Showing papers on "QR decomposition published in 1998"

Numerical Linear Algebra for Applications in Statistics

Abstract: 1 Computer Storage and Manipulation of Data.- 1.1 Digital Representation of Numeric Data.- 1.2 Computer Operations on Numeric Data.- 1.3 Numerical Algorithms and Analysis.- Exercises.- 2 Basic Vector/Matrix Computations.- 2.1 Notation, Definitions, and Basic Properties.- 2.1.1 Operations on Vectors Vector Spaces.- 2.1.2 Vectors and Matrices.- 2.1.3 Operations on Vectors and Matrices.- 2.1.4 Partitioned Matrices.- 2.1.5 Matrix Rank.- 2.1.6 Identity Matrices.- 2.1.7 Inverses.- 2.1.8 Linear Systems.- 2.1.9 Generalized Inverses.- 2.1.10 Other Special Vectors and Matrices.- 2.1.11 Eigenanalysis.- 2.1.12 Similarity Transformations.- 2.1.13 Norms.- 2.1.14 Matrix Norms.- 2.1.15 Orthogonal Transformations.- 2.1.16 Orthogonalization Transformations.- 2.1.17 Condition of Matrices.- 2.1.18 Matrix Derivatives.- 2.2 Computer Representations and Basic Operations.- 2.2.1 Computer Representation of Vectors and Matrices.- 2.2.2 Multiplication of Vectors and Matrices.- Exercises.- 3 Solution of Linear Systems.- 3.1 Gaussian Elimination.- 3.2 Matrix Factorizations.- 3.2.1 LU and LDU Factorizations.- 3.2.2 Cholesky Factorization.- 3.2.3 QR Factorization.- 3.2.4 Householder Transformations (Reflections).- 3.2.5 Givens Transformations (Rotations).- 3.2.6 Gram-Schmidt Transformations.- 3.2.7 Singular Value Factorization.- 3.2.8 Choice of Direct Methods.- 3.3 Iterative Methods.- 3.3.1 The Gauss-Seidel Method with Successive Overrelaxation.- 3.3.2 Solution of Linear Systems as an Optimization Problem Conjugate Gradient Methods.- 3.4 Numerical Accuracy.- 3.5 Iterative Refinement.- 3.6 Updating a Solution.- 3.7 Overdetermined Systems Least Squares.- 3.7.1 Full Rank Coefficient Matrix.- 3.7.2 Coefficient Matrix Not of Full Rank.- 3.7.3 Updating a Solution to an Overdetermined System.- 3.8 Other Computations for Linear Systems.- 3.8.1 Rank Determination.- 3.8.2 Computing the Determinant.- 3.8.3 Computing the Condition Number.- Exercises.- 4 Computation of Eigenvectors and Eigenvalues and the Singular Value Decomposition.- 4.1 Power Method.- 4.2 Jacobi Method.- 4.3 QR Method for Eigenanalysis.- 4.4 Singular Value Decomposition.- Exercises.- 5 Software for Numerical Linear Algebra.- 5.1 Fortran and C.- 5.1.1 BLAS.- 5.1.2 Fortran and C Libraries.- 5.1.3 Fortran 90 and 95.- 5.2 Interactive Systems for Array Manipulation.- 5.2.1 Matlab.- 5.2.2 S, S-Plus.- 5.3 High-Performance Software.- 5.4 Test Data.- Exercises.- 6 Applications in Statistics.- 6.1 Fitting Linear Models with Data.- 6.2 Linear Models and Least Squares.- 6.2.1 The Normal Equations and the Sweep Operator.- 6.2.2 Linear Least Squares Subject to Linear Equality Constraints.- 6.2.3 Weighted Least Squares.- 6.2.4 Updating Linear Regression Statistics.- 6.2.5 Tests of Hypotheses.- 6.2.6 D-Optimal Designs.- 6.3 Ill-Conditioning in Statistical Applications.- 6.4 Testing the Rank of a Matrix.- 6.5 Stochastic Processes.- Exercises.- Appendices.- A Notation and Definitions.- B Solutions and Hints for Selected Exercises.- Literature in Computational Statistics.- World Wide Web, News Groups, List Servers, and Bulletin Boards.- References.- Author Index.

Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics

Abstract: Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation' for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.

Computing rank-revealing QR factorizations of dense matrices

Abstract: We develop algorithms and implementations for computing rank-revealing QR (RRQR) factorizations of dense matrices. First, we develop an efficient block algorithm for approximating an RRQR factorization, employing a windowed version of the commonly used Golub pivoting strategy, aided by incremental condition estimation. Second, we develop efficiently implementable variants of guaranteed reliable RRQR algorithms for triangular matrices originally suggested by Chandrasekaran and Ipsen and by Pan and Tang. We suggest algorithmic improvements with respect to condition estimation, termination criteria, and Givens updating. By combining the block algorithm with one of the triangular postprocessing steps, we arrive at an efficient and reliable algorithm for computing an RRQR factorization of a dense matrix. Experimental results on IBM RS/6000 SGI R8000 platforms show that this approach performs up to three times faster that the less reliable QR factorization with column pivoting as it is currently implemented in LAPACK, and comes within 15% of the performance of the LAPACK block algorithm for computing a QR factorization without any column exchanges. Thus, we expect this routine to be useful in may circumstances where numerical rank deficiency cannot be ruled out, but currently has been ignored because of the computational cost of dealing with it.

A method for real-time nonlinear system state estimation and control

Abstract: A method for the estimation of the state variables of nonlinear systems with exogenous inputs is based on improved extended Kalman filtering (EKF) type techniques. The method uses a discrete-time model, based on a set of nonlinear differential equations describing the system, that is linearized about the current operating point. The time update for the state estimates is performed using integration methods. Integration, which is accomplished through the use of matrix exponential techniques, avoids the inaccuracies of approximate numerical integration techniques. The updated state estimates and corresponding covariance estimates use a common time-varying system model for ensuring stability of both estimates. Other improvements include the use of QR factorization for both time and measurement updating of square-root covariance and Kalman gain matrices and the use of simulated annealing for ensuring that globally optimal estimates are produced.

New Serial and Parallel Recursive QR Factorization Algorithms for SMP Systems

Abstract: We present a new recursive algorithm for the QR factorization of an m by n matrix A. The recursion leads to an automatic variable blocking that allow us to replace a level 2 part in a standard block algorithm by level 3 operations. However, there are some additional costs for performing the updates which prohibits the efficient use of the recursion for large n. This obstacle is overcome by using a hybrid recursive algorithm that outperforms the LAPACK algorithm DGEQRF by 78% to 21% as m=n increases from 100 to 1000. A successful parallel implementation on a PowerPC 604 based IBM SMP node based on dynamic load balancing is presented. For 2, 3, 4 processors and m=n=2000 it shows speedups of 1.96, 2.99, and 3.92 compared to our uniprocessor algorithm.

Low power architecture of the soft-output Viterbi algorithm

Abstract: An important technique for reducing pow er consumption in VLSI systems is strength reduction, the substitution of a less-costly operation such as a shift, for a more-costly operation such a multiplication. Using a logarithmic number represen tation provides sev eral opportunities for strength reductions; in particular, m ultiplicationis performed as the fixed-point addition of logarithms, and extracting a square root is implemented via a shift. These reductions occur transparently at the hardware level; consequently relativ ely little algorithmic modification is required, and they are readily applicable to adaptive filtering. For performing Givens rotations in the QR decomposition recursiv e least squares adaptive filter, logarithmic arithmetic is shown to compare favorably to other strength reduction techniques, such as CORDIC arithmetic, in terms of switched capacitance and numerical accuracy.

Phase unwrapping by regions using least-squares approach

Abstract: An algebraic method based on finding least-squares solutions from a set of simultaneous linear equations (SLEs) is applied to phase unwrapping by regions in optical interferometry By transforming the phase unwrapping problem into a matrix equation, the characteristics of the phase map can be observed entirely from the properties, such as the rank or the null space, of the matrix that represents the phase unwrap- ping problem The familiarity of solving SLEs using common numerical routines such as QR decomposition means that the algebraic method is easily formulated and computed The robustness of the phase- unwrapping method is then tested computationally and experimentally and its enhancement is proven through the simulation and experimental results © 1998 Society of Photo-Optical Instrumentation Engineers (S0091-3286(98)02311-3)

A Fast Stable Solver for Nonsymmetric Toeplitz and Quasi-Toeplitz Systems of Linear Equations

Abstract: We derive a stable and fast solver for nonsymmetric linear systems of equations with shift structured coefficient matrices (e.g., Toeplitz, quasi-Toeplitz, and product of two Toeplitz matrices). The algorithm is based on a modified fast QR factorization of the coefficient matrix and relies on a stabilized version of the generalized Schur algorithm for matrices with displacement structure. All computations can be done in O(n2) operations, where n is the matrix dimension, and the algorithm is backward stable.

Prediction of Chaotic Time-Series with a Resource-Allocating RBF Network

Abstract: One of the main problems associated with artificial neural networks on-line learning methods is the estimation of model order. In this paper, we report about a new approach to constructing a resource-allocating radial basis function network exploiting weights adaptation using recursive least-squares technique based on Givens QR decomposition. Further, we study the performance of pruning strategy we introduced to obtain the same prediction accuracy of the network with lower model order. The proposed methods were tested on the task of Mackey-Glass time-series prediction. Order of resulting networks and their prediction performance were superior to those previously reported by Platt [12].

Multichannel fast QRD-LS adaptive filtering: new technique and algorithms

Abstract: A direct, unified approach for deriving fast multichannel QR decomposition (QRD) least squares (LS) adaptive algorithms is introduced. The starting point of the new methodology is the efficient update of the Cholesky factor of the input data correlation matrix. Using the new technique, two novel fast multichannel algorithms are developed. Both algorithms comprise scalar operations only and are based exclusively oh numerically robust orthogonal Givens rotations. The first algorithm assumes channels of equal orders and processes them all simultaneously. It is highly modular and provides enhanced pipelinability, with no increase in computational complexity, when compared with other algorithms of the same category. The second multichannel algorithm deals with the general case of channels with different number of delay elements and processes each channel separately. A modification of the algorithm leads to a scheme that can be implemented on a very regular systolic architecture. Moreover, both schemes offer substantially reduced computational complexity compared not only with the first algorithm but also with previously derived multichannel fast QRD schemes. Experimental results in two specific application setups as well as simulations in a finite precision environment are also included.

The logarithmic number system for strength reduction in adaptive filtering

Abstract: An important technique for reducing pow er consumption in VLSI systems is strength reduction, the substitution of a less-costly operation such as a shift, for a more-costly operation such a multiplication. Using a logarithmic number represen tation provides sev eral opportunities for strength reductions; in particular, m ultiplicationis performed as the fixed-point addition of logarithms, and extracting a square root is implemented via a shift. These reductions occur transparently at the hardware level; consequently relativ ely little algorithmic modification is required, and they are readily applicable to adaptive filtering. For performing Givens rotations in the QR decomposition recursiv e least squares adaptive filter, logarithmic arithmetic is shown to compare favorably to other strength reduction techniques, such as CORDIC arithmetic, in terms of switched capacitance and numerical accuracy.

New fast QR decomposition least squares adaptive algorithms

Abstract: This paper presents two new, closely related adaptive algorithms for LS system identification. The starting point for the derivation of the algorithms is the inverse Cholesky factor of the data correlation matrix, obtained via a QR decomposition (QRD). Both algorithms are of O(p) computational complexity, with p being the order of the system. The first algorithm is a fixed order QRD scheme with enhanced parallelism. The second is an order recursive lattice type algorithm based exclusively on orthogonal Givens rotations, with lower complexity compared to previously derived ones. Both algorithms are derived following a new approach, which exploits efficient the and order updates of a specific state vector quantity.

The QR Decomposition and the Singular Value Decomposition in the Symmetrized Max-Plus Algebra

Abstract: In this paper we discuss matrix decompositions in the symmetrized max-plus algebra. The max-plus algebra has maximization and addition as basic operations. In contrast to linear algebra, many fundamental problems in the max-plus algebra still have to be solved. In this paper we discuss max-algebraic analogues of some basic matrix decompositions from linear algebra. We show that we can use algorithms from linear algebra to prove the existence of max-algebraic analogues of the QR decomposition, the singular value decomposition (SVD), the Hessenberg decomposition, the LU decomposition, and so on.

Wrap-around partitioning for block bidiagonal linear systems

Abstract: A stable vector algorithm for the solution of block bidiagonal linear systems is obtained by a permutation of the unknowns called wrap-around partitioning combined with standard QR factorization. Wrap-around partitioning uses blocking and selects the unknowns in the blocks in turns. After a suitable orthogonal elimination step one ends up with a reduced system which is again block bidiagonal and so wrap-around partitioning can be applied again. Using a simple model for vectorization overhead it is shown that small block sizes give best performance. The minimal block size 2, which corresponds to cyclic reduction, is suboptimal due to memory bank conflicts.

A new QRD-based block adaptive algorithm

Abstract: In this paper we present a new robust adaptive algorithm. It is derived from the standard QR decomposition based RLS (QRD-RLS) algorithm by introducing a non-orthogonal transform into the update recursion. Instead of updating an upper triangular matrix, as it is the case for the QRD-RLS, we adapt an upper triangular block diagonal matrix. The complexity of the algorithm, thus obtained, varies from O(N/sup 2/) to O(N) when the size of the diagonal blocks decreases. Simulations of the new algorithm have shown a better robustness than the standard QRD-based algorithm in the context of multichannel adaptive filtering with highly inter-correlated channels.

The Relation Between the QR and LR Algorithms

Abstract: For an Hermitian matrix the QR transform is diagonally similar to two steps of the LR transforms. Even for non-Hermitian matrices the QR transform may be written in rational form.

Accurate Determination of Object Position from Imprecise Data

Abstract: The problem of accurate determination of object position from imprecise and excess measurement data arises in kinematics, biomechanics, robotics, CAD/CAM and flight/vehicle simulator design. Several methods described in the literature are reviewed. Two new methods which take advantage of the modern matrix oriented software (e.g., MATLAB, IMSL, EISPACK) are presented and compared with a basic method. It is found that both of the proposed decomposition methods (I: SVD/QR and II: SVD/QS) give better absolute results than a basic method available from the text books. On a relative basis, the second method (SVD/QS Decomposition) gives slightly better results than the first method (SVD/QR Decomposition). Examples are presented for the cases when the points chosen are nearly dependent and when the independent points have small the points chosen their coordinates.

Transducer Placement for Broadband Active Vibration Control Using a Novel Multidimensional QR Factorization

Abstract: This paper advances the state of the art in the selection of minimal configurations ofsensors and actuators for active vibration control with smart structures. The method extends previous transducer selection work by (1) presenting a unified treatment of the selection and placement of large numbers of both sensors and actuators in a smart structure, (2) developing computationally efficient techniques to select the best sensor-actuator pairs for multiple unknown force disturbances exciting the structure, (3) selecting the best sensors and actuators over multiple frequencies, and (4) providing bounds on the performance of the transducer selection algorithms. The approach is based on a novel, multidimensional extension of the Householder QR factorization algorithm applied to the frequency response matrices that define the vibration control problem. The key features of the algorithm are its very low computational complexity, and a computable bound that can be used to predict whether the transducer selection algorithm will yield an optimal configuration before completing the search. Optimal configurations will result from the selection method when the bound is tight, which is the case for many practical vibration control problems. This paper presents the development of the method, as well as its application in active vibration control of a plate.

A QR-based least mean squares algorithm for adaptive parameter estimation

Abstract: The optimum nonlinearly modified least-mean-square (ONM-LMS) algorithm has been shown to perform better than both the LMS and the normalized LMS algorithms. This paper proposes a QR-LMS adaptive parameter estimation algorithm that can perform significantly better than ONM-LMS. The performances of QR-LMS, including its numerical stability, error propagation property, and tracking ability, are analyzed. These properties are also verified with numerical examples.

A comparison of initialization schemes for blind adaptive beamforming

Abstract: Many blind adaptive beamforming algorithms require the selection of one or more non-zero initial weight vectors. Proper selection of the initial weight vectors can speed algorithm convergence and help ensure convergence to the desired solutions. Three alternative initialization approaches are compared here, all of which depend only on second order statistics of the observed data. These methods are based on Gram-Schmidt orthogonalization, eigendecomposition, and QR decomposition of the observed data covariance matrix. We show through computer simulation that the eigendecomposition approach yields the best performance.

Complex scaled tangent rotations (CSTAR) for fast space-time adaptive equalization of wireless TDMA

Abstract: A new update algorithm for space-time equalization of wireless time-division multiple access signals is presented. The method is based on a modified QR factorization that reduces the computational complexity of the traditional QR-decomposition based recursive least squares method and maintains numerical stability. Square roots operations are avoided due to the use of an approximately orthogonal transformation, defined complex scaled tangent rotation.

Fault Tolerant QR-Decomposition Algorithm and Its Parallel Implementation

Abstract: A fault tolerant algorithm based on Givens rotations and a modified weighted checksum method is proposed for the QR-decomposition of matrices. The algorithm enables us to correct a single error in each row or column of an input M × N matrix A occurred at any among N steps of the algorithm. This effect is obtained at the cost of 2.5N 2+O(N) multiply-add operations (M=N). A parallel version of the proposed algorithm is designed, dedicated for a fixed-size linear processor array with fully local communications and low I/O requirements.

Pipelined implementation of Cordic-based QRD-MVDR adaptive beamforming

Abstract: Cordic-based QR decomposition-based minimum variance distortionless response (QRD-MVDR) adaptive beamforming algorithms possess desirable properties for VLSI implementation such as regularity and good finite-word length behavior. But this algorithm suffers from a speed limitation constraint due to the presence of recursive operations in the algorithm. A fine-grain pipelined Cordic-based QRD-MVDR adaptive beamforming algorithm is developed using the matrix lookahead technique. The proposed architecture can operate at arbitrarily high sample rates, and consists of only Givens rotations which can be mapped onto a Jacobi specific dataflow processor. It requires a complexity of O(M(p/sup 2/+Kp)) Givens rotations per sample time, where p is the number of antenna elements, K is the number of look direction constraints, and M is the pipelining level.

On the implementation of adaptive equalizers for wireless ATM with an extended QR-decomposition-based RLS-algorithm

Abstract: An extended quadratic residue (QR)-decomposition (QRD)-based RLS-algorithm is introduced, applicable for equalization, allowing an estimate of the transmitted symbol, when it is a-priori unknown. To achieve low-cost implementations strength reduction and square-root free Givens-rotations are applied. Several QRD-based RLS-algorithms are compared in terms of the number of mathematical operations. The QR-RLS-algorithm indicates lowest complexity and is always a good candidate, when a huge training overhead has to be avoided, resulting in an equalizer of less than 15 taps. Finally, the hardware complexity of a 13 MBaud-DFE for wireless ATM is estimated.

Iterative Identification Methods for Ill-Conditioned Processes

Abstract: Some ill-conditioned processes are very sensitive to small elementwise uncertainties arising in classical element-by-element model identifications. For such processes, accurate identification of singular values and right singular vectors are more important than those of the elements themselves. Singular values and right singular vectors can be found by iterative identification methods that implement the input and output transformations iteratively. Methods based on SVD decomposition, QR decomposition, and LU decomposition are proposed and compared with Kuong and MacGregor's method. Convergence proofs are given. These SVD and QR methods use orthogonal matrices for the transformations that cannot be calculated analytically in general, and so they are hard to apply to dynamic processes, whereas the LU method uses simple analytic transformations and can be directly applied to dynamic processes.

A Cubically Convergent Parallelizable Method for the Hermitian Eigenvalue Problem

Abstract: We propose a cubically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. The building blocks of the algorithm are matrix--matrix multiplication and QR decomposition which are highly parallelizable. We present a detailed convergence analysis and explore the so-called mixed convergence phenomenon, the understanding of which will be very helpful in devising convergence improvement. We also discuss a number of implementation details and demonstrate convergence properties of the algorithm using several numerical examples.

On Deriving the Quasi-Minimal Residual Method

Abstract: A new derivation of the quasi-minimal residual (QMR) method for the solution of a system of linear equations is presented. This derivation differs from the original one in solving a particular least-squares problem by applying simple recurrences rather than computing a QR decomposition by means of Givens rotations.