on positive semidefinite modification schemes for incomplete cholesky factorization incomplete cholesky factorizations have long been important as preconditioners for use in solving large scale symmetric positive definite linear systems in this paper we focus on the relationship between two important positive semidefinite modification schemes that were introduced to avoid factorization breakdown namely the approach of jennings and malik and that of tismenetsky we present a novel view of the relationship between the two schemes and implement them in combination with a limited memory approach we explore their effectiveness using extensive numerical experiments involving a large set of test problems arising from a wide range of practical applications

experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

parilut a new parallel threshold ilu factorization we propose a parallel algorithm for computing a threshold incomplete lu ilu factorization the main idea is to interleave a parallel fixed point iteration that approximates an incomplete factoriz

on rigidity of grauert tubes over riemannian manifolds of constant curvature it is well known that a real analytic manifold x admits a complexification xc a complex manifold that contains x as the fixed point set of an antiholomorphic involution this can be seen as follows the transition functions defining the manifold x are real analytic local diffeomorphisms of rn the taylor expansions of these transition functions can be considered as local biholomorphisms of cn hence they serve as transition functions of a complex manifold the germ of the complexification xc is unique every sufficiently small tubular neighborhood of x in the tangent bundle tx admits a real analytic diffeomorphism into xc that fixes x therefore a sufficiently small tubular neighborhood ofx in tx has a complex structure and can be considered as a complexification of x in general the complex structure on the tubular neighborhood is not unique since there are manyways to embed it intoxc there have been a lot of interest in finding canonical complex structures on tubular neighborhoods ofx in tx with additional datum of a real analytic riemannian metric g on x a canonical complex structure can be specified for sufficiently small tubular neighborhoods of x in tx see gs ls s1 there is a unique complex structure on such that the map f i i c is holomorphic wherever it is defined

incomplete cholesky factorizations with limited memory we propose an incomplete cholesky factorization for the solution of large scale trust region subproblems and positive definite systems of linear equations this factorization depends on a parameter p that specifies the amount of additional memory in multiples of n the dimension of the problem that is available there is no need to specify a drop tolerance our numerical results show that the number of conjugate gradient iterations and the computing time are reduced dramatically for small values of p we also show that in contrast with drop tolerance strategies the new approach is more stable in terms of number of iterations and memory requirements

symmetric permutations for i matrices to delay and avoid small pivots during factorization in this article we present several new permutations for i matrices making these more suitable for incomplete lu factorization preconditioners used in solving linear systems by iterative methods a general matrix can be transformed by row permutation as well as row and columns scaling into an i matrix i e a matrix having elements of modulus 1 on the diagonal and elements of modulus of no more than 1 elsewhere reordering rows and columns by the same permutation clearly preserves i matrices in this article we consider such reordering techniques which make the permuted matrix more suitable for an incomplete lu factorization preconditioner than the original i matrix we use a multilevel iluc an incomplete lu factorization preconditioner using crout s implementation of gaussian elimination without pivoting to test these reorderings the combination of i matrix preprocessing with the various algorithms presented here and the multilevel incomplete lu factorizations forms a powerful preconditioning method for unsymmetric highly indefinite problems the c code has been made available in the software package ilu

an algorithm for profile and wavefront reduction of sparse matrices summary an algorithm for reducing the profile and wavefront of a sparse matrix is described the scheme is applicable to any sparse matrix which has a symmetric pattern of zeros and may be used to generate efficient labellings for finite element grids in particular it is suitable for generating efficient labellings for profile and frontal solution schemes empirical evidence obtained from analysis of the 30 test problems collected by everstine suggests that the new algorithm is superior to existing methods for profile and wavefront reduction it is fast requires only a small amount of memory and is simple to program

block ilu factorization preconditioners for a block tridiagonal h matrix abstract we propose new parallelizable block ilu incomplete lu factorization preconditioners for a nonsymmetric block tridiagonal h matrix theoretical properties of these block ilu preconditioners are compared with those of block ilu preconditioners for the corresponding comparison matrix numerical results of the bicgstab using these block preconditioners are compared with those of the bicgstab using a standard ilu factorization preconditioner to see the effectiveness of the block ilu preconditioners lastly parallel computations of the block ilu preconditioners are carried out and evaluated on the cray c90

incomplete blockwise factorizations of block h matrices the use of incomplete blockwise factorizations as preconditioners in conjugate gradient like methods has become more and more popular in recent years most of the theory concerning existence and applicability of these factorizations has been limited to m matrices so far here we introduce a more general definition of block h matrices robert 8 and we extend the theory to this class of matrices

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

a generalized block fsai preconditioner for nonsymmetric linear systems the efficient solution to nonsymmetric linear systems is still an open issue especially on parallel computers in this paper we generalize to the unsymmetric case the block factorized sparse approximate inverse block fsai preconditioner which has already proved very effective on symmetric positive definite spd problems block fsai is a hybrid approach combining an inner preconditioner with the aim of transforming the system matrix structure to block diagonal with an outer one a block diagonal incomplete or exact factorization intended to improve the conditioning of each block the proposed algorithm is experimented with in a number of large size matrices showing both a good robustness and scalability

stability of block ldlt factorization of a symmetric tridiagonal matrix for symmetric indefinite tridiagonal matrices block ldl t factorization without interchanges is shown to have excellent numerical stability when a pivoting strategy of bunch is used to choose the dimension 1 or 2 of the pivots 1999 elsevier science

a new pivoting strategy for gaussian elimination this paper discusses a method for determining a good pivoting sequence for gaussian elimination based on an algorithm for solving assignment problems the worst case complexity is o n3 in practice o n2 25 operations are sufficient

the incomplete factorization multigraph algorithm we present a new family of multigraph algorithms ilu mg based upon an incomplete sparse matrix factorization using a particular ordering and allowing a limited amount of fill in while much of the motivation for multigraph comes from multigrid ideas ilu mg is distinctly different from algebraic multilevel methods the graph of the sparse matrix a is recursively coarsened by eliminating vertices using a graph model similar to gaussian elimination incomplete factorizations are obtained by allowing only the fill in generated by the vertex parents associated with each vertex multigraph is numerically compared with algebraic multigrid on some examples arising from discretizations of partial differential equations on unstructured grids

a second order accurate projection method for the incompressible navier stokes equations on non graded adaptive grids we present an unconditionally stable second order accurate projection method for the incompressible navier stokes equations on non graded adaptive cartesian grids we employ quadtree and octree data structures as an efficient means to represent the grid we use the supra convergent poisson solver of c h min f gibou h ceniceros a supra convergent finite difference scheme for the variable coefficient poisson equation on fully adaptive grids cam report 05 29 j comput phys in press a second order accurate semi lagrangian method to update the momentum equation an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non graded grids we sample all the variables at the grid nodes producing a scheme that is straightforward to implement we propose two and three dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the l1 and l norms

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

convergence of two stage and multisplitting methods using various types of splittings we first study the convergence of both two stage iterative method and multisplitting methods using various types of splittings for solving a linear system whose coefficient matrix is an m matrix or an h matrix we next study a parallel implementation of two stage multisplitting method associated with a special multisplitting and an application of the two stage multisplitting method to parallel preconditioner of krylov subspace methods lastly we provide parallel performance results of both two stage multisplitting method and krylov subspace method with a parallel preconditioner which is derived from two stage multisplitting method on the ibm p690 supercomputer

on the stability of the incomplete lu factorizations and characterizations of h matrices meijerink and van der vorst 8 have shown that the incomplete lu factorizations are numerically stable for m matrices varga saff and mehrmann 16 gave some characterizations of the h matrices by using the incomplete lu factorizations of them the purpose of this paper is to show that the incomplete lu factorizations of an h matrix are at least as stable as the complete lu factorizations of its comparison matrix we give also some new characterizations of the h matrices in connection with their incomplete lu factorizations

convergence behaviors of multisplitting methods with k 1 relaxed parameters in this paper two multisplitting methods with k 1 relaxed parameters are established for solving a linear system whose coefficient matrix is a large sparse m matrix or h matrix and the corresponding convergence behaviors are studied then the implementation of these two methods with ilu factorizations as inner splittings is investigated finally some numerical experiments are presented to illustrate the effectiveness of the preconditioners obtained from our methods when combined with bicgstab

enhancing performance and robustness of ilu preconditioners by blocking and selective transposition incomplete factorization is one of the most effective general purpose preconditioning methods for krylov subspace solvers for large sparse systems of linear equations two techniques for enhancing the robustness and performance of incomplete lu factorization for sparse unsymmetric systems are described a block incomplete factorization algorithm based on the crout variation of lu factorization is presented the algorithm is suitable for incorporating threshold based dropping as well as unrestricted partial pivoting and it overcomes several limitations of existing incomplete lu factorization algorithms with and without blocking it is shown that blocking has a three pronged impact it speeds up the computation of incomplete factors and the solution of the associated triangular systems it permits denser and more robust factors to be computed economically and it permits a trade off with the restart parameter of gmres to further improve the overall speed and robustness a highly effective heuristic for imp

iterative solution methods for certain sparse linear systems with a non symmetric matrix arising from pde problems abstract in this paper methods are described for the solution of certain sparse linear systems with a non symmetric matrix the power of these methods is demonstrated by extensive numerical experiments application of the methods is limited to problems where the matrix has only eigenvalues with positive real part an important class of this type of matrix arises from discretisation of second order partial differential equations with first order derivative terms

on using cholesky based factorizations and regularization for solving rank deficient sparse linear least squares problems by examining the performance of modern parallel sparse direct solvers and exploiting our knowledge of the algorithms behind them we perform numerical experiments to study how they can be used to efficiently solve rank deficient sparse linear least squares problems arising from practical applications the cholesky factorization of the normal equations breaks down when the least squares problem is rank deficient while applying a symmetric indefinite solver to the augmented system can give an unacceptable level of fill in the factors to try to resolve these difficulties we consider a regularization procedure that modifies the diagonal of the unregularized matrix this leads to matrices that are easier to factorize we consider both the regularized normal equations and the regularized augmented system we employ the computed factors of the regularized systems as preconditioners with an iterative solver to obtain the solution of the original unregularized problem furthermore we look at using limited me

weighted matchings for preconditioning symmetric indefinite linear systems maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems especially in direct solvers in this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for incomplete mathrm ldl t factorizations the reorderings are constructed such that the matched entries form 1 times 1 or 2 times 2 diagonal blocks in order to increase the diagonal dominance of the system during the incomplete factorization only tridiagonal pivoting is used we report results for this approach and comparisons with other solution methods for a diverse set of symmetric indefinite matrices ranging from nonlinear elasticity to interior point optimization

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

linear time approximation for maximum weight matching the maximum cardinality and maximum weight matching problems can be solved in o m n time a bound that has resisted improvement despite decades of research here m and n are the number of edges and vertices in this article we demonstrate that this m n barrier can be bypassed by approximation for any e 0 we give an algorithm that computes a 1 e approximate maximum weight matching in o me 1 log e 1 time that is optimal linear time for any fixed e our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error

large scale sparse inverse covariance matrix estimation the estimation of large sparse inverse covariance matrices is a ubiquitous statistical problem in many application areas such as mathematical finance geology health and many others the ell 1

acceleration techniques for reduced order models based on proper orthogonal decomposition this paper presents several acceleration techniques for reduced order models based on the proper orthogonal decomposition pod method the techniques proposed herein are i an algorithm for splitting the database of snapshots generated by the full order model ii a method for solving quasi symmetrical matrices iii a strategy for reducing the frequency of the projection the acceleration techniques were applied to a pod based reduced order model of the two phase flows in fluidized beds this reduced order model was developed using numerical results from a full order computational fluid dynamics model of a two dimensional fluidized bed using these acceleration techniques the computational time of the pod model was two orders of magnitude shorter than the full order model

self similarity of binary quasiperiodic sequences self similarity in binary quasiperiodic sequences generated by a projection method is shown to exist when and only when it is associated with quadratic irrational numbers and the explicit self similarity transformation for an arbitrary quadratic number is obtained the self similarity transformation is shown not to be reducible to the simplest form for a class of quadratic numbers

adaptive pattern research for block fsai preconditioning an adaptive algorithm is presented to generate automatically the nonzero pattern of the block factored sparse approximate inverse bfsai preconditioner it is demonstrated that in symmetric positive definite spd problems bfsai minimizes an upper bound to the kaporin number of the preconditioned matrix the mathematical structure of this bound suggests an efficient and easily parallelizable strategy for improving the given nonzero pattern of bfsai thus providing a novel adaptive bfsai abf preconditioner numerical experiments performed on large sized finite element problems show that abf coupled with a block incomplete cholesky ic outperforms bfsai ic even by a factor of 4 preserving the same preconditioner density and exhibiting an excellent parallelization degree

a finite size scaling analysis of the localization properties of one dimensional quasiperiodic systems the authors show that the phenomenological renormalization group analysis based on the finite size scaling hypothesis is very effective in analysing the critical properties of the localization delocalization transition ldt of the wavefunctions in quasiperiodic systems they have applied it successfully to the ldt of the discrete schrodinger equation with an incommensurate modulation potential characterized by a quadratic irrational showing that different models with a common incommensurate ratio belong to the same universality class if their potential functions are smooth moreover there exists a one to one correspondence between the whole universality classes of the ldt and the whole equivalence classes of quadratic irrationals with respect to the modular transformation

mixed constraint preconditioning in computational contact mechanics contact mechanics can be addressed numerically by finite elements using either a penalty formulation or the lagrange multipliers the penalty approach leads to a linearized symmetric positive definite system which can prove severely ill conditioned with the iterative solution to large 3d problems requiring expensive preconditioners to accelerate or even to allow for convergence in the present paper a mixed constraint preconditioner mcp is developed and implemented for the iterative solution of contact problems based on the penalty approach a theoretical bound for the eigenspectrum of the preconditioned matrix is provided depending on the user specified parameters that control the preconditioner fill in the performance of mcp combined with the preconditioned conjugate gradient is studied in two real 3d test cases describing the geomechanics of faulted porous media and compared to a well established preconditioner such as the incomplete cholesky decomposition illt with optimal fill in numerical results show that typically mcp outperforms illt especially in the most challenging problems where very large penalty parameters must be used finally a modification aimed at simplifying the mcp implementation and increasing its robustness is developed and experimented with showing that the modified mcp can be viewed as an efficient and viable alternative to both mcp and illt especially in problems where the size of the contact surfaces is large with respect to the global domain

an algebraic multilevel multigraph algorithm we describe an algebraic multilevel multigraph algorithm many of the multilevel components are generalizations of algorithms originally applied to general sparse gaussian elimination indeed general sparse gaussian elimination with minimum degree ordering is a limiting case of our algorithm our goal is to develop a procedure which has the robustness and simplicity of use of sparse direct methods yet offers the opportunity to obtain the optimal or near optimal complexity typical of classical multigrid methods

performance of ilu factorization preconditioners based on multisplittings in this paper we study the convergence of multisplitting methods associated with a multisplitting which is obtained from the ilu factorizations of a general h matrix and then we propose parallelizable ilu factorization preconditioners based on multisplittings for a block tridiagonal h matrix we also describe a parallelization of preconditioned krylov subspace methods with the ilu preconditioners based on multisplittings on distributed memory computers such as the cray t3e lastly parallel performance results of the preconditioned bicgstab are provided to evaluate the efficiency of the ilu preconditioners based on multisplittings on the cray t3e

the asymptotic expansion of a cr invariant and grauert tubes cr manifolds the abstract models of real hypersurfaces in complex manifolds are 2n 1 dimensional manifolds m with a codimension one subbundle h of the tangent bundle which carries a complex structure the cr refers to cauchy riemann because for m c c n l the subbundle h consists of induced cauchy riemann operators there is a wealth of geometry and analysis associated with these structures especially when the cr manifolds are strictly pseudoconvex for example two strictly pseudoconvex domains are biholomorphically equivalent if and only if their boundaries are cr equivalent a fundamental problem in cr geometry is to find computable invariants associated with the cr structures the global cr invariant we will consider in this paper is the chem simons type invariant discovered by bums and epstein b e 1 it is a real valued global cr invariant of a compact 3dimensional strictly pseudo convex cr manifold whose holomorphic tangent bundle is trivial cheng and lee independently found this invariant and extend the definition of b e invariant to a relative invariant on an arbitrary compact 3 dimensional cr manifold cf c l we will evaluate this g asymptotically on the boundary of small grauert tubes before posing the question in a more precise form we will first say a few workds about grauert tubes let x be a real analytic manifold then every coordinate patch u c r n can be thickened to obtain an open set r c c n since the coordinate changes of x are real analytic maps by taking power series expansions and by shrinking c u to get convergence they can be extended holomorphically to such enlarged domains and thus they can be used as holomorphic transition

multilevel preconditioners constructed from inverse based ilus this paper analyzes dropping strategies in a multilevel incomplete lu decomposition context and presents a few strategies for obtaining related ilus with enhanced robustness the analysis shows that the incomplete lu factorization resulting from dropping small entries in gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large as a consequence a few strategies are developed whose goal is to achieve this feature a number of templates for enabling implementations of these factorizations are presented numerical experiments show that the resulting ilus offer a good compromise between robustness and efficiency

multilevel ilu with reorderings for diagonal dominance this paper presents a preconditioning method based on combining two sided permutations with a multilevel approach the nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix the method can be regarded as a complete pivoting version of the incomplete lu factorization this leads to an effective incomplete factorization preconditioner for general nonsymmetric irregularly structured sparse linear systems

shishkin mesh simulation a new stabilization technique for convection diffusion problems abstract a new stabilization procedure is presented it is based on a simulation of the interaction between the coarse and fine parts of a shishkin mesh but can be applied on coarse and irregular meshes and on domains with nontrivial geometries the technique which does not require adjusting any parameter can be applied to different stabilized and non stabilized methods numerical experiments show it to obtain oscillation free approximations on problems with boundary and internal layers on uniform and nonuniform meshes and on domains with curved boundaries

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

a stable method for the incomplete factorization of h matrices abstract ahac and olesky have shown that by using the most diagonal dominat column as a pivot element for m matrices was numerically stable the purpose of this paper is to show that this method of selecting a pivoting is stable when finding the incomplete factorization of an h matrix

parallel relaxed multisplitting methods for a symmetric positive definite matrix we first study convergence of both the relaxed multisplitting method and the relaxed nonstationary two stage multisplitting method using incomplete factorizations for solving a linear system whose coefficient matrix is a large sparse symmetric positive definite matrix we next study a parallel implementation of the relaxed two stage multisplitting method using incomplete factorizations as inner splittings and an application of the relaxed two stage multisplitting method to parallel preconditioner of the cg conjugate gradient method lastly we provide parallel performance results of both the relaxed two stage multisplitting method and the cg method with a parallel preconditioner which is derived from the relaxed two stage multisplitting method on the ibm p690 supercomputer

cell centered finite volume method for heterogeneous anisotropic poromechanics problem abstract the article introduces a finite volume method for the poromechanics problem for heterogeneous anisotropic media the media are characterized by the full tensors of elasticity permeability and biot coefficient the pressure and displacement unknowns are collocated at centers of cells poromechanics represents a multi physics problem of darcy flow and linear elasticity coupled via biot terms based on the flux and the displacement and pressure continuity conditions we derive the coupled block expression for the flux the flux expression is split into a block harmonic part a transversal part and an additional contribution due to gravity the block harmonic part is equivalent to the two point flux approximation method for the darcy problem for the approximation of the transversal part we derive an interpolation method for the coupled pressure and displacement fields valid over discontinuities of the tensors finally we derive the flux expression for generalized boundary conditions the flux expression does not require introduction of auxiliary unknowns at the boundary we evaluate the performance of the proposed method on a number of numerical tests and apply the method to a problem on a grid corresponding to a real oil field

a qmr based interior point algorithm for solving linear programs a new approach for the implementation of interior point methods for solving linear programs is proposed its main feature is the iterative solution of the symmetric but highly indefinite 2 2 block systems of linear equations that arise within the interior point algorithm these linear systems are solved by a symmetric variant of the quasi minimal residual qmr algorithm which is an iterative solver for general linear systems the symmetric qmr algorithm can be combined with indefinite preconditioners which is crucial for the efficient solution of highly indefinite linear systems yet it still fully exploits the symmetry of the linear systems to be solved to support the use of the symmetric qmr iteration a novel stable reduction of the original unsymmetric 3 3 block systems to symmetric 2 2 block systems is introduced and a measure for a low relative accuracy for the solution of these linear systems within the interior point algorithm is proposed some indefinite preconditioners are discussed finally we report results of a few preliminary numerical experiments to illustrate the features of the new approach

robust incomplete factorization for nonsymmetric matrices in this paper a new incomplete lu factorization preconditioner for nonsymmetric matrices is being considered which is also breakdown free no zero pivots occurs for positive definite matrices to construct this preconditioner only the information of matrix a is used and just one of the factors of the ainv process is computed the l factor is extracted as a by product of the ainv process the pivots of the ainv process are used as diagonal entries of u the new preconditioner has left and right looking versions to improve the efficiency of the preconditioner we have used the inverse based dropping strategies for both l and u factors numerical experiments show that the left looking version of the preconditioner is significantly faster than its right looking version in terms of preconditioning time and both are equally effective to reduce the number of iterations comparisons of the new preconditioner with ainv and ilut preconditioners are also presented

convergence of two stage iterative methods using incomplete factorization we first study the convergence of two stage iterative methods using the incomplete factorization for solving a linear system whose coefficient matrix is an h matrix and then we study the convergence of two stage iterative methods using the incomplete factorization for solving a linear system whose coefficient matrix is a symmetric positive definite matrix lastly numerical experiments are provided to analyze theoretical results

analysis of iterative methods for saddle point problems a unified approach in this paper two classes of iterative methods for saddle point problems are considered inexact uzawa algorithms and a class of methods with symmetric preconditioners in both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems the obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature a few numerical tests are given which confirm the sharpness of the estimates

solving symmetric indefinite systems in an interior point method for linear programming we describe an implementation of a primal dual path following method for linear programming that solves symmetric indefinite augmented systems directly by bunch parlett factorization rather than reducing these systems to the positive definite normal equations that are solved by cholesky factorization in many existing implementations the augmented system approach is seen to avoid difficulties of numerical instability and inefficiency associated with free variables and with dense columns in the normal equations approach solving the indefinite systems does incur an extra overhead whose median is about 40 in our tests but the augmented system approach proves to be faster for a minority of cases in which the normal equations have relatively dense cholesky factors a detailed analysis shows that the augmented system factorization is reliable over a fairly large range of the parameter settings that control the tradeoff between sparsity and numerical stability

convergence of nonstationary multisplitting methods using ilu factorizations in this paper we first study convergence of nonstationary multisplitting methods associated with a multisplitting which is obtained from the ilu factorizations for solving a linear system whose coefficient matrix is a large sparse h matrix we next study a parallel implementation of the relaxed nonstationary two stage multisplitting method called algorithm 2 in this paper using ilu factorizations as inner splittings and an application of algorithm 2 to parallel preconditioner of krylov subspace methods lastly we provide parallel performance results of both algorithm 2 using ilu factorizations as inner splittings and the bicgstab with a parallel preconditioner which is derived from algorithm 2 on the ibm p690 supercomputer

a robust ilu with pivoting based on monitoring the growth of the inverse factors abstract an incomplete lu decomposition with pivoting is presented that progressively monitors the growth of the inverse factors of l u the information on the growth of the inverse factors is used as feedback for dropping entries in l and u this method often yields a robust and effective preconditioner especially when the system is highly indefinite numerical examples demonstrate the effectiveness of this approach

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

stable lu factorization of h matrices abstract results are given concerning the lu factorization of h matrices and gaussian elimination with column diagonal dominant pivoting is shown to be applicable to h matrices this algorithm which uses a symmetric permutation to exchange the most diagonally dominant column of the unreduced submatrix into the pivotal position is shown to be numerically stable by deriving an upper bound on the growth factor associated with the backward error analysis for gaussian elimination

on incomplete block factorization methods of generalized ssor type for h matrices abstract two types of modified incomplete block factorization methods are considered and the existence and convergence of the related splittings are established for h matrices this is done by studying in parallel the similar preconditioner for an m matrix a which satisfies a m a we show also for the m matrix case somewhat more general results for the type of preconditioners considered than previously found in the literature

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

a new deflated block gcrot m k method for the solution of linear systems with multiple right hand sides linear systems with multiple right hand sides arise in many applications to solve such systems efficiently a new deflated block gcrot m k method is explored in this paper by exploiting a modified block arnoldi deflation incorporating this modified block arnoldi deflation the new algorithm can address the possible linear dependence at each iteration during the block arnoldi procedure and reduces expensive computational operations moreover as a block version of gcrot m k the new method inherits the property of easy operability finally some numerical examples also illustrate the effectiveness of the proposed method

hopdm version 2 12 a fast lp solver based on a primal dual interior point method hopdm is an implementation of the primal dual interior point method for solving large scale linear programming lp problems hopdm stands for higher order primal dual method its newest version 2 12 of may 1995 is a robust and efficient code that compares favourably with today s commercial lp solvers

analysis of parallel incomplete point factorizations abstract incomplete point factorizations are analyzed regarding their parallelism and the norm of the error matrix and the relation between the two

ilu preconditioners for nonsymmetric saddle point matrices with application to the incompressible navier stokes equations motivated by the numerical solution of the linearized incompressible navier stokes equations we study threshold incomplete lu factorizations for nonsymmetric saddle point matrices the resulting preconditioners are used to accelerate the convergence of a krylov subspace method applied to finite element discretizations of fluid dynamics problems in three space dimensions the paper presents and examines an extension for nonsymmetric matrices of the tismenetsky kaporin incomplete factorization it is shown that in numerically challenging cases of higher reynolds number flows one benefits from using this two parameter modification of a standard threshold ilu preconditioner the performance of the ilu preconditioners is studied numerically for a wide range of flow and discretization parameters and the efficiency of the approach is shown if threshold parameters are chosen suitably the practical utility of the method is further demonstrated for the haemodynamic problem of simulating blood flow in a right c

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

symmetric quasidefinite matrices it is stated here that a symmetric matrix k is quasidefinite if it has the form k begin bmatrix e a t a f end bmatrix where e and f are symmetric positive definite matrices although such matrices are indefinite it is shown that any symmetric permutation of a quasidefinite matrix yields a factorization ldl t this result is applied to obtain a new approach for solving the symmetric indefinite systems arising in interior point methods for linear and quadratic programming these systems are typically solved either by reducing to a positive definite system or by performing a bunch parlett factorization of the full indefinite system at every iteration this is an intermediate approach based on reducing to a quasidefinite system this approach entails less fill in than further reducing to a positive definite system but is based on a static ordering and is therefore more efficient than performing bunch parlett factorizations of the original indefinite system