a quadratically convergent parallel jacobi process for diagonally dominant matrices with nondistinct eigenvalues abstract this article presents a new jacobi like eigenvalue algorithm for non hermitian almost diagonal n n matrices in each step n 2 submatrices of order 2 are diagonalized the precautions for the multiple eigenvalues are based on theorems of fan and hoffman 1954 and wilkinson 1961 the proof of the quadratic convergence generalizes our previous result for distinct eigenvalues the convergence theorem is pessimistic concerning the region of attraction to a diagonal the local information structure makes the process suitable for parallelization on a hypercube or a systolic array

joint angle and delay estimation using shift invariance techniques in a multipath communication scenario it is often relevant to estimate the directions and relative delays of each multipath ray we derive a closed form subspace based method for the simultaneous estimation of these parameters from an estimated channel impulse response using knowledge of the transmitted pulse shape function the algorithm uses a two dimensional 2 d esprit like shift invariance technique to separate and estimate the phase shifts due to delay and direction of incidence with automatic pairing of the two parameter sets improved resolution is obtained by enlarging the data matrix with shifted and conjugated copies of itself

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

an inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems we discuss an inverse free highly parallel spectral divide and conquer algorithm it can compute either an invariant subspace of a nonsymmetric matrix a or a pair of left and right deflating subspaces of a regular matrix pencil a lambda b this algorithm is based on earlier ones of bulgakov godunov and malyshev but improves on them in several ways this algorithm only uses easily parallelizable linear algebra building blocks matrix multiplication and qr decomposition but not matrix inversion similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function which requires matrix inversion and is faster but can be less stable than the new algorithm

an analytical constant modulus algorithm iterative constant modulus algorithms such as godard 1980 and cma have been used to blindly separate a superposition of cochannel constant modulus cm signals impinging on an antenna array these algorithms have certain deficiencies in the context of convergence to local minima and the retrieval of all individual cm signals that are present in the channel we show that the underlying constant modulus factorization problem is in fact a generalized eigenvalue problem and may be solved via a simultaneous diagonalization of a set of matrices with this new analytical approach it is possible to detect the number of cm signals present in the channel and to retrieve all of them exactly rejecting other non cm signals only a modest amount of samples is required the algorithm is robust in the presence of noise and is tested on measured data collected from an experimental set up

a quadratically convergent parallel jacobi process for diagonally dominant matrices with distinct eigenvalues abstract this paper discusses a generalization for non hermitian matrices of the jacobi eigenvalue process 1846 in each step 1 2 n pairs of nondiagonal elements are annihilated in an almost diagonal matrix with distinct eigenvalues we prove that the recursively constructed sequence of matrices converges to a diagonal matrix as in the classical jacobi method the convergence is quadratic and the process is adapted to parallel implementation on an array processor or a hypercube

s numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions an important consequence of the kato rosenblum theorem in abstract scattering theory 5 81 is the invariance up to unitary equivalence of absolutely continuous spectra under trace class perturbations the extension of this fact to higher integral dimensions was given in lo a careful inspection of trace class scattering theory reveals the crucial role of l2 boundedness of the hilbert transform in the one dimensional case in the n dimensional generalization this is replaced by the asymptotics of fourier coefficients of 2 1 x k c n 1 11 2 on the torus or equivalently by the asymptotics of s numbers of corresponding singular integrals the general functional analysis machinery developed in 9 lo for studying normed ideal perturbations of systems of hilbert space operators reduces this kind of operator theory questions to questions about the existence or non existence of quasicentral approximate units relative to normed ideals the existence of quasicentral approximate units in turn is related to commutator questions 11 j for related considerations in the quasidiagonality context see 7 the natural operators to examine are then singular integral operators in this paper we consider the fractional dimensional case the natural singular integral problem in this case is to estimate the s numbers of singular integrals with kernels yjl lx 1 in l p where 1 is a compactly supported probability measure on w of fractional dimension like hausdorff measure restricted to suitable compacta the results we

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

a symplectic acceleration method for the solution of the algebraic riccati equation on a parallel computer abstract we give a cubic acceleration method for improving the current symplectic jacobi like algorithm for computing the hamiltonian schur decomposition of a hamiltonian matrix and finding the positive semidefinite solution of the riccati equation the acceleration method can speed up the rate of convergence at the end of the symplectic jacobi like process when the norm of the current strictly j lower triangle has become sufficiently small it has high parallelism and takes o n computational time when implemented on a mesh connected n n array processor system a quantitative analysis of convergence and numerical comparisons of one jacobi sweep versus one correction step are presented

a generalization of voiculescu s theorem for normal operators to semifinite von neumann algebras abstract in this paper we provide a generalized version of voiculescu s theorem for normal operators by showing that in a von neumann algebra with separable pre dual and a faithful normal semifinite tracial weight each normal operator is an arbitrarily small max 2 norm perturbation of a diagonal operator furthermore in a countably decomposable properly infinite von neumann algebra with a faithful normal semifinite tracial weight we prove that each self adjoint operator can be diagonalized modulo normed ideals satisfying a natural condition an analogue for nuclear c algebras of voiculescu s absorption theorem 33 theorem 2 4 is also proved in the case of semifinite factors

a condition for diagonalization modulo arbitrary norm ideals abstract we give a spectral condition which is sufficient for the simultaneous diagonalization of a commuting tuple of self adjoint operators modulo a given norm ideal for diagonalization modulo certain norm ideals this condition is also necessary while for other norm ideals this condition seems to be close to being necessary moreover this condition is easy to verify in applications

diagonalization modulo a class of orlicz ideals abstract we consider a class of norm ideals c g which are non commutative analogues of orlicz spaces and which satisfy a previously introduced condition called qk we give a spectral condition which is necessary and sufficient for a commuting tuple of self adjoint operators a a 1 a n to be simultaneously diagonalizable modulo c g

on measures of nonnormality of matrices abstract this paper discusses several measures of nonnormality of matrices i e functions v cn n r plus where v a 0 iff a is normal besides measures already in the literature we introduce new ones and give comparisons between them at length some of these comparisons e g c14 c15 and c17 manifest well known phenomena of ill conditioned eigenproblems

a parallel implementation of the nonsymmetric qr algorithm for distributed memory architectures one approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the qr algorithm not long ago this was widely considered to be a hopeless task recent efforts have led to significant advances although the methods proposed up to now have suffered from scalability problems this paper discusses an approach to parallelizing the qr algorithm that greatly improves scalability a theoretical analysis indicates that the algorithm is ultimately not scalable but the nonscalability does not become evident until the matrix dimension is enormous experiments on the intel paragon system the ibm sp2 supercomputer the sgi origin 2000 and the intel asci option red supercomputer are reported

on the irreducible representations of the specializations in characteristics 2 and 3 of the generic hecke algebra of type f4 abstract a complete determination of the irreducible modules of specialized hecke algebras of type f4 with respect to specializations with equal parameters has been obtained by m geck and k lux 1991 manuscripta math 70 285 306 for all characteristics a similar determination for specializations with v u2 and v u4 has been obtained by k bremke 1994 manuscripta math 83 331 346 in another paper 1999 j algebra 218 654 671 the authors determined the irreducible modules for all remaining specializations other than those into fields of characteristic 2 or 3 obtaining en route decompositions of the generic irreducible modules under such specializations in this paper the corresponding results for characteristic 2 or 3 are obtained again it is found that the decomposition matrices may be expressed in lower uni triangular form in all these cases and that the splitting fields are those generated by the images of the parameters

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

on the irreducible representations of the specializations of the generic hecke algebra of type f4 abstract a complete determination of the irreducible modules of specialized hecke algebras of type f4 with respect to specializations with equal parameters was obtained by m geck and k lux 1991 manuscripta math 70 285 306 a similar determination for specializations with v u2 and v u4 was obtained by k bremke 1994 manuscripta math 83 331 346 in this paper we determine the irreducible modules for all remaining specializations other than those into fields of characteristic 2 or 3 obtaining en route decompositions of the generic irreducible modules under such specializations we find that the decomposition matrices may be expressed in lower uni triangular form in all these cases

diagonalization modulo norm ideals with lipschitz estimates abstract suppose that x d is a compact metric space and that is a regular borel measure on x let lip x denote the collection of lipschitz functions on x in this paper we investigate the problem of simultaneous diagonalization of the multiplication operators m f f lip x on l 2 x modulo norm ideals of compact operators

an acceleration method for computing the generalized eigenvalue problem on a parallel computer abstract we give a cubic correction step for improving the current eigenvalue algorithms for computing the generalized schur decomposition of a regular pencil b a using a jacobi like method the correction method can be used to speed up the convergence at the end of the jacobi like process when the strictly lower triangular elements of the matrix pair a b have become sufficiently small it can be implemented in parallel on an n n square array of mesh connected processors in o n computational time a quantitative analysis of the convergence and a comparison of the complexity of one jacobi sweep versus one correction step are presented

parallel vlsi matrix pencil algorithm for high resolution direction finding algorithms to find the directions of arrival doas of multiple signals from measurements on an array of antenna doublets esprit method and their parallel vlsi implementation are discussed in particular algorithms that allow large scale pipelining and use only robust unitary transformations are identified this problem is solved by a matrix pencil approach in which the generalized eigenvalues of a pair of data matrices are determined a modified stewart jacobi approach is used for which convergence is improved and parameter computations are simplified the resulting architecture is a two layer jacobi array that can handle all the subproblems two qr factorizations two svds and a single generalized schur decomposition the mapping of the subproblems on a single parallel array of cordic processors is considered

the decomposition numbers of the hecke algebra of type letw be the finite coxeter group of typef4 andhr q be the associated hecke algebra with parameter a prime powerq defined over a valuation ringr in a large enough extension field ofq with residue class field of characteristicr in this paper ther modular decomposition numbers ofhr q are determined for allq andr such thatr does not divideq the methods of the proofs involve the study of the generic hecke algebra of typef4 over the ringa u1 2 u 1 2 of laurent polynomials in an indeterminateu1 2 and its specializations onto the ring of integers in various cyclotomic number fields substancial use of computers and computer program systems gap maple meat axe has been made

on the schur decomposition of a matrix for parallel computation an algorithm to solve the eigenproblem for nonsymmetric matrices on an n n array of mesh connected processors isomorphic to the architecture described by brent and luk for symmetric matrices is presented this algorithm is a generalization of the classical jacobi method and as such holds promise for parallel architectures the rotational parameters for the nonsymmetric case are carefully analyzed many examples of a working program simulating the parallel architecture are given with experimental evidence of quadratic convergence

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

singular integral operators and norm ideals satisfying a quantitative variant of kuroda s condition a norm ideal c is said to satisfy condition qk if there exist constants 0 t 1 and 0 b such that x k c bkt x c for every finite rank operator x and every k n where x k denotes the direct sum of k copies of x let be a regular borel measure whose support is contained in a unit cube q in rn and let kj be the singular integral operator on l2 rn with the kernel function xj yj x y 2 1 j n let qw w w be the usual dyadic decomposition of q i e qw w l is the dyadic partition of q by cubes of the size 2 l 2 l we show that if c satisfies qk and if w w2 w qw w w c where c is the dual of c 0 and w w w is any orthonormal set then k1 kn c this is a very general obstruction result for the problem of simultaneous diagonalization of commuting tuples of self adjoint operators modulo c

murnaghan nakayama rules for characters of iwahori hecke algebras of classical type in this paper we give murnaghan nakayama type formulas for computing the irreducible characters of the iwahori hecke algebras of types an 1 bn and dn our method is a generalization of a derivation of the murnaghan nakayama formula for the irreducible characters of the symmetric group given by curtis greene greene s approach is to sum up the diagonal entries of the matrices of certain cycle permutations in young s seminormal representations the analogues of the young seminormal representations for the iwahori hecke algebras of types an 1 bn and dn were given by hoefsmit

diagonalization and unitary equivalence modulo schatten p classes we give a necessary and sufficient condition for a commuting tuple a a1 an of self adjoint operators to be simultaneously diagonalizable modulo cp 1 p