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

adaptive estimation of eigensubspace in a recent work we recast the problem of estimating the minimum eigenvector eigenvector corresponding to the minimum eigenvalue of a symmetric positive definite matrix into a neural network framework we now extend this work using an inflation technique to estimate all or some of the orthogonal eigenvectors of the given matrix based on these results we form a cost function for the finite data case and derive a newton based adaptive algorithm the inflation technique leads to a highly modular and parallel structure for implementation the computational requirement of the algorithm is o n sup 2 n being the size of the covariance matrix we also present a rigorous convergence analysis of this adaptive algorithm the algorithm is locally convergent and the undesired stationary points are unstable computer simulation results are provided to compare its performance with that of two adaptive subspace estimation methods proposed by yang and kaveh 1988 and an improved version of one of them for stationary and nonstationary signal scenarios the results show that the proposed approach performs identically to one of them and is significantly superior to the remaining two

filtering and frequency estimation using perturbation formulas a new method for updating the svd is introduced based on perturbation formulas the complexity of the method is o n sup 2 applications are made to frequency estimation and filtering

orthogonal eigensubspace estimation using neural networks we present a neural network nn approach for simultaneously estimating all or some of the orthogonal eigenvectors of a symmetric nonindefinite matrix corresponding to its repeated minimum in magnitude eigenvalue this problem has its origin in the constrained minimization framework and has extensive applications in signal processing we recast this problem into the nn framework by constructing an appropriate energy function which the nn minimizes the nn is of feedback type with the neurons having sigmoidal activation function the proposed approach is analyzed to characterize the nature of the minimizers of the energy function the main result is that the matrix w is a minimizer of the energy function if and only if the columns of w are the orthogonal eigenvectors with a given norm corresponding to the smallest eigenvalue of the given matrix further all minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of differential equations which define the dynamics of the nn have also been derived results of computer simulations are presented to support our analysis

deflated decomposition of solutions of nearly singular systems when solving the linear system ax b where a may be nearly singular and b is not consistent with a one is often interested in computing a deflated solution i e a unique solution to a nearby singular but consistent system a s x d tilde b kelley 14 15 has considered deflated solutions with a s corresponding to setting a small pivot of the lu factorization of a to zero stewart 22 proposed an iterative algorithm for computing a deflated solution with a s corresponding to setting the smallest singular value of a to zero kelley s approach explicitly uses submatrices of the lu factors whereas stewart s approach is implicit in that it only involves solving systems with a in this paper we generalize the concept of a deflated solution to that of a deflated decomposition which expresses the solution x in terms of x d and the null vectors of a s we treat such decompositions in a uniform framework that includes the approaches of kelley and stewart and introduce some new deflated

a convergence proof for bird s direct simulation monte carlo method for the boltzmann equation bird s direct simulation monte carlo method for the boltzmann equation is considered the limit as the number of particles tends to infinity of the random empirical measures associated with the bird algorithm is shown to be a deterministic measure valued function satisfying an equation close in a certain sense to the boltzmann equation a markov jump process is introduced which is related to bird s collision simulation procedure via a random time transformation convergence is established for the markov process and the random time transformation these results together with some general properties concerning the convergence of random measures make it possible to characterize the limiting behavior of the bird algorithm

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

qr factorization based blind channel identification with second order statistics most eigenstructure based blind channel identification and equalization algorithms with second order statistics need svd or evd of the correlation matrix of the received signal in this paper we address new algorithms based on qr factorization of the received signal directly without calculating the correlation matrix this renders the qr factorization based algorithms more robust against ill conditioned channels i e those channels with almost common zeros among the subchannels first we present a block algorithm that performs the qr factorization of the received data matrix as a whole then a recursive algorithm is developed based on the qr factorization by updating a rank revealing ulv decomposition compared with existing algorithms in the same category our algorithms are computationally more efficient the computation in each recursion of the recursive algorithm is on the order of o m sup 2 if only equalization is required where m is the dimension of the received signal vector our recursive algorithm preserves the fast convergence property of the subspace algorithms thus converging faster than other adaptive algorithms such as the super exponential algorithm with comparable computational complexities moreover our proposed algorithms do not require noise variance estimation numerical simulations demonstrate the good performance of the proposed algorithms

application of degenerate perturbation theory to subspace tracking first order perturbation theory has been applied to subspace tracking this previous work did not solve the problem of signal subspace near degeneracy the signal subspace is nearly degenerate when two or more signal subspace eigenvalues are nearly equal which can occur under a number of circumstances failure to correctly handle the near degeneracy leads to errors in the subspace decomposition we demonstrate here the application of degenerate perturbation theory to signal subspace tracking under conditions where the eigenvalues may be degenerate the problems of noise subspace degeneracy and signal subspace near degeneracy are both solved through diagonalization of the perturbation matrix over the degenerate subspace the degenerate subspace diagonalization markedly reduces the subspace error with a minimal increase in computational complexity similarly it is shown that using a second order perturbation update reduces the subspace error by a substantial margin again with a minimal increase in computational complexity the effect of increasing the integration time is also examined by using the second order perturbation update and the degenerate subspace diagonalization the resulting subspace decomposition is extremely accurate robust and fast

a robust past algorithm for subspace tracking in impulsive noise the past algorithm is an effective and low complexity method for adaptive subspace tracking however due to the use of the recursive least squares rls algorithm in estimating the conventional correlation matrix like other rls algorithms it is very sensitive to impulsive noise and the performance can be degraded substantially to overcome this problem a new robust correlation matrix estimate based on robust statistics concept is proposed in this paper it is derived from the maximum likelihood ml estimate of a multivariate gaussian process in contaminated gaussian noise cg similar to the m estimates in robust statistics this new estimator is incorporated into the past algorithm for robust subspace tracking in impulsive noise furthermore a new restoring mechanism is proposed to combat the hostile effect of long burst of impulses which sporadically occur in communications systems the convergence of this new algorithm is analyzed by extending a previous ordinary differential equation ode based method for past both theoretical and simulation results show that the proposed algorithm offers improved robustness against impulsive noise over the past algorithm the performance of the new algorithm in nominal gaussian noise is very close to that of the past algorithm

bi iterative least square method for subspace tracking subspace tracking is an adaptive signal processing technique useful for a variety of applications in this paper we introduce a simple bi iterative least square bi ls method which is in contrast to the bi iterative singular value decomposition bi svd method we show that for subspace tracking the bi ls method is easier to simplify than the bi svd method the linear complexity algorithms based on bi ls are computationally more efficient than the existing linear complexity algorithms based on bi svd although both have the same performance for subspace tracking a number of other existing subspace tracking algorithms of similar complexity are also compared with the bi ls algorithms

an updating algorithm for subspace tracking in certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components the usual tool for doing this is the singular value decomposition however the singular value decomposition has the drawback that it requires o p sup 3 operations to recompute when a new sample arrives it is shown that a different decomposition called the urv decomposition is equally effective in exhibiting the null space and can be updated in o p sup 2 time the updating technique can be run on a linear array of p processors in o p time

a random discrete velocity model and approximation of the boltzmann equation an approximation procedure for the boltzmann equation based on random choices of collision pairs from a fixed velocity set and on discrete velocity models is designed in a suitable limit the procedure is shown to converge to the time discretized and spatially homogeneous boltzmann equation

l 1 stability of multi dimensional discrete velocity boltzmann equations we study the l 1 stability of multi dimensional discrete velocity boltzmann equations under suitable smallness assumption on initial data we show that bounded mild solutions are l 1 stable for a stability estimate we employ bony s multi dimensional analysis for total interactions over characteristic planes

blind adaptive multiuser detection in multipath cdma channels based on subspace tracking a blind adaptive technique for signal demodulation in multipath code division multiple access cdma communication channels is proposed this technique is based on signal subspace estimation the receiver employs a bank of linear filters decorrelating filters or linear mmse filters at the front end to mitigate the multiple access interference and the multipath interference a channel estimator is used to estimate the channel state for diversity combining it is shown that through the use of signal subspace estimation both the decorrelating filterbank and the linear mmse filterbank can be obtained blindly i e they can be estimated from the received signal with the prior knowledge of only the signature waveform of the desired user two forms of the subspace based linear filterbanks are developed and their equivalence in terms of the interference suppression capability is established these subspace based blind adaptive interference suppression techniques require at each symbol epoch the eigenvalues and the eigenvectors of an appropriate signal subspace which ran be obtained using computationally efficient sequential adaptive eigendecomposition subspace tracking algorithms moreover a blind adaptive method for estimating the channel state is developed which also produces the postcombining decision statistic as a byproduct

plane rotation based evd updating schemes for efficient subspace tracking we present new algorithms based on plane rotations for tracking the eigenvalue decomposition evd of a time varying data covariance matrix these algorithms directly produce eigenvectors in orthonormal form and are well suited for the application of subspace methods to nonstationary data after recasting evd tracking as a simplified rank one evd update problem computationally efficient solutions are obtained in two steps first a new kind of parametric perturbation approach is used to express the eigenvector update as an unimodular orthogonal transform which is represented in exponential matrix form in terms of a reduced set of small unconstrained parameters second two approximate decompositions of this exponential matrix into products of plane or givens rotations are derived one of which being previously unknown these decompositions lead to new plane rotation based evd updating schemes proteus whose main feature is the use of plane rotations for updating the eigenvectors thereby preserving orthonormality finally the proteus schemes are used to derive new evd trackers whose convergence and numerical stability are investigated via simulations one algorithm can track all the signal subspace evd components in only o lm operations where l and m respectively denote the data vector and signal subspace dimensions while achieving a performance comparable to an exact evd approach and maintaining perfect orthonormality of the eigenvectors the new algorithms show no signs of error buildup

parallel codes for computing the numerical rank abstract in this paper we present an experimental comparison of several numerical tools for computing the numerical rank of dense matrices the study includes the well known svd the urv decomposition and several rank revealing qr factorizations the qr factorization with column pivoting and two qr factorizations with restricted column pivoting two different parallel programming methodologies are analyzed in our paper first we present block partitioned algorithms for the urv decomposition and rank revealing qr factorizations which provide efficient implementations on shared memory environments furthermore we also present parallel distributed algorithms based on the message passing paradigm for computing rank revealing qr factorizations on multicomputers

efficient solution of the rank deficient linear least squares problem in this paper we present several new fast and reliable algorithms for solving rank deficient linear least squares problems by means of the complete orthogonal decomposition experimental comparison of our algorithms with the existing implementations in lapack on a wide range of platforms shows considerable performance improvements moreover for full rank matrices the performance of the new algorithm is very close to that of the fast method based on qr factorization thus providing a valuable general tool for full rank matrices rank deficient matrices and those matrices with unknown rank

a survey of condition number estimation for triangular matrices we survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix and make recommendations concerning the use of the estimates in applications each of the methods is shown to bound the condition number the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms for each bound we examine by how much at worst it can overestimate or underestimate the condition number numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators

stabilized hyperbolic householder transformations a modification of the hyperbolic householder scheme is introduced which is demonstrably stable theoretically according to an established stability criterion and which exhibits superior numerical behavior in simulations the modified transform scheme effects downdating by applying conventional orthonormal rather than hyperbolic householder transformations to the data the latter have preferable numerical properties however the construction of these orthonormal operators itself requires hyperbolic computations thus the proposed method is in some sense half hyperbolic and half orthonormal there is no computational penalty incurred with these stabilized hyperbolic householder transforms they enjoy an operation count identical to their conventional counterparts

approximation of the boltzmann equation by discrete velocity models two convergence results related to the approximation of the boltzmann equation by discrete velocity models are presented first we construct a sequence of deterministic discrete velocity models and prove convergence as the number of discrete velocities tends to infinity of their solutions to the solution of a spatially homogeneous boltzmann equation second we introduce a sequence of markov jump processes interpreted as random discrete velocity models and prove convergence as the intensity of jumps tends to infinity of these processes to the solution of a deterministic discrete velocity model

a blas 3 version of the qr factorization with column pivoting the qr factorization with column pivoting qrp originally suggested by golub numer math 7 1965 206 216 is a popular approach to computing rank revealing factorizations using level 1 blas it was implemented in linpack and using level 2 blas in lapack while the level 2 blas version delivers superior performance in general it may result in worse performance for large matrix sizes due to cache effects we introduce a modification of the qrp algorithm which allows the use of level 3 blas kernels while maintaining the numerical behavior of the linpack and lapack implementations experimental comparisons of this approach with the linpack and lapack implementations on ibm rs 6000 sgi r8000 and dec axp platforms show considerable performance improvements

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

implementation of adaptive array algorithms some new efficient and numerically stable algorithms for the recursive solution of matrix problems arising in optimal beam forming and direction finding are described and analyzed the matrix problems considered are systems of linear equations and spectral decomposition while recursive solution procedures based on the matrix inversion lemma may be unstable ours are stable furthermore these algorithms are extremely fast

investigations in the numerical behavior of the adaptive rank revealing qr factorization we present a tracking procedure based on the rank revealing qr rrqr factorization and investigate its numerical properties by applying it to the direction of arrival problem we address numerical issues raised by the related work proposed earlier by prasad et al 1991 and we compare the performance of the proposed algorithm to that obtained using an evd based technique

galerkin petrov approach for the boltzmann equation abstract in this work we propose a new galerkin petrov method for the numerical solution of the classical spatially homogeneous boltzmann equation this method is based on an approximation of the distribution function by associated laguerre polynomials and spherical harmonics and test in a variational manner with globally defined three dimensional polynomials a numerical realisation of the algorithm is presented the algorithmic developments are illustrated with the help of several numerical tests

computing approximate fekete points by qr factorizations of vandermonde matrices we propose a numerical method implemented in matlab for computing approximate fekete points on compact multivariate domains it relies on the search of maximum volume submatrices of vandermonde matrices computed on suitable discretization meshes and uses a simple greedy algorithm based on qr factorization with column pivoting the method gives also automatically an algebraic cubature formula provided that the moments of the underlying polynomial basis are known numerical tests are presented for the interval and the square which show that approximate fekete points are well suited for polynomial interpolation and cubature

nonlinear functionals of multi d discrete velocity boltzmann equations in this paper we study nonlinear functionals measuring potential interactions and l1 distance between two mild solutions for the multi dimensional discrete velocity boltzmann equations when the initial data are a small perturbation of a vacuum we employ bony s dispersion estimates to show that these functionals satisfy lyapunov type estimates which are useful for the study of time asymptotics and l1 stability of mild solutions

unitary esprit how to obtain increased estimation accuracy with a reduced computational burden esprit is a high resolution signal parameter estimation technique based on the translational invariance structure of a sensor array previous esprit algorithms do not use the fact that the operator representing the phase delays between the two subarrays is unitary the authors present a simple and efficient method to constrain the estimated phase factors to the unit circle if centro symmetric array configurations are used unitary esprit the resulting closed form algorithm has an esprit like structure except for the fact that it is formulated in terms of real valued computations throughout since the dimension of the matrices is not increased this completely real valued algorithm achieves a substantial reduction of the computational complexity furthermore unitary esprit incorporates forward backward averaging leading to an improved performance compared to the standard esprit algorithm especially for correlated source signals like standard esprit unitary esprit offers an inexpensive possibility to reconstruct the impinging wavefronts signal copy these signal estimates are more accurate since unitary esprit improves the underlying signal subspace estimates simulations confirm that even for uncorrelated signals the standard esprit algorithm needs twice the number of snapshots to achieve a precision comparable to that of unitary esprit thus unitary esprit provides increased estimation accuracy with a reduced computational burden

on line subspace estimation using a schur type method a recently developed schur type matrix approximation technique is applied to subspace estimation the method is applicable if an upper bound of the noise level is approximately known the main feature of the algorithm is that updating and downdating is straightforward and efficient and that the subspace dimension is elegantly tracked as well

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

analysis of a recursive least squares hyperbolic rotation algorithm for signal processing abstract the application of hyperbolic plane rotations to the least squares downdating problem arising in windowed recursive least squares signal processing is studied a forward error analysis is given to show that this algorithm can be expected to perform well in the presence of rounding errors provided that the problem is not too ill conditioned the hyperbolic rotation algorithm is shown to be forward weakly stable and in fact comparable to an orthogonal downdating method shown to be backward stable by stewart it is shown in detail how the method s accuracy depends upon the conditioning numerical comparisons are made with the usual method based upon orthogonal rotations as implemented in linpack both methods have the important advantage over the classical normal equations appraoch that they can be effectively implemented on special purpose signal processing devices requiring shorter word lengths however the hyperbolic rotation requires n 2 fewer multiplications and additions for each downdating step than the orthogonal method where n is the number of least squares filter coefficients in addition it is more amenable to implementation on a variety of vector and parallel machines in many signal processing applications n is not large and if n processors are available then the downdating process can be accomplished in 2 n time steps by the hyperbolic rotation method

a quasi newton adaptive algorithm for generalized symmetric eigenvalue problem we first recast the generalized symmetric eigenvalue problem where the underlying matrix pencil consists of symmetric positive definite matrices into an unconstrained minimization problem by constructing an appropriate cost function we then extend it to the case of multiple eigen vectors using an inflation technique based on this asymptotic formulation we derive a quasi newton based adaptive algorithm for estimating the required generalized eigen vectors in the data case the resulting algorithm is modular and parallel and it is globally convergent with probability one we also analyze the effect of inexact inflation on the convergence of this algorithm and that of inexact knowledge of one of the matrices in the pencil on the resulting eigenstructure simulation results demonstrate that the performance of this algorithm is almost identical to that of the rank one updating algorithm of karasalo 1986 further the performance of the proposed algorithm has been found to remain stable even over 1 million updates without suffering from any error accumulation problems

accuracy of tsvd solutions computed from rank revealing decompositions rank revealing decompositions are favorable alternatives to the singular value decomposition svd because they are faster to compute and easier to update although they do not yield all the information that the svd does they yield enough information to solve various problems because they provide accurate bases for the relevant subspaces in this paper we consider rank revealing decompositions in computing estimates of the truncated svd tsvd solution to an overdetermined system of linear equations a x approx b where a is numerically rank deficient we derive analytical bounds which show how the accuracy of the solution is intimately connected to the quality of the subspaces

sliding window adaptive svd algorithms the singular value decomposition svd is an important tool for subspace estimation in adaptive signal processing we are especially interested in tracking the svd of a recursively updated data matrix this paper introduces a new tracking technique that is designed for rectangular sliding window data matrices this approach which is derived from the classical bi orthogonal iteration svd algorithm shows excellent performance in the context of frequency estimation it proves to be very robust to abrupt signal changes due to the use of a sliding window finally an ultra fast tracking algorithm with comparable performance is proposed

grobner shirshov bases for lie superalgebras and their universal enveloping algebras abstract we show that a set of monic polynomials in a free lie superalgebra is a grobner shirshov basis for a lie superalgebra if and only if it is a grobner shirshov basis for its universal enveloping algebra we investigate the structure of grobner shirshov bases for kac moody superalgebras and give explicit constructions of grobner shirshov bases for classical lie superalgebras

composition diamond lemma for associative conformal algebras abstract we prove the composition diamond lemma for associative conformal algebras as some corollaries we prove that the word problem for some homogeneous associative conformal algebras is solvable while it is unsolvable in general

state space system identification approach to radar data processing space time adaptive processing stap algorithms typically consist of a data transformation step to reduce the number of degrees of freedom and a sampling step wherein radar returns from adjacent range bins are used to estimate interference statistics the reduction in the number of degrees of freedom inadequate sample support presence of target in sampled data and range dependence of interference are some of the main reasons for stap performance loss in this paper we present an approach to target detection and localization that mitigates these performance losses using the well known stochastic realization algorithm from system identification theory we first identify a state space model from the radar return data in range pulse domain for a given range bin and then perform detection and localization using the identified state space matrices as interference statistics are not directly computed and since there is no sampling from adjacent range bins this approach is more robust to sample support issues target in training and range dependence of clutter a numerical comparison of the approach with beam space post doppler stap using simulated data is given

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

blind multiuser detection a subspace approach a new multiuser detection scheme based on signal subspace estimation is proposed it is shown that under this scheme both the decorrelating detector and the linear minimum mean square error mmse detector can be obtained blindly i e they can be estimated from the received signal with the prior knowledge of only the signature waveform and timing of the user of interest the consistency and asymptotic variance of the estimates of the two linear detectors are examined a blind adaptive implementation based on a signal subspace tracking algorithm is also developed it is seen that compared with the previous minimum output energy blind adaptive multiuser detector the proposed subspace based blind adaptive detector offers lower computational complexity better performance and robustness against signature waveform mismatch two extensions are made within the framework of signal subspace estimation first a blind adaptive method is developed for estimating the effective user signature waveform in the multipath channel secondly a multiuser detection scheme using spatial diversity in the form of an antenna array is considered a blind adaptive technique for estimating the array response for diversity combining is proposed it is seen that under the proposed subspace approach blind adaptive channel estimation and blind adaptive array response estimation can be integrated with blind adaptive multiuser detection with little attendant increase in complexity

beam tracking for interference alignment in slowly fading mimo interference channels a perturbations approach under a linear framework in this paper the beam design for signal space interference alignment in slowly fading multiuser multiple input multiple output mimo interference channels is considered based on a linear formulation for interference alignment a predictive beam tracking algorithm is proposed using matrix perturbation theory the proposed algorithm based on a mixture of iteration and update computes interference aligning beamforming vectors at the current time by updating the previous beam vectors based on the channel difference between the two time steps during the predictively updating phase and yields significant reduction in computational complexity compared with existing methods recalculating beams at each time step the tracking performance of the algorithm is analyzed in terms of mean square error and sum rate loss between the predictively updating approach and the recalculating approach and the impact of imperfect channel knowledge is also investigated under the state space channel model numerical results show that the proposed algorithm has almost the same performance as non predictive methods in sum rate thus the proposed algorithm provides a very efficient way to realize interference alignment in a realistic slowly fading mimo channel environment

on sigma lossless transfer functions and related questions abstract this paper is concerned with a systematic approach to the properties of lossless rational transfer functions in the discrete as well as in the continuous time case as a result a unifying framework is revealed where several known results fit naturally special attention is given to the embedding problem of the lyapunov equation in view of its direct application to generalized levinson algorithms

equations de type de boltzmann spatialement homogenes we study the construction of a nonhomogeneous n valued markov process whose laws at fixed times evolve according to a boltzmann type equation spatially homogeneous we then consider the problem of the asymptotic propagation of chaos for a system of a large number of interacting particles

grobner shirshov bases for irreducible sln 1 modules abstract we determine the grobner shirshov bases for finite dimensional irreducible representations of the special linear lie algebra sl n 1 and construct explicit monomial bases for these representations we also show that each of these monomial bases is in 1 1 correspondence with the set of semistandard young tableaux of a given shape

fast approximated power iteration subspace tracking this paper introduces a fast implementation of the power iteration method for subspace tracking based on an approximation that is less restrictive than the well known projection approximation this algorithm referred to as the fast approximated power iteration api method guarantees the orthonormality of the subspace weighting matrix at each iteration moreover it outperforms many subspace trackers related to the power iteration method such as past nic np3 and opast while having the same computational complexity the api method is designed for both exponential windows and sliding windows our numerical simulations show that sliding windows offer a faster tracking response to abrupt signal variations

qr factorization with complete pivoting and accurate computation of the svd a new algorithm of demmel et al for computing the singular value decomposition svd to high relative accuracy begins by computing a rank revealing decomposition rrd demmel et al analyse the use of gaussian elimination with complete pivoting gecp for computing the rrd we investigate the use of qr factorization with complete pivoting that is column pivoting together with row sorting or row pivoting as an alternative to gecp since this leads to a faster svd algorithm we derive a new componentwise backward error result for householder qr factorization and combine it with the theory of demmel et al to show that high relative accuracy in the computed svd can be expected for matrices that are diagonal scalings of a well conditioned matrix an a posteriori error bound is derived that gives useful estimates of the relative accuracy of the computed singular values numerical experiments confirm the theoretical predictions

on updating signal subspaces the authors develop an algorithm for adaptively estimating the noise subspace of a data matrix as is required in signal processing applications employing the signal subspace approach the noise subspace is estimated using a rank revealing qr factorization instead of the more expensive singular value or eigenvalue decompositions using incremental condition estimation to monitor the smallest singular values of triangular matrices the authors can update the rank revealing triangular factorization inexpensively when new rows are added and old rows are deleted experiments demonstrate that the new approach usually requires o n sup 2 work to update an n n matrix and that it accurately tracks the noise subspace

wild s solution of the nonlinear boltzmann equation the relaxation to equilibrium of a spatially uniform pseudo maxwellian gas is considered a modified wild expansion is defined for solving the nonlinear boltzmann equation the positivity and asymptotic conditions as well as the conservation rules are maintained at each truncation order some particular examples are evaluated the comparison with exact solutions illustrates the very fast convergence of this method

on the existence and computation of lu factorizations with small pivots let a be an n by n matrix which may be singular with a one dimensional null space and consider the lu factorization of a when a is exactly singular we show conditions under which a pivoting strategy will produce a zero n th pivot when a is not singular we show conditions under which a pivoting strategy will produce an nth pivot that is o g or o k a where is the smallest singular value of a and k a is the condition number of a these conditions are expressed in terms of the elements of a in general but reduce to conditions on the elements of the singular vectors corresponding to when a is nearly or exactly singular they can be used to build a 2 pass factorization algorithm which is guaranteed to produce a small n th pivot for nearly singular matrices as an example we exhibit an lu factorization of the n by n upper triangular matrix

blind channel identification subspace tracking method without rank estimation subspace ss methods are an effective approach for blind channel identification however these methods also have two major disadvantages 1 they require accurate channel length estimation and or rank estimation of the correlation matrix which is difficult with noisy channels and 2 they require a large amount of computation for the singular value decomposition svd which makes it inconvenient for adaptive implementation although many adaptive subspace tracking algorithms can be applied the computational complexity is still o m sup 3 where m is the data vector length in this paper we introduce new recursive subspace algorithms using ulv updating and successive cancellation techniques the new algorithms do not need to estimate the rank of the correlation matrix furthermore the channel length can be overestimated initially and be recovered at the end by a successive cancellation procedure which leads to more convenient implementations the adaptive algorithm has computations of o m sup 2 in each recursion the new methods can be applied to either the single user or the multiuser cases simulations demonstrate their good performance

on the computation of the jordan canonical form of regular matrix polynomials in this paper an algorithm for the computation of the jordan canonical form of regular matrix polynomials is proposed the new method contains rank conditions of suitably defined block toeplitz matrices and it does not require the computation of the jordan chains or the smith form the segre and weyr characteristics are also considered

rank and null space calculations using matrix decomposition without column interchanges abstract the most widely used stable methods for numerical determination of the rank of a matrix a are the singular value decomposition and the qr algorithm with column interchanges here two algorithms are presented which determine rank and nullity in a numerically stable manner without using column interchanges one algorithm makes use of the condition estimator of cline moler stewart and wilkinson and relative to alternative stable algorithms is particularly efficient for sparse matrices the second algorithm is important in the case that one wishes to test for rank and nullity while sequentially adding columns to a matrix

four algorithms for the the efficient computation of truncated pivoted qr approximations to a sparse matrix in this paper we propose four algorithms to compute truncated pivoted qr approximations to a sparse matrix three are based on the gram schmidt algorithm and the other on householder triangularization all four algorithms leave the original matrix unchanged and the only additional storage requirements are arrays to contain the factorization itself thus the algorithms are particularly suited to determining low rank approximations to a sparse matrix

linear algebraic approach to grobner shirshov basis theory we construct a new efficient algorithm for finding grobner shirshov bases for noncommutative algebras and their representations this algorithm uses the macaulay matrix f s macaulay on some formula in elimination proc london math soc 33 1 1902 3 27 and can be viewed as a representation theoretic analogue of the f4 algorithm developed by j c faugere we work out some examples of universal enveloping algebras of lie algebras and of their representations to illustrate the algorithm

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed