an iterative algorithm for computing mean first passage times of markov chains abstract mean first passage times are an essential ingredient in both the theory and the applications of markov chains in the literature they have been expressed in elegant closed form formulas these formulas involve explicit full matrix inversions and if computed directly may incur numerical instability in this paper we present a new iterative algorithm for computing mean first passage times in a manner that does not rely on explicit full matrix inversions results regarding the convergence behavior of this algorithm are also developed

on sparse block factorization iterative methods a synthetic formalism is proposed for the description of sparse block factorization iterative methods it is used to develop existence convergence and comparison theorems and to compare block factorization schemes against point factorization schemes

scaled pivots and scaled partial pivoting strategies scaled pivots for gaussian elimination of an n n matrix are introduced they are used to obtain bounds for the skeel condition number of the resulting upper triangular matrix and for a growth factor which has been introduced by amodio and mazzia bit 39 1999 pp 385 402 a bound of this growth factor for row scaled partial pivoting strategies is also included it is proved that the skeel condition number of an n n upper triangular matrix which is strictly diagonally dominant by rows is bounded above by a number which is independent of n it is also shown that the calculation of the n scaled pivots associated with a pivoting strategy presenting nice properties when applied to nonsingular n n m matrices adds mathcal o n elementary operations to the complete cost of this pivoting strategy

very weak solutions of degenerate parabolic systems with non standard p x t growth abstract we study higher integrability of very weak solutions to certain degenerate parabolic systems whose model is the parabolic p x t laplacian system t u div d u p x t 2 d u div f p x t 2 f under natural assumptions on the exponent function p 0 t 2 we prove that any very weak solution u 0 t r n with d u p 1 e l 1 belongs to the natural energy space i e d u p l loc 1 provided e 0 is small enough

lu decompositions of generalized diagonally dominant matrices using the simple vehicle ofm matrices the existence and stability oflu decompositions of matricesa which can be scaled to diagonally dominant possibly singular matrices are investigated bounds on the growth factor for gaussian elimination ona are derived motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsax 0 arising in markov queuing networks input output models in economics and compartmental systems wherea or a is an irreducible singularm matrix this paper extends earlier work by funderlic and plemmons and by varga and cai

block lu factorizations of m matrices it is well known that any nonsingular m matrix admits an lu factorization into m matrices with l and u lower and upper triangular respectively and any singular m matrix is permutation similar to an m matrix which admits an lu factorization into m matrices varga and cai establish necessary and sufficient conditions for a singular m matrix without permutation to allow an lu factorization with l nonsingular we generalize these results in two directions first we find necessary and sufficient conditions for the existence of an lu factorization of a singular m matrix where l and u are both permitted to be singular second we establish the minimal block structure that a block lu factorization of a singular m matrix can have when l and u are m 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

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

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

compressed suffix arrays and suffix trees with applications to text indexing and string matching the proliferation of online text such as found on the world wide web and in online databases motivates the need for space efficient text indexing methods that support fast string searching we model this scenario as follows consider a text t consisting of n symbols drawn from a fixed alphabet sigma the text t can be represented in n lg sigma bits by encoding each symbol with lg sigma bits the goal is to support fast online queries for searching any string pattern p of m symbols with t being fully scanned only once namely when the index is created at preprocessing time the text indexing schemes published in the literature are greedy in terms of space usage they require omega n lg n additional bits of space in the worst case for example in the standard unit cost ram suffix trees and suffix arrays need omega n memory words each of omega lg n bits these indexes are larger than the text itself by a multiplicative factor of omega smash lg sigma n which is significant when sigma is of constant size such as in textsc ascii or textsc unicode on the other hand these indexes support fast searching either in o m lg sigma time or in o m lg n time plus an output sensitive cost o mathit occ for listing the mathit occ pattern occurrences we present a new text index that is based upon compressed representations of suffix arrays and suffix trees it achieves a fast smash o m lg sigma n lg sigma epsilon n search time in the worst case for any constant 0 epsilon leq 1 using at most smash bigl epsilon 1 o 1 bigr n lg sigma bits of storage our result thus presents for the first time an efficient index whose size is provably linear in the size of the text in the worst case and for many scenarios the space is actually sublinear in practice as a concrete example the compressed suffix array for a typical 100 mb textsc ascii file can require 30 40 mb or less while the raw suffix array requires 500 mb our theoretical bounds improve emph both time and space of previous indexing schemes listing the pattern occurrences introduces a sublogarithmic slowdown factor in the output sensitive cost giving o mathit occ smash lg sigma epsilon n time as a result when the patterns are sufficiently long we can use auxiliary data structures in o n lg sigma bits to obtain a total search bound of o m lg sigma n mathit occ time which is optimal

a general incomplete block matrix factorization method abstract a general method for incomplete factorization of m matrices in block matrix form is presented the method generalizes previous methods for pointwise incomplete factorization methods of varga 1960 of wo nicki 1973 and of meijerink and van der vorst 1977 as well as a method for tridiagonal block matrices also we present a new and more general result on the existence of so called modified incomplete factorizations which satisfy a generalized rowsum criterion

the l dominance concept and its applications to multivariable system designs a new type of dominance measure based on triangular factorizations is developed and is named the l dominance measure this new measure is tighter than any existing dominance measures and in particular is tighter than those based on generalized dominance and m matrix theory the l dominance condition on matrices with non positive off diagonal elements is proved to be equivalent to the m matrix condition but in contrast to the m matrix condition the testing of the l dominance condition is easier and can be carried out in sequential order furthermore these special properties enable us to construct a synthesis based design procedure superior to those based on generalized dominance or m matrix theory the last two methods are generally regarded as post mortem analytical tools only

block incomplete factorization preconditioners for a symmetric block tridiagonal m matrix we propose new block incomplete factorization preconditioners for a symmetric block tridiagonal m matrix which can be computed in parallel and then theoretical properties for these block preconditioners are studied spectral properties of the transformed coefficient matrices with the block incomplete factorization preconditioners are also examined to see the convergence rate of the preconditioned cg pcg method lastly numerical results of the pcg using the block incomplete factorization preconditioners are compared with those of the pcg using a standard incomplete factorization preconditioner to see how effective the block incomplete factorization preconditioners are

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

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

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

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

entrywise perturbation theory and error analysis for markov chains grassmann taksar and heyman introduced a variant of gaussian climination for computing the steady state vector of a markov chain in this paper we prove that their algorithm is stable and that the problem itself is well conditioned in the sense of entrywise relative error thus the algorithm computes each entry of the steady state vector with low relative error even the small steady state probabilities are computed accurately the key to our analysis is to focus on entrywise relative error in both the data and the computed solution rather than making the standard assessments of error based on norms our conclusions do not depend on any condition numbers for the problem

the accuracy of solutions to triangular systems triangular systems play a fundamental role in matrix computations it has been prominently stated in the literature but is perhaps not widely appreciated that solutions to triangular systems are usually computed to high accuracy higher than the traditional condition numbers for linear systems suggest this phenomenon is investigated by use of condition numbers appropriate to the componentwise backward error analysis of triangular systems results of wilkinson are unified and extended among the conclusions are that the conditioning of a triangular system depends on the right hand side as well as the coefficient matrix that use of pivoting in lu qr and cholesky factorisations can greatly improve the conditioning of a resulting triangular system and that a triangular matrix may be much more or less ill conditioned than its transpose

proximity in group inverses of m matrices and inverses of diagonally dominant m matrices in this paper we connect generalize and broaden properties of matrices related to i the triangular inequality for mean first passage times in finite homogeneous ergodic markov chains ii the triangle inequality for proximities in laplacian matrices of undirected weighted graphs and iii the metzler property of the column entrywise diagonal dominance of inverses of diagonally dominant m matrices

a direct projection method for markov chains a direct method based on oblique projections is adapted to compute the stationary distribution vector of a finite markov chain the algorithm can also be used to compute the group inverse of the corresponding generator matrix it is shown how to update the stationary vector and other quantities of interest when one row of the transition probability matrix is modified a gth like variant that appears to compute the stationary probabilities to high relative accuracy is developed

very weak solutions of subquadratic parabolic systems with non standard p x t growth abstract the aim of this paper is to establish a higher integrability result for very weak solutions of certain parabolic systems whose model is the parabolic p x t laplacian system under assumptions on the exponent function p t 0 t 2 n n 2 2 it is shown that any very weak solution u t r n with d u p 1 e l 1 t belongs to the natural energy spaces i e d u p l loc 1 t provided e 0 is small enough this extends the main result of bogelein and li 2014 to the subquadratic case

space efficient algorithms for computing minimal shortest unique substrings abstract given a string t of length n a substring u t i j of t is called a shortest unique substring sus for an interval s t if a u occurs exactly once in t b u contains the interval s t i e i s t j and c every substring v of t with v u containing s t occurs at least twice in t given a query interval s t 1 n the interval sus problem is to output all the suss for the interval s t in this article we propose a 4 n o n bits data structure answering an interval sus query in output sensitive o o c c time where occ is the number of returned suss additionally we focus on the point sus problem which is the interval sus problem for s t here we propose a log 2 3 1 n o n bits data structure answering a point sus query in the same output sensitive time we also propose space efficient algorithms for computing the minimal unique substrings of t

some notes on z matrices abstract an inequality is obtained relating the eigenvalue of minimum modulus of an n0 matrix a to the eigenvalue of minimum modulus of a a11 the schur complement of a11 in a where a11 is an m matrix similar results are obtained for irreducible m matrices and irreducible nonsingular f0 matrices circulant n0 matrices of order three are characterized in terms of their spectrum matrices whose inverses are f0 matrices are shown to have nonpositive almost principal minors

interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents abstract we prove boundary higher integrability for the spatial gradient of very weak solutions of quasilinear parabolic equations of the form u t div a x t u 0 on t t u 0 on t t where the non linear structure a x t u is modeled after the variable exponent p x t laplace operator given by u p x t 2 u to this end we prove that the gradients satisfy a reverse holder inequality near the boundary by constructing a suitable test function which is lipschitz continuous and preserves the boundary values in the interior case such a result was proved in verena bogelein and qifan li 2014 provided p x t p 2 holds and was then extended to the singular case 2 n n 2 p p x t p 2 in qifan li 2017 this restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case p 2 and p 2 in this paper we develop a new approach using which we are able to extend the results of verena bogelein and qifan li 2014 qifan li 2017 to the full range 2 n n 2 p p x t p and also obtain analogous results up to the boundary the main novelty of this paper is that we make use of a unified intrinsic scaling using which our methods are able to handle both the singular case and degenerate case simultaneously our techniques improve and simplify many aspects of the method of parabolic lipschitz truncation even in the constant exponent case studied extensively in existing literature to simplify the exposition we will only prove the higher integrability result near the boundary provided the domain satisfies a uniform measure density condition and are non perturbative in nature hence we make no regularity assumptions for the coefficients of the nonlinear operator our techniques are also applicable to higher order equations as well as systems

on matrix majorants and minorants with applications to differential equations abstract some tools of linear algebra are collected and developed for potential use in the analysis of stiff differential equations bounds for the triangular factors of a large matrix are given in terms of the triangular factors of an associated minorant matrix of lower order minorants are also used to produce estimates of solutions of systems of ordinary differential equations which may be sharper than those obtained by the use of logarithmic norms

exact convergence and divergence domains for the symmetric successive overrelaxation iterative ssor method applied to h matrices abstract in this paper exact convergence and divergence domains for the ssor iterative method as applied to the class of h matrices are obtained the theory of regular splittings and the recent results of varga niethammer and cai are used as tools in establishing these convergence and divergence domains

an iterative method for computing the approximate inverse of a square matrix and the moore penrose inverse of a non square matrix in this paper an iterative scheme is proposed to find the roots of a nonlinear equation it is shown that this iterative method has fourth order convergence in the neighborhood of the root based on this iterative scheme we propose the main contribution of this paper as a new high order computational algorithm for finding an approximate inverse of a square matrix the analytical discussions show that this algorithm has fourth order convergence as well next the iterative method will be extended by theoretical analysis to find the pseudo inverse also known as the moore penrose inverse of a singular or rectangular matrix numerical examples are also made on some practical problems to reveal the efficiency of the new algorithm for computing a robust approximate inverse of a real or complex matrix

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

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

suffix arrays a new method for on line string searches a new and conceptually simple data structure called a suffix array for on line string searches is introduced in this paper constructing and querying suffix arrays is reduced to a sort and search paradigm that employs novel algorithms the main advantage of suffix arrays over suffix trees is that in practice they use three to five times less space from a complexity standpoint suffix arrays permit on line string searches of the type is w a substring of a to be answered in time o p log n where p is the length of w and n is the length of a which is competitive with and in some cases slightly better than suffix trees the only drawback is that in those instances where the underlying alphabet is finite and small suffix trees can be constructed in o n time in the worst case versus o n log n time for suffix arrays however an augmented algorithm is given that regardless of the alphabet size constructs suffix arrays in o n expected time albeit with lesser space efficiency it is

inverse m matrix inequalities and generalized ultrametric matrices abstract we use weighted directed graphs to introduce a class of nonnegative matrices which under a simple condition are inverse m matrices we call our class the generalized ultrametic matrices since it contains the class of symmetric ultrametric matrices and some unsymmetric matrices we show that a generalized ultrametric matrix is the inverse of a row and column diagonally dominant m matrix if and only if it contains no zero row and no two of its rows are identical this theorem generalizes the known result that a symmetric strictly ultrametric matrix is the inverse of a strictly diagonally dominant m matrix we also present inequalities and conditions for equality among the entries of the inverse of a row diagonally dominant m matrix some of these inequalities and conditions for equality generalize results of stieltjes on inverses of symmetric diagonally dominant m matrices

on approximate factorization methods for block matrices suitable for vector and parallel processors abstract some existence results for methods based on the approximate factorization of block matrices are proven these methods are based on recursive computations of diagonal block matrices and the approximation of their inverses to preserve sparsity we also discuss a recently proposed 1 inverse free factorization method and present some numerical tests for it

lu decomposition of m matrices by elimination without pivoting abstract it is shown that if a or a is a singular m matrix satisfying the generalized diagonal dominance condition y t a 0 for some vector y 0 then a can be factored into a lu by a certain elimination algorithm where l is a lower triangular m matrix with unit diagonal and u is an upper triangular m matrix the existence of lu decomposition of symmetric permutations of a and for irreducible m matrices and symmetric m matrices follow as colollaries this work is motivated by applications to the solution of homogeneous systems of linear equations ax 0 where a or a is an m matrix these applications arise e g in the analysis of markov chains input output economic models and compartmental systems a converse of the theorem metioned above can be established by considering the reduced normal form of a

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

space time trade offs for finding shortest unique substrings and maximal unique matches abstract given a string x 1 n and a position k 1 n a shortest unique substring of x covering k denoted by s k is a substring x i j of x which satisfies the following conditions i i k j ii i is the only position where there is an occurrence of x i j and iii j i is minimized current best known algorithms for finding s k require n words of working space and o n time let be a given parameter we present the following new results given a k 1 n we can compute s k in o n 2 log n time using x and an additional o n words of working space for every k 1 n we can compute s k in o n 2 log n time using x and an additional o n words and 4 n o n bits of working space we present an o n log c 1 n time randomized algorithm that uses n log c n words in addition to that mentioned above where c 0 is an arbitrary constant in this case the reported string is unique and covers k but with probability at most n o 1 may not be the shortest by choosing 1 our results imply the first sub linear space in addition to the input string solution to these problems we also present the following two results an algorithm that finds s k for every k 1 n using o n log bits of working space in o n log n time where n is the number of distinct symbols in x a 4 n o n bit index that can report s k for any k in o 1 time as a consequence of our techniques we also obtain similar space and time tradeoffs for a related problem of finding maximal unique matches of two strings delcher et al 1999 14

stability of block lu factorization for block tridiagonal block h matrices by a block representation of lu factorization for a general matrix introduced by amodio and mazzia p amodio f mazzia a new approach to the backward error analysis in the lu factorization algorithm bit 39 1999 385 402 a block representation of block lu factorization for block tridiagonal block h matrices is obtained and some properties on the factors of the factorization are presented perturbation theory for the block lu factorization of block tridiagonal block h matrices is also considered then a rounding error analysis of the block lu factorization for block tridiagonal block h matrices is given and some bounds for the growth factor are proposed finally a numerical example is presented to illustrate our theoretical results

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

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

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

computable error bounds for aggregated markov chains abstract this paper describes a method for computing the steady state probability vector of a nearly completely decomposable markov chain the method is closely related to one proposed by simon and ando and developed by courtois however the method described here does not require the determination of a completely decomposable stochastic approximation to the transition matrix and hence it is applicable to matrices other than stochastic an error analysis of the procedure is given which results in effectively computable error bounds author

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 thelu factorization ofm matrices in this paper we give in theorem 1 a characterization based on graph theory of when anm matrixa admits anlu factorization intom matrices wherel is a nonsingular lower triangularm matrix andu is an upper triangularm matrix this result generalizes earlier factorization results of fiedler and ptak 1962 and kuo 1977 as a consequence of theorem 1 we show in theorem 3 that the conditionx t a 0 t for somex 0 for anm matrixa is both necessary and sufficient forpap t to admit such anlu factorization for everyn n permutation matrixp this latter result extends recent work of funderlic and plemmons 1981 finally theorem 1 is extended in theorem 5 to give a characterization similarly based on graph theory of when anm matrixa admits anlu factorization intom matrices

forward error analysis of gaussian elimination part ii establishes a priori error and residual estimates for the solutions of linear algebraic systems whose matrices are totally nonnegative or signregular 2 orm matrices 3 in addition stability estimates and a priori error estimates for the computation of pivot elements and of the determinant are given particular attention is paid to stability constants and growth factors of band matrices

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

chebyshev type methods and preconditioning techniques recently a newton s iterative method is attracting more and more attention from various fields of science and engineering this method is generally quadratically convergent in this paper some chebyshev type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system ax b theoretic analysis and numerical experiments show that chebyshev s method is more effective than newton s one in the case of constructing approximate inverse preconditioners

breaking a time and space barrier in constructing full text indices suffix trees and suffix arrays are the most prominent full text indices and their construction algorithms are well studied in the literature the fastest algorithm runs in o n time while it requires o n log n bit working space where n denotes the length of the text on the other hand the most space efficient algorithm requires o n bit working space while it runs in o n log n time it was open whether these indices can be constructed in both o n log n time and o n log n bit working space this paper breaks the above time and space barrier under the unit cost word ram we give an algorithm for constructing the suffix array which takes o n time and o n bit working space for texts with constant size alphabets note that both the time and the space bounds are optimal for constructing the suffix tree our algorithm requires o n log epsilon n time and o n bit working space for any 0 epsilon 1 apart from that our algorithm can also be adopted to build other existing full text indices such as compressed suffix tree compressed suffix arrays and fm index we also study the general case where the size of the alphabet sigma is not constant our algorithm can construct a suffix array and a suffix tree using optimal o n log sigma bit working space while running in o n log log sigma time and o n log epsilon n log sigma time respectively these are the first algorithms that achieve o n log n time with optimal working space moreover for the special case where log sigma o log log n 1 epsilon we can speed up our suffix array construction algorithm to the optimal o n

componentwise analysis of direct factorization of real symmetric and hermitian matrices abstract we derive componentwise error bound for the factorization h gjg t where h is a real symmetric matrix g has full column rank and j is diagonal with 1 s on the diagonal we also derive a componentwise forward error bound that is we bound the difference between the exact and the computed factor g in the cases where such a bound is possible we extend these results to the hermitian case and to the well known bunch parlett factorization finally we prove bounds for the scaled condition of the matrix g and show that the factorization can have the rank revealing property

pivoting strategies leading to diagonal dominance by rows pivoting strategies for gaussian elimination leading to upper triangular matrices which are diagonally dominant by rows are studied forward error analysis of triangular systems whose coefficient matrices are diagonally dominant by rows is performed we also obtain small bounds of the backward errors for the pivoting strategies mentioned above our examples of matrices include h matrices and some generalizations of diagonally dominant matrices and scaled partial pivoting for the 1 norm is an example of these pivoting strategies in the case of an n times n m matrix a pivoting strategy of computational complexity o n 2 is proposed which satisfies all the results of the paper