Showing papers on "QR decomposition published in 1984"

On the Method of Weighting for Equality Constrained Least Squares Problems

Abstract: The generalized singular value decomposition is used to analyze the problem of minimizing $||Ax b||_{2}$ subject to the constraint Bx = d. A byproduct of the analysis is a new iterative procedure that can be used to improve an approximate solution obtained via the method of weights. All that is required to implement the procedure is a single QR factorization. These developments turn out to be of interest when A and B are sparse and for the case when systolic architectures are used to carry out the computations.

On the Asymptotic Behavior of Scaled Singular Value and QR Decompositions

Abstract: Asymptotic expressions are derived for the singular value decompositon of a matrix some of whose columns approach zero. Expressions are also derived for the QR factorization of a matrix some of whose rows approach zero. The expressions give insight into the method of weights for approximating the solutions of constrained least squares problems.

A Parallel QR Method Using Fast Givens' Rotations.

Abstract: : Given a prescribed order in which to introduce zeroes, and constraints on the architecture it is shown how to develop a parallel QR factorization based on fast Givens' rotations for a rectangular array of processors, suitable to VLSI implementation. Unlike designs based on standard Givens' transformations, the present one requires no square root computations. Assuming each processor performs the elementary operations (+,*,/), less than O(w sub 2) processors can achieve the decomposition of a w-banded, order n matrix in time O(n). Application is made to a variant of Bareiss' G-Algorithm for the solution of weighted multiple linear least squares problems. Given k different right hand side vectors, (w sub 2) processors compute the factorization in O(n + k) steps. (Author)

On the relation between triangular matrix decomposition and linear interpolation

Abstract: The December 1983 letter by C. W. Therrien concerning the relation between triangular matrix decomposition and linear prediction is extended to include linear interpolation.

Concurrent array processor for fast eigenvalue computations

Abstract: We present in this paper an algorithm that doubles-up on Francis's QR algorithm. By this we mean that we can obtain the QR iterates at step 1, 2, 4, 8, 16, etc for every sweep over the matrix. We also show that the algorithm can be implemented on a highly regular array of computing elements with only neighborhood communication between processors. Simulations are presented which suggest that algorithm is stable.

Analysis of Some Matrix Problems Using the CS Decomposition

Abstract: The gist of the CS decomposition is that the blocks of a partitioned orthogonal matrix have related singular value decompositions. In this paper we develop a perturbation theory for the CS decomposition and use it to analyze (a) the total least squares problem, (b) the Golub-Klema-Stewart subset selection algorithm, (c) the algebraic Riccati equation, and (d) the generalized singular value decomposition.

A Jacobi-like Algorithm for Computing the QR-Decomposition

Abstract: A parallel Jacobi-like method for computing the QR-decomposition of an $n \times n$ matrix is proposed. It requires $O(n^{2})$ processors and $O(n)$ units of time. The method can be extended to handle an $m \times n$ matrix $(m \geq n)$. The requirements become $O(n^{2})$ processors and $O(m)$ time.

A building blocks approach to statistical databases

Abstract: In recent years, considerable attention has been given to various data management issues for statistical databases. However, most of the research in statistical databases has concentrated on retrieval, sampling, and aggregation type statistical queries. Data management issues associated with the complex and computational statistical queries have been ignored. This dissertation endorses an integrated approach to statistical databases. In an integrated system, the database management software supports both data retrieval and computational queries. As a first step towards an integrated system we identify and analyze the performance of three important statistical computational methods: X'X, the QR decomposition, and the Singular Value Factorization. Our performance evaluation compares two alternative secondary storage organizations: the transposed and relational organizations. We also propose several implementation strategies for each computational method. The alternative implementations correspond to vector building block, vector-matrix, and direct algorithms. We develop buffer management algorithms for each implementation strategy, and, evaluate corresponding I/O and total execution time cost functions in terms of data and system parameters for a general system architecture. The variable parameters in our performance curves are the number of active columns, the size of main storage, and the time per floating point operation. We also propose special purpose multiprocessor architectures for each computational method. A multistage shuffle/exchange interconnection network (the ZETA network) is proposed for X'X. A multidisk/multiprocessor organization is proposed for the QR decomposition, and a cyclic shift multiprocessor interconnection is proposed for the Singular Value Factorization. Three main conclusions which can be drawn from our extensive performance evaluations of X'X, the QR decomposition, and the Singular Value Factorization. The first conclusion is that integrated statistical database management systems which support the computational methods through vector building blocks or vector-matrix operations have significantly inferior performance compared to direct implementations of the computational methods. The second conclusion is that for those statistical computational methods whose algorithms involve an iterative decrease in the number of active columns, the preferable underlying storage structure is the fully transposed secondary storage organization. Finally, our results indicate that developing a general-purpose statistical database machine appears to be a difficult task.

On QR factorization updatings

Abstract: In recent years several algorithms have appeared in the literature for modifying the factors of a matrix following a rank-1 change. These methods have always been given in the context of specific applications and this has probably inhibited their use over a wider field. In this report a method is given for obtaining theQR factors of a matrix and its updatings after a row or a column has been deleted.

Studies in Statistical Signal Processing.

Abstract: : The primary objective of our research is to develop efficient and numerically stable algorithms for nonstationary signal processing problems by understanding and exploiting special structures, both deterministic and stochastic, in the problems We also strive to establish and broaden links with related disciplines, such as cascade filter synthesis, scattering theory, numerical linear algebra, and mathematical operator theory for the purpose of cross fertilization of ideas and techniques These explorations have led to new results both in estimation theory and in these other fields, eg, to new orthogonal cascade digital filter structures, new algorithms for triangular and QR factorization of structured matrices and new techniques for stability testing For several years, the guiding principle in these studies has been the concept of (Toeplitz-oriented) displacement structure (Kailath, Kung and Morf, (1979)), which generalized and subsumed our earlier work on fast (Chandrasekhar) control and estimation algorithms for state-space models (Morf, Sidhu and Kailath, (1974)) Several authors have since picked up these ideas in a number of fields A notable such work is a recent book by Heinig and K Rost of East Germany, entitled 'Algebraic Methods for Toeplitz-Like Matrices and Operators'