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

trends in systolic and cellular computation abstract a profile is given of current research as it pertains to computational mathematics on very large scale integration vlsi array processors in this type of parallel computers the cells of the array operate in single instruction multiple data simd mode and algorithms are executed in systolic or cellular fashion the focus of the presentation is on linear algebraic techniques a systolization process is illustrated by matching an arbitrary gaxpy operation onto a fixed size square array processor two recent systolic methods for solving systems of linear equations one iterative and the other direct are described a cellular algorithm for a fast fourier transform based on a new implementation on rectangular array processors of the perfect shuffle permutation is then derived using annotated lists of recent references snapshots of active research areas are given on systolic linear solvers the singular value decomposition artificial neural networks and the simulated annealing algorithm

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

certified reduced basis method for electromagnetic scattering and radar cross section estimation we study nontrivial applications of the reduced basis method rbm for electromagnetic applications with an emphasis on scattering and the estimation of radar cross section rcs the method and several extensions are explained with two examples with different characteristics parameters that are allowed to vary within the model include frequency incident angle and measurement angle as well as the geometry of the scatterers with appropriate applications of the empirical interpolation method eim transformation of the domain configuration of perfectly matched layer exponential convergence of the reduced basis solution over the entire parameter domain is achieved moreover we demonstrate that this approach allows for the effective capture of the critical behavior in this case through shapes that minimize scattering this further highlights the robustness and quality of the greedy approximation and the reduced basis method approach

a systolic implementation of the chan foster rrqr algorithm the rank revealing qr factorization is a useful tool in many signal processing applications since it explicitly yields all the necessary information to solve rank deficient least squares problems and subset selection problems to compute signal and noise subspaces etc the authors present a systolic algorithm for computing a rank revealing qr factorization and consider the performance of the algorithm

numerical solution of nonlinear hypersingular integral equations of the peierls type in dislocation theory in this paper we analyze hypersingular integral equations of the peierls type specified on the whole real axis the method of approximate approximations maz ya proceedings of the eighth conference on the mathematics of finite elements and applications viii mafelap 1993 brunel university 1994 pp 107 104 is employed in order to produce numerical solutions of high accuracy and the results are applied to problems in the theory of dislocations the critical peierls stress is evaluated for different configurations of dislocations and for different types of the interatomic force law

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

on stable solutions for boundary reactions a de giorgi type result in dimension 4 1 we prove that every bounded stable solution of begin aligned delta 1 2 u f u 0 qquad text in mathbb r 3 end aligned is a 1d profile i e u x phi e cdot x for some e in mathbb s 2 where phi mathbb r rightarrow mathbb r is a nondecreasing bounded stable solution in dimension one equivalently stable critical points of boundary reaction problems in mathbb r d 1 mathbb r d 1 cap x d 1 ge 0 of the form begin aligned int x d 1 ge 0 frac 1 2 nabla u 2 dx dx d 1 int x d 1 0 f u dx end aligned are 1d when d 3 these equations have been studied since the 1940 s in crystal dislocations also as it happens for the allen cahn equation the associated energies enjoy a gamma convergence result to the perimeter functional in particular when f u u 3 u or equivalently when f u frac 1 4 1 u 2 2 our result implies the analogue of the de giorgi conjecture for the half laplacian in dimension 4 namely that monotone solutions are 1d note that our result is a pde version of the fact that stable embedded minimal surfaces in mathbb r 3 are planes it is interesting to observe that the corresponding statement about stable solutions to the allen cahn equation namely when the half laplacian is replaced by the classical laplacian is still unknown for d 3

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

accuracy of two svd algorithms for 2 2 triangular matrices a new algorithm for the accurate computation of the singular value decomposition of 2x2 triangular matrices is proposed the algorithm is based on voevodin formulas sharp accuracy bounds are derived by using a subtle error analysis which tracks the signs of the errors of intermediate quantities and does not neglect the non linear parts of the errors the analysis is fine tuned for the case of almost diagonal matrices the same analysis is also used to analyze the errors for the xlasv2 computational routine of lapack the error estimates of the new algorithm compare favorably to those of the lapack routine

accelerated a posteriori error estimation for the reduced basis method with application to 3d electromagnetic scattering problems we propose a new method for fast estimation of error bounds for outputs of interest in the reduced basis context efficiently applicable to real world 3d problems geometric parameterizations of complicated 2d or even simple 3d structures easily leads to affine expansions consisting of a high number of terms propto100 1000 application of state of the art techniques for computation of error bounds becomes practically impossible as a way out we propose a new error estimator inspired by the subdomain residuum method which leads to substantial savings orders of magnitude regarding online and offline computational times and memory consumption we apply certified reduced basis techniques with the newly developed error estimator to 3d electromagnetic scattering problems on unbounded domains a numerical example from computational lithography demonstrates the good performance and effectivity of the proposed estimator

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

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

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

the hyperbolic schur decomposition we propose a hyperbolic counterpart of the schur decomposition with the emphasis on the preservation of structures related to some given hyperbolic scalar product we give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices

rounding error and perturbation bounds for the indefinite qr factorization abstract indefinite qr factorization is a generalization of the well known qr factorization where q is a unitary matrix with respect to the given indefinite inner product matrix j this factorization can be used for accurate computation of eigenvalues of the hermitian matrix a g jg where g and j are initially given or naturally formed from initial data the classical example of such a matrix is a b b c c with given b and c in this paper we present the rounding error and perturbation bounds for the so called triangular case of the indefinite qr factorization these bounds fit well into the relative perturbation theory for hermitian matrices given in factorized form

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

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

on the stability of sequential updates and downdates the updating and downdating of cholesky decompositions has important applications in a number of areas there is essentially one standard updating algorithm based on plane rotations which is backward stable three downdating algorithms have been treated in the literature the linpack algorithm the method of hyperbolic transformations and chambers 1971 algorithm although none of these algorithms is backward stable the first and third satisfy a relational stability condition it is shown that relational stability extends to a sequence of updates and downdates in consequence other things being equal if the final decomposition in the sequence is well conditioned it will be accurately computed even though intermediate decompositions may be almost completely inaccurate these results are also applied to the two sided orthogonal decompositions such as the urv decomposition

a note on optimal block scaling of matrices after pointing out that two recent results on optimal blockscaling are equivalent a new short and simple proof of both results is given

relative residual bounds for indefinite hermitian matrices abstract we prove several residual bounds for relative perturbations of the eigenvalues of indefinite hermitian matrix the bounds fall into two categories the weyl type bounds and the hofmann wielandt type bounds the bounds are expressed in terms of sines of acute principal angles between certain subspaces associated with the indefinite decomposition of the given matrix the bounds are never worse than the classical residual bounds and can be much sharper in some cases the bounds generalize the existing relative residual bounds for positive definite matrices to indefinite case

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

hyperbolic householder transformations a class of transformation matrices analogous to the householder matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least squares problems these matrices which we term hyperbolic householder are shown to effect deletion or simultaneous addition and deletion of data with much less sensitivity to rounding errors than for techniques based on normal equations when the addition deletion sets are large this numerical robustness is obtained at the expense of only a modest increase in computations and when only a relatively small fraction of the data set is modified there is a decrease in required computations two applications to signal processing problems are considered first these transformations are used to obtain a square root algorithm for windowed recursive least squares filtering second the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array

redundant and on line cordic for unitary transformations two sided unitary transformations of arbitrary 2 spl times 2 matrices are needed in parallel algorithms based on jacobi like methods for eigenvalue and singular value decompositions of complex matrices this paper presents a two sided unitary transformation structured to facilitate the integrated evaluation of parameters and application of the typically required transformations using only the primitives afforded by cordic thus enabling significant speedup in the computation of these transformations on special purpose processor array architectures implementing jacobi like algorithms we discuss implementation in nonredundant cordic to motivate and lead up to implementation in the redundant and on line enhancements to cordic both variable and constant scale factor redundant cfr cordic approaches are detailed and it is shown that the transformations may be computed in 10n spl delta time where n is the data precision in bits and spl delta is a constant accounting for accumulated on line delays a more area intensive approach using a novel on line cordic encoded angle summation difference scheme reduces computation time to 6n spl delta the area time complexities involved in the various approaches are detailed

bistable boundary reactions in two dimensions in a bounded domain omega subset mathbb r 2 with smooth boundary we consider the problem delta u 0 quad rm in omega qquad frac partial u partial nu frac1 varepsilon f u quad rm on partial omega where is the unit normal exterior vector e 0 is a small parameter and f is a bistable nonlinearity such as f u sin u or f u 1 u2 u we construct solutions that develop multiple transitions from 1 to 1 and vice versa along a connected component of the boundary we also construct an explicit solution when is a disk and f u sin u

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

ground states solutions for a non linear equation involving a pseudo relativistic schrodinger operator in this paper we prove the existence regularity and symmetry of a ground state for a nonlinear equation in the whole space involving a pseudo relativistic schrodinger operator

a sin2θ theorem for graded indefinite hermitian matrices this paper gives double angle theorems that bound the change in an invariant subspace of an indefinite hermitian matrix in the graded form h d ad subject to a perturbation h h d a a d these theorems extend recent results on a definite hermitian

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

the peierls nabarro and benjamin ono equations abstract an intimate connection between the peierls nabarro equation in crystal dislocation theory and the travelling wave form of the benjamin ono equation in hydro dynamics is uncovered it is used to prove the essential uniqueness of peierls solution of the peierls nabarro equation and to give in closed form all solutions of the analogous periodic problem the latter problem is shown to be an example of global bifurcation with no secondary symmetry breaking bifurcations for a nonlinear neumann boundary value problem or equivalently for an equation involving the conjugate operator which is the hilbert transform of functions on the unit circle

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

parallel singular value decomposition of complex matrices using multidimensional cordic algorithms the singular value decomposition svd of complex matrices is computed in a highly parallel fashion on a square array of processors using kogbetliantz s analog of jacobi s eigenvalue decomposition method to gain further speed new algorithms for the basic svd operations are proposed and their implementation as specialized processors is presented the algorithms are 3 d and 4 d extensions of the cordic algorithm for plane rotations when these extensions are used in concert with an additive decomposition of 2 spl times 2 complex matrices which enhances parallelism and with low resolution rotations early on in the svd process which reduce operation count a fivefold speedup can be achieved over the fastest alternative approach

an overview of relative sin t theorems for invariant subspaces of complex matrices abstract relative perturbation bounds for invariant subspaces of complex matrices are reviewed with emphasis on bounding the sines of the largest principal angle between two subspaces i e sin theorems the goal is to provide intuition as well as an idea for why the bounds hold and why they look the way they do relative bounds have the advantage of being better at exploiting structure in a perturbation than absolute bounds therefore the reaction of subspaces to relative perturbations can be different than to absolute perturbations in particular there are certain classes of relative perturbations to which subspaces of indefinite hermitian matrices can be more sensitive than subspaces of definite matrices

unifying unitary and hyperbolic transformations abstract in this paper we describe unified formulas for unitary and hyperbolic reflections and rotations and show how these unified transformations can be used to compute a hermitian triangular decomposition r h d r of a strongly nonsingular indefinite matrix a given in the form a x 1 h x 1 x 2 h x 2 1 the unification is achieved by the introduction of signature matrices which determine whether the applicable transformations are unitary hyperbolic or their generalizations we derive formulas for the condition numbers of the unified transformations propose pivoting strategies for lowering the condition number of the transformations and present a unified stability analysis for applying the transformations to a matrix

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

g reflectors analogues of householder transformations in scalar product spaces we characterize the analogues of householder transformations in matrix groups associated with scalar products and precisely delimit their mapping capabilities given a matrix group image and vectors x y necessary and sufficient conditions are derived for the existence of a householder like analogue image such that gx y when g exists we show how it can be constructed from x and y examples of matrix groups to which these results apply include the symplectic and pseudo unitary groups

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

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

nonlinear fractional laplacian problems with nonlocal gradient terms let n 2 be a smooth bounded domain for s 1 2 1 we consider a problem of the form where 0 is a real parameter f belongs to a suitable lebesgue space and is a nonlocal gradient square term given by depending on the real parameter 0 we derive existence and non existence results the proof of our existence result relies on sharp calderon zygmund type regularity results for the fractional poisson equation with low integrability data we also obtain existence results for related problems involving different nonlocal diffusion terms

nonlinear equations for fractional laplacians ii existence uniqueness and qualitative properties of solutions this paper which is the follow up to part i concerns the equation delta s v g v 0 in mathbb r n with s in 0 1 where delta s stands for the fractional laplacian the infinitesimal generator of a l evy process when n 1 we prove that there exists a layer solution of the equation i e an increasing solution with limits pm 1 at pm infty if and only if the potential g has only two absolute minima in 1 1 located at pm 1 and satisfying g 1 g 1 0 under the additional hypothesis g 1 0 and g 1 0 we also establish its uniqueness and asymptotic behavior at infinity furthermore we provide with a concrete almost explicit example of layer solution for n geq 1 we prove some results related to the one dimensional symmetry of certain solutions in the spirit of a well known conjecture of de giorgi for the standard laplacian

long time asymptotics for evolutionary crystal dislocation models abstract we consider a family of evolution equations that generalize the peierls nabarro model for crystal dislocations they can be seen as semilinear parabolic reaction diffusion equations in which the diffusion is regulated by a fractional laplace operator of order 2 s 0 2 acting in one space dimension and the reaction is determined by a 1 periodic multi well potential we construct solutions of these equations that represent the typical propagation of n 2 equally oriented dislocations of size 1 for large times the dislocations occur around points that evolve according to a repulsive dynamical system when s 1 2 1 these solutions are shown to be asymptotically stable with respect to odd perturbations

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

parallel and adaptive high resolution direction finding recently the new class of so called subspace methods for high resolution direction finding has received a great deal of attention in the literature when a real time implementation is aimed at the computational complexity involved is known to represent a serious impediment in this paper an esprit type algorithm is developed which is fully adaptive and therefore particularly suited for real time processing furthermore a systolic array is described which allows the processing of incoming data at a rate which is independent of the problem size the algorithm is based on orthogonal transformations only estimates are computed for the angles of arrival as well as for the source signals our aim is not so much to develop yet another esprit type algorithm but rather to show that it is indeed possible to develop an algorithm that is fully parallel and adaptive this is something that has not been done before

a systolic array for recursive least squares computations a triangular processor array has been developed for implementing recursive least squares computations the throughput is o n sup 0 which means that data vectors are fed in at a rate that is independent of the problem size the corresponding least squares solutions then run out of the array at the same rate the overall efficiency is roughly 67 this array is believed to be the first to implement a complete rls algorithm in a stable and efficient manner

a few remarks about the hilbert transform a generalized integral similar to integrability b is used to study the hilbert transform onx 1 p lp r with a view to obtaining i a mollifier which commutes with the hilbert transform onxand coincides with the friedrichs mollifier onlloc1 r ii estimates for nonlinear equations iii an integral representation for the hilbert transform of a regular schwartz distribution iv a generalized multiplier representation of the hilbert transform onl1 r v an elementary proof of the injectivity of h onx

multiplicity results for a non local problem with concave and convex nonlinearities abstract we shall prove a multiplicity result for a non local problem with a super critical nonlinearity of the form 1 s u u u p 2 u u q 2 in u 0 on r n where n is a bounded domain 0 s 1 and 1 q 2 p as a consequence of our results for each p 2 there exists 0 such that for each 0 problem 1 has a sequence of solutions with a negative energy this result is new for super critical values of p we shall also explore the existence of symmetric solutions for symmetric domains our methodology is based on a variational principle established by one of the authors that allows one to deal with super critical problems when the standard euler lagrange functional is restricted to certain convex sets

a systolic array for recursive least squares computations mapping directionally weighted rls on an svd updating array a systolic algorithm array is described for recursive least squares rls estimation which achieves an o n sup 0 throughput rate with o n sup 2 parallelism the array is also useful for several other applications such as e g svd updating and kalman filtering an additional advantage is that unlike with other rls arrays it is now possible to incorporate alternative data weighting strategies such as directional weighting without sacrificing speed

a unified algebraic transformation approach for parallel recursive and adaptive filtering and svd algorithms in this paper a unified algebraic transformation approach is presented for designing parallel recursive and adaptive digital filters and singular value decomposition svd algorithms the approach is based on the explorations of some algebraic properties of the target algorithms representations several typical modern digital signal processing examples are presented to illustrate the applications of the technique they include the cascaded orthogonal recursive digital filter the givens rotation based adaptive inverse qr algorithm for channel equalization and the qr decomposition based svd algorithms all three examples exhibit similar throughput constraints there exist long feedback loops in the algorithms signal flow graph representation and the critical path is proportional to the size of the problem applying the proposed algebraic transformation techniques parallel architectures are obtained for all three examples for cascade orthogonal recursive filter retiming transformation and orthogonal matrix decompositions or pseudo commutativity are applied to obtain parallel filter architectures with critical path of five givens rotations for adaptive inverse qr algorithm the commutativity and associativity of the matrix multiplications are applied to obtain parallel architectures with critical path of either four givens rotations or three givens rotations plus two multiply add operations whichever turns out to be larger for svd algorithms retiming and associativity of the matrix multiplications are applied to derive parallel architectures with critical path of eight givens rotations the critical paths of all parallel architectures are independent of the problem size as compared with being proportional to the problem size in the original sequential algorithms parallelism is achieved at the expense of slight increase or the same for the svd case in the algorithms computational complexity

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

minimum rank skew hermitian solutions to the matrix approximation problem in the spectral norm in this paper we discuss the following two minimum rank matrix approximation problems in the spectral norm i for given a a h c m i m b c m i n determining x s 1 such that rank x min y s 1 rank y s 1 y y h c n i n a b y b h 2 min ii for given a a h c m i m b c m i n determining x s 2 such that rank x min y s 2 rank y s 2 y y h c n i n a b y b h 2 min by applying the norm preserving dilation theorem the hermitian type skew hermitian type generalized singular value decomposition hgsvd shgsvd we characterize the expressions of the minimum rank and derive a general form of minimum rank skew hermitian solutions to the matrix approximation problem

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

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

compensated optimal grids for elliptic boundary value problems a method is proposed which allows to efficiently treat elliptic problems on unbounded domains in two and three spatial dimensions in which one is only interested in obtaining accurate solutions at the domain boundary the method is an extension of the optimal grid approach for elliptic problems based on optimal rational approximation of the associated neumann to dirichlet map in fourier space it is shown that using certain types of boundary discretization one can go from second order accurate schemes to essentially spectrally accurate schemes in two dimensional problems and to fourth order accurate schemes in three dimensional problems without any increase in the computational complexity the main idea of the method is to modify the impedance function being approximated to compensate for the numerical dispersion introduced by a small finite difference stencil discretizing the differential operator on the boundary we illustrate how the method can be efficiently applied to nonlinear problems arising in modeling of cell communication

a recursive three stage least squares method for large scale systems of simultaneous equations a new numerical method is proposed that uses the qr decomposition and its variants to derive recursively the three stage least squares 3sls estimator of large scale simultaneous equations models sem the 3sls estimator is obtained sequentially once the underlying model is modified by adding or deleting rows of data a new theoretical pseudo sem is developed which has a non positive definite dispersion matrix and is proved to yield the 3sls estimator that would be derived if the modified sem was estimated afresh in addition the computation of the iterative 3sls estimator of the updated observations sem is considered the new recursive method utilizes efficiently previous computations exploits sparsity in the pseudo sem and uses as main computational tool orthogonal and hyperbolic matrix factorizations this allows the estimation of large scale sems which previously could have been considered computationally infeasible to tackle numerical trials have confirmed the effectiveness of the new estimation procedures the new method is illustrated through a macroeconomic application

strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions we present some comparison results for solutions to certain non local elliptic and parabolic problems that involve the fractional laplacian operator and mixed boundary conditions given by a zero dirichlet datum on part of the complementary of the domain and zero neumann data on the rest these results represent a non local generalization of a hopf s lemma for elliptic and parabolic problems with mixed conditions in particular we prove the non local version of the results obtained by davila and davila and dupaigne for the classical case s 1 in 23 24 respectively

stable subspace tracking algorithm based on a signed urv decomposition subspace estimation and tracking are of fundamental importance in many signal processing algorithms the class of schur subspace estimators provides a complete parametrization of all principal subspace estimates defined as the column spans of corresponding low rank matrix approximants that lie within a specified 2 norm distance of a given matrix the parametrization is found in terms of a two sided hyperbolic decomposition hyperbolic urv or hurv which can be computed using hyperbolic rotations unfortunately such rotations are commonly associated with numerical instabilities in this paper we present a numerically stable non iterative algorithm to compute the hurv called the signed urv surv algorithm we show that this algorithm implicitly imposes certain constraints on the hurv such that important norm bounds that guarantee stability are satisfied the constraints also restrict the parametrization of the subspace estimate such that it becomes close to the principal subspace provided by the svd which is a special case within this class the complexity of the algorithm is of the same order as that of a qr update updating and downdating are of the same complexity and are both numerically stable surv is proven to provide rank estimates consistent with the svd with the same rank threshold it can replace an svd where only subspace estimation is needed typical applications would e g be the detection of the number of signals in array signal processing and subspace estimation for source separation and interference mitigation such as the first step in music and esprit type algorithms simulation results demonstrate the numerical stability and confirm that this algorithm provides exact rank estimates and good principal subspace estimates as compared to the svd

certified dual corrected radiation patterns of phased antenna arrays by offline online order reduction of finite element models this paper presents a fast numerical method for computing certified far field patterns of phased antenna arrays over broad frequency bands as well as wide ranges of steering and look angles the proposed scheme combines finite element analysis dual corrected model order reduction and empirical interpolation to assure the reliability of the results improved a posteriori error bounds for the radiated power and directive gain are derived both the reduced order model and the error bounds algorithm feature offline online decomposition a real world example is provided to demonstrate the efficiency and accuracy of the suggested approach

on the qr algorithm and updating the svd and the urv decomposition in parallel abstract a jacobi type updating algorithm for the svd or the urv decomposition is developed which is related to the qr algorithm for the symmetric eigenvalue problem the algorithm employs one sided transformations and therefore provides a cheap alternative to earlier developed updating algorithms based on two sided transformations the present algorithm as well as the corresponding systolic implementation is therefore rpughly twice as fast as the former method while the tracking properties are preserved the algorithm is also extended to the two matrix q svd or qurv case finally the differences with a number of closely related algorithms that have recently been proposed are discussed

optimal block diagonal scaling of block 2 cyclic matrices abstract we describe a class of optimal block diagonal scalings preconditionings of a symmetric positive definite block 2 cyclic matrix generalizing a result of forsythe and strauss 1 for point 2 cyclic matrices

on efficient implementations of kogbetliantz s algorithm for computing the singular value decomposition in this paper we compare several implementations of kogbetliantz s algorithm for computing the svd on sequential as well as on parallel machines comparisons are based on timings and on operation counts the numerical accuracy of the different methods is also analyzed

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

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

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

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

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

order recursive factorization of the pseudoinverse of a covariance matrix a numerically reliable algorithm is developed to recursively update the square root of a pseudoinverse matrix using the square root of a lower dimension pseudoinverse matrix the numerical computations are based on a generalized singular value decomposition which is used to do a canonical correlation analysis an operation count is given for sequential and parallel implementation of a partitioned order recursive algorithm these methods are useful for covariance analysis to determine the contributions due to various modeling errors in the design of a kalman filter

computing the generalized singular value decomposition we present a new numerical method for computing the gsvd 36 27 of two matrices a and b this method is a variation on paige s method 30 it differs from previous algorithms in guaranteeing both backward stability and con vergence there are two innovations the first is a new pre processing step which reduces a and b to upper triangular forms satisfying certain rank conditions the second is a new 2 by 2 triangular gsvd algorithm which constitutes the inner loop of paige s method we present proofs of stability and convergence of our method and demonstrate examples on which all previous algorithms fail

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

linear convergence of the row cyclic jacobi and kogbetliantz methods linear convergence of the row cyclic jacobi and kogbetliantz methods can be guaranteed if certain constraints concerning the angles of rotations are implemented unlike the results of forsythe and henrici convergence can be achieved without under rotations