computational methods in linear algebra abstract this paper is the written version of a talk given at the workshop on computational seismology salishan lodge oregon march 24 26 1981 state of the art and recent developments in computational linear algebra including linear systems least squares techniques the singular value decomposition and eigenvalue problems are reviewed briefly

algorithm for direct numerical simulation of emulsion flow through a granular material a multipole accelerated 3d boundary integral algorithm capable of modelling an emulsion flow through a granular material by direct multiparticle multidrop simulations in a periodic box is developed and tested the particles form a random arrangement at high volume fraction rigidly held in space including the case of an equilibrium packing in mechanical contact deformable drops with non deformed diameter comparable with the particle size squeeze between the particles under a specified average pressure gradient the algorithm includes recent boundary integral desingularization tools especially important for drop solid and drop drop interactions the hebeker representation for solid particle contributions and unstructured surface triangulations with fixed topology multipole acceleration with two levels of mesh node decomposition entire drop solid surfaces and patches is a significant improvement over schemes used in previous purely multidrop simulations it remains efficient at very high resolutions 10 4 10 5 triangular elements per surface and has no lower limitation on the number of particles or drops such resolutions are necessary in the problem to alleviate lubrication difficulties especially for near critical squeezing conditions as well as using 10 4 time steps and an iterative solution at each step both for contrast and matching viscosities examples are shown for squeezing of 25 40 drops through an array of 9 14 solids with the total volume fraction of 70 for particles and drops the flow rates for the drop and continuous phases are calculated extensive convergence testing with respect to program parameters triangulation multipole truncation etc is made

a graph theoretic model of symmetric givens operations and its implications abstract symmetric givens operations a gag t g a givens rotation matrix are a basic tool in many matrix computations especially for eigenvalue eigenvector computations a graph theoretic model of these operations is given for symmetric matrices analogous to the graph theoretic model of cholesky factorization using this model it is shown that unless there is accidental cancellation it is impossible to reduce a range of different matrix classes to tridiagonal form in o n 2 givens operations these classes include arrowhead matrices pentadiagonal matrices and cyclic tridiagonal matrices

minimum fill in inapproximability and almost tight lower bounds abstract given an n n sparse symmetric matrix with m nonzero entries performing gaussian elimination may turn some zeroes into nonzero values so called fill ins the minimum fill in problem asks whether it is possible to perform the elimination with at most k fill ins we exclude the existence of polynomial time approximation schemes for this problem assuming p np and the existence of 2 o n 1 time approximation schemes for any positive assuming the exponential time hypothesis we also give a 2 o k 1 2 n o 1 parameterized lower bound all these results come as corollaries of a new reduction from vertex cover to the minimum fill in problem which might be of its own interest all previous reductions for similar problems start from some kind of graph layout problems and hence have limited use in understanding their fine grained complexity

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

optimizing quadratic forms of adjacency matrices of trees and related eigenvalue problems abstract let a be an adjacency matrix of a tree t with n vertices conditions are determined for the existence of a fixed permutation matrix p that maximizes the quadratic form x t p t apx over all nonnegative vectors x with entries arranged in nondecreasing order this quadratic form problem is completely solved and its answer leads to a corresponding solution for the problem of determining conditions for the existence of a fixed permutation matrix p that maximizes the largest eigenvalue of matrices of the form pdp t a over all real diagonal matrices d with nondecreasing diagonal entries it is shown that there is a tree with six vertices for which neither of the problems has a solution and all other trees with six or fewer vertices have solutions for both problems by duality the results also apply to the analogous problem of minimizing the smallest eigenvalue of matrices of the form pdp t a

an efficient algorithm for calculating multiparticle thermal interaction in a concentrated dispersion of spheres abstract the boundary value problem of heat conduction through a multiparticle system in a medium of a different conductivity e is considered the particle set is a random arrangement of n equisized spheres of conductivity in a cubic cell continued periodically into all space the mean temperature gradient is given the problem is reduced to an infinite set of equations for the coefficients of the temperature expansion into spherical harmonics on the interfaces the algorithm exploits the idea that only the interaction of the low frequency harmonics is long ranged to construct the economical truncation as well as the rotational transformations of spherical harmonics to promote the efficient iterative solution the method is capable of providing high resolution for arbitrary conductivity ratio e and enabling large scale simulations of the effective conductivity of highly concentrated dispersions with various microstructures even on a small computer

robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming we have formulated the problem of generating dense packings of nonoverlapping nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive shrinking cell asc formulation s torquato and y jiao phys rev e 80 041104 2009 because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d dimensional euclidean space r d it is most suitable and natural to solve the corresponding asc optimization problem using sequential linear programming slp techniques we implement an slp solution to produce robustly a wide spectrum of jammed sphere packings in r d for d 2 3 4 5 and 6 with a diversity of disorder and densities up to the respective maximal densities a novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures locally maximally dense and mechanically stable packings besides the usual disordered ones such as the maximally random jammed state with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere for example in three dimensions we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range 0 6 0 7408 where 18 0 7408 corresponds to the density of the densest packing we also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d 2 4 5 and 6 our jammed sphere packings are characterized and compared to the corresponding packings generated by the well known lubachevsky stillinger ls molecular dynamics packing algorithm compared to the ls procedure our slp protocol is able to ensure that the final packings are truly jammed produces disordered jammed packings with anomalously low densities and is appreciably more robust and computationally faster at generating maximally dense packings especially as the space dimension increases

interval completion is fixed parameter tractable we present an algorithm with runtime o k 2k n 3m for the following np complete problem m garey and d johnson computers and intractability a guide to the theory of np completeness w h freeman and co san francisco 1979 problem gt35 given an arbitrary graph g on n vertices and m edges can we obtain an interval graph by adding at most k new edges to g this resolves the long standing open question h kaplan r shamir and r e tarjan siam j comput 28 1999 pp 1906 1922 r g downey and m r fellows parameterized complexity springer verlag new york 1999 m serna and d thilikos bull eur assoc theory comput sci eatcs 86 2005 pp 41 65 g gutin s szeider and a yeo in proceedings iwpec 2006 lecture notes in comput sci 4169 springer verlag berlin 2006 pp 60 71 first posed by kaplan shamir and tarjan of whether this problem was fixed parameter tractable the problem has applications in profile minimization for sparse matrix computations j a george and j w h liu computer solution of large sparse positive definite systems prentice hall englewood cliffs nj 1981 r e tarjan in sparse matrix computations j r bunch and d j rose eds academic press 1976 pp 3 22 and our results show tractability for the case of a small number k of zero elements in the envelope our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search

a symmetric formulation of coupled bem fem in solving responses of submerged elastic structures for large degrees of freedom this work presents a symmetric formulation for the coupling of boundary element method bem with finite element method fem to compute responses of submerged elastic structures in a heavy acoustic medium the acoustic loading derived from bem is formulated in a symmetric complex matrix the symmetry of the acoustic loading matrix is proven by an acoustic reciprocal principle which relate two sets of arbitrary surface normal velocities and surface pressures in terms of surface integrals the structural equation is represented by fem the fact that the acoustic loading and the structural equation are symmetric allows us to take the storage and computational advantages of the banded matrix which has been widely used in conventional finite element methods consequently the computational efficiency of the proposed coupled fem bem is significantly increased the proposed method involving 20 000 degrees of freedom can be implemented on a personal computer numerical results are verified by analytical solutions of a constant thickness spherical shell subject to a point alternating force as well as the previously published variational solution of a constant thickness spheroidal shell excited by a concentrated force

a direct method for sparse least squares problems with lower and upper bounds a direct method is developed for solving linear least squares problems mathop min left ax b right 2 limits x wherea is large and sparse and the solution is subject to lower and upper boundsl x u the problem is initially transformed to upper triangular form by a sparseqr factorization an active set algorithm is then used the key step is the stable updating of ther factor associated with the columns ofa corresponding to the free variables when theq factor is not available for this a new method is developed which uses the semi normal equations and iterative refinement

diversity of order and densities in jammed hard particle packings recently the conventional notion of random close packing has been supplanted by the more appropriate concept of the maximally random jammed mrj state this inevitably leads to the necessity of distinguishing the mrj state among the entire collection of jammed packings while the ideal method of addressing this question would be to enumerate and classify all possible jammed hard sphere configurations practical limitations prevent such a method from being employed instead we generate numerically a large number of representative jammed hard sphere configurations primarily relying on a slight modification of the lubachevsky stillinger algorithm to do so and evaluate several commonly employed order metrics for each of these packings our investigation shows that even in the large system limit jammed systems of hard spheres can be generated with a wide range of packing fractions from phi approximately 0 52 to the fcc limit phi approximately 0 74 moreover at a fixed packing fraction the variation in the order can be substantial indicating that the density alone does not uniquely characterize a packing interestingly each order metric evaluated yielded a relatively consistent estimate for the packing fraction of the maximally random jammed state phi mrj approximately 0 63 this estimate however is compromised by the weaknesses in the order metrics available and we propose several guiding principles for future efforts to define more broadly applicable metrics

large scale estimation of variance and covariance components this paper concerns matrix computations within algorithms for variance and covariance component estimation hemmerle and hartley technometrics 15 1973 pp 819 831 showed how to compute the objective function and its derivatives for maximum likelihood estimation of variance components using matrices with dimensions of the order of the number of coefficients rather than that of the number of observations their approach was extended by corbeil and searle technometrics 18 1976 pp 31 38 for restricted maximum likelihood estimation a similar reduction in dimension is possible using expectation maximization em algorithms in most cases variance components are assumed to be strictly positive we advocate the use of a modification that is numerically stable even if variance component estimates are small in magnitude for problems in which the number of coefficients is large fellner proc statistical computing section american statistical association 1984 pp 150 154 comm statist simulat

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

a unified set of resequencing algorithms a general algorithm for resequencing equations and finite element meshes is presented this algorithm incorporates in a unified form the most successful resequencing schemes namely the sloan gibbs king gk and gibbs poole stockmeyer gps algorithms as well it provides a context within which the development of new algorithms may take place with a minimum of change it is easily programmed and has been shown to speed up the execution of the original gk algorithm

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

homotopy commutativity of finite loop spaces let p be a fixed prime we say that x is a finite loop space if x has the homotopy type of a 1 connected p localized finite cw complex and x has a classifying space bx as examples of finite loop spaces we can consider 1 the p localizations of the compact 1 connected lie groups 2 for each irreducible complex reflection group g of order prime to p whose unitary representation restricts to the p adics we can construct a space x g where mod p cohomology is the ring of invariants of g and it is therefore a polynomial algebra since by 2 this space can be pulled back to a space of finite type then 2 x g p is a finite loop space these irreducible types are listed in clark ewing list 3 in 9 quillen showed that there is a space x such that h x f fp x i xn where deg xl 2mi and m is any divisor of p 1 these spaces are called p adic grasmannians and also by 2 f2 x p are finite loop spaces 4 zabrodsky constructed in 13 two more spaces with polynomial mod p cohomology which also produce finite loop spaces in the same way as above 5 in 1 aguad6 proved that the types 29 and 34 of the clark ewing list are realizables for p 5 and p 7 respectively and consequently their rood p cohomology is the ring of invariants furthermore for the type 29 and p 5 aguad6 has also proved that its m o d p cohomology is a polynomial algebra hence its loop space is a finite loop space since for the type 34 and p 7 the proof of the same is in progress we also include its loop space as a finite loop space 6 any product of the spaces above is also a finite loop space

the dual polyhedral product cocategory and nilpotence abstract the notion of a dual polyhedral product is introduced as a generalization of hovey s definition of lusternik schnirelmann cocategory properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if x is a simply connected space then the weak cocategory of x equals the homotopy nilpotency class of x this answers a fifty year old problem posed by ganea the methods are applied to determine the homotopy nilpotency class of quasi p regular exceptional lie groups and sporadic p compact groups

a hybrid algorithm for reducing matrix bandwidth a hybrid algorithm for reducing the bandwidth of symmetric matrices is described in terms of a finite element grid the new algorithm produces significantly lower bandwidths than either the commonly used gibbs poole stockmeyer gps or cuthill mckee cm algorithms with run times comparable to the original cm algorithm the new hybrid algorithm uses the gps algorithm as a preprocessor to provide a good initial node numbering for the author s node shuffling algorithm which accounts for variable degrees of freedom per node dof node the hybrid algorithm was tested on the 30 benchmark problems that were compiled by everstine and on 10 supplemental problems with variable dof node bandwidths and cpu times are presented for all problems

substructuring preconditioners for the q_1 mortar element method the q 1 mortar element method is a non conforming finite element method with q 1 elements based on domain decomposition for the laplace equation it yields an ill conditioned linear system for solving the linear system the so called preconditioned conjugate gradient method in a subspace is used preconditioners are proposed and estimates on condition numbers and arithmetical complexity are given finally numerical experiments are presented

how good is recursive bisection the most commonly used p way partitioning method is recursive bisection rb it first divides a graph or a mesh into two equal sized pieces by a good bisection algorithm and then recursively divides the two pieces ideally we would like to use an optimal bisection algorithm because the optimal bisection problem that partitions a graph into two equal sized subgraphs to minimize the number of edges cut is np complete practical rb algorithms use more efficient heuristics in place of an optimal bisection algorithm most such heuristics are designed to find the best possible bisection within allowed time we show that the rb method even when an optimal bisection algorithm is assumed may produce a p way partition that is very far way from the optimal one our negative result is complemented by two positive ones first we show that for some important classes of graphs that occur in practical applications such as well shaped finite element and finite difference meshes rb is within a constant factor of the optimal one almost always second we show that if the balance condition is relaxed so that each block in the p way partition is bounded by 2n p where n is the number of vertices of the graph then a modified rb finds an approximately balanced p way partition whose cost is within an o log p factor of the cost of the optimal p way partition

a fast direct solver for elliptic partial differential equations on adaptively refined meshes we present a new class of fast direct solvers for elliptic partial differential equations on adaptively refined meshes these solvers rely on a combination of standard fast solvers for uniform grids and potential theory unlike standard iterative approaches they have a well defined operation count they also preserve the order of accuracy of the uniform grid solver despite the presence of coarse fine interfaces

a robust and efficient ilu that incorporates the growth of the inverse triangular factors in this paper we present a new ilu decomposition which is based on an existing sparse direct solver in contrast to many incomplete lu decompositions this ilu incorporates information about the inverse factors l 1 and u 1 which have direct influence on the dropping strategy we demonstrate in several large scale examples that this implementation constructs a robust preconditioner

log determinant relaxation for approximate inference in discrete markov random fields graphical models are well suited to capture the complex and non gaussian statistical dependencies that arise in many real world signals a fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes this paper proposes a novel method applicable to discrete valued markov random fields mrfs on arbitrary graphs for approximately solving this marginalization problem the foundation of our method is a reformulation of the marginalization problem as the solution of a low dimensional convex optimization problem over the marginal polytope exactly solving this problem for general graphs is intractable for binary markov random fields we describe how to relax it by using a gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope this combination leads to a log determinant maximization problem that can be solved efficiently by interior point methods thereby providing approximations to the exact marginals we show how a slightly weakened log determinant relaxation can be solved even more efficiently by a dual reformulation when applied to denoising problems in a coupled mixture of gaussian model defined on a binary mrf with cycles we find that the performance of this log determinant relaxation is comparable or superior to the widely used sum product algorithm over a range of experimental conditions

an efficient algorithm for hydrodynamical interaction of many deformable drops an efficient and accurate 3d algorithm for dynamical simulations of many deformable drops with strong hydrodynamical interactions at zero reynolds numbers is developed the drop to medium viscosity ratio and the bond number are arbitrary and the drops are subject to gravity with stationary triply periodic boundary conditions the algorithm at each step is a hybrid of boundary integral and economical multipole techniques with extensive use of rotational transformations and economical truncation of multipole expansions to optimize near field interactions a significant part of the code is the new best paraboloid spline technique for calculating the normal vectors and curvatures on drop surfaces which greatly improves the quality of long time simulations examples show the phenomenon of clustering in a concentrated sedimenting emulsion for 0 25 and 1 which leads to an increase in the average sedimentation velocity with time a high efficiency of the method is demonstrated with two orders of magnitude gains over the standard o n2n2 boundary integral technique for n 102 drops in a periodic cell with n 103 triangular boundary elements per drop so that typical long time dynamical simulations can be performed in a few days or weeks on a standard workstation as compared to the several years which would be required for the same simulations using standard boundary integral techniques the effects of drop triangulation and truncation of multipole expansions on dynamical simulations are assessed

efficient recursions for general factorisable models let n s valued categorical variables be jointly distributed according to a distribution known only up to an unknown normalising constant for an unnormalised joint likelihood expressible as a product of factors we give an algebraic recursion which can be used for computing the normalising constant and other summations a saving in computation is achieved when each factor contains a lagged subset of the components combining in the joint distribution with maximum computational efficiency as the subsets attain their minimum size if each subset contains at most r 1 of the n components in the joint distribution we term this a lag r model whose normalising constant can be computed using a forward recursion in o sr 1 computations as opposed to o sn for the direct computation we show how a lag r model represents a markov random field and allows a neighbourhood structure to be related to the unnormalised joint likelihood we illustrate the method by showing how the normalising constant of the ising or autologistic model can be computed

quantitative description and numerical simulation of random microstructures of composites and their effective elastic moduli abstract a digital image processing technique is used for measurement of centroid coordinates of fibers with forthcoming estimation of statistical parameters and functions describing the stochastic structure of a real fiber composite comparative statistical analysis of the real and numerically simulated structure are performed accompanying of known methods of the generation of random configurations by the random shaking procedure allows creating of the most homogenized and mixed structures that do not depend on the initial protocol of particle generation we consider a linearly elastic composite medium which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions the multiparticle effective field method see for references buryachenko appl mech rev 2001a 54 1 47 based on the theory of functions of random variables and green s functions is used for demonstration of the dependence of effective elastic moduli of fiber composites on the radial distribution functions estimated from the real experimental data as well as from the ensembles generated by the method proposed

improving the run time and quality of nested dissection ordering when performing sparse matrix factorization the ordering of matrix rows and columns has a dramatic impact on the factorization time this paper describes an approach to the reordering problem that produces significantly better orderings than prior methods the algorithm is a hybrid of nested dissection and minimum degree ordering and combines an assortment of different algorithmic advances new or improved algorithms are described for graph compression multilevel partitioning and separator improvement when these techniques are combined the resulting orderings average 39 better than minimum degree over a suite of test matrices while requiring roughly 2 7 times the run time of liu s multiple minimum degree

a pipelined givens method for computing the qr factorization of a sparse matrix this paper discusses an extension of the pipelined givens method for computing the qr factorization of a real m x n matrix to the case in which the matrix is sparse when restricted to one process the algorithm performs the same computation as the serial sparse givens algorithm of george and heath our implementation is compatible with the data structures used in sparspak the pipelined algorithm is well suited to parallel computers having globally shared memory and low overhead synchronization primitives such as the denelcor hep for which computational results are presented we point out certain synchronization problems that arise in the adaptation to the sparse setting and discuss the effect on parallel speedup of accessing a serial data file 15 references 7 figures

implicit multigrid algorithms for incompressible turbulent flows on unstructured grids an implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described the foundation of the code is a backward euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned generalized minimal residual gmres technique for the gmres calculations several techniques are investigated for forming the matrix vector product convergence acceleration is achieved through a multigrid scheme that uses nonnested coarse grids that are generated using a technique described in the present paper convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four element airfoil viscous results which are compared with experimental data include the turbulent flow over a naca 4412 airfoil a three element airfoil for which mach number effects are investigated and three dimensional flow over a wing with a partial span flap

a linear programming algorithm to test for jamming in hard sphere packings jamming in hard particle packings has been the subject of considerable interest in recent years in a paper by torquato and stillinger j phys chem b 105 2001 a classification scheme of jammed packings into hierarchical categories of locally collectively and strictly jammed configurations has been proposed they suggest that these jamming categories can be tested using numerical algorithms that analyze an equivalent contact network of the packing under applied displacements but leave the design of such algorithms as a future task in this work we present a rigorous and practical algorithm to assess whether an ideal hard sphere packing in two or three dimensions is jammed according to the aforementioned categories the algorithm is based on linear programming and is applicable to regular as well as random packings of finite size with hard wall and periodic boundary conditions if the packing is not jammed the algorithm yields representative multi particle unjamming motions furthermore we extend the jamming categories and the testing algorithm to packings with significant interparticle gaps we describe in detail two variants of the proposed randomized linear programming approach to test for jamming in hard sphere packings the first algorithm treats ideal packings in which particles form perfect contacts another algorithm treats the case of jamming in packings with significant interparticle gaps this extended algorithm allows one to explore more fully the nature of the feasible particle displacements we have implemented the algorithms and applied them to ordered as well as random packings of circular disks and spheres with periodic boundary conditions some representative results for large disordered disk and sphere packings are given but more robust and efficient implementations as well as further applications e g non spherical particles are anticipated for the future

decomposed block cholesky factorization in the karmarkar algorithm solving a class of super large lp problems abstract with the introduction of the karmarkar algorithm hope for optimally solving super large lp problems often encountered in the communication industry had risen high however a general code of the original karmarkar algorithm takes a long computation time to solve such problems employing proper factorization schemes substantially reduces the computation time of the karmarkar algorithm for solving super large linear programming lp problems in this paper a modified version of the karmarkar algorithm is described and the complexities involved in applying the algorithm to a class of super large lp problems are identified a factorization scheme decomposed block cholesky factorization is proposed in order to reduce the complexities of a matrix inversion operation that occurs within every iteration of the karmarkar algorithm the factorization scheme capitalizes on the generally well behaved block structure of the constraint matrix of the super large lp problems the computation time required for solving a class of super large lp problems using this scheme is estimated to be reduced by a factor of two orders of magnitude the proposed factorization scheme is even more promising in solving the super large lp problems on parallel processors using the karmarkar algorithm

a survey of the advances in the exploitation of the sparsity in the solution of large problems abstract the numerical treatment of many mathematical models which arise for example in physics chemistry biology or economics leads very often to huge algebraic problems so that it is difficult both with regard to the storage needed and with regard to the computing time spent to handle these problems even on the big modern computers however the matrices that occur in the algebraic problems are fortunately sparse many of their elements are equal to zero the exploitation of the sparsity leads to savings in both storage and computer time so that problems which can not be handled numerically when the zero elements are stored in the computer memory and when the arithmetic operations involving zero elements are performed become tractable if the sparsity is exploited in a proper way there are two basic groups of storage schemes for exploiting the sparsity if a scheme of the first group is in use then the non zero elements have permanent locations in the computer memory during the whole computational process therefore the schemes of the first group are called static a non zero element may be moved from one location to another when a scheme from the second group is applied such schemes are called dynamic the advantages and the limitations of the schemes from these two groups are discussed the advances achieved after 1980 in the efforts to improve the performance of the schemes belonging to both groups are given in a systematic way some questions that are still open are briefly discussed the advances achieved in some other stages in the exploitation of the sparsity which are not directly connected with the storage schemes used are outlined in the last section

finite element network approximation of conductivity in particle composites a new finite element method computes conductivity in some unstructured particle reinforced composite material the 2 phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed the mathematical model is a poisson type problem with discontinuous coefficients the discontinuities are huge in contrast and quantity the proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure important geometric parameters such as the volume fraction are preserved exactly the computational complexity of the method is almost proportional to the number of inclusions this is minimal in the sense that the representation of the underlying geometry via the positions and radii of the inclusions is of the same complexity the discretization error is proportional to the distance of neighboring inclusions and independent of the conductivity contrast in the medium

why finite element discretizations can be factored by triangular hierarchical matrices although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second order elliptic partial differential operators is far from being optimal these methods are often preferred over modern iterative methods this is mainly due to their robustness in this article it is shown that an approximate lu decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical lu decomposition low precision approximations may be used for preconditioning iterative solvers as a byproduct we prove that schur complements of stiffness matrices can be approximated with almost linear complexity

homotopy nilpotency in p regular loop spaces we consider the problem how far from being homotopy commutative is a loop space having the homotopy type of the p completion of a product of finite numbers of spheres we determine the homotopy nilpotency of those loop spaces as an answer to this problem

the factorability of symmetric matrices and some implications for statistical linear models abstract general conditions where a symmetric matrix is factorable by cholesky decomposition are described while numerical stability is a remaining issue whenever the cholesky decomposition is used to factor indefinite matrices the existence of such factors is demonstrated for matrix structures that are commonly found in statistics kalman filtering for example is rediscovered in the cholesky decomposition of an indefinite matrix moreover the cholesky decomposition uniquely defines the likelihood function in linear statistical models and this includes situations when the variance matrix is singular or when the cholesky decomposition does not run to completion alternative methods of likelihood evaluation which may involve for example the bunch parlett factorization are available only when the cholesky decomposition exists suggestions are made for computing an adaptive precision cholesky decomposition when numerical stability is an issue

refining an approximate inverse direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades the advances in graph theoretic algorithms have not received enough attention from the iterative methods community for example we demonstrate how symbolic factorization algorithms from direct methods can accelerate the computation of a factored approximate inverse preconditioner for very sparse preconditioners however a reformulation of the algorithm with outer products can exploit even more zeros to good advantage we also explore the possibilities of improving cache efficiency in the application of the preconditioner through reorderings the article ends by proposing a block version of the algorithm for further gains in efficiency and robustness

on the realization of invariant subgroups of _ let p be a prime and t x x a self map let a be a multiplicatively closed subset of the algebraic closure of fp denote by vta the set of characteristic values of t t x fp lying in a it is proved that under certain conditions vta is realizable by a pair x t there exist a space x maps t x x and f x x so that fottof 7r f is modp injective and im v f fp vta this theorem yields among others examples of spaces whose mod p cohomology rings are polynomial algebras introduction the technique of using self maps t x x to obtain mod p splittings retracts and realizable subalgebras of the mod p cohomology of a given space has been well exploited by now freyd used idempotents to split spectra nishida used the 4x maps defined on bu n by sullivan to obtain a mod p splitting of su n wilkerson produced mod p retracts of h and ho spaces and cooke and smith splitted co h spaces all using self maps self maps were used to obtain geometric realizations of subalgebras of the mod p cohomology of spaces e g clark and ewing cooke stasheff and zabrodsky j the two main methods used can be described briefly as follows the direct limit one constructs the infinite telescope of t x x i e tel t x x i x n n the set of natural numbers where is the equivalence relation generated by x 1 n t x o n 1 t n t n if hm t m is an idempotent i e hm t2 m hm t m then h tel t m im h t m and one obtains a realization of a submodule of h x m or h x m the orbit space if g is a finite group acting freely on a topological space x the orbit space x g has the property h x g z mz hg x z mz x e h x z mz g x x for everyg e g provided m is prime to the order of g the two methods do not have obvious eckmann hilton duals a third method described in zabrodskyll does have such a dual and this dual construction is the main subject of this paper because zabrodsky 1 has not been published yet we received by the editors october 2 1979 and in revised form august 12 1980 and december 29 1981 1980 mathematics subject classification primary 55q52 secondary 55s45 partially supported by nsf grant no mcs77 01593 1984 american mathematical society 0002 9947 84 1 00 25 per page

optimal placements of flexible objects i analytical results for the unbounded case the authors consider optimal placements of two dimensional flexible elastic deformable objects the objects are discs of equal size placed within a rigid boundary the paper is divided into two parts in the first part analytical results for three types of regular periodic arrangements the hexagonal square and triangular placements are presented the regular arrangements are analyzed for rectangular boundaries and radii of discs that are small compared to the area of the placement region because in this case the influence of boundary conditions can be neglected this situation is called the unbounded case they show that for the unbounded case among the three regular placements the type of hexagonal arrangements provides the largest number of placed units for the same deformation depth furthermore it can be proved that these regular placements are not too far from the truly optimal arrangements for example hexagonal placements differ at most by the factor of 1 1 from the largest possible number of generally shaped units in arbitrary arrangements these analytical results are used as guidances for testing stochastic algorithms optimizing placements of flexible objects in the second part mainly two problems are considered the underlying physical model and a simulated annealing algorithm maximizing the number of flexible discs in equilibrium placements along with the physical model an approximate formula is derived reflecting the deformation force relationship for a large range of deformations

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

lower eigenvalue bounds for singular pencils of matrices abstract beauwens procedure for obtaining lower eigenvalue bounds for regular pencils of matrices a b is simplified and extended to the singular case the theory is then compared through a particular perturbed modified incomplete factorization with notay s generalization of another approach initiated by gustafsson and developed by axelsson and barker and by wilmet

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

block diagonalization analysis of symmetric plates abstract this paper presents a block diagonalization method to solve stiffness equations of isotropic symmetric plates by means of a suitable local coordinate transformation chosen based on group theory the stiffness matrix is decomposed into a block diagonal form the stiffness equation in the local coordinate is solved block by block thus realizing numerical efficiency and greatly reducing the requisite amount of computer memory the efficiency has been further upgraded with the aid of the concept of augmented orbit this method is applied to a square isotropic plate subject to asymmetric loads to show its usefulness

three level bddc in two dimensions balancing domain decomposition by constraints bddc methods are non overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems they are similar to the balancing neumann neumann algorithm however in bddc methods a small number of continuity constraints are enforced across the interface and these constraints form a new coarse global component an important advantage of using such constraints is that the schur complements that arise in the computation will all be strictly positive definite the matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation however this problem can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately in two dimensional space while still maintaining a good convergence rate estimates of the condition numbers are provided for the two three level bddc methods and numerical experiments are also discussed copyright 2006 john wiley sons ltd

inverse free recursive multiresolution algorithms for a data approximation problem we present inverse free recursive multiresolution algorithms for data approximation problems based on energy functionals minimization during the multiresolution process a linear system needs to be solved at each different resolution level which can be solved with direct or iterative methods numerical results are reported using the sparse cholesky factorization for two applications one concerning the localization of regions in which the energy of a given surface is mostly concentrated and another one regarding noise reduction of a given dataset in addition for large scale data approximation problems that require a very fine resolution we discuss the use of the preconditioned conjugate gradient pcg iterative method coupled with a specialized monolithic preconditioner for which one preconditioner is built for the highest resolution level and then the corresponding blocks of that preconditioner are used as preconditioners for the forthcoming lower levels

on large scale generalized inverses in solving two by two block linear systems abstract the goal is to analyze a role of generalized inverses in the solution of two by two block linear systems by means of an algorithm combining the schur complement reduction with the null space method the efficient procedures for computing actions of generalized inversions to rectangular large scale matrices are presented with a special attention to the moore penrose inverse a regularization technique for matrices with the known null space basis is introduced that enables to get generalized inverses without necessity to recognize zero pivots the theoretical results are confirmed by numerical examples

an algorithm to compute a sparse basis of the null space leta be a realm n matrix with full row rankm in many algorithms in engineering and science such as the force method in structural analysis the dual variable method for the navier stokes equations or more generally null space methods in quadratic programming it is necessary to compute a basis matrixb for the null space ofa hereb isn r r n m of rankr withab 0 in many instancesa is large and sparse and often banded the purpose of this paper is to describe and test a variation of a method originally suggested by topcu and called the turnback algorithm for computing a banded basis matrixb two implementations of the algorithm are given one using gaussian elimination and the other using orthogonal factorization by givens rotations the fortran software was executed on an ibm 3081 computer with an fps 164 attached array processor at the triangle universities computing center and on a cyber 205 vector computer test results on a variety of structural analysis problems including two and three dimensional frames plane stress plate bending and mixed finite element problems are discussed these results indicate that both implementations of the algorithm yielded a well conditioned banded basis matrixb whena is well conditioned however the orthogonal implementation yielded a better conditionedb for large illconditioned problems

packing hyperspheres in high dimensional euclidean spaces we present a study of disordered jammed hard sphere packings in four five and six dimensional euclidean spaces using a collision driven packing generation algorithm we obtain the first estimates for the packing fractions of the maximally random jammed mrj states for space dimensions d 4 5 and 6t o be mrj0 46 0 31 and 0 20 respectively to a good approximation the mrj density obeys the scaling form mrj c1 2 d c2d 2 d where c1 2 72 and c2 2 56 which appears to be consistent with the highdimensional asymptotic limit albeit with different coefficients calculations of the pair correlation function g2r and structure factor sk for these states show that short range ordering appreciably decreases with increasing dimension consistent with a recently proposed decorrelation principle which among other things states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d as in three dimensions where mrj0 64 the packings show no signs of crystallization are isostatic and have a power law divergence in g2r at contact with power law exponent 0 4 across dimensions the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2r has its first minimum additionally we obtain estimates for the freezing and melting packing fractions for the equilibrium hard sphere fluid solid transition f0 32 and m0 39 respectively for d 4 and f0 20 and m0 25 respectively for d 5 although our results indicate the stable phase at high density is a crystalline solid nucleation appears to be strongly suppressed with increasing dimension

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

distributing the kalman filter for large scale systems this paper presents a distributed kalman filter to estimate the state of a sparsely connected large scale n dimensional dynamical system monitored by a network of n sensors local kalman filters are implemented on nl dimensional subsystems nl lt n obtained by spatially decomposing the large scale system the distributed kalman filter is optimal under an lth order gauss markov approximation to the centralized filter we quantify the information loss due to this lth order approximation by the divergence which decreases as l increases the order of the approximation l leads to a bound on the dimension of the subsystems hence providing a criterion for subsystem selection the approximated centralized riccati and lyapunov equations are computed iteratively with only local communication and low order computation by a distributed iterate collapse inversion dici algorithm we fuse the observations that are common among the local kalman filters using bipartite fusion graphs and consensus averaging algorithms the proposed algorithm achieves full distribution of the kalman filter nowhere in the network storage communication or computation of n dimensional vectors and matrices is required only nl lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors in other words knowledge of the state is itself distributed

multiply rooted multiscale models for large scale estimation divide and conquer or multiscale techniques have become popular for solving large statistical estimation problems the methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems each more straightforward than the original however this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound leading to problems of computational complexity and numerical stability in this paper we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided and the problem is instead addressed piece by piece the approach possesses promising attributes it is not a local method the estimate at every point is based on all measurements it is numerically stable for problems of arbitrary size and the approach retains the benefits of the multiscale framework on which it is based a broad class of statistical models a stochastic realization theory an algorithm to calculate statistical likelihoods and the ability to fuse local and nonlocal measurements

lu decomposition and numerical structure for solving large sparse nonsymmetric linear systems abstract in this work the solution of a large sparse linear system of equations with an arbitrary sparsity pattern is obtained by using lu decomposition method as well as numerical structure approach the lu decomposition method is based on doolittle s method while the numerical structure approach is based on cramer s rule the numerical structure approach produces direct solution without facing fill in problems as encountered in lu decomposition in order to reduce the fill ins in the decomposition the powers of a boolean matrix obtained from the coefficient matrix a are taken so that the fill ins in the structure of a can be known in advance the position of fill ins in a are thus determined in the best choice manner that is it is very effective and memory wise cheap we also outline a method by using numerical structure with reduced computation efforts finally experiments are performed on eight examples to compare the efficiency of the proposed methods the results obtained are reported in a table it is found that the lu decomposition method is much better than numerical structure the usefulness of numerical structure approach is also discussed

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

instantaneous shrinking of the support of energy solutions the cauchy problem for a class of nonlinear diffusion reaction equations is studied the equations may be classified as being of degenerate parabolic type it is shown that under certain conditions solutions of the problem exhibit instantaneous shrinking this is to say at any positive time the spatial support of the solution is bounded above although the support of the initial data function is not we also provide some estimates of the behavior of the free boundary

algebraic two level preconditioners for the schur complement method the solution of elliptic problems is challenging on parallel distributed memory computers since their green s functions are global to address this issue we present a set of preconditioners for the schur complement domain decomposition method they implement a global coupling mechanism through coarse space components similar to the one proposed in bramble pasciak and shatz math comp 47 1986 pp 103 134 the definition of the coarse space components is algebraic they are defined using the mesh partitioning information and simple interpolation operators these preconditioners are implemented on distributed memory computers without introducing any new global synchronization in the preconditioned conjugate gradient iteration the numerical and parallel scalability of those preconditioners are illustrated on two dimensional model examples that have anisotropy and or discontinuity phenomena

a second order accurate immersed boundary lattice boltzmann method for particle laden flows abstract a new immersed boundary lattice boltzmann method ib lbm is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles the immersed boundary method ibm recently developed by breugem 2012 19 is adopted in the present method development including the retraction technique the multi direct forcing method and the direct account of the inertia of the fluid contained within the particles the present ib lbm is however formulated with further improvement with the implementation of the high order runge kutta schemes in the coupled fluid particle interaction the major challenge to implement high order runge kutta schemes in the lbm is that the flow information such as density and velocity cannot be directly obtained at a fractional time step from the lbm since the lbm only provides the flow information at an integer time step this challenge can be however overcome as given in the present ib lbm by extrapolating the flow field around particles from the known flow field at the previous integer time step the newly calculated fluid particle interactions from the previous fractional time steps of the current integer time step are also accounted for in the extrapolation the ib lbm with high order runge kutta schemes developed in this study is validated by several benchmark applications it is demonstrated for the first time that the ib lbm has the capacity to resolve the translational and rotational motion of particles with the second order accuracy the optimal retraction distances for spheres and tubes that help the method achieve the second order accuracy are found to be around 0 30 and 0 47 times of the lattice spacing respectively simulations of the stokes flow through a simple cubic lattice of rotational spheres indicate that the lift force produced by the magnus effect can be very significant in view of the magnitude of the drag force when the practical rotating speed of the spheres is encountered this finding may lead to more comprehensive studies of the effect of the particle rotation on fluid solid drag laws it is also demonstrated that when the third order or the fourth order runge kutta scheme is used the numerical stability of the present ib lbm is better than that of all methods in the literature including the previous ib lbms and also the methods with the combination of the ibm and the traditional incompressible navier stokes solver

algorithm for random close packing of spheres with periodic boundary conditions the isotropic algorithm is constructed for random close packing of equisized spheres with triply periodic boundary conditions all previously published packing methods with periodic boundaries were kinetics determined i e they contained a densification rate as an arbitrary parameter in contrast the present algorithm is kinetic independent and demonstrates an unambiguous convergence to the experimental results to suppress crystallization the main principles of our algorithm are 1 to form a contact network at an early stage and 2 retain contacts throughout the densification as far as possible the particles are allowed to swell by the numerical solution of the differential equations of densification the rhs of these equations is calculated efficiently from a linear system by a combination of conjugate gradient iterations and exact sparse matrix technology when an excessive contact occurs and one of the existing bonds should be broken to continue the densification an efficient criterion based on multidimensional simplex geometry is used for searching the separating bond the algorithm has a well defined termination point resulting in a perfect contact network with the average coordination number six for particle number n 1 and a system of normal reactions between the spheres maintaining the structure these forces are the counterpart of the algorithm and can be used to calculate small elastic particle deformations in a granular medium extensive calculations are presented for 50 n 5 400 and demonstrate very good agreement with the experimental packing density about 0 637 and structure

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

a parallel multithreaded sparse triangular linear system solver abstract we propose a parallel sparse triangular linear system solver based on the spike algorithm sparse triangular systems are required to be solved in many applications often they are a bottleneck due to their inherently sequential nature furthermore typically many successive systems with the same coefficient matrix and with different right hand side vectors are required to be solved the proposed solver decouples the problem at the cost of extra arithmetic operations as in the banded case compared to the banded case there are extra savings due to the sparsity of the triangular coefficient matrix we show the parallel performance of the proposed solver against the state of the art parallel sparse triangular solver in intel s math kernel library mkl on a multicore architecture we also show the effect of various sparse matrix reordering schemes numerical results show that the proposed solver outperforms mkl s solver in 80 of cases by a factor of 2 47 on average

modified block approximate factorization strategies two variants of modified incomplete block matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations pdes even in the case of quasi singular problems

stochastic preconditioning for diagonally dominant matrices this paper presents a new stochastic preconditioning approach for large sparse matrices for the class of matrices that are rowwise and columnwise irreducibly diagonally dominant we prove that an incomplete rm ldl t factorization in a symmetric case or an incomplete ldu factorization in an asymmetric case can be obtained from random walks and used as a preconditioner it is argued that our factor matrices have better quality i e better accuracy size trade offs than preconditioners produced by existing incomplete factorization methods therefore a resulting preconditioned krylov subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance the advantage increases for larger and denser matrices these claims are verified by numerical tests and we provide techniques that can potentially extend the theory to non diagonally dominant matrices

cauchy problem for a system of dynamics of biological groups abstract in this paper we establish the existence and uniqueness of weak solutions for a class of degenerate diffusion systems which arise from modelling interacting evolution of two biological groups especially some interesting properties such as non extinction coexistence expansion of the living spaces etc are proved

robust stability analysis of sparsely interconnected uncertain systems in this paper we consider robust stability analysis of large scale sparsely interconnected uncertain systems by modeling the interconnections among the subsystems with integral quadratic constraints we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities we also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness the sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem

effective preconditioning through minimum degree ordering interleaved with incomplete factorization in this paper we study a kind of effective preconditioning technique which interleaves the incomplete cholesky ic factorization with an approximate minimum degree ordering an ic factorization algorithm derived from ikj version gaussian elimination is proposed and some details on implementation are presented then we discuss the ways to compute the degrees of the unnumbered nodes exactly and approximately using the concept of element absorbing when used in conjunction with conjugate gradient algorithm the new preconditioners usually lead to fast convergence the numerical experiments show that the interleaving of symbolic ordering and numerical ic factorization will generate better preconditioners than those generated by the ic factorization without ordering or with purely symbolic ordering ahead of the factorization