on high relative accuracy of the kogbetliantz method the relative accuracy of the kogbetliantz method for computing the singular value decomposition of real triangular matrices is considered sharp relative error bounds for the computed singular values are derived the results are obtained by estimating the norms of scaled perturbation matrices arising from errors generated in one step and one batch of commuting transformations perturbation matrices are scaled either from the left or right side by the inverse of the diagonal matrix

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

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

energy norm a posteriori error estimates for mixed finite element methods this paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations it is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method the analysis is performed in two different ways under a saturation assumption and using a helmholtz decomposition for vector fields

a parallel algorithm for the eigenvalues and eigenvectors of a general complex matrix a new parallel jacobi like algorithm is developed for computing the eigenvalues of a general complex matrix most parallel methods for this problem typically display only linear convergence sequential norm reducing algorithms also exist and they display quadratic convergence in most cases the new algorithm is a parallel form of the norm reducing algorithm due to eberlein it is proven that the asymptotic convergence rate of this algorithm is quadratic numerical experiments are presented which demonstrate the quadratic convergence of the algorithm and certain situations where the convergence is slow are also identified the algorithm promises to be very competitive on a variety of parallel architectures in particular the algorithm can be implemented usingn 2 4 processors takingo n log2 n time for random matrices

a priori and a posteriori error analyses of a high order unfitted mixed fem for stokes flow abstract we propose and analyze a high order unfitted mixed finite element method for the pseudostress velocity formulation of the stokes problem with dirichlet boundary condition on a fluid domain with curved boundary the method consists of approximating by a polyhedral subdomain d h with boundary h where a galerkin method is applied to approximate the solution and on a transferring technique based on integrating the extrapolated discrete gradient of the velocity to approximate the dirichlet boundary data on the computational boundary h the associated galerkin scheme is defined by raviart thomas elements of order k 0 for the pseudostress and discontinuous polynomials of degree k for the velocity provided suitable hypotheses on the mesh near the boundary we prove well posedness of the galerkin scheme by means of the babuska brezzi theory and establish the corresponding optimal convergence of o h k 1 moreover for the case when h is constructed through a piecewise linear interpolation of we propose a reliable and quasi efficient residual based a posteriori error estimator its definition makes use of a postprocessed velocity with enhanced accuracy to achieve the same rate of convergence of the method when the solution is smooth enough numerical experiments illustrate the performance of the scheme show the behavior of the associated adaptive algorithm and validate the theory

the partition function of the extended r reduced kadomtsev petviashvili hierarchy we derive a particular solution of the extended r reduced kp hierarchy which is specified by a generalized string equation the work is a generalization to arbitrary of buryak s recent results of a solution to the extended open kdv hierarchy which corresponds to r 2

on quadratic convergence bounds of the j symmetric jacobi method this paper deals with quadratic convergence estimates for the serial j symmetric jacobi method recently proposed by veselic the method is characterized by the use of orthogonal and hyperbolic plane rotations using a new technique recently introduced by hari we prove sharp quadratic convergence bounds in the general case of multiple eigenvalues

accurate computation of the product induced singular value decomposition with applications we present a new algorithm for floating point computation of the singular value decomposition svd of the product b tau c where b and c are full row rank matrices the algorithm replaces the pair b c with an equivalent pair b c and then it uses the jacobi svd algorithm to compute the svd of the explicitly computed matrix b tau c in this way each nonzero singular value sigma is approximated with some sigma delta sigma where the relative error delta sigma sigma is up to a factor of the dimensions of order boldmath varepsilon unboldmath min delta in cal d kappa 2 delta b min delta in cal d kappa 2 delta c where cal d denotes the set of diagonal nonsingular matrices kappa 2 cdot denotes the spectral condition number and boldmath varepsilon unboldmath is the roundoff unit of floating point arithmetic the new algorithm is applied to the eigenvalue problem hm x lambda x with symmetric positive definite h and m it is shown that each eigenvalue lambda is computed with high relative accuracy and that the relative error delta lambda lambda of the computed approximation lambda delta lambda is up to a factor of the dimension of order boldmath varepsilon unboldmath min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta the new algorithm can also be used for accurate svd computation of a single matrix g that admits an accurate factorization g b tau c

on the global and cubic convergence of a quasi cyclic jacobi method in this paper we consider the global and the cubic convergence of a quasi cyclic jacobi method for the symmetric eigenvalue problem the method belongs to a class of quasi cyclic methods recently proposed by w mascarenhas mascarenhas showed that the methods from his class asymptotically converge cubically per quasi sweep one quasi sweep is equivalent to 1 25 cyclic sweeps provided the eigenvalues are simple here we prove the global convergence of our method and derive very sharp asymptotic convergence bounds in the general case of multiple eigenvalues we discuss the ultimate cubic convergence of the method and present several numerical examples which all well comply with the theory

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

computing the singular value decomposition with high relative accuracy we analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy this means that each computed singular value is guaranteed to have some correct digits even if the singular values have widely varying magnitudes this is in contrast to the absolute accuracy provided by conventional backward stable algorithms which in general only guarantee correct digits in the singular values with large enough magniturds it is of interest to compute the tiniest singular values with several correct digits because in some cases such as finite element problems and quantum mechanics it is the smallest singular values that have physical meaning and should be determined accurately by the data many recent papers have identified special classes of matrices where high relative accuracy is possible since it is not possible in general the perturbation theory and algorithms for these matrix classes have been quite different motivating us to seek a common perturbation theory and common algorithm we provide these in this paper and show that high relative accuracy is possible in many new cases as well the briefest way to describe our results is that we can compute the svd to high relative accuracy provided we can compute a high accuracy pivoted ldu decomposition we provide many examples of matrix classes permitting such an ldu decomposition

quadratic convergence estimate of scaled iterates by j symmetric jacobi method abstract this paper estimates the quadratic convergence reduction of scaled iterates by j symmetric jacobi method numer math 64 1993 241 although the method is well defined for a general definite pair h j h ht j diag im in m the paper considers the most important case when h is positive definite in that case the method is an accurate floating point eigensolver for the pair h j as such it is used in a compound algorithm for accurate floating point computation of eigenvalues and eigenvectors of a non singular indefinite symmetric matrix the new result is proved for scaled diagonally dominant matrices in the general case of multiple eigenvalues it uses frobenius norm of the off diagonal part of symmetrically scaled iteration matrix and a relative gap in the spectrum of h j it can be effectively used in connection with stopping criterion of the method especially with its one sided version

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

full block j jacobi method for hermitian matrices abstract the paper considers convergence accuracy and efficiency of a block j jacobi method the method is a proper blas 3 generalization of the known method of veselic for computing the hyperbolic singular value decomposition of rectangular matrices at each step the proposed algorithm diagonalizes the block pivot submatrix the convergence is proved for cyclic strategies which are weakly equivalent to the row cyclic strategy the relative accuracy is proved under the standard conditions numerical tests show improved performance with respect to the block oriented generalization of the original method of veselic combined with the hermitian indefinite factorization the proposed method becomes accurate and efficient eigensolver for hermitian indefinite matrices

perturbation theory for the eigenvalues of factorised symmetric matrices abstract we obtain eigenvalue perturbation results for a factorised hermitian matrix h gjg where j 2 i and g has full row rank and is perturbed into g g where g is small with respect to g this complements the earlier results on the easier case of g with full column rank applied to square factors g our results help to identify the so called quasidefinite matrices as a natural class on which the relative perturbation theory for the eigensolution can be formulated in a way completely analogous to the one already known for positive definite matrices

a local post processing technique for improving the accuracy in mixed finite element approximations a general simple post processing technique is given in this paper under certain conditions if an algorithm approximating a function u also provides a good approximation for its gradient and for some locally defined projection pu then by this method a better approximation for u can be easily obtained essentially the leading term of the error between u and its original approximation can be computed it is shown how the method may be applied to various mixed type finite element methods

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

triangular processor array for computing singular values a triangular processor array for computing the singular values of an m n m n matrix is proposed a jacobi type algorithm is used to first triangularize the given matrix and then diagonalize the resultant triangular form the requirements are 14n2 o n processors and o m ns time where s denotes the number of sweeps the triangular array can be extended to a rectangular one with 12mn o m processors for the computational of singular vectors

singular values doubly stochastic matrices and applications abstract the hadamard square of any square matrix a is bounded above and below by some doubly stochastic matrices times the square of the largest and the smallest singular values of a applications to graphs permanents and eigenvalue perturbations are discussed

a posteriori error estimates for mixed finite element approximations of elliptic problems we derive residual based a posteriori error estimates of the flux in l 2 norm for a general class of mixed methods for elliptic problems the estimate is applicable to standard mixed methods such as the raviart thomas nedelec and brezzi douglas marini elements as well as stabilized methods such as the galerkin least squares method the element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

an overview of parallel algorithms for the singular value and symmetric eigenvalue problems abstract solving dense symmetric eigenvalue problems and computing singular value decompositions continue to be two of the most dominating tasks in numerous scientific applications with the advent of multiprocessor computer systems the design of efficient parallel algorithms to determine these solutions becomes of paramount importance in this paper we discuss two fundamental approaches for determining the eigenvalues and eigenvectors of dense symmetric matrices on machines such as the alliant fx 8 and cray x mp one approach capitalizes upon the inherent parallelism offered by jacobi methods while the other relies upon an efficient reduction to tridiagonal form via householder s transformations followed by a multisectioning technique to obtain the eigenvalues and eigenvectors of the corresponding symmetric tridiagonal matrix for the singular value decomposition we discuss an efficient method for rectangular matrices in which the number of rows is substantially larger or smaller than the number of columns this scheme performs an initial orthogonal factorization using block householder transformation followed by a parallel one sided jacobi method to obtain the singular values and singular vectors of the resulting upper triangular matrix exceptional performance for this svd scheme is demonstrated for tall matrices of full or deficient rank having clustered or multiple singular values a hybrid method that combines one and two sided jacobi schemes is also discussed performance results for each of the above algorithms on the alliant fx 8 and cray x mp computer systems will be presented with particular emphasis given to speedups obtained over such classical eispack algorithms

relative perturbation bound for invariant subspaces of graded indefinite hermitian matrices abstract we give a bound for the perturbations of invariant subspaces of graded indefinite hermitian matrix h d ad which is perturbed into h h d a a d such relative perturbations include an important case where h is given with an element wise relative error application of our bounds requires only the knowledge of the size of relative perturbation a and not the perturbation a itself this typically occurs when data are given with relative uncertainties when the matrix is being stored into computer memory and when analyzing some numerical algorithms subspace perturbations are measured in terms of perturbations of angles between subspaces and our bound is therefore a relative variant of the well known davis kahan sin theorem our bounds generalize some of the recent relative perturbation results

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

relative perturbation theory for hyperbolic eigenvalue problem we give relative perturbation bounds for eigenvalues and perturbation bounds for eigenspaces of a hyperbolic eigenvalue problem hx jx where h is a positive definite matrix and j is a diagonal matrix of signs we consider two types of perturbations when a graded matrix h d ad is perturbed in a graded sense to h h d a a d and the multiplicative perturbations of the form h h i e h i e our bounds are simple to compute compare well to the classical results and can be used when analyzing numerical algorithms

two sided hyperbolic svd abstract in this paper we propose the two sided hyperbolic svd 2hsvd for square matrices i e a u v where u and v are j unitary j diag 1 and is a real diagonal matrix of double hyperbolic singular values we show that with some natural conditions such decomposition exists without the use of hyperexchange matrices in other words u and v are really j unitary with regard to j and not some matrix j which is permutationally similar to matrix j we provide full characterization of 2hsvd and completely relate it to the semidefinite j polar decomposition

j orthogonal matrices properties and generation a real square matrix q is j orthogonal if qtjq j where the signature matrix j diag pm 1 j orthogonal matrices arise in the analysis and numerical solution of various matrix problems involving indefinite inner products including in particular the downdating of cholesky factorizations we present techniques and tools useful in the analysis application and construction of these matrices giving a self contained treatment that provides new insights first we define and explore the properties of the exchange operator which maps j orthogonal matrices to orthogonal matrices and vice versa then we show how the exchange operator can be used to obtain a hyperbolic cs decomposition of a j orthogonal matrix directly from the usual cs decomposition of an orthogonal matrix we employ the decomposition to derive an algorithm for constructing random j orthogonal matrices with specified norm and condition number we also give a short proof of the fact that j orthogonal matrices are optimally scaled und

highly accurate symmetric eigenvalue decomposition and hyperbolic svd let g be a m n real matrix with full column rank and let j be a n n diagonal matrix of signs jii 1 1 the hyperbolic singular value decomposition hsvd of the pair g j is defined as g u v 1 where u is orthogonal is positive definite diagonal and v is j orthogonal matrix vtjv j we analyze when it is possible to compute the hsvd with high relative accuracy this essentially means that each computed hyperbolic singular value is guaranteed to have some correct digits even if they have widely varying magnitudes we show that one sided j orthogonal jacobi method method computes the hsvd with high relative accuracy more precisely let b gd 1 where d is diagonal such that the columns of b have unit norms essentially we show that the computed hyperbolic singular values of the pair g j will have log10 e min b correct decimal digits where e is machine precision we give the necessary relative perturbation bounds and error analysis of the algorithm our numerical tests confirmed all theoretical results for the symmetric non singular eigenvalue problem hx x we analyze the two step algorithm which consists of factorization h gjgt followed by the computation of the hsvd of the pair g j here g is square and non singular let b dg where d is diagonal such that the rows of b have unit norms and let b be defined as above essentially we show that the computed eigenvalues of h will have log10 e 2min b e min b correct decimal digits this accuracy can be much higher then the one obtained by the classical qr and jacobi methods applied to h where the accuracy depends on the spectral condition number of h particularly if the matrices b and b are well conditioned and we are interested in the accurate computation of tiny eigenvalues again we give the perturbation and error bounds and our theoretical predictions are confirmed by a series of numerical experiments we also give the corresponding results for eigenvectors and hyperbolic singular vectors

rounding error and perturbation bounds for the cholesky and ldl t factorizations abstract componentwise rounding error and perturbation bounds for the cholesky and ldl t factorizations are derived the precision of the computed cholesky and ldl t factorizations is given

floating point perturbations of hermitian matrices abstract we consider the perturbation properties of the eigensolution of hermitian matrices for the matrix entries and the eigenvalues we use the realistic floating point error measure a a recently demmel and veselic considered the same problem for a positive definite matrix h showing that the floating point perturbation theory holds with constants depending on the condition number of the matrix a dhd where aii 1 and d is a diagonal scaling we study the general hermitian case along the same lines thus obtaining new classes of well behaved matrices and matrix pairs our theory is applicable to the already known class of scaled diagonally dominant matrices as well as to matrices given by factors like those in symmetric indefinite decompositions we also obtain norm estimates for the perturbations of the eigenprojections and show that some of our techniques extend to non hermitian matrices however unlike in the positive definite case we are still unable to describe simply the set of all well behaved hermitian matrices

on computing accurate singular values and eigenvalues of matrices with acyclic graphs abstract it is known that small relative perturbations in the entries of a bidiagonal matrix only cause small relative perturbations in its singular values independentof the values of the matrix entries in this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic we also show how to compute the singular values of such a matrix to high relative accuracy the same algorithm can compute eigenvalues of symmetric matrices with acyclic graphs with tiny componentwise relative backward error this class includes tridiagonal matrices arrow matrices and exponentially many others

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

an a posteriori error estimate for the variable degree raviart thomas method we propose a new a posteriori error analysis of the variable degree hybridized version of the raviart thomas method for second order elliptic problems on conforming meshes made of simplexes we establish both the reliability and eciency of the estimator for the l2 norm of the error of the ux we also nd the explicit dependence of the estimator on the order of the local spaces k 0 the only constants that are not explicitly computed are those depending on the shape regularity of the simplexes in particular the constant of the local eciency in equality is proven to behave like k 2 3 2 however we present numer ical experiments suggesting that such constant is actually independent of k

the hyberbolic singular value decomposition and applications a new generalization of the singular value decomposition svd the hyperbolic svd is advanced and its existence is established under mild restrictions the hyperbolic svd accurately and efficiently finds the eigenstructure of any matrix that is expressed as the difference of two matrix outer products signal processing applications where this task arises include the covariance differencing algorithm for bearing estimation in sensor arrays sliding rectangular windowing and array calibration two algorithms for effecting this decomposition are detailed one is sequential and follows a similar pattern to the sequential bidiagonal based svd algorithm the other is for parallel implementation and mimics hestenes svd algorithm 1958 numerical examples demonstrate that like its conventional counterpart the hyperbolic svd exhibits superior numerical behavior relative to explicit formation and solution of the normal equations furthermore the hyperbolic svd applies in problems where the conventional svd cannot be employed

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

a jacobi eigenreduction algorithm for definite matrix pairs we propose a jacobi eigenreduction algorithm for symmetric definite matrix pairsa j of small to medium size real symmetric matrices withj 2 i j diagonal neitherj nora itself need be definite our jacobi reduction works only on one matrix and usesj orthogonal elementary congruences which include both trigonometric and hyperbolic rotations and preserve the symmetry throughout the process for the rotation parameters only the pivotal elements of the current matrix are needed which facilitates parallelization we prove the global convergence of the method the quadratic convergence was observed in all experiments we apply our method in two situations i eigenreducing a single real symmetric matrix and ii eigenreducing an overdamped quadratic matrix pencil in both cases our method is preceded by a symmetric indefinite decomposition and performed in its one sided variant on the thus obtained factors our method outdoes the standard methods like standard jacobi orqr ql in accuracy in spite of the use of hyperbolic transformations which are not orthogonal a theoretical justification of this behaviour is made elsewhere the accuracy advantage of our method can be particularly drastic if the eigenvalues are of different order in addition in working with quadratic pencils our method is shown to either converge or to detect non overdampedness

an algebro geometric proof of witten s conjecture we present a new proof of witten s conjecture the proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and hurwitz numbers enumerating ramified coverings of the 2 sphere

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

on quadratic convergence bounds for thej symmetric jacobi method this paper deals with quadratic convergence estimates for the serialj symmetric jacobi method recently proposed by veselic the method is characterized by the use of orthogonal and hyperbolic plane rotations using a new technique recently introduced by hari we prove sharp quadratic convergence bounds in the general case of multiple eigenvalues

non uniform order mixed fem approximation implementation post processing computable error bound and adaptivity the present work provides a straightforward and focused set of tools and corresponding theoretical support for the implementation of an adaptive high order finite element code with guaranteed error control for the approximation of elliptic problems in mixed form the work contains details of the discretisation using non uniform order mixed finite elements of arbitrarily high order a new local post processing scheme for the primary variable the use of the post processing scheme in the derivation of new fully computable bounds for the error in the flux variable and an hp adaptive refinement strategy based on the a posteriori error estimator numerical examples are presented illustrating the results obtained when the procedure is applied to a challenging problem involving a ten pole electric motor with singularities arising from both geometric features and discontinuities in material properties the procedure is shown to be capable of producing high accuracy numerical approximations with relatively modest numbers of unknowns

error estimators for a mixed method we present two a posteriori error estimators for elliptic partial differential equations when we use a mixed method one is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem we prove their equivalence with the norm of the error when the data is locally smooth

a note on the existence of the hyperbolic singular value decomposition abstract in this note we present a new matrix decomposition for a matrix pair a b with a hermitian the hyperbolic svd comes as a special case of the decomposition with a set to be the signature matrix

relative perturbation theory for hyperbolic singular value problem we give relative perturbation bounds for singular values and perturbation bounds for singular subspaces of a hyperbolic singular value problem for the pair g j where g is a full rank matrix and j is a diagonal matrix of signs we consider two types of relative perturbations g g b b d and g g d b b depending whether g has full column or full row rank respectively in both cases we also consider relative element wise perturbations of g which typically occur in numerical computations

polynomial averages in the kontsevich model we obtain in closed form averages of polynomials taken over hermitian matrices with the gaussian measure involved in the kontsevich integral and prove a conjecture of witten enabling one to express analogous averages with the full cubic potential measure as derivatives of the partition function with respect to traces of inverse odd powers of the external argument the proofs are based on elementary algebraic identities involving a new set of invariant polynomials of the linear group closely related to the general schur functions

on pairs of almost diagonal matrices abstract the known results on almost diagonal hermitian matrices are generalized to deal with pairs of almost diagonal hermitian matrices the pairs are assumed to be definite or positive definite the obtained estimates are used to justify the failure of the quadratic convergence of jacobi methods for the generalized eigenvalue problem ax bx

a note on the hyperbolic singular value decomposition abstract a procedure is presented for constructing the hyperbolic singular value decomposition of a matrix a with respect to a signature matrix when the rank of a is less than the rank of a h a the decomposition preserves the structure of the isotropic subspace of the column space of a

weil petersson volumes and intersection theory on the moduli space of curves in this paper we establish a relationship between the weil petersson volume vgin b of the moduli space mg n b of hyperbolic riemann surfaces with geodesic boundary components of lengths b bn and the intersection numbers of tauto logical classes on the moduli space mg n of stable curves as a result by using the recursive formula for vg n b obtained in 22 we derive a new proof of the virasoro constraints for a point this result is equivalent to the witten kontsevich formula 14

a note on one sided jacobi algorithm we propose a one sided or implicit variant of the jacobi diagonalization algorithm for positive definite matrices the variant is based on a previous cholesky decomposition and currently uses essentially one square array which on output contains the matrix of eigenvectors thus reaching the storage economy of the symmetric ql algorithm the current array is accessed only columnwise which makes the algorithm attractive for various parallelized and or vectorized implementations even on a serial computer our algorithm shows improved efficiency in particular if the cholesky step is made with diagonal pivoting on matrices of ordern 25 200 our algorithm is about 2 3 5 times slower than ql thus being almost on the halfway between the standard jacobi and ql algorithms the previous cholesky decomposition can be performed with higher precision without extra time or storage costs thus offering considerable gains in accuracy with highly conditioned input matrices

on the constants in hp finite element trace inverse inequalities we derive inverse trace inequalities for hp finite elements utilizing orthogonal polynomials we show how to derive explicit expressions for the constants when considering triangular and tetrahedral elements we also discuss how to generalize this technique to the general d simplex

a jacobi like method for solving algebraic riccati equations on parallel computers an algorithm to solve continuous time algebraic riccati equations through the hamiltonian schur form is developed it is an adaption for hamiltonian matrices of an asymmetric jacobi method of eberlein 1987 it uses unitary symplectic similarity transformations and preserves the hamiltonian structure of the matrix each iteration step needs only local information about the current matrix thus admitting efficient parallel implementations on certain parallel architectures convergence performance of the algorithm is compared with the hamiltonian jacobi algorithm of byers 1990 the numerical experiments suggest that the method presented here converges considerably faster for non hermitian hamiltonian matrices than byers hamiltonian jacobi algorithm besides that numerical experiments suggest that for the method presented here the number of iterations needed for convergence can be predicted by a simple function of the matrix size

a posteriori error estimation for lowest order raviart thomas mixed finite elements for the lowest order raviart thomas mixed finite element we derive an a posteriori error estimator that provides actual guaranteed computable upper bounds on the error in the flux variable regardless of jumps in the material coefficients across interfaces moreover the estimator is efficient in that it provides a local lower bound on the error up to a constant that is independent of the solution and the local mesh size the estimator may be evaluated at virtually no additional cost compared to the evaluation of the finite element approximation itself

quadratic convergence of scaled matrices in jacobi method a quadratic convergence bound for scaled jacobi iterates is proved provided the initial symmetric positive definite matrix has simple eigenvalues the bound is expressed in terms of the off norm of the scaled initial matrix and the minimum relative gap in the spectrum the obtained result can be used to predict the stopping moment in the two sided and especially in the one sided jacobi method

absolute and relative perturbation bounds for invariant subspaces of matrices absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices in the two norm and in the frobenius norm the absolute bounds can be considered extensions of davis and kahan s sin theorem to general matrices and invariant subspaces of any dimension the relative bounds are the most general relative bounds for invariant subspaces because they place no restrictions on the matrix or the perturbation when the perturbed subspace has dimension one the relative bound is implied by the absolute bound

a new look at pencils of matrix valued functions abstract matrix pencils depending on a parameter and their canonical forms under equivalence are discussed the study of matrix pencils or generalized eigenvalue problems is often motivated by applications from linear differential algebraic equations daes based on the weierstrass kronecker canonical form of the underlying matrix pencil one gets existence and uniqueness results for linear constant coefficients daes in order to study the solution behavior of linear daes with variable coefficients one has to look at new types of equivalence transformations this then leads to new canonical forms and new invariances for pencils of matrix valued functions we give a survey of recent results for square pencils and extend these results to nonsquare pencils furthermore we partially extend the results for canonical forms of hermitian pencils and give new canonicalforms there too based on these results we obtain new existence and uniqueness theorems for differential algebraic systems which generalize the classical results of weierstrass and kronecker

a hierarchically blocked jacobi svd algorithm for single and multiple graphics processing units we present a hierarchically blocked one sided jacobi algorithm for the singular value decomposition svd targeting both single and multiple graphics processing units gpus the blocking structure reflects the levels of the gpus memory hierarchy the algorithm may outperform magma s textttdgesvd while retaining high relative accuracy to this end we developed a family of parallel pivot strategies on the gpu s shared address space but applicable also to inter gpu communication unlike common hybrid approaches our algorithm in a single gpu setting needs a cpu for the controlling purposes only while utilizing the gpu s resources to the fullest extent permitted by the hardware when required by the problem size the algorithm in principle scales to an arbitrary number of gpu nodes the scalability is demonstrated by more than twofold speedup for sufficiently large matrices on a tesla s2050 system with four gpus versus a single fermi card

on sharp quadratic convergence bounds for the serial jacobi methods using a new technique we derive sharp quadratic convergence bounds for the serial symmetric and svd jacobi methods for the symmetric jacobi method we consider the cases of well and poorely separated eigenvalues our result implies the result proposed but not correctly proved by van kempen it also extends the well known result of wilkinson to the case of multiple eigenvalues

residual bounds for some or all singular values abstract for a given real or complex matrix we derive perturbation bounds for some or all singular values the bounds are of the same quality as corresponding residual bounds for the hermitian eigenvalue problem they are suitable for numerical computation

convergence to diagonal form of block jacobi type methods we provide sufficient conditions for the general sequential block jacobi type method to converge to the diagonal form for cyclic pivot strategies which are weakly equivalent to the column cyclic strategy given a block matrix partition a ij a i j of a square matrix mathbf a a the paper analyzes the iterative process of the form mathbf a k 1 mathbf p k mathbf a k mathbf q k a k 1 p k a k q k k ge 0 k 0 mathbf a 0 mathbf a a 0 a where mathbf p k p k and mathbf q k q k are elementary block matrices which differ from the identity matrix in four blocks two diagonal and the two corresponding off diagonal blocks in our analysis of convergence a promising new tool is used namely the theory of block jacobi operators typical applications lie in proving the global convergence of block jacobi type methods for solving standard and generalized eigenvalue and singular value problems

fully mass conservative impes schemes for incompressible two phase flow in porous media abstract in this paper we consider fully mass conservative numerical schemes for the simulation of incompressible and immiscible two phase flow in porous media with capillary pressure compared with two kinds of conventional implicit pressure explicit saturation impes schemes the new schemes deserve a merit that the conservation of mass of both phases can be obtained the total conservation equation is obtained by the summation of the discretized conservation equation for each phase this approach is quite different from the conventional impes schemes we present two kinds of fully mass conservative impes schemes to solve the coupled systems for pressure auxiliary velocity and saturation of each phase the upwind mixed finite element methods are used to solve the pressure velocity systems which can be decoupled and the problems in the decoupled systems can be proved to be well posed moreover the new schemes are unbiased and the saturation of each phase can be proved to be bounds preserving if the time step size is smaller than a certain value the new schemes can also be applied to approximate the incompressible and immiscible two phase flow in heterogeneous porous media with different capillarity pressures several interesting examples of incompressible and immiscible two phase flow in porous media are presented to demonstrate the robustness of the new algorithms

an implicit jacobi like method for computing generalized hyperbolic svd abstract in this paper we introduce a joint hyperbolic orthogonal decomposition of two matrices which we call a generalized hyperbolic singular value decomposition or ghsvd this decomposition can be used for finding the eigenvalues and eigenvectors of a symmetric definite pencil x t x y t y where diag 1 we also present an implicit jacobi like method for computing this ghsvd

convergence of scaled iterates by the jacobi method abstract a quadratic convergence bound for scaled iterates by the serial jacobi method for hermitian positive definite matrices is derived by scaled iterates we mean the matrices diag h k 1 2 h k diag h k 1 2 where h k k 0 are matrices generated by the method the bound is obtained in the general case of multiple eigenvalues it depends on the minimum relative separation of the eigenvalues

computing the singular value decomposition on the connection machine consideration is given to the computation of the singular value decomposition svd on the connection machine cm brief descriptions of the lisp language and some typical matrix manipulating functions are given implementation details of various jacobi svd algorithms on an 8192 processor cm are presented for n n matrices where n

a priori and computable a posteriori error estimates for an hdg method for the coercive maxwell equations abstract in this paper we present and analyze a hybridizable discontinuous galerkin hdg method for a mixed curl curl formulation of the steady state coercive maxwell equations with a discrete sobolev embedding type estimates for the discontinuous polynomials we provide a priori error estimates for the electric field and the lagrange multiplier in the energy norm with the smooth or minimal regularity assumption on the exact solution we have optimal convergence rate for the electric field and the lagrange multiplier in the energy norm the a priori error estimate for the electric field in the l 2 norm is also obtained by the duality argument and the approximation is also optimal for the electric field in the l 2 norm moreover by employing suitable helmholtz decompositions of the error together with the upper bound estimate for the lagrange multiplier we provide a computable residual based a posteriori error estimator which is derived based on the error measured in terms of a mesh dependent energy norm the efficiency of the a posteriori error estimator is also established three dimensional numerical results testing the performance of the a priori and a posteriori error estimates for the maxwell equations are given

accuracy of the kogbetliantz method for scaled diagonally dominant triangular matrices the paper proves that kogbetliantz method computes all singular values of a scaled diagonally dominant triangular matrix which can be well scaled from both sides symmetrically to high relative accuracy special attention is paid to deriving sharp accuracy bounds for one step one batch and one sweep of the method by a simple numerical test it is shown that the methods based on bidiagonalization are generally not accurate on that class of well behaved matrices

convergence of adaptive discontinuous galerkin approximations of second order elliptic problems a residual type a posteriori error estimator is introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem with dirichlet neumann type boundary conditions an adaptive algorithm using this estimator together with specific marking and refinement strategies is constructed and shown to achieve any specified error level in the energy norm in a finite number of cycles the convergence rate is in effect linear with a guaranteed error reduction at every cycle results of numerical experiments are presented

a bound for the condition of a hyperbolic eigenvector matrix the hyperbolic eigenvector matrix is a matrix x which simultaneously diagonalizes the pair h j where h is hermitian positive definite and j diag 1 such that x hx and x jx j we prove that the spectral condition of x x is bounded byk x mink d hd where the minimum is taken over all non singular matrices d which commute with j this bound is attainable and it can be simply computed similar results hold for other signature matrices j like in the discretized klein gordon equation

block oriented j jacobi methods for hermitian matrices the paper describes a way how one sided jacobi type algorithm of veselic for computing the hyperbolic singular value decomposition of rectangular matrices can be modified to work with blocks the proposed modification preserves the relative accuracy property of the original algorithm and essentially improves its performance special attention is devoted to proving the global convergence of the method under some important classes of block oriented pivot strategies as numerical tests indicate the block oriented j jacobi methods combined with the hermitian indefinite factorization become efficient and accurate eigensolvers for hermitian indefinite matrices

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

existence of the hyperbolic singular value decomposition abstract the hyperbolic singular value decomposition is defined on a general n m matrix and an m m signature matrix pair it is employed in finding the eigenstructure of any matrix that is expressed as the difference of two matrix outer products such differences arise in signal processing applications in the context of the covariance differencing the hyperbolic svd applies in problems where the conventional svd cannot be employed the existence of the hyperbolic singular value decomposition is here extended to the most general case where neither the general matrix nor the matrix product is assumed full rank

a parallel jacobi diagonalization algorthm for a loop multiple processor system a parallel algorithm for the solution of the eigen value problem by the jacobi method is described at each step of the algorithm n 2 off diagonal elements of the n xn matrix are annihilated simultaneously furthermore because of the regular pattern of interprocessor communication this algorithm is suitable for efficient implementation on a simple loop multiple processor system

matrix models and a proof of the open analog of witten s conjecture in a recent work r pandharipande j p solomon and the second author have initiated a study of the intersection theory on the moduli space of riemann surfaces with boundary they conjectured that the generating series of the intersection numbers satisfies the open kdv equations in this paper we prove this conjecture our proof goes through a matrix model and is based on a kontsevich type combinatorial formula for the intersection numbers that was found by the second author

implicit standard jacobi gives high relative accuracy we prove that the jacobi algorithm applied implicitly on a decomposition a xdx t of the symmetric matrix a where d is diagonal and x is well conditioned computes all eigenvalues of a to high relative accuracy the relative error in every eigenvalue is bounded by o epsilon kappa x where epsilon is the machine precision and kappa x equiv x 2 cdot x 1 2 is the spectral condition number of x the eigenvectors are also computed accurately in the appropriate sense we believe that this is the first algorithm to compute accurate eigenvalues of symmetric indefinite matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations

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

efficient implementation of jacobi s diagonalization method on the dap the dap architecture brings into consideration the jacobi method where several non interacting rotations can be performed in parallel however the design of the algorithm is crucial in a parallel environment in this paper we shall consider two techniques specifically designed to reduce the organisation and the arithmetic components of the parallel jacobi method

a proof of convergence for two parallel jacobi svd algorithms the authors consider two parallel jacobi algorithms due to r p brent et al j vlsi comput syst vol 1 p 242 70 1985 and f t luk 1986 j lin alg applic vol 77 p 259 73 for computing the singular value decomposition of an n n matrix by relating the algorithms to the cyclic by rows jacobi method they prove convergence of the former for odd n and of the latter for any n the authors also give a nonconvergence example for the former method for all even n or 4