communication complexity of the gaussian elimination algorithm on multiprocessors abstract this paper proposes a few lower bounds for communication complexity of the gaussian elimination algorithm on multiprocessors three types of architectures are considered a bus architecture a nearest neighbor ring network and a nearest neighbor grid network it is shown that for the bus and the ring architectures the minimum communication time is o n2 independent of the number of processors while for the grid it is reduced to o k built 1 2 n2 o k built1 2 n for a lock step gaussian elimination algorithm and to o k built 1 2 n2 o k built1 2 for any pipelined gaussian elimination algorithm where k is the total number of processors the practical implications of these bounds are discussed

preconditioning by fast direct methods for nonself adjoint nonseparable elliptic equations we consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself adjoint elliptic boundary value problems we derive bounds on convergence rates that are independent of discretization mesh size for two dimensional problems on rectangular domains discretized on an n times n grid these bounds lead to asymptotic operation counts of o n 2 log n log varepsilon 1 to achieve relative error varepsilon and o n 2 log n 2 to reach truncation error

a spectral multiplier theorem for a sublaplacian on su 2 we prove a hormander type spectral multiplier theorem for a sublaplacian on su 2 with critical index determined by the euclidean dimension of the group this result is the analogue for su 2 of the result for the heisenberg group obtained by d muller and e m stein and by w hebisch

numerical linear algebra algorithms and software abstract the increasing availability of advanced architecture computers has a significant effect on all spheres of scientific computation including algorithm research and software development in numerical linear algebra linear algebra in particular the solution of linear systems of equations lies at the heart of most calculations in scientific computing this paper discusses some of the recent developments in linear algebra designed to exploit these advanced architecture computers we discuss two broad classes of algorithms those for dense and those for sparse matrices

parallel vector solution of large scale structural analysis problems on supercomputers this direct method is based on the choleski factorization procedure and uses a skyline storage scheme unique features of the method include its parallel computation at the outermost do loop and vector computation at the innermost do loop

solving linear systems on a vector computer abstract this paper gives a classification for the triangular factorization of square matrices these factorizations are used for solving linear systems efficient algorithms for vector computers are presented on basis of criteria for optimal algorithms moreover the gauss jordan elimination algorithm in a version which admits efficient implementation on a vector computer is described comparative experiments in fortran 77 with fortran 200 extensions for the cyber 205 are reported

a parallel algorithm for the sylvester observer equation we present a new algorithm for solving the sylvester observer equation arising in the context of the luenberger observer the algorithm embodies two main computational phases the solution of several independent equation systems and a series of matrix matrix multiplications the algorithm is thus well suited for parallel and high performance computing by reducing the coefficient matrix a to lower hessenberg form one can implement the algorithm efficiently with few floating point operations and little workspace the algorithm has been successfully implemented on a cray c90 a comparison both theoretical and experimental has been made with the well known hessenberg schur algorithm which solves an arbitrary sylvester equation our theoretical analysis and experimental results confirm the superiority of the proposed algorithm both in efficiency and speed over the hessenberg schur algorithm

comparisons of gaussian elimination algorithms on a cray y mp abstract the results of various implementations of gaussian elimination on full matrices on a single processor cray y mp are presented and discussed it is shown that when the manufacturer supplied blas kernels are used the difference between thebest versions of level 2 blas and level 3 blas blocked implementations is almost negligible it is suggested that to improve the performance of blocked gaussian elimination it is possible to utilize strassen s matrix multiplication algorithm

about riesz transforms on the heisenberg groups see also m1 subsequently several other proofs have been given of probabilistic b ba see also g v or analytic d r p nature also p a meyer has given in m2 a similar result in the gaussian setting the aim of this work is to prove analogous inequalities for the family of heisenberg groups h n instead of the family of euclidean spaces r n the euclidean proofs do not adapt to the non commutative situation of h however our method is reminiscent of some ideas of the proofs by duoandikoetxearubio de francia d r and pisier p a new feature is however the central r6te played by the heat kernel and our proof heavily relies on its explicit expression in h n given by gavean g and hulanicki h let us recall finally that apart from any question of uniformity the boundedness of the riesz transforms on every nilpotent lie group is a well known issue see l v

on closed form solutions to the generalized sylvester conjugate matrix equation abstract in this paper several explicit parametric solutions to the generalized sylvester conjugate matrix equation are given the proposed solutions can provide all the degrees of freedom which are represented by a free parameter matrix in addition the approaches adopted in this paper do not require that the coefficient matrices are in any canonical forms

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

parallel and large scale matrix computations in control some ideas abstract the design and analysis of time invariant linear control systems give rise to a variety of interesting linear algebra problems numerically effective methods now exist for several of these problems however algorithms for large scale computations and efficient parallel algorithms for these problems are virtually nonexistent in this paper we propose several efficient general purpose parallel algorithms for single input and multiinput eigenvalue assignment problems a desirable feature of these algorithms is that they are composed of simple linear algebraic operations such as matrix vector multiplication solution of a linear system and computations of the eigensystem and singular values of a symmetric matrix for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms the proposed algorithms thus have potential for implementations on some existing and future parallel processors we also propose a numerical method for the sylvester matrix equation arising in the construction of a luenberger observer the method does not need reduction to condensed forms and is thus suitable for large and sparse matrices the method also exhibits a certain parallelism

an optimal schedule for gaussian elimination on an mimd architecture in this paper we propose an algorithm for allocating the tasks of the well known gaussian elimination algorithm on an mimd architecture and prove that the schedule is optimal in order of magnitude up to a polylog factor

a gaussian estimate for the heat kernel on differential forms and application to the riesz transform let m g be a complete riemannian manifold which satisfies a sobolev inequality of dimension n and on which the volume growth is comparable to the one of mathbb r n for big balls if there is no non zero l 2 harmonic 1 form and the ricci tensor is in l frac n 2 varepsilon cap l infty for an varepsilon 0 then we prove a gaussian estimate on the heat kernel of the hodge laplacian acting on 1 forms this allows us to prove that under the same hypotheses the riesz transform d vardelta 1 2 is bounded on l p for all 1 3

bellman function and linear dimension free estimates in a theorem of bakry by using an explicit bellman function we prove a bilinear embedding theorem for the laplacian associated with a weighted riemannian manifold m having the bakry emery curvature bounded from below the embedding acting on the cartesian product of lp m and lq t m 1 p 1 q 1 involves estimates which are independent of the dimension of the manifold and linear in p as a consequence we obtain linear dimension free estimates of the lp norms of the corresponding shifted riesz transform all our proofs are analytic

skewness for multivariate distributions two approaches this paper presents two approaches for qualitative quantitative and comparative concepts of skewness to be defined with respect to the spatial median for multivariate distributions they extend the known quantile based notions defined for real distributions the main tool for such extensions consists of a family of central parts that provide suitable generalizations of the real interquantile intervals

robust stability via sign definite decomposition this paper considers the problem of robust stability of a polynomial family whose coefficients are polynomial functions of the parameters of interests the problem occurs in the design of a fixed order or fixed structure multivariable feedback controller parametrized by a real design parameter vector for a plant containing a vector of uncertain parameters the characteristic polynomials of such systems often contain coefficients which depend polynomially on and using results on sign definite decomposition a new stability test is developed that gives a sufficient condition for hurwitz stability of the family of closed loop systems that result when and vary over prescribed boxes this test is reminiscent of kharitonov s theorem even though the family of polynomials considered here is certainly not restricted to be interval or even convex moreover the test does reduce to kharitonov s theorem for the special case of interval polynomials using this criterion recursively and modularly sets of controllers that stabilize the family of uncertain plants are determined

the impact of high performance computing in the solution of linear systems trends and problems we review the influence of the advent of high performance computing on the solution of linear equations we will concentrate on direct methods of solution and consider both the case when the coefficient matrix is dense and when it is sparse we will examine the current performance of software in this area and speculate on what advances we might expect in the early years of the next century

parallel cholesky factorization on a shared memory multiprocessor abstract a parallel algorithm is developed for cholesky factorization on a shared memory multiprocessor the algorithm is based on self scheduling of a pool of tasks the subtasks in several variants of the basic elimination algorithm are analyzed for potential concurrency in terms of precedence relations work profiles and processor utilization this analysis is supported by simulation results the most promising variant which we call column cholesky is identified and implemented for the denelcor hep multiprocessor experimental results are given for this machine

letter to the editor on the ijk forms of the truncated lu decomposition matrix lu decomposition has six ijk forms different forms have different computational complexities and storage requirements particularly on vector and parallel computers other factors governing the choice of a particular form are considered for treating fredholm integral equations of the first kind the truncated lu decomposition of the resulting system matrix is recommended required modifications to selected known ijk forms are presented

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

a new block algorithm for full rank solution of the sylvester observer equation a new block algorithm for computing a full rank solution of the sylvester observer equation arising in state estimation is proposed the major computational kernels of this algorithm are 1 solutions of standard sylvester equations in each case of which one of the matrices is of much smaller order than that of the system matrix and furthermore this small matrix can be chosen arbitrarily and 2 orthogonal reduction of small order matrices there are numerically stable algorithms for performing these tasks including the krylov subspace methods for solving large and sparse sylvester equations the proposed algorithm is also rich in level 3 basic linear algebra subroutine blas 3 computations and is thus suitable for high performance computing furthermore the results on numerical experiments on some benchmark examples show that the algorithm has better accuracy than that of some of the existing block algorithms for this problem

orthogonal error methods in this paper we discuss the conjugate gradient method generalized to nonsymmetric matrices constructing a class of algorithms that includes both the conjugate gradient algorithms as described in 12 and the orthogonal residual algorithms cf elman 10 we call this class orthogonal error algorithms and characterize the class of matrices for which finite term orthogonal error algorithms exist

optimal scheduling algorithms for parallel gaussian elimination this paper uses a graph theoretic approach to derive asymptotically optimal algorithms for parallel gaussian elimination on simd mimd couputers with a shared memory system given a problem of sinze n and using p n processors where 2 43 0 305 the asymptotically optimal algorithms are of efficiency e 1 1 3 0 972 this evidences the high degree of parallelism that can be achieved

an iterative algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices a matrix email protected r n x n is said to be a symmetric orthogonal matrix if p p t p 1 a matrix email protected r n x n is said to be generalized centro symmetric generalized central anti symmetric with respect to p if a pap a pap the generalized centro symmetric matrices have wide applications in information theory linear estimate theory and numerical analysis in this paper we propose a new iterative algorithm to compute a generalized centro symmetric solution of the linear matrix equations ayb e cyd f we show when the matrix equations are consistent over generalized centro symmetric matrix y for any initial generalized centro symmetric matrix y 1 the sequence y k generated by the introduced algorithm converges to a generalized centro symmetric solution of matrix equations ayb e cyd f the least frobenius norm generalized centro symmetric solution can be derived when a special initial generalized centro symmetric matrix is chosen furthermore the optimal approximation generalized centro symmetric solution to a given generalized centro symmetric matrix can be derived several numerical examples are given to show the efficiency of the presented method

torus data flow for parallel computation of missized matrix problems abstract a data flow approach is used to solve dense symmetric systems of equations on a torus connected 2 d mesh of processors a torus mapping of the matrix onto this processor array allows the cholesky decomposition to be completed in 3 n 2 time steps using only n 2 4 processors less than half the number needed in previously reported results new definitions for missized problems and parallel algorithm performance are given along with various time step efficiency and processor utilization plots

assignment and scheduling in parallel matrix factorization abstract we consider the problem of factoring a dense n n matrix on a network consisting of p mimd processors with no shared memory when the network is smaller than the number of elements in the matrix p n 2 the specific example analyzed is a computational network that arises in computing the lu qr or cholesky factorizations we prove that if the nodes of the network are evenly distributed among processors and if computations are scheduled by a round robin or a least recently executed scheduling algorithm then optimal order of speedup is achieved however such speedup is not necessarily achieved for other scheduling algorithms or if the computation for the nodes is inappropriately split across processors and we give examples of these phenomena lower bounds on execution time for the algorithm are established for two important node assignment strategies

on gradient bounds for the heat kernel on the heisenberg group it is known that the couple formed by the two dimensional brownian motion and its levy area leads to the heat kernel on the heisenberg group which is one of the simplest sub riemannian space the associated diffusion operator is hypoelliptic but not elliptic which makes difficult the derivation of functional inequalities for the heat kernel however driver and melcher and more recently h q li have obtained useful gradient bounds for the heat kernel on the heisenberg group we provide in this paper simple proofs of these bounds and explore their consequences in terms of functional inequalities including cheeger and bobkov type isoperimetric inequalities for the heat kernel

the gauss huard algorithm and lu factorization abstract in this paper we analyze the gauss huard algorithm from a description of the algorithm in terms of matrix vector operations we reveal a close relation between the gauss huard algorithm and an lu factorization as constructed in an ikj variant

gundy varopoulos martingale transforms and their projection operators on manifolds and vector bundles this paper proves the l p boundedness of generalized first order riesz transforms obtained as conditional expectations of martingale transforms a la gundy varopoulos for quite general diffusions on manifolds and vector bundles several specific examples and applications are presented lie groups of compact type the heisenberg group mathbb su 2 and riesz transforms on forms and spinors

blocking neville elimination algorithm for exploiting cache memories neville elimination is a method for solving a linear system of equations that introduces zeros in a matrix column by adding to each row an adequate multiple of the previous one in this paper we explore block algorithms for neville elimination which take into account the memory hierarchies of a computer these algorithms try to manage the memory movements to optimize them thus the matrix of the system is divided following three different strategies blocking by rows columns or submatrices in each case we study the performance of the algorithm according to the ratio of floating point operations to memory references q theoretical estimations show that q depends on data partitioning being submatrix blocks the best choice

closed form solutions to sylvester conjugate matrix equations some explicit closed form solutions of homogeneous and nonhomogeneous sylvester conjugate matrix equations are provided in this paper one of the solutions is expressed in terms of controllability matrices and observability matrices the proposed approach does not require all the coefficient matrices to be in any canonical forms and the solutions provide a significant degree of freedom which is represented by an arbitrarily chosen parameter matrix by specifying the solutions of the homogeneous sylvester conjugate equation some new expressions of the solutions of the normal sylvester normal sylvester conjugate and sylvester equations are given this fact reveals that the sylveter conjugate matrix equations are a more general class of some previously investigated matrix equations

minimal faithful upper triangular matrix representations for solvable lie algebras the existence of matrix representations for any given finite dimensional complex lie algebra is a classic result on lie theory in particular such representations can be obtained by means of an isomorphic matrix lie algebra consisting of upper triangular square matrices unfortunately there is no general information about the minimal order for the matrices involved in such representations in this way our main goal is to revisit debug and implement an algorithm which provides the minimal order for matrix representations of any finite dimensional solvable lie algebra when inserting its law as well as returning a matrix representative of such an algebra by using the minimal order previously computed in order to show the applicability of this procedure we have computed minimal representatives not only for each solvable lie algebra with dimension less than 6 but also for some solvable lie algebras of arbitrary dimension

reverse poincare inequalities isoperimetry and riesz transforms in carnot groups abstract we prove an optimal reverse poincare inequality for the heat semigroup generated by the sub laplacian on a carnot group of any step as an application we give new proofs of the isoperimetric inequality and of the boundedness of the riesz transform in carnot groups

the hodge de rham laplacian and lp boundedness of riesz transforms on non compact manifolds let m be a complete non compact riemannian manifold satisfying the volume doubling property as well as a gaussian upper bound for the corresponding heat kernel we study the boundedness of the riesz transform d 12 on both hardy spaces hp and lebesgue spaces lp under two different conditions on the negative part r of the ricci curvature first we prove that if r is subcritical for some 0 1 then the riesz transform d 12 on differential 1 forms is bounded from the associated hardy space h p 1t m to lp m for all p 1 2 as a consequence d 12 is bounded on lp for all p 1 p0 where p0 2 depends on and the constant appearing in the doubling property second we prove that if 01 r 12v t 1p1 p1dtt 1 r 12v t 1p2 p2dtt for some p1 2 and p2 3 then the riesz transform d 12 is bounded on lp for all 1 2 under conditions on r and the potential v we prove both positive and negative results on the boundedness of da 12 on lp

skewness and asymmetry measures and orderings recent interest in skewness has tended to separate two aspects of the concept two distributions may be compared with respect to skewness or a distribution may be self compared that is the distributions of the random variables of x and x may be compared this paper uses the unification of these two aspects to attemp to complete a skewness structure of orderings that identifies the roles of various skewness and scale measures and enables classification of the skewness properties of any distribution the structure is also used to propose measures of asymmetry some skewness properties of the weibull and johnson systems are examined

riesz transform and l p cohomology for manifolds with euclidean ends let m be a smooth riemannian manifold which is the union of a compact part and a finite number of euclidean ends rr n setminus b 0 r for some r 0 each of which carries the standard metric our main result is that the riesz transform on m is bounded from l p m to l p m t m for 1 n the result is new for 2 2 for a more general class of manifolds assume that m is a n dimensional complete manifold satisfying the nash inequality and with an o r n upper bound on the volume growth of geodesic balls we show that boundedness of the riesz transform on l p for some p 2 implies a hodge de rham interpretation of the l p cohomology in degree 1 and that the map from l 2 to l p cohomology in this degree is injective

clustered gauss huard algorithm for the solution of ax b abstract in this paper we present and evaluate a new parallel algorithm for gauss huard method with scaled partial pivoting strategy for the solution of linear systems of equations the algorithm is generic in the sense that matrix distribution methods are decoupled from the algorithm details which makes it applicable to a wide spectrum of distribution functions and hence saving the additional efforts that would be needed to develop a new algorithm for every instance of matrix distribution functions the proposed algorithm is first analyzed then evaluated using a cluster of networked workstations a timing model is developed and verified to accurately estimate the execution time of the proposed algorithm the obtained results reveal two important observations first cluster computing is a viable and low cost alternative for solving computationally intensive problems such as achieving fast and stable solutions for systems of linear equations a problem often encountered in many real life applications second the obtained results reveal that gauss huard performs better than the well known gauss jordan algorithm in cluster environments

hamilton and jacobi come full circle jacobi algorithms for structured hamiltonian eigenproblems we develop jacobi algorithms for solving the complete eigenproblem for hamiltonian and skew hamiltonian matrices that are also symmetric or skew symmetric based on the direct solution of 4 4 and in one case 8 8 subproblems these structure preserving algorithms produce symplectic orthogonal bases for the invariant subspaces associated with a matrix in any one of the four classes under consideration the key step in the construction of the algorithms is a quaternion characterization of the 4 4 symplectic orthogonal group and the subspaces of 4 4 hamiltonian skew hamiltonian symmetric and skew symmetric matrices in addition to preserving structure these algorithms are inherently parallelizable numerically stable and show asymptotic quadratic convergence

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

a structure preserving algorithm for the quaternion cholesky decomposition in this paper we discuss the cholesky decomposition of the hermitian positive definite quaternion matrix for the first time the structure preserving gauss transformation is defined and then a novel structure preserving algorithm which is applied to its real representation matrix is proposed our algorithm needs only real number operations does not depend on the quaternion toolbox for matlab qtfm and has more portability although the flops of our algorithm are theoretically about the same as those based on quaternion arithmetic operations or qtfm numerical experiments show that our algorithm runs faster

reduced order observers a new algorithm and proof this paper gives a new recursive algorithm to construct a reduced order observer with prescribed spectrum of a given observable system the method is based on the staircase form and implicitly gives a new proof for the existence of such reduced order observers

real structure preserving algorithms of householder based transformations for quaternion matrices in this paper we survey three different forms of householder based transformations for quaternion matrices in the literature and propose a new form of quaternion householder based transformation we propose real structure preserving algorithms of these householder based transformations which make the procedure computationally more flexible and efficient we compare the computation counts and assignment numbers of these algorithms we also compare the effectiveness of these real structure preserving algorithms applying to the quaternion qrd and the quaternion svd all these four real structure preserving algorithms are more efficient comparing to the algorithms which apply quaternion toolbox using quaternion arithmetics or algorithms which directly performs real householder transformations on the real representation of a quaternion matrix among these four real structure preserving algorithms the most efficient ones are based on quaternion householder reflection and new proposed householder based transformation

sharp martingale inequalities and applications to riesz transforms on manifolds lie groups and gauss space abstract we prove new sharp l p logarithmic and weak type inequalities for martingales under the assumption of differential subordination the l p estimates are feynman kac type versions of burkholder s celebrated martingale transform inequalities from the martingale l p inequalities we obtain that riesz transforms on manifolds of nonnegative bakry emery ricci curvature have exactly the same l p bounds as those known for riesz transforms in the flat case of r d from the martingale logarithmic and weak type inequalities we obtain similar inequalities for riesz transforms on compact lie groups and spheres combining the estimates for spheres with poincare s limiting argument we deduce the corresponding results for riesz transforms associated with the ornstein uhlenbeck semigroup thus providing some extensions of p a meyer s l p inequalities

hardy s inequality for fractional powers of the sublaplacian on the heisenberg group we prove hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the heisenberg group h n we prove two versions of such inequalities depending on whether the weights involved are non homogeneous or homogeneous in the first case the constant arising in the hardy inequality turns out to be optimal in order to get our results we will use ground state representations the key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by m cowling and u haagerup the approach to prove the integral representations is via the language of semigroups as a consequence of the hardy inequalities we also obtain versions of heisenberg uncertainty inequality for the fractional sublaplacian c 2016 elsevier inc all rights reserved

lp estimates and existence theorems for the operator on complete kahler manifolds abstract in this paper we establish some l p estimates and existence theorems for the operator on a holomorphic hermitian vector bundle over a complete kahler manifold as applications we prove some vanishing theorems of the l p dolbeault cohomology on complete kahler manifolds with suitable geometric conditions

martingale transforms and lp norm estimates of riesz transforms on complete riemannian manifolds under the condition that the bakry emery ricci curvature is bounded from below we prove a probabilistic representation formula of the riesz transforms associated with a symmetric diffusion operator on a complete riemannian manifold using the burkholder sharp l p inequality for martingale transforms we obtain an explicit and dimension free upper bound of the l p norm of the riesz transforms on such complete riemannian manifolds for all 1 p in the euclidean and the gaussian cases our upper bound is asymptotically sharp when p 1 and when p

rehabilitation of the gauss jordan algorithm in this paper a gauss jordan algorithm with column interchanges is presented and analysed we show that in contrast with gaussian elimination the gauss jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability for solutions obtained by gauss jordan with column interchanges a more satisfactory bound for the residual norm can be given the analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by gaussian elimination and that in most practical situations the residuals are equally small this is confirmed by numerical experiments moreover timing experiments on a cyber 205 vector computer show that the algorithm presented has good vectorisation properties

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

arnoldi methods for large sylvester like observer matrix equations and an associated algorithm for partial spectrum assignment abstract we propose an arnoldi based numerical method for solving a sylvester type equation arising in the construction of the luenberger observer given an n n matrix a and an n m matrix g the method simultaneously constructs an m m hessenberg matrix h with a preassigned spectrum and an n m orthonormal matrix x such that ax xh g we consider the case when a is large and sparse so that the standard techniques such as the well known hessenberg schur method for solving a sylvester equation cannot be easily applied as a byproduct we propose an algorithm for the partial pole assignment problem for large matrices

linear inverse problems in wave motion nonsymmetric first kind integral equations we present a general framework to study the solution of first kind integral equations the integral operator is assumed to be compact and nonself adjoint and the integral equation can possess a nonempty null space an approach is presented for adding contributions from the null space to the minimum energy solution of the integral equation through the introduction of weighted hilbert spaces stability accuracy and nonuniqueness of the solution are discussed through the use of model resolution data fit and model covariance operators the application of this study is to inverse problems that exhibit nonuniqueness

parallel algorithms for solving large linear systems the solution of linear systems continues to play an important role in scientific computing the problems to be solved often are of very large size so that solving them requires large computer resources to solve these problems at least supercomputers with large shared memory or massive parallel computer systems with distributed memory are needed this paper gives a survey of research on parallel implementation of various direct methods to solve dense linear systems in particular are considered gaussian elimination gauss jordan elimination and a variant due to huard 19791 and an algorithm due to enright 19781 designed in relation to solving stiff odes such that stepsize and other method parameters can easily be varied some theoretical results are mentioned including a new result on error analysis of huard s algorithm moreover practical considerations and results of experiments on supercomputers and on a distributed memory computer system are presented

implementing linear algebra algorithms for dense matrices on a vector pipeline machine this paper examines common implementations of linear algebra algorithms such as matrix vector multiplication matrix matrix multiplication and the solution of linear equations the different versions are examined for efficiency on a computer architecture which uses vector processing and has pipelined instruction execution by using the advanced architectural features of such machines one can usually achieve maximum performance and tremendous improvements in terms of execution speed can be seen over conventional computers

neville elimination a study of the efficiency using checkerboard partitioning abstract it is well known that checkerboard partitioning can exploit more concurrency than striped partitioning because the matrix computation can be divided among more processors than in the case of striping in this work we analyze the performance of neville method when a checkerboard partitioning is used focusing on the special case of block cyclic checkerboard partitioning this method is an alternative to gaussian elimination and it has been proved to be very useful for some classes of matrices such as totally positive matrices the performance of this parallel system is measured in terms of the efficiency the fraction of time for which a processor is usefully employed which in our model is close to one when the optimum block size is used also we have executed our algorithms on a parallel pc cluster observing that both efficiencies theoretical and empirical are quite similar

the subelliptic heat kernel on su 2 representations asymptotics and gradient bounds the lie group su 2 endowed with its canonical subriemannian structure appears as a three dimensional model of a positively curved subelliptic space the goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities

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

concavity and estimation on donne des conditions simplifiees pour la consistance et la normalite asymptotique d estimations m deduites par maximisation de moyennes de fonctions concaves aleatoires identiquement distribuees et independantes des applications sont donnees pour l estimation de maximum de vraisemblance

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance