Showing papers on "QR decomposition published in 2003"

MMSE extension of V-BLAST based on sorted QR decomposition

Abstract: In rich-scattering environments, layered space-time architectures like the BLAST system may exploit the capacity advantage of multiple antenna systems. We present a novel, computationally efficient algorithm for detecting V-BLAST architectures with respect to the MMSE criterion. It utilizes a sorted QR decomposition of the channel matrix and leads to a simple successive detection structure. The new algorithm needs only a fraction of computational effort compared to the standard V-BLAST algorithm and achieves the same error performance.

Multiuser diversity for a dirty paper approach

Abstract: Multiuser diversity has attracted significant attention recently. In this letter, we propose a greedy algorithm based on QR decomposition of the channel and dirty paper precoding to exploit the multiuser diversity of the Gaussian vector broadcast channel. Simulations show the approach provides performance which is extremely close to a well-known upper bound on the sum rate. Further, exploiting multiuser diversity can provide large gain over approaches ignoring this resource.

Reduced complexity MMSE detection for BLAST architectures

Abstract: Theoretical and experimental studies have shown that layered space-time architectures like the BLAST system can exploit the capacity advantage of multiple antenna systems in rich-scattering environments. We present a new efficient algorithm for detecting such architectures with respect to the MMSE criterion. This algorithm utilizes a sorted QR decomposition of the channel matrix and leads to a simple successive detection structure. The algorithm needs only a fraction of the computational effort compared to the standard V-BLAST algorithm and achieves the same bit error performance.

Subspace methods for multimicrophone speech dereverberation

Abstract: A novel approach for multimicrophone speech dereverberation is presented. The method is based on the construction of the null subspace of the data matrix in the presence of colored noise, using the generalized singular-value decomposition (GSVD) technique, or the generalized eigenvalue decomposition (GEVD) of the respective correlation matrices. The special Silvester structure of the filtering matrix, related to this subspace, is exploited for deriving a total least squares (TLS) estimate for the acoustical transfer functions (ATFs). Other less robust but computationally more efficient methods are derived based on the same structure and on the QR decomposition (QRD). A preliminary study of the incorporation of the subspace method into a subband framework proves to be efficient, although some problems remain open. Speech reconstruction is achieved by virtue of the matched filter beamformer (MFBF). An experimental study supports the potential of the proposed methods.

Algorithms for the QR-Decomposition

Abstract: In this report we review the algorithms for the QR decomposition that are based on the Schmidt orthonormalization process and show how an accurate decomposition can be obtained using modified Gram Schmidt and reorthogonalization. We also show that the modified Gram Schmidt algorithm may be derived using the representation of the matrix product as a sum of matrices of rank one.

Derivation of eigenvectors for spatial processing in MIMO communication systems

Abstract: Techniques for deriving eigenvectors based on steered reference and used for spatial processing A steered reference is a pilot transmission on one eigenmode of a MIMO channel per symbol period using a steering vector for that eigenmode The steered reference is used to estimate both a matrix SIGMA of singular values and a matrix U of left eigenvectors of a channel response matrix H A matrix U with orthogonalized columns may be derived based on the estimates of SIGMA and U, eg, using QR factorization, minimum square error computation, be used for matched filtering of data transmission received via a first link The estimate U of or the matrix U may also be used for spatial processing of data transmission on a second link or polar decomposition The estimates of SIGMA and U (or the estimate of SIGMA and the matrix U) may (for reciprocal first and second links)

Scaled and decoupled Cholesky and QR decompositions with application to spherical MIMO detection

Abstract: Motivated by the need for the Cholesky factorization in implementing a spherical MIMO detector, this paper considers Cholesky and QR decompositions suitable for fixed-point implementation. In particular, we reformulate the decompositions to avoid the many square-root and division operations required in their natural form. This is achieved by decoupling the numerator and denominator calculations and applying scaling by powers of 2 (corresponding to bit shifts) to ensure numerical stability in the recursions. We consider the impact on the spherical detector formulation.

A Robust Preconditioner with Low Memory Requirements for Large Sparse Least Squares Problems

Abstract: This paper describes a technique for constructing robust preconditioners for the CGLS method applied to the solution of large and sparse least squares problems. The algorithm computes an incomplete LDLT factorization of the normal equations matrix without the need to form the normal matrix itself. The preconditioner is reliable (pivot breakdowns cannot occur) and has low intermediate storage requirements. Numerical experiments illustrating the performance of the preconditioner are presented. A comparison with incomplete QR preconditioners is also included.

QR factorization with Morton-ordered quadtree matrices for memory re-use and parallelism

Abstract: Quadtree matrices using Morton-order storage provide natural blocking on every level of a memory hierarchy. Writing the natural recursive algorithms to take advantage of this blocking results in code that honors the memory hierarchy without the need for transforming the code. Furthermore, the divide-and-conquer algorithm breaks problems down into independent computations. These independent computations can be dispatched in parallel for straightforward parallel processing.Proof-of-concept is given by an algorithm for QR factorization based on Givens rotations for quadtree matrices in Morton-order storage. The algorithms deliver positive results, competing with and even beating the LAPACK equivalent.

Parallel algorithms for computing all possible subset regression models using the QR decomposition

Abstract: Efficient parallel algorithms for computing all possible subset regression models are proposed. The algorithms are based on the dropping columns method that generates a regression tree. The properties of the tree are exploited in order to provide an efficient load balancing which results in no inter-processor communication. Theoretical measures of complexity suggest linear speedup. The parallel algorithms are extended to deal with the general linear and seemingly unrelated regression models. The case where new variables are added to the regression model is also considered. Experimental results on a shared memory machine are presented and analyzed.

Design of a parameterizable silicon intellectual property core for QR-based RLS filtering

Abstract: The availability of an intellectual property core for recursive least squares (RLS) filtering could enable the RLS algorithm to replace the least mean squares algorithm in a wide range of applications. The goal of this study is to develop a parameterizable generic architecture for RLS filtering in the form of a hardware description language (HDL) description, which can be used to generate highly efficient silicon layout. The key issue is to develop a family of circuit architectures that are 100% efficient and locally connected. This paper presents a generic mapping for RLS filtering and circuit architectures that can be mapped to a range of application requirements. It outlines the transition from array to architecture covering detailed design issues such as timing and control generation. The result is a family of QR designs, which are parameterized in terms of architecture size, wordlength, performance, and arithmetic processor timing.

A new method of computation of current distribution maps in bulk high-temperature superconductors: analysis and validation

Abstract: We present a test and analysis of our method of computation of the distribution of currents in bulk superconducting samples, which is sensitive to current contribution in the deep layers of the sample. The procedure is based on measurements of the magnetic field with a Hall probe, inverted by linearization and orthogonal triangularization, known as QR decomposition, of the matrix in the resulting linear system. No assumptions on the number or geometry of domains are required. The only constraint on the method is that the critical current must be homogeneous along the c-axis. Our method is applied to real 3d samples with size in the cm range and different geometries of technological interest. The propagation of errors in the general case is analysed, and we also supply a method to estimate the error in every computation, which is applied to the computed J(Jx, Jy) in the above samples.

An algorithm-based error detection scheme for the multigrid method

Abstract: Algorithm-based fault tolerance (ABFT) is a technique to provide system level error detection and correction on array processors as well as multiprocessors at a low cost. Since the early 1980s the technique has been extensively applied to several linear algebraic algorithms, e.g., matrix multiplication, Gaussian elimination, QR factorization, and singular value decompositions, etc. An important class of problems in numerical linear algebra dealing with the iterative solution of linear algebraic equations arising due to the finite difference discretization or the finite element discretization of a partial differential equation, however, has been overlooked. The only exception is the recent application of algorithm based error detection (ABED) encodings to the successive overrelaxation algorithm for Laplace's equation. In this paper, ABED is applied to a multigrid algorithm for the iterative solution of a Poisson equation in two dimensions. Invariants are created to implement checking in the relaxation, the restriction, and the interpolation operators. Modifications to invariants due to roundoff errors accumulated within the operators, which often lead to a situation known as false alarms, have been addressed by deriving the expressions for the roundoff errors in the algebraic processes in the operators and correcting the invariants accordingly. The ABED encoded multigrid algorithm is shown to be insensitive to the size and the range of the input data besides providing excellent error coverage at a low latency for floating-point, integer, and memory errors.

Subspace methods for multi microphone speech dereverberation

Abstract: A novel approach for multi–microphone speech dereverberation is presented. The method is based on the construction of the null subspace of the data matrix in the presence of colored noise, using the generalized singular value decomposition (GSVD) technique, or the generalized eigenvalue decomposition (GEVD) of the respective correlation matrices. The special Silvester structure of the filtering matrix, related to this subspace, is exploited for deriving a total least squares (TLS) estimate for the acoustical transfer functions (ATFs). Other, less robust but computationally more efficient methods are derived based on the same structure and on the QR decomposition (QRD). A preliminary study of the incorporation of the subspace method into a This research work was partly carried out at the ESAT laboratory of the Katholieke Universiteit Leuven, in the frame of the Interuniversity Attraction Pole IUAP P4-02, Modeling, Identification, Simulation and Control of Complex Systems, the Concerted Research Action Mathematical Engineering Techniques for Information and Communication Systems (GOA-MEFISTO-666) of the Flemish Government and the IT-project Multi-microphone Signal Enhancement Techniques for handsfree telephony and voice controlled systems (MUSETTE-2) of the I.W.T., and was partially sponsored by Philips-ITCL.

Solving Rank-Deficient and Ill-posed Problems Using UTV and QR Factorizations

Abstract: The algorithm of Mathias and Stewart [Linear Algebra Appl., 182 (1993), pp. 91--100] is examined as a tool for constructing regularized solutions to rank-deficient and ill-posed linear equations. The algorithm is based on a sequence of QR factorizations. If it is stopped after the first step, it produces the same solution as the complete orthogonal decomposition used in LAPACK's xGELSY. However, we show that for low-rank problems a careful implementation can lead to an order of magnitude improvement in speed over xGELSY as implemented in LAPACK. We prove, under assumptions similar to assumptions used by others, that if the numerical rank is chosen at a gap in the singular value spectrum and if the initial factorization is rank-revealing, then, even if the algorithm is stopped after the first step, approximately half the time its solutions are closer to the desired solution than are the singular value decomposition (SVD) solutions. Conversely, the SVD will be closer approximately half the time, and in this case overall the two algorithms are very similar in accuracy. We confirm this with numerical experiments. Although the algorithm works best for problems with a gap in the singular value spectrum, numerical experiments suggest that it may work well for problems with no gap.

The Equality Constrained Indefinite Least Squares Problem: Theory and Algorithms

Abstract: We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.

Direct and inverse eigenvalue problems for diagonal-plus-semiseparable matrices

Abstract: In this paper we study both direct and inverse eigenvalue problems for diagonal-plus-semiseparable (dpss) matrices. In particular, we show that the computation of the eigenvalues of a symmetric dpss matrix can be reduced by a congruence transformation to solving a generalized symmetric definite tridiagonal eigenproblem. Using this reduction, we devise a set of recurrence relations for evaluating the characteristic polynomial of a dpss matrix in a stable way at a linear time. This in turn allows us to apply divide-and-conquer eigenvalue solvers based on functional iterations directly to dpss matrices without performing any preliminary reduction into a tridiagonal form. In the second part of the paper, we exploit the structural properties of dpss matrices to solve the inverse eigenvalue problem of reconstructing a symmetric dpss matrix from its spectrum and some other informations. Finally, applications of our results to the computation of a QR factorization of a Cauchy matrix with real nodes are provided.

Generic SoC QR array processor for adaptive beamforming

Abstract: A generic architecture for implementing a QR array processor in silicon is presented. This improves on previous research by considerably simplifying the derivation of timing schedules for a QR system implemented as a folded linear array, where account has to be taken of processor cell latency and timing at the detailed circuit level. The architecture and scheduling derived have been used to create a generator for the rapid design of System-on-a-Chip (SoC) cores for QR decomposition. This is demonstrated through the design of a single-chip architecture for implementing an adaptive beamformer for radar applications.

A multilevel implementation of the QR compression for method of moments

Abstract: We present a multilevel version of a previously reported compression technique for the method of moments (Poirier, J.-R. et al., ibid., vol.46, p.1175-6, 1998). This method is based on QR compression of the impedance matrix off-diagonal blocks. The block structure of the matrix is built up to optimize the compression rate. It results from the study that the compression threshold is the unique adjustment parameter, which makes this method very easy to handle. Some results are presented, which stress the improvement in memory storage and computation time obtained through this approach with respect to the mono-level algorithm.

Convergence of the unitary QR algorithm with a unimodular Wilkinson shift

Abstract: In applying the QR algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the QR iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.

Numerically Robust Learning Algorithms for Feed Forward Neural Networks

Abstract: In recent years several neural networks learning algorithms have been developed by making use of the RLS recursion. These algorithms are based on the matrix inversion lemma and in some cases can be numerically ill-conditioned. For example the rounding errors can accumulate and cause errors to occur in both the estimated parameters and the covariance matrix. In this paper we present numerically robust learning algorithms based on the QR decomposition — a well known technique in the linear prediction theory.

Globally convergent state estimation via the trust region method

Abstract: This paper introduces trust region methods into power system state estimation. The traditional Newton (Gauss-Newton) method is not always reliable particularly in the presence of bad data, topological or parameter errors. The motivation was to enhance convergence properties of the state estimator under those conditions, and together with QR factorization to make a globally convergent and reliable algorithm. The trust region formulation shows that such a model is more robust then the traditional Newton (Gauss-Newton). The algorithm has been programmed and applied to representative power networks, and the computational requirement has been found.

Dimensionality reduction for text-independent speaker identification using Gaussian mixture model

Abstract: Reducing the dimensionality of the training and testing data is crucial for text-independent speaker identification tasks. In this paper, the performance of various dimensionality reduction techniques is evaluated for speaker identification systems using Gaussian mixture model (GMM) as the statistical classifier. An enhancement of the standard linear discriminant analysis (LDA) is proposed in which class distributions are assumed to follow Gaussian mixture distribution. This assumption is more appropriate for asymmetric and multimodal class conditional densities. In addition, a new feature selection technique based on the QR factorization method is introduced. Computer simulation results reveal that the proposed modification to the LDA outperforms the standard algorithm in terms of classification accuracy. Moreover, the QR-based selection technique produces comparable results to other prominent dimensionality reduction techniques

Efficient Algorithms for Maximum Covariance Analysis of Datasets with Many Variables and Fewer Realizations: A Revisit

Abstract: Fast and reliable algorithms for maximum covariance analysis (MCA) are investigated. The traditional algorithm based on the direct singular value decomposition (SVD) of a covariance matrix is computationally expensive for large datasets. An alternate algorithm proposed in this study uses the QR factorization technique to reduce the computational burden of the MCA of datasets with many variables and fewer realizations. It is slightly slower but more reliable, as indicated in an example, than an existing alternate algorithm based on the eigenvalue decomposition of a quadruple matrix product. It is faster than another alternate algorithm that uses the principal component analyses of the datasets as the preliminary step of the MCA.

Block algorithms for orthogonal symplectic factorizations

Abstract: On the basis of a new WY-like representation block algorithms for orthogonal symplectic matrix factorizations are presented. Special emphasis is placed on symplectic QR and URV factorizations. The block variants mainly use level 3 (matrix-matrix) operations that permit data reuse in the higher levels of a memory hierarchy. Timing results show that our new algorithms outperform standard algorithms by a factor 3-4 for sufficiently large problems.