complexity of solving linear systems in different models of computation the computational complexity of solving an n times n system of linear equations depends on whether the computational model is a sequential or parallel and b fixed precision or variable precision we survey known complexity results for each of the four cases and introduce a new algorithm for the parallel variable precision case that is based on newton s method if n 3 processors are available this algorithm has complexity asymptotically equivalent to one scalar multiplication independent of n if only n 2 processors are available the complexity is proportional to but still competitive with known alternatives

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

fast parallel solvers for elliptic problems in an earlier work 1 we have considered the use of t 2 preconditioners for the solution of dirichlet boundary value problems by means of the preconditioned conjugate gradients methods in this paper we propose the use of toeplitz matrices for the fast parallel solution of differential problems we propose some algorithms to solve preconditioned systems which are especially suitable in parallel environments we also present several numerical results to prove the effectiveness of our technique

transformations of matrix structures work again abstract matrices with the structures of toeplitz hankel vandermonde and cauchy types are omnipresent in modern computations in sciences engineering and signal and image processing these four matrix classes have distinct features but in our 1990 paper in mathematics of computation we showed that vandermonde and hankel multipliers can be applied to transform each class into the others and then we demonstrated how by using these transforms we can readily extend any successful matrix inversion algorithm from one of these classes to all the others the power of this approach was widely recognized later when novel numerically stable algorithms solved nonsingular toeplitz linear systems of equations in quadratic versus classical cubic arithmetic time based on transforming toeplitz into cauchy matrix structures more recent papers combined the same transformation with a link of the cauchy matrices to the hierarchical semiseparable matrix structure which is a specialization of matrix representations employed by the fast multipole method this produced numerically stable algorithms that approximated the solution of a nonsingular toeplitz linear system of equations in nearly linear arithmetic time we first revisit the successful method of structure transformation covering it comprehensively then we analyze the latter approximation algorithms for toeplitz linear systems and extend them to approximate multiplication of vandermonde and cauchy matrices by a vector and to approximate solution of vandermonde and cauchy linear systems of equations provided that they are nonsingular and well conditioned we decrease the arithmetic cost of the known numerical approximation algorithms for these tasks from quadratic to nearly linear and similarly for the computations with the matrices having structures that generalize the structures of vandermonde and cauchy matrices and for polynomial and rational evaluation and interpolation we also accelerate a little further the known numerical approximation algorithms for a nonsingular toeplitz or toeplitz like linear system by employing distinct transformations of matrix structures and we briefly comment on some natural research challenges particularly some promising applications of our techniques to high precision computations

fractional hadamard powers of positive semidefinite matrices we consider the class sn of all real positive semidefinite n n matrices and the subclass sn of all a sn with non negative entries for a positive non integer number and some a sn when will the fractional hadamard power a again belong to sn it is known that for a specific this holds for all a sn if and only if n 2 now let a sn be of the form a t v where t sn has rank 1 and v sn has rank p 1 if the hadamard quotient of t and v is hadamard independent in general position and v has sufficently small entries then a complete answer is given depending on n p and special attention is given to the case that p 1

algorithmique et calculs de complexite pour un solveur de type dissections emboitees we consider the nested dissection method based on separator theorems introduced by gilbert tarjan and roman used for solving large sparse systems of linear equations more precisely we study a block storage scheme such as proposed by george for regular square grids and we prove the following results first for families of graphs of bounded degree withn separator theorem 1 2 1 the overhead storage of the block data structure for the factored matrix is linear in system dimension on the other hand by adding a non restrictive assumption on the separation this structure can be constructed by a block symbolic factorization which runs in time linear in matrix dimension numerical experiments illustrating these theoretical results are provided on considere la methode de dissections emboitees basee sur des theoremes de separation introduite par gilbert tarjan et roman utilisee pour la resolution par elimination de gauss de grands systemes lineaires creux plus precisemment on etudie une structure de donnees par blocs similaire a celle proposee par george dans le cadre des graphes en grille et on demontre les proprietes suivantes d une part pour des familles de graphes a degre borne admettant unn theoreme de separation 1 2 1 le stockage secondaire de la structure par blocs contenant la matrice factorisee est lineaire par rapport a la taille du systeme d autre part en rajoutant une hypothese non restrictive sur la maniere d effectuer la separation la structure peut etre construite en temps lineaire par une factorisation logique par blocs des exemples numeriques illustrent ces resultats theoriques

new resultant inequalities and complex polynomial factorization the author deduces some new probabilistic estimates on the distances between the zeros of a polynomial p x by using some properties of the discriminant of p x and applies these estimates to improve the fastest deterministic algorithm for approximating polynomial factorization over the complex field namely given a natural n positive epsilon such that log 1 epsilon o n log n and the complex coefficients of a polynomial p x sum n i 0 p ix i such that p n ne 0 sum i p i le 1 a factorization of p x within the error norm epsilon is computed as a product of factors of degrees at most n 2 by using o log 2n time and n 3 processors under the pram arithmetic model of parallel computing or by using o n 2 log 2 n arithmetic operations the algorithm is randomized of las vegas type allowing a failure with a probability at most delta for any positive delta 1 such that log 1 delta o log n except for a narrow class of polynomials p x these results can be also obtained for epsilon such that log 1 epsilon o n 2 log n

successive matrix squaring algorithm for computing the drazin inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 this paper derives a successive matrix squaring sms algorithm to approximate the drazin inverse which can be expressed in the form of successive squaring of a composite matrix t given an n by n matrix a the study shows that the drazin inverse of a can be computed in parallel time ranging from o logn to o log 2n provided that there are enough processors to support matrix multiplication in time o logn the sms algorithm is generalized to higher order schemes where the composite matrix is repeatedly raised to an integer power l 2 this form of expression leads to a simplified notation compared to that of earlier methods we argue that there is no obvious advantage in choosing l other than 2 our derived error bound for the approximation of a d is new

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

hypergraph partitioning based fill reducing ordering for symmetric matrices a typical first step of a direct solver for the linear system mx b is reordering of the symmetric matrix m to improve execution time and space requirements of the solution process in this work we propose a novel nested dissection based ordering approach that utilizes hypergraph partitioning our approach is based on the formulation of graph partitioning by vertex separator gpvs problem as a hypergraph partitioning problem this new formulation is immune to deficiency of gpvs in a multilevel framework and hence enables better orderings in matrix terms our method relies on the existence of a structural factorization of the input m matrix in the form of m aa t or m ad 2a t we show that the partitioning of the row net hypergraph representation of the rectangular matrix a induces a gpvs of the standard graph representation of matrix m in the absence of such factorization we also propose simple yet effective structural factorization techniques that are based on finding an edge clique cover of the standard graph representation of matrix m and hence applicable to any arbitrary symmetric matrix m our experimental evaluation has shown that the proposed method achieves better ordering in comparison to state of the art graph based ordering tools even for symmetric matrices where structural m aa t factorization is not provided as an input for matrices coming from linear programming problems our method enables even faster and better orderings

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

an involutory pascal matrix abstract an involutory upper triangular pascal matrix u n is investigated eigenvectors of u n and of u n t are considered a characterization of u n is obtained and it is shown how the results can be extended to matrices over a commutative ring with unity

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

algebraic complexity of computing polynomial zeros abstract using the power sum techniques of turan we evaluate all the complex zeros of an n th degree univariate polynomial with relative errors using o n 2 log n n log n log 1 arithmetic operations o n log n log 1 operations suffice to approximate with relative error a single complex zero using turan s method and all the zeros if they are real using graeffe s method in all cases n processors suffice for cn times parallel acceleration where c is a positive constant incorporating turan s techniques into another more recent algorithm gives a single complex zero with absolute error using o log 2 n log log 1 arithmetic parallel steps n 2 processors which places the problem in nc and also gives all the complex zeros in o n 2 log n log log 1 arithmetic operations where 1 is the absolutely largest zero computations with o log n 1 binary bits support the latter estimates for the arithmetic complexity which leads to a simple proof of the current best estimate for the boolean circuit complexity bit operation complexity of computing all the complex zeros of a polynomial announced by a schonhage in 1982 but not proven yet

work preserving speed up of parallel matrix computations brent s scheduling principle provides a general simulation scheme when fewer processors are available than specified by the fastest parallel algorithm such a scheme preserves under slow down the actual number of executed operations also called work in this paper we take the complementary viewpoint and rather than consider the work preserving slow down of some fast parallel algorithm we investigate the problem of the achievable speed ups of computation while preserving the work of the best known sequential algorithm for the same problem the proposed technique eminently applicable to problems of matrix computational flavor achieves its result through the interplay of two algorithms with significantly different features analogous but structurally different interplays have been used previously to improve the algorithmic efficiency of graph computations selection and list ranking we demonstrate the efficacy of our technique for the computation of path algebras in graphs and digraphs and various fundamental computations in linear algebra some of the fundamental new algorithms may have practical value for instance we substantially improve the algorithmic performance of the parallel solution of triangular and toeplitz linear systems of equations and the computation of the transitive closure of digraphs

efficient approximate solution of sparse linear systems abstract we consider the problem of approximate solution x of of a linear system ax b over the reals such that a x b e b for a given e 0 e a a a 1 be the condition number of the n n input matrix a sparse diagonally dominant dd linear systems appear very frequently in the solution of linear systems associated with pdes and stochastic systems and generally have polynomial condition number while there is a vast literature on methods for approximate solution of sparse dd linear systems most of the results are empirical and to date there are no known proven linear bounds on the complexity of this problem using iterative algorithms and building on the work of vaidya 1 and gremban et al 2 4 we provide the best known sequential work bounds for the solution of a number of major classes of dd sparse linear systems let log a e the sparsity graph of a is a graph whose nodes are the indices and whose edges represent pairs of indices of a with nonzero entries the following results hold for a dd matrix a with nonzero off diagonal entries of bounded magnitude 1 1 if a has a sparsity graph which is a regular d dimensional grid for constant d then our work is o n 2 2 2 if a is a stochastic matrix with fixed s n separable graph as its sparsity graph then our work is o n s n 2 the following results hold for a dd matrix a with entries of unbounded magnitude 1 3 if a is sparse i e o n nonzeros our work is less than o n log n 1 5 2 4 if a has a sparsity graph in a family of graphs with constant size forbidden graph minors e g planar graphs then our work is bounded by o n log n 1 o 1 in the case log n o log and o n log n 1 o 1 in the case log o log n we use approximate preconditioned iterations to compute a sequence of iterative improvements to the preconditioned linear system for class 1 of matrices and class 2 with s n o n we construct in n work preconditioners which reduce the condition number of the resulting preconditioned linear system condition number to o 1 and our resulting total work bounds to approximately solve the linear systems are o n if o 1 for class 4 we are within a small increasing factor of these bounds

computing the sign or the value of the determinant of an integer matrix a complexity survey computation of the sign of the determinant of a matrix and the determinant itself is a challenge for both numerical and exact methods we survey the complexity of existing methods to solve these problems when the input is an n n matrix a with integer entries we study the bit complexities of the algorithms asymptotically in n and the norm of a existing approaches rely on numerical approximate computations on exact computations or on both types of arithmetic in combination

parallel complexity of computations with general and toeplitz like matrices filled with integers and extensions computations with toeplitz and toeplitz like matrices are fundamental for many areas of algebraic and numerical computing the list of computational problems reducible to toeplitz and toeplitz like computations includes in particular the evaluation of the greatest common divisor gcd the least common multiple lcm and the resultant of two polynomials computing pade approximation and the berlekamp massey recurrence coefficients as well as numerous problems reducible to these transition to toeplitz and toeplitz like computations is currently the basis for the design of the parallel randomized nc rnc algorithms for these computational problems our main result is in constructing nearly optimal randomized parallel algorithms for toeplitz and toeplitz like computations and consequently for numerous related computational problems including the computational problems listed above where all the input values are integers and all the output values are computed exactly this includes randomized parallel algorithms for computing the rank the determinant and a basis for the null space of an n n toeplitz or toeplitz like matrix a filled with integers as well as a solution x to a linear system a x f if the system is consistent our algorithms use o log n log n log a parallel time and o n log n processors each capable of performing in unit time an arithmetic operation a comparision or a rounding of a rational number to a closest integer the cost bounds cover the cost of the verification of the correctness of the output the computations by these algorithms can be performed with the precision of o n log a bits which matches the precision required in order to represent the output except for the rank computation where the precision of the computation decreases the algorithms involve either a single random parameter or at most 2n 1 parameters the cited processor bounds are less by roughly factor n than ones supported by the known algorithms that run in polylogarithmic arithmetic time and do not use rounding to the closest integers technically we first devise new algorithms supporting our old nearly optimal complexity estimates for parallel computations with general matrices filled with integers then we decrease dramatically by roughly factor n1 376 the processor bounds required in these algorithms in the case where the input matrix is toeplitz like our algorithms exploit and combine some new techniques which may be of independent interest e g in the study of parallel and sequential computation of recursive factorization of integer matrices as well as our earlier techniques of variable diagonal relating to each other several known algebraic and numerical methods stream contraction and the truncation of displacement generators in toeplitz like computations our development and application of these techniques may be of independent

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

complexity of computations with matrices and polynomials this paper reviews the complexity of polynomial and matrix computations as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms

search space size in contraction hierarchies contraction hierarchies are a speed up technique to improve the performance of shortest path computations which works very well in practice despite convincing practical results there is still a lack of theoretical explanation for this behavior in this paper we develop a theoretical framework for studying search space sizes in contraction hierarchies we prove the first bounds on the size of search spaces that depend solely on structural parameters of the input graph that is they are independent of the edge lengths to achieve this we establish a connection with the well studied elimination game our bounds apply to graphs with treewidth k and to any minor closed class of graphs that admits small separators for trees we show that the maximum search space size can be minimized efficiently and the average size can be approximated efficiently within a factor of 2 we show that under a worst case assumption on the edge lengths our bounds are comparable to those in the recent paper vc dimension and shortest path algorithms of abraham et al 1 whose analysis depends also on the edge lengths as a side result we link their notion of highway dimension a parameter that is conjectured to be small but is unknown for all practical instances with the notion of pathwidth this is the first relation of highway dimension with a well known graph parameter

parallel computation of polynomial gcd and some related parallel computations over abstract fields several fundamental problems of computations with polynomials and structured matrices are well known for their resistance to effective parallel solution this includes computing the gcd the 1cm and the resultant of two polynomials as well as any selected entry of either the extended euclidean scheme for these polynomials or the pade approximation table associated to a fixed power series the solution of the berlekamp massey problem of recovering the n coefficients of a linear recurrence sequence from its first 2n terms the solution of a toeplitz linear system of equations computing the ranks and the null spaces of toeplitz matrices and several other computations with toeplitz hankel vandermonde and cauchy generalized hilbert matrices and with matrices having similar structures our algorithms enable us to reach new record estimates for randomized parallel arithmetic complexity of all these computations over any field of constants that is o log n h time and o n2 log log n log n h g arithmetic processors where n is the input size g varies from 1 to 2 and h varies from 2 to 4 depending on the computational problem and on the fixed field of constants the estimates include the cost of correctness tests for the solution the results have further applications to many related computations over abstract fields and rely on some new techniques in particular on recursive pade approximation and regularization of pade approximation via randomization which might be of some independent technical interest in particular an auxiliary algorithm computes over any field the coefficients of a polynomial from the power sums of its zeros which is an important problem of algebraic coding

complexity of parallel matrix computations abstract we estimate parallel complexity of several matrix computations under both boolean and arithmetic machine models using deterministic and probabilistic approaches those computations include the evaluation of the inverse the determinant and the characteristic polynomial of a matrix recently processor efficiency of the previous parallel algorithms for numerical matrix inversion has been substantially improved in pan and reif 1987 reaching optimum estimates up to within a logarithmic factor that work however applies neither to the evaluation of the determinant and the characteristic polynomial nor to exact matrix inversion nor to the numerical inversion of ill conditioned matrices we present four new approaches to the solution of those latter problems having several applications to combinatorial computations in order to extend the suboptimum time and processor bounds of pan and reif 1987 to the case of computing the inverse determinant and characteristic polynomial of an arbitrary integer input matrix in addition processor efficient algorithms using polylogarithmic parallel time are devised for some other matrix computations such as triangular and qr factorizations of a matrix and its reduction to hessenberg form

the linear algebra of the pascal matrix abstract pascal s triangle can be represented as a square matrix in two basically different ways as a lower triangular matrix pn or as a full symmetric matrix qn it has been found that the pn pnt is the cholesky factorization of qn pn can be factorized by special summation matrices it can be shown that the inverses of these matrices are the operators which perform the gaussian elimination steps for calculating cholesky s factorization by applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems finally an explicit formula for the sum of the kth powers is given

parallel algorithms for certain matrix computations abstract the complexity of performing matrix computations such as solving a linear system inverting a nonsingular matrix or computing its rank has received a lot of attention by both the theory and the scientific computing communities in this paper we address some nonclassical matrix problems that find extensive applications notably in control theory more precisely we study the matrix equations ax xa t c and ax xb c the inverse of the eigenvalue problem called pole assignment and the problem of testing whether the matrix b ab a n 1 b has full row rank for these problems we show two kinds of pram algorithms on one side very fast i e polylog time algorithms and on the other side almost linear time and processor efficient algorithms in the latter case the algorithms rely on basic matrix computations that can be performed efficiently also on realistic machine models

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

an efficient memetic algorithm for themax bisection problem the max bisection problem consists in partitioning the vertices of a weighted undirected graph into two equally sized subsets so as to maximize the sum of the weights of crossing edges it is an np hard combinatorial optimization problem that arises in many applications in this paper we present a memetic algorithm for the max bisection problem which integrates a new fast local search procedure a crossover operator and a pool updating strategy these strategies achieve a balance between intensification and diversification extensive experiments were performed on a number of benchmark instances with 800 to 10 000 vertices from the literature the proposed memetic algorithm improved the best known solutions for all benchmark instances tested in this paper the improvement in terms of cut value over the circut by burer et al ranging from 0 02 to 4 15 percent and the average time of our proposed memetic algorithm is much lower than that of circut it shows that the proposed memetic algorithm can find high quality solutions in an acceptable running time

the analysis of a nested dissection algorithm nested dissection is an algorithm invented by alan george for preserving sparsity in gaussian elimination on symmetric positive definite matrices nested dissection can be viewed as a recursive divide and conquer algorithm on an undirected graph it usesseparators in the graph which are small sets of vertices whose removal divides the graph approximately in half george and liu gave an implementation of nested dissection that used a heuristic to find separators lipton and tarjan gave an algorithm to findn1 2 separators in planar graphs and two dimensional finite element graphs and lipton rose and tarjan used these separators in a modified version of nested dissection guaranteeing bounds ofo n logn on fill ando n3 2 on operation count we analyze the combination of the original george liu nested dissection algorithm and the lipton tarjan planar separator algorithm this combination is interesting because it is easier to implement than the lipton rose tarjan version especially in the framework of existing sparse matrix software using some topological graph theory we proveo n logn fill ando n3 2 operation count bounds for planar graphs twodimensional finite element graphs graphs of bounded genus and graphs of bounded degree withn1 2 separators for planar and finite element graphs the leading constant factor is smaller than that in the lipton rose tarjan analysis we also construct a class of graphs withn1 2 separators for which our algorithm does not achieve ano n logn bound on fill

algebraic and numerical techniques for the computation of matrix determinants we review modify and combine together several numerical and algebraic techniques in order to compute the determinant of a matrix or the sign of such a determinant the resulting algorithms enable us to obtain the solution by using a lower precision of computations and relatively few arithmetic operations the problem has important applications to computational geometry

parallel algorithms for matrix polynomial division abstract in this paper several algorithms for the matrix polynomial division are taken into consideration such algorithms represent extensions of known parallel algorithms for the scalar polynomial division with remainder the interest resides in the comparison of the parallel computational cost of these algorithms in the general non scalar case

on practical algorithms for accelerated matrix multiplication abstract we present several bilinear algorithms for the acceleration of multiplication of n x n matrices that are superior to both the classical and strassen s algorithm for moderate n starting with n 20 the bini lotti result on the weak stability of bilinear algorithms applies to these algorithms we present them in two equivalent versions bilinear and trilinear we also apply one of these algorithms over finite fields or rings of constants surprisingly this enables us to decrease the bilinear complexity of n x n matrix multiplication for 20 n 1020 below the current record upper bound for the same computation over the infinite fields of complex real or rational numbers

solution of sparse positive definite systems on a hypercube the solution of large sparse positive definite systems of equations typically involves four steps ordering data structure set up symbolic factorization numerical factorization and triangular solution this article describes how these four phases are implemented on a hypercube multiprocessor the role of elimination trees in the exploitation of sparsity and the identification of parallelism is explained and pseudo code algorithms are provided for some of the important algorithms numerical experiments run on an intel ipsc multiprocessor are presented in order to provide some indication of the performance of the various algorithms

improved parallel polynomial division the authors compute the first n coefficients of the reciprocal r x r x p x 1 bmod x n given a natural n and a polynomial p x p 0 ne 0 by using o h log n arithmetic steps and o n h 1 2 h log h n processors for any h h 1 2 ldots log n under the pram arithmetic models provided that o log m steps and m processors suffice to perform discrete fourier transforms on m points and that log 0 n n log h n log 2 log h 1 n h 1 ldots log n log n max h log h n 0 the same estimates apply to some other computations such as the division with a remainder of two polynomials of degrees o n and the inversion of an n times n triangular toeplitz matrix this improves the known estimates of reif tate and georgiev the presented techniques are extended to parallel implementation of other recursive processes such as the evaluation modulo x n of the mth root p x 1 m of p x

parallel algorithms for algebraic problems fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials computing all entries of the extended euclidean scheme of two polynomials over an arbitrary field gcd and 1cm of many polynomials factoring polynomials over finite fields and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields for the following estimates assume that the input polynomials have degree at most n and the finite field has p d elements the euclidean algorithm is deterministic and runs in parallel time o log 2 n all the other algorithms are probabilistic las vegas in the general case but when applicable to bf q or bf r they can be implemented deterministically over these fields the algorithms for gcd and lcm use parallel time o log 2 n the factoring algorithm runs in parallel time o log 2 n log 2 d 1 log p the algorithm for squarefree decomposition runs in parallel time o log 2 n

computing matrix eigenvalues and polynomial zeros where the output is real surprisingly simple corollaries from the courant fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set s interleaving the set e of the eigenvalues of an n x n real symmetric tridiagonal rst matrix tn as well as a point that splits e into two subsets of comparable cardinalities furthermore we extend this algorithm so as to approximate all the n eigenvalues of tn at nearly optimal sequential and parallel cost that is at the cost of staying within polylogarithmic factors from the straightforward lower bounds the resulting improvement of the known processor bound in nc algorithms for the rst eigenproblem is roughly by factor n our approach extends the previous works m ben or and p tiwari j complexity 6 1990 pp 417 442 and m ben or et al siam j comput 17 1988 pp 1081 1092 for the approximation of the zeros of a polynomial having only real zeros and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem

parametrization of newton s iteration for computations with structured matrices and applications abstract we apply a new parametrized version of newton s iteration in order to compute over any field f of constants the solution or at least squares solution to linear system b x v with an n n toeplitz or toeplitz like matrix b as well as the determinant of b and the coefficients of its characteristic polynomial det i b dramatically improving the processor efficiency of the known fast parallel algorithms our algorithms together with some previously known and some recent results of 1 5 as well as with our new techniques for computing polynomial god s and lcm s imply respective improvement of the known estimates for parallel arithmetic complexity of several fundamental computations with polynomials and with both structured and general matrices

signal scaling by centered discrete dilated hermite functions we present a novel method to generate centered discrete dilated hermite functions by utilizing hermite gaussian differential equation and dft commuting matrix proposed by pei compared with the conventional methods our method provides much higher accuracy and wider working range based on expansion of compact support signals into discrete dilated hermite basis several examples of signal approximation and scaling are demonstrated our results in ecg signal compression can achieve lesser approximation error in comparison with discrete daubechies wavelet transform also our proposed discrete dilated hermite functions have better ability to do signal scaling than the existing method

nearly optimal solution of rational linear systems of equations with symbolic lifting and numerical initialization abstract hensel s symbolic lifting is a highly effective method for the solution of a general or structured e g toeplitz or hankel linear system of equations with integer or rational coefficients of bounded length it can handle ill conditioned inputs for which numerical methods become costly lifting amounts to recursive multiplications by vectors of the input coefficient matrices and its precomputed inverse modulo a fixed integer s such multiplications only involve small numbers of data movements and arithmetic operations with bounded precision the known methods for precomputation of the inverse are more costly however in particular they involve more data movements as our remedy for this bottleneck stage we create an auxiliary matrix sharing its inverse modulo s with the input matrix and we readily compute this inverse by applying numerical iterative refinement which is a numerical counterpart of lifting in the case of general unstructured as well as toeplitz hankel and other popular structured inputs our hybrid algorithms involve a small number of data movements and optimal number of boolean that is bitwise operations up to a logarithmic factor we extend the algorithms to nearly optimal computation of polynomial greatest common divisors gcds least common multiples lcms and pade approximations as well as the berlekamp massey reconstruction of linear recurrences we also cover newton s lifting for matrix inversion specialize it to the case of structured input and combine it with hensel s to enhance the overall efficiency our initialization techniques work for newton s lifting as well furthermore we extend all our lifting algorithms to allow their initialization modulo powers of two thus implementing them in the binary base

on parallel computations with banded matrices we devise parallel algorithms for solving a banded linear system of equations and for computing the determinant of a banded matrix substantially improving the previous record computational complexity estimates of e our algorithms are in nc or rnc and support new record bounds on the parallel time complexity of these computations and simultaneously the bounds on their potential work the product of time and processor bounds which match within constant or logarithmic factors the record sequential time bounds for the same computations moreover these algorithms are in nc1 or rnc1 if the bandwidth is a constant

eigenvalues of hadamard powers of large symmetric pascal matrices abstract let s n be the positive real symmetric matrix of order n with i j entry equal to i j 2 j 1 and let x be a positive real number eigenvalues of the hadamard or entry wise power s n x are considered in particular for k a positive integer it is shown that both the perron root and the trace of s n k are approximately equal to 4 k 4 k 1 2 n 2 n 1 k

matrix sparsification and nested dissection over arbitrary fields the generalized nested dissection method developed by lipton et al 1979 is a seminal method for solving a linear system ax b where a is a symmetric positive definite matrix the method runs extremely fast whenever a is a well separable matrix such as matrices whose underlying support is planar or avoids a fixed minor in this work we extend the nested dissection method to apply to any nonsingular well separable matrix over any field the running times we obtain essentially match those of the nested dissection method an important tool is a novel method for matrix sparsification that preserves determinants and minors and that guarantees that constant powers of the sparsified matrix remain sparse

randomized preprocessing of homogeneous linear systems of equations abstract our randomized preprocessing enables pivoting free and orthogonalization free solution of homogeneous linear systems of equations in the case of toeplitz inputs we decrease the estimated solution time from quadratic to nearly linear and our tests show dramatic decrease of the cpu time as well we prove numerical stability of our approach and extend it to solving nonsingular linear systems inversion and generalized moore penrose inversion of general and structured matrices by means of newton s iteration approximation of a matrix by a nearby matrix that has a smaller rank or a smaller displacement rank matrix eigen solving and root finding for polynomial and secular equations and for polynomial systems of equations some by products and extensions of our study can be of independent technical intersest e g our extensions of the sherman morrison woodbury formula for matrix inversion our estimates for the condition number of randomized matrix products and preprocessing via augmentation

analysis of coppersmith s block wiedemann algorithm for the parallel solution of sparse linear systems by using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems we analyze such a scheme proposed by coppersmith for wiedemann s coordinate recurrence algorithm which is based in part on the krylov subspace approach we prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors our analysis is valid for fields of entries that have sufficiently large cardinality our analysis also deals with an arising subproblem of solving a singular block toeplitz system by use of the theory of toeplitz like matrices

high probability parallel transitive closure algorithms there is a straightforward algorithm for computing the transitive closure of an n node graph in o log 2 n time on an erew pram using n 3 log n processors or indeed with m n log n processors if serial matrix multiplication in m n time can be done this algorithm is within a log factor of optimal in work processor time product for solving the all pairs transitive closure problem for dense graphs however this algorithm is far from optimal when either a the graph is sparse or b we want to solve the single source transitive closure problem it would be ideal to have an mathcal nc algorithm for transitive closure that took about e processors for the single source problem on a graph with n nodes and e geqq n arcs or about en processors for the all pairs problem on the same graph while an algorithm that good cannot be offered algorithms with the following performance can be offered 1 for single source tilde o n varepsilon time with tilde o en 1 2 varepsil

error estimates for approximate solutions of the riccati equation with real or complex potentials a method is presented for obtaining rigorous error estimates for approximate solutions of the riccati equation with real or complex potentials our main tool is to derive invariant region estimates for complex solutions of the riccati equation we explain the general strategy for applying these estimates and illustrate the method in typical examples where the approximate solutions are obtained by gluing together wkb and airy solutions of corresponding one dimensional schrodinger equations our method is motivated by and has applications to the analysis of linear wave equations in the geometry of a rotating black hole

fast and efficient parallel solution of sparse linear systems this paper presents a parallel algorithm for the solution of a linear system a bf x bf b with a sparse n times n symmetric positive definite matrix a associated with the graph g a that has n vertices and has an edge for each nonzero entry of a if g a has an s n separator family and a known s n separator tree then the algorithm requires only o log 3 n time and e m s n log n processors for the evaluation of the solution vector bf x a 1 bf b where e is the number of edges in g a and m n is the number of processors sufficient for multiplying two n times n rational matrices in time o log n furthermore for this computational cost the algorithm computes a recursive factorization of a such that the solution of any other linear system a bf x bf b with the same matrix a requires only o log 2 n time and e log n s n 2 processors

calculs de complexite relatifs a une methode de dissection emboitee nous presentons une numerotation de type nested dissection des inconnues d un systeme lineaireax b pour des ensembles de matricescreuses symetriques definies positives correspondant a des famille de graphes non orientes a degre borne et admettant un sqrt n thereme de separation comparativement aux methodes et resultats de rose 9 l algorithme presente est plus simple mais les theoremes de complexite moins generaux en raison de l hypothese restrictive de degre borne en outre les demonstrations font appel en permanence a la structure d arbre sur la famille des separateurs qui constitue par ailleurs une partition de l ensemble des sommets du graphe initial nous presentons ensuite le schema general d implementation dans le cadre du code d elements finis modulef pourdes problemes plans d elements finis et nous donnons quelques mesures comparatives avec la numerotation plus classique qui tend a minimiser le profil de la matrice a nested dissection ordering is given for solving any system of linear equationsax b for the family ofsparse symmetric positive definite matrices corresponding to the class of undirected graphs withbounded degree and satisfyinga sqrt n separator theorem if we compare the methods and results presented by rose 9 our algorithm is more simple but the complexity results are less general because of the restriction of bounded degree besides our proofs use continually the arborescent structure on the family of separators which is by another way a partition of the set of vertices for the initial graph then the general implementation scheme in the finite element package modulef fortwo dimensional finite element problems is presented and numerical comparisons between our ordering and the standard envelope method are given

fast and efficient parallel solution of dense linear systems the most efficient previously known parallel algorithms for the inversion of a nonsingular n n matrix a or solving a linear system ax b over the rational numbers require o log2n time and m n n processors provided that m n processors suffice in order to multiply two n n rational matrices in time o logn furthermore the known polylog arithmetic time algorithms for those problems are numerically unstable in this paper we apply newton s iteration and initially choose an approximate inverse matrix by following ben israel this quadratically convergent and numerically stable iterative method takes o log2n parallel time using m n processors to compute the inverse within the relative precision 2 nc for a positive constant c of an n n rational matrix a with the condition number at most nd for a constant d this is the optimum processor bound and by a factor of n improvement of the previously known processor bounds for polylogarithmic time matrix inversion the algorithm does not require to precompute the condition number of the input matrix but it just converges slower for ill conditioned input matrices

fast and efficient parallel evaluation of the zeros of a polynomial having only real zeros abstract given a polynomial p x of degree n with integer coefficients between 2 m and 2 m it suffices to use o log 2 n log b parallel arithmetic steps and n 2 processors in order to compute all the zeros of p x with absolute errors at most 2 h provided that all the zeros are real and b m h the algorithm combines the recent techniques and results of 1 3 in order to improve the previous record processor bound of 1 at least on the factor of n preserving its upper bound on parallel time

fast parallel computation of polynomials using few processors it is shown that any multivariate polynomial of degree d that can be computed sequentially in c steps can be computed in parallel in o log d log c log d steps using only cd o 1 processors

a note on a toeplitz inversion formula abstract it is shown that a formula of heinig for the inverse of a toeplitz matrix follows from a formula of the author for the inverse of a hankel matrix

sequential and parallel complexity of approximate evaluation of polynomial zeros our new sequential and parallel algorithms establish new record upper bounds on both arithmetic and boolean complexity of approximating to complex polynomial zeros o n2 log b log n arithmetic operations or o n log n log bn parallel steps and n log b log bn processors suffice in order to approximate with absolute errors 2m b to all the complex zeros of an nth degree polynomial p x whose coefficients have mod 2m if we only need such an approximation to a single zero of p x then o n log b log n arithmetic operations or o log2 n log bn steps and n log n log b log bn processors suffice which places the latter problem in nc that is in the class of problems that can be solved using polylogarithmic parallel time and a polynomial number of processors those estimates are reached in computations with o bn binary bits where the polynomial has integer coefficients we also reach the sequential boolean time bounds o bn3 log bn log log bn for approximating to all the zeros very minor improvement of the bound announced in 1982 by schonhage and o bn2log log n log bn log log bn for approximating to a single zero among further implications are the improvements of the known algorithms and complexity estimates for computing matrix eigenvalues for polynomial factorization over the field of complex numbers and for solving systems of polynomial equations the computations rely on recursive application of turan s proximity test of 1968 on its more recent extensions to root radii computations on contour integration via fast fourier transform fft within geometric constructions for search and exclusion and for the final minor improvements of the complexity bounds on the recursive factorization of p x over discs on the complex plane via numerical integration and newton s iterations

spectral estimates and non selfadjoint perturbations of spheroidal wave operators we derive a spectral representation for the oblate spheroidal wave operator which is holomorphic in the aspherical parameter in a neighborhood of the real line for real estimates are derived for all eigenvalue gaps uniformly in the proof of the gap estimates is based on detailed estimates for complex solutions of the riccati equation the spectral representation for complex is obtained using the theory of slightly non selfadjoint perturbations

parallel complexity of numerically accurate linear system solvers we prove a number of negative results about practical i e work efficient and numerically accurate algorithms for computing the main matrix factorizations in particular we prove that the popular householder and givens methods for computing the qr decomposition are p complete and hence presumably inherently sequential under both real and floating point number models we also prove that gaussian elimination ge with a weak form of pivoting which aims only at making the resulting algorithm nondegenerate is likely to be inherently sequential as well finally we prove that ge with partial pivoting is p complete over gf 2 or when restricted to symmetric positive definite matrices for which it is known that even standard ge no pivoting does not fail altogether the results of this paper give further formal support to the widespread belief that there is a tradeoff between parallelism and accuracy in numerical algorithms