations of the painleve transcendentspv andvi

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

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

new perturbation results for equality constrained least squares problems abstract we present some perturbation results for least squares problems with equality constraints relative errors are obtained on perturbed solutions and lagrange multipliers of the problem based on the equivalence of the problem to a consistent system of linear equations

coherent state theory of the noncompact symplectic group an extended coherent state theory is presented for the noncompact sp 3 r group which reveals a simple relationship between the sp 3 r algebra and its contracted u 3 boson limit the relationship is used to derive a remarkably accurate analytic expression for sp 3 r matrix elements for the generic lowest weight representations the expression is shown to be exact whenever the states involved are multiplicity free with respect to the u 3 subalgebra it is further shown how exact matrix elements are easily calculated in general dyson and holstein primakoff type u 3 boson expansions are given

corrected uzawa methods for solving large nonsymmetric saddle point problems abstract in this paper we consider the solution of linear systems of large nonsymmetric saddle point problems by two nonlinear iterative methods which are similar in some respects to the uzawa type methods and are called corrected uzawa methods their convergence rates are analyzed the results of numerical experiments are presented when we apply them to navier stokes equations discretized by mixed finite elements

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

generalized two mode harmonic oscillator dynamical group and squeezed states in this paper we generalize our previous results of the generalized one mode harmonic oscillator to the generalized two mode case systematic use is made of the so 3 2 dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes a complete classification of the exact solutions is made and we derive them explicitly whenever possible the relevant results on algebraic decomposition coherent state generators and classification of the solutions are shown in tables a plethora of soluble hamiltonians already treated in the literature which appear to be particular cases of the general formalism presented herein are analysed as well as new cases which to the authors knowledge have not yet been considered

existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods given a nonsingular matrix a and a matrix t of the same order under certain very mild conditions there is a unique splitting a b c such that t b 1 c moreover all properties of the splitting are derived directly from the iteration matrix t these results do not hold when the matrix a is singular in this case given a matrix t and a splitting a b c such that t b 1 c there are infinitely many other splittings corresponding to the same matrices a and t and different splittings can have different properties for instance when t is nonnegative some of these splittings can be regular splittings while others can be only weak splittings analogous results hold in the symmetric positive semidefinite case given a singular matrix a not for all iteration matrices t there is a splitting corresponding to them necessary and sufficient conditions for the existence of such splittings are examined as an illustration of the theory developed the convergence of certain alternating iterations is analyzed different cases where the matrix is monotone singular and positive semi definite are studied

d functions and immanants of unitary matrices and submatrices motivated by recent results in multiphoton interferometry we expand a result of kostant on immanants of an arbitrary m m unitary matrix to the submatrices of t specifically we show that immanants of principal submatrices of a unitary matrix t are a sum of the diagonal d functions of group element with t determined by the choice of submatrix and the irrep determined by the immanant under consideration we also provide evidence that this result extends to some submatrices that are not principal diagonal and we discuss how this result can be extended to cases where t carries an su m representation that is different from the defining representation

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

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

a modified generalized shift splitting method for nonsymmetric saddle point problems abstract in this paper we propose a modified generalized shift splitting denoted by mgssp preconditioned method for solving large sparse saddle point problems by theoretical analyses we verify the mgssp iteration method unconditionally converges to the unique solution of the saddle point problems estimate the sharp eigenvalue bounds of the related iteration matrix and point out the corresponding preconditioned matrix is positive real finally we perform some numerical computations to show the efficiency and the feasibility of the mgssp preconditioner

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

from klein to painleve via fourier laplace and jimbo we describe a method for constructing explicit algebraic solutions to the sixth painleve equation generalising that of dubrovin and mazzocco there are basically two steps first we explain how to construct finite braid group orbits of triples of elements of rm sl 2 rm c out of triples of generators of three dimensional complex reflection groups this involves the fourier laplace transform for certain irregular connections then we adapt a result of jimbo to produce the painleve vi solutions in particular this solves a riemann hilbert problem explicitly each step is illustrated using the complex reflection group associated to klein s simple group of order 168 this leads to a new algebraic solution with seven branches we also prove that unlike the algebraic solutions of dubrovin and mazzocco and hitchin this solution is not equivalent to any solution coming from a finite subgroup of rm sl 2 rm c the results of this paper also yield a simple proof of a recent theorem of inaba iwasaki and saito on the action of okamoto s affine rm d 4 symmetry group as well as the correct connection formulae for generic painleve vi equations

the spectral properties of the hermitian and skew hermitian splitting preconditioner for generalized saddle point problems in this paper we consider the hermitian and skew hermitian splitting hss preconditioner for generalized saddle point problems with nonzero 2 2 blocks the spectral property of the preconditioned matrix is studied in detail under certain conditions all eigenvalues of the preconditioned matrix with the original system being non hermitian will form two tight clusters one is near 0 0 and the other is near 2 0 as the iteration parameter approaches to zero from above so do all eigenvalues of the preconditioned matrix with the original system being hermitian numerical experiments are given to demonstrate the results

a generalization of preconditioned parameterized inexact uzawa method for indefinite saddle point problems the parameterized inexact uzawa methods have been used to solve some of the symmetric saddle point problems in this paper a new preconditioned parameterized inexact uzawa method is presented to solve indefinite saddle point problems after preconditioning theoretical analyses show that the iteration method converges under certain conditions so we propose three new algorithms based on these conditions numerical experiments are provided to show the effectiveness of the proposed preconditioner and all these algorithms have fantastic convergence rates by choosing optimal parameters

regularized dpss preconditioners for generalized saddle point linear systems abstract by introducing a regularization matrix and an additional iteration parameter a new class of regularized deteriorated positive definite and skew hermitian splitting rdpss preconditioners are proposed for generalized saddle point linear systems compared with the well known hermitian and skew hermitian splitting hss preconditioner and the regularized hss preconditioner bai 2019 studied recently the new rdpss preconditioners have much better computing efficiency especially when the 1 1 block matrix is non hermitian it is proved that the corresponding rdpss stationary iteration method is unconditionally convergent in addition clustering property of the eigenvalues of the rdpss preconditioned matrix is studied in detail two numerical experiments arising from the meshfree discretization of a static piezoelectric equation and the finite element discretization of the navier stokes equation show the effectiveness of the new proposed preconditioners

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

a practical formula for computing optimal parameters in the hss iteration methods in the hss iteration methods proposed by bai golub and ng z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 the determination of the optimal parameter is a tough task when solving a non hermitian positive definite linear system in this paper a new and simple strategy for obtaining the optimal parameter is proposed which computes the optimal parameter by solving a cubic polynomial equation the coefficients of this polynomial are determined by several traces of some matrices related to the symmetric and skew symmetric parts of the coefficient matrix of the real linear system numerical experiments show that our new strategy is very effective for approximating the optimal parameter in the hss iteration methods as it leads to fast convergence of the method

regular flat structure and generalized okubo system we study a relationship between regular flat structures and generalized okubo systems we show that the space of variables of isomonodromic deformations of a regular generalized okubo system can be equipped with a flat structure as its consequence we introduce flat structures on the spaces of independent variables of generic solutions to classical painleve equations except for pi in our framework the painleve equations pvi pii can be treated uniformly as just one system of differential equations called the four dimensional extended wdvv equation then the well known coalescence cascade of the painleve equations corresponds to the degeneration scheme of the jordan normal forms of a square matrix of rank four

the deligne simpson problem a survey abstract the deligne simpson problem dsp respectively the weak dsp is formulated like this give necessary and sufficient conditions for the choice of the conjugacy classes c j gl n c or c j gl n c so that there exist irreducible respectively with trivial centralizer p 1 tuples of matrices m j c j or a j c j satisfying the equality m 1 m p 1 i or a 1 a p 1 0 the matrices m j and a j are interpreted as monodromy operators of regular linear systems and as matrices residua of fuchsian ones on riemann s sphere the present paper offers a survey of the results known up to now concerning the dsp

analysis on block diagonal and triangular preconditioners for a pml system of an electromagnetic scattering problem abstract we shall propose several block triangular preconditioners for a pml system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems when the pml system is discretized by edge element methods it results in a discrete system with its stiffness matrix being complex symmetric but indefinite which can be formulated into a real symmetric but indefinite saddle point system in order to preserve the symmetry of the coefficient matrix we present block triangular preconditioners with two sided preconditioning for the discrete pml system we will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively on the other hand one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric to this end we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems in addition we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems

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

multiplicative preprojective algebras middle convolution and the deligne simpson problem abstract we introduce a family of algebras which are multiplicative analogues of preprojective algebras and their deformations as introduced by m p holland and the first author we show that these algebras provide a natural setting for the middle convolution operation introduced by n m katz in his book rigid local systems and put in an algebraic setting by m dettweiler and s reiter and h volklein we prove a homological formula relating the dimensions of hom and ext spaces study varieties of representations of multiplicative preprojective algebras and use these results to study simple representations we apply this work to the deligne simpson problem obtaining a sufficient and conjecturally necessary condition for the existence of an irreducible solution to the equation a 1 a 2 a k 1 with the a i in prescribed conjugacy classes in gl n c

some new estimates on the complex eigenvalues of the hss preconditioned matrix in this paper we derive some new bounds on the complex eigenvalues of the hermitian and skew hermitian splitting hss preconditioner for saddle point problems and improve those stated in previously published works

optimal parameters of the generalized symmetric sor method for augmented systems for the augmented system of linear equations zhang and lu 2008 recently studied the generalized symmetric sor method gssor with two parameters in this note the optimal parameters of the gssor method are obtained and numerical examples are given to illustrate the corresponding results

an inexact relaxed dpss preconditioner for saddle point problem based on the relaxed deteriorated positive definite and skew hermitian splitting dpss preconditioner in this paper we proposed a class of relaxed deteriorated positive definite and skew hermitian splitting rdpss preconditioner for solving the saddle point problem the proposed rdpss preconditioner is a technical modification of the deteriorated positive definite and skew hermitian splitting dpss preconditioner 36 the pss preconditioner is a straightforward application of the positive definite and skew hermitian splitting pss iteration method for solving non hermitian positive definite linear systems initially established by bai et al 37 numerical results have shown that the proposed rdpss preconditioner is advantageous over the existing dpss preconditioner

su 1 1 intelligent states we construct all the intelligent states of the non compact generators of su 1 1 for every positive discrete representation of this lie algebra and discuss some of the properties of these states

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

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

algorithms for su n boson realizations and d functions boson realizations map operators and states of groups to transformations and states of bosonic systems we devise a graph theoretic algorithm to construct the boson realizations of the canonical su n basis states which reduce the canonical subgroup chain for arbitrary n the boson realizations are employed to construct d functions which are the matrix elements of arbitrary irreducible representations of su n in the canonical basis we demonstrate that our d function algorithm offers significant advantage over the two competing procedures namely factorization and exponentiation

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

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

two block triangular preconditioners for asymmetric saddle point problems in this paper two block triangular preconditioners for the asymmetric saddle point problems with singular 1 1 block are presented the spectral characteristics of the preconditioned matrices are discussed in detail theoretical analysis shows that all the eigenvalues of the preconditioned matrices are strongly clustered numerical experiments are reported to the efficiency of the proposed preconditioners

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

moduli spaces of meromorphic connections and quiver varieties abstract we describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the riemann sphere with at most one unramified irregular singularity and arbitrary number of simple poles as nakajima s quiver varieties this result enables us to solve partially the additive irregular deligne simpson problem

the spectral properties of the preconditioned matrix for nonsymmetric saddle point problems in this paper on the basis of matrix splitting two preconditioners are proposed and analyzed for nonsymmetric saddle point problems the spectral property of the preconditioned matrix is studied in detail when the iteration parameter becomes small enough the eigenvalues of the preconditioned matrices will gather into two clusters one is near 0 0 and the other is near 2 0 for the ppss preconditioner no matter whether a is hermitian or non hermitian and for the phss preconditioner when a is a hermitian or real normal matrix numerical experiments are given to illustrate the performances of the two preconditioners

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

a note on parameterized block triangular preconditioners for generalized saddle point problems in this paper based on the results presented by jiang et al m q jiang y cao l q yao on parameterized block triangular preconditioners for generalized saddle point problems appl math comput 216 2010 1777 1789 we consider an indefinite parameterized block triangular preconditioner for symmetric saddle point problems the eigenvalue bound of the preconditioned matrix is discussed some presented results are improved numerical experiments are given to confirm our theoretical results

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

a numerical algorithm for the explicit calculation of su n and sl n c clebsch gordan coefficients we present an algorithm for the explicit numerical calculation of su n and sl n c clebsch gordan coefficients based on the gelfand tsetlin pattern calculus our algorithm is well suited for numerical implementation we include a computer code in an appendix our exposition presumes only familiarity with the representation theory of su 2

the generalized cholesky factorization method for saddle point problems over the past 10 years a variety of iterative methods for saddle point problems have been proposed in this paper we present a class of direct methods the so called generalized cholesky factorization method for solving linear systems arising from saddle point problems or discretization of the stokes equations numerical results illustrate the efficiency of new methods given in this paper

clebsch gordan coefficients of su 3 in su 2 and so 3 bases new algorithms are developed for the purpose of optimizing the efficient calculation of su 3 clebsch gordan coefficients in both su 2 and so 3 coupled bases the new algorithms make use of the fact that highest weight states in a tensor product space are easily identified by vector coherent state methods the methods are developed for su 3 but apply to other compact semi simple lie groups

middle convolution of fuchsian systems and the construction of rigid differential systems abstract in m dettweiler s reiter an algorithm of katz and its application to the inverse galois problem j symbolic comput 30 2000 761 798 a purely algebraic analogon of katz middle convolution functor see n m katz rigid local systems ann of math stud vol 139 princeton university press 1997 is given in this paper we find an explicit riemann hilbert correspondence for this functor this leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the riemann hilbert correspondence

a generalized relaxed positive definite and skew hermitian splitting preconditioner for non hermitian saddle point problems for non hermitian saddle point problems with the non hermitian positive definite 1 1 block zhang et al 2014 presented a relaxed positive definite and skew hermitian splitting rpss preconditioner to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the convergence property of the grpss iteration method is proved and a generalized rpss grpss preconditioner is proposed the grpss preconditioner is much closer to the coefficient matrix than the rpss preconditioner in certain norm which straightforwardly results in an grpss iteration method we employing the grpss preconditioner to accelerate some krylov subspace methods like gmres the spectral distribution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the grpss preconditioner