Showing papers on "Matrix decomposition published in 1985"

Numerical linear algebra aspects of control design computations

Abstract: The interplay between recent results and methodologies in numerical linear algebra and mathematical software and their application to problems arising in systems, control, and estimation theory is discussed. The impact of finite precision, finite range arithmetic [including the implications of the proposed IEEE floating point standard(s)] on control design computations is illustrated with numerous examples as are pertinent remarks concerning numerical stability and conditioning. Basic tools from numerical linear algebra such as linear equations, linear least squares, eigenproblems, generalized eigenproblems, and singular value decomposition are then outlined. A selected list of applications of the basic tools then follows including algorithms for solution of problems such as matrix exponentials, frequency response, system balancing, and matrix Riccati equations. The implementation of such algorithms as robust mathematical software is then discussed. A number of issues are addressed including characteristics of reliable mathematical software, availability and evaluation, language implications (Fortran, Ada, etc.), and the overall role of mathematical software as a component of computer-aided control system design.

Identification of linear systems with input and output noise: the Koopmans-Levin method

Abstract: The Koopmans-Levin (KL) method of parameter estimation of discrete-time linear systems with input and output noise is based on the spectral decomposition of a covariance matrix, which gives approximately maximum likelihood estimates (MLE) if the noise is white Gaussian. In the paper, three robust algorithms, namely the batch method, the sequential updating of the batch solution and the sequential square-root estimation using an information matrix, are developed, based on the singular-value decomposition of matrices. Coding of these algorithms is relatively straightforward using matrix routines available in standard program libraries. The procedures and the properties of the methods are illustrated using published examples.

Multiple source location--A matrix decomposition approach

Abstract: In this paper we present a new method, without using the eigenstructure of the sensor covariance matrix, for estimating the number of sources d and the direction-of-arrival of sources based on the matrix decomposition. This method uniformly handles coherent and noncoherent sources alike. In our procedure a number p is calculated and used for determination of the maximum number of distinguishable sources by a given linear array system.

Computing the CS and the generalized singular value decompositions

Abstract: If the columns of a matrix are orthonormal and it is partitioned into a 2-by-1 block matrix, then the singular value decompositions of the blocks are related. This is the essence of the "CS decomposition". The computation of these related SVD's requires some care. Stewart has given an algorithm that uses the LINPACK SVD algorithm together with a Jacobitype "clean-up" operation on a cross-product matrix. Our technique is equally stable and fast but avoids the cross product matrix. The simplicity of our technique makes it more amenable to parallel computation on systolic-type computer architectures. These developments are of interest because a good way to compute the generalized singular value decomposition of a matrix pair (A, B) is to compute the CS decomposition of a certain orthogonal column matrix related toA andB.

A fast algorithm for solving a Toeplitz system of equations

Abstract: A fast algorithm for the solution of a Toeplitz system of equations is presented. The algorithm requires order N(\log N)^{2} computations where N is the number of equations. For banded Toeplitz matrices the order of computations is reduced to only N \log N + m(\log m)^{2} where 2m is the maximum number of nonzero principal subdiagonals of the Toeplitz matrix. The algorithm is in essence a fast implementation of the Trench algorithm in reverse. Thus, the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast HD (half-divisor) algorithm to compute the first row of the inverse matrix. The desired solution is then obtained directly from the first row by applying fast Fourier transform techniques in order N \log N computations. Finally, the extension of the algorithm to block Toeplitz matrices is also presented.

The LU decomposition theorem and its implications to the realization of two-dimensional digital filters

Abstract: A new approach to the realization of two-dimensional (2- D) FIR and IIR digital filters is introduced based on the so-called "lower-upper triangular (LU) decomposition" of matrix coefficients of their two-dimensional polynomials. It is demonstrated that the LU realization scheme enjoys a number of attributes for VLSI implementation including high parallelism, modularity, and regularity. This paper also shows that the computational requirements of the LU realization structure are much smaller than those of the Jordan (JD), singular value (SV), and canonical (direct) realizations. Furthermore, it is shown that if a canonical 2-D IIR filter is realizable, then a realizable LU decomposition form can be always obtained, a property that is not shared by the JD and SV decomposition forms. Finally, the extension of the LU decomposition approach to the realization of m-D digital filters is discussed.

On descrete band-limited signal extrapolation

Abstract: Two generalized discrete band-limited signal extrapolation methods are presented. These methods are capable of altering the structure of the eigenvectors as well as the distribution of the eigenvalues of the linear matrix operation, and thus improving the extrapolation process.

Solution of the embedding problem and decomposition of symmetric matrices.

Abstract: A solution of the problem of calculating cartesian coordinates from a matrix of interpoint distances (the embedding problem) is reported. An efficient and numerically stable algorithm for the transformation of distances to coordinates is then obtained. It is shown that the embedding problem is intimately related to the theory of symmetric matrices, since every symmetric matrix is related to a general distance matrix by a one-to-one transformation. Embedding of a distance matrix yields a decomposition of the associated symmetric matrix in the form of a sum over outer products of a linear independent system of coordinate vectors. It is shown that such a decomposition exists for every symmetric matrix and that it is numerically stable. From this decomposition, the rank and the numbers of positive, negative, and zero eigenvalues of the symmetric matrix are obtained directly.

High resolution bearing estimation without eigen decomposition

Abstract: We consider the bearing estimation problem as a matrix approximation problem The columns of a matrix X are embedded with the snapshot vectors from an N element array The matrix X is approximated by a matrix X M in the least square sense The rank, as well as the structure of the space spanned by columns of X M , are prespecified After X M is computed, the bearings of the sources, the spatial correlation of the source signals can be estimated Our technique is then compared with other methods such as MUSIC and SVD processing When the number of snapshot vectors available for processing is large a simpler adaptive algorithm is suggested

Vectorized LU Decomposition Algorithms for Large-Scale Circuit Simulation

Abstract: Proposed here are two kinds of vectorized LU decomposition algorithms for an unstructured sparse matrix arising from large scale circuit simulation. Either algorithm implemented on our supercomputer S810 improves efficiency 11 to 82 times for LU decomposition and 2.1 to 8.9 times in total simulation, as compared with a conventional algorithm. Both algorithms detect operational parallelism in the irregularity of a matrix. While one of them limits the scope of parallelism detection to each set of consecutive columns so as to take advantage of the dense matrix method applicable to the lower right corner, the other tries to locate parallelism all over the matrix in every step without switching to a faster linear-index vector.

Properties of a Representation of a Basis for the Null Space.

Abstract: Given a rectangular matrix A(x) that depends on the independent variables x, many constrained optimization methods involve computations with Z(x), a matrix whose columns form a basis for the null space of A/sup T/(x). When A is evaluated at a given point, it is well known that a suitable Z (satisfying A/sup T/Z = 0) can be obtained from standard matrix factorizations. However, Coleman and Sorensen have recently shown that standard orthogonal factorization methods may produce orthogonal bases that do not vary continuously with x; they also suggest several techniques for adapting these schemes so as to ensure continuity of Z in the neighborhood of a given point. This paper is an extension of an earlier note that defines the procedure for computing Z. Here, we first describe how Z can be obtained by updating an explicit QR factorization with Householder transformations. The properties of this representation of Z with respect to perturbations in A are discussed, including explicit bounds on the change in Z. We then introduce regularized Householder transformations, and show that their use implies continuity of the full matrix Q. The convergence of Z and Q under appropriate assumptions is then proved. Finally, we indicate why themore » chosen form of Z is convenient in certain methods for nonlinearly constrained optimization.« less

A parallel method for computing the generalized singular value decomposition

Abstract: We describe a new parallel algorithm for computing the generalized singular value decomposition of two n × n matrices, one of which is nonsingular. Our procedure requires O (n) time and one triangular array of O(n2) processors.

Efficient algorithms for use in probabilistic finite element analysis

Abstract: This paper investigates the use of Fast Probability Integration (FPI) algorithms in a Finite Element environment. A method allowing the representation of correlated fields in terms of a vector of uncorrelated transformed variables, based on the spectral decomposition of the variance-covariance matrix is developed. The response of the deterministic model corresponding to selected perturbations of these uncorrelated variables is then obtained via a Newton-type iterative scheme. The results of the perturbed problems are used to construct a local representation of the model's behavior in the neighborhood of the deterministic state, which the FPI algorithm will use to estimate the reliability of the system. Although the proposed strategy has thus far only been applied to linear elastostatics, the extension of the method to a broader class of problems appears to be feasible.

Decomposition of Fuzzy Matrices

Abstract: A problem of decomposition of fuzzy rectangular matrices is examined and some properties of decomposition are shown. Any fuzzy matrix can be factored into a product of a square matrix and a rectangular matrix of the same dimension. This square matrix has reflexivity and transitivity. The decomposition of fuzzy matrices is closely related to fuzzy databases and fuzzy retrieval models.

Signal synthesis from Wigner distribution

Abstract: The problem of synthesizing a signal from a modified or specified Wigner distribution function is considered. This problem arises in signal separation, time frequency filtering, window and filter designs. The modified or specified time frequency function may not correspond to a valid Wigner function, thus our synthesis problem is formulated as approximately a two-dimensional function as a product of two one-dimensional function by using least squares procedure. The first procedure involves expressing a time-frequency function as a bilinear combination of basis auto and cross Wigner functions. The least squares approximation leads to an eigenvalue-eigenvector decomposition of a symmetric matrix. The second procedure involves the approximation of a pre-computed matrix as an outer product of two vectors.

The weight matrix in asymptotic distribution-free methods

Abstract: An important problem with asymptotic distribution-free (ADF) methods is the size of the weight matrix. Whereas under the assumption of normality of the observed variables the weight matrix can nicely be decomposed into two matrices of smaller order, under non-normality this cannot be done straightforwardly. In this paper we propose a method in which the weight matrix can be decomposed into matrices of smaller order, which makes inverting the matrix computationally less heavy and extends the usefulness of ADF methods to applications with a larger number of variables. An additional advantage of our method is that the weight matrix is formulated in terms of model parameters. As a consequence, one should expect the weight matrix to be more stable than in cases in which the weight matrix is computed from the data itself. In addition, estimates of the parameters may be less biased, a problem which often arises in ADF methods.

Sparse matrix factor modification in structural reanalysis

Abstract: Structural reanalysis problems, such as in nonlinear finite element analysis or optimum design, involve progressive changes in the global stiffness matrix and its matrix factors. Although many studies have been devoted to the subject of matrix factor modification, most investigations have dealt with the problem separately from sparse matrix methods. This paper introduces a graph-theoretic model for the forward solution procedure which is applicable for identifying the modified entries of the matrix factors due to changes in the original matrix. Applications of this graph-theoretic model to existing refactorization methods are presented. The relation between substructuring and sparse matrix ordering strategies, and their effects on reanalysis are discussed. Modification of a sparse matrix associated with an n × n finite element grid ordered by the nested dissection scheme is analysed.

On the factorization of scattering transfer matrices of multidimensional lossless two-ports

Abstract: The problem of cascade decomposition of arbitrary bivariate lossless two-ports is addressed via factorization of the associated scattering transfer matrix. Necessary and sufficient conditions for the existence of solution to the factorization problem so formulated, are obtained. It is shown that the criterion for factorability can be expressed in terms of solvability of a set of linear simultaneous equations, which is overdetermined in general. Since no restrictions are imposed on the type of transmission zeros of the given scattering transfer matrix, earlier results in the literature can, in principle, be viewed as special cases of the results obtained here.

The Singular-Value Decomposition as a Tool for Solving Estimability Problems

Abstract: The intent of this article is to present a straightforward method of investigating the estimability problems of linear models As a teaching tool the value is twofold: (a) the problems can be structured and solved by standard matrix multiplication, and (b) the uniqueness (or absence of uniqueness) of a solution is explicitly demonstrated to the student The approach is a direct application of the singular-value decomposition of matrix As an intermediate step, a useful representation of the generalized inverse of a matrix is formulated

Error analysis of an algorithm for solving an underdetermined linear system

Abstract: The accumulation of rounding errors in a method used to compute the solution of an underdetermined system of linear equations at the least distance from a given point is being studied. The method used is based on orthogonal matrix decompositions.

Maximum Likelihood Angular Resolution of Multiple Sources

Abstract: normal. If dn denotes the distance of the n The maximum likelihood technique is developed for estimating the directions Of multiple, ClOSelYspaced signal sources from data obtained at the elements of an array. The technique makes use of a certain matrix decomposition which results in some computational simp1 ification. Statistical written as test criteria, which do not require the determination of subjective thresholds, are also applied to the maximum likelihood technique for estimation of the number of sources. The Performance of the maximum likelihood technique and comparison with an eigenvector decomposition technique and the Minimum-Energy technique is determined by computer Simulation. element from some arbitrary first element (origin) and ak(i) denotes the complex amplitude of the ith signal at the origin at time instant tky the observations at the nth element can be

Combinatorial Problems in Matrix Computation

Abstract: When a matrix A is factored, using either a LU decomposition or a QR factorization, its nonzero structure will in general change. I have studied the problem of accurately predicting the changes in structure in the matrix. A framework for categorizing structure-predicting algorithms as "correct" or not is introduced. A combinatorial algorithm that cor- rectly predicts changes in structure is called exact. Previously sug- gested structure-predicting algorithms are shown not to be exact. Precise conditions are given under which structural Gaussian elimi- nation is exact. For a subclass of matrix structures, having the so called "strong Hall property", it is shown that structural QR factorization is exact. The problem of minimizing the filling in of zeros with nonzeros in the factor of the matrix is known to be NP-complete for LU decompo- sition. One suggested heuristic approach for this problem is Quotient Tree Partitioning. Three different measures of what constitutes a good partitioning are suggested, and for each measure the problem is shown to be NP-complete. An algorithm for finding a maximal Quotient Tree Partitioning is given, and shown to have a linear running time. The NP-completeness of minimizing fill in QR factorization is shown, and an algorithm for QR factorization, that avoids the problem of predicting a too large structure for the factored matrix, is given. Chordal graphs are shown to have separators of size O(SQRT.(m)) that can be found in linear time. And chordal graph recognition is shown to be in NC, the class of problems with fast (O(log('k) n)) parallel algorithms.

A dantzig-wolfe decomposition variant equivalent to basis factorization

Abstract: A variant of Dantzig-Wolfe decomposition and basis factorization are compared as solution techniques for block angular systems. It is shown that the two methods follow the same solution path to the optimum. The result has implications for the use of decomposition and factorization algorithms together.

A note on winner-hopf matrix factorization

Abstract: This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Quarterly Journal of Mechanics and Applied Mathematics following peer review. The definitive publisher-authenticated version Rawlins, A D (1985). A note on Wiener-Hopf matrix factorisation. Quarterly Journal of Mechanics and Applied Mathematics. 38 (3) 433-437 is available online at: http://qjmam.oxfordjournals.org/cgi/reprint/38/3/433.pdf

Rank Reduction For Broadband Waves Incident On A Linear Receiving Aperture

Abstract: following form: Express a vector function a(t) (or a matrix A) as the product of a constant, rectangular Vandermonde matrix V and a vector function c(t) (or a matrix C) Transformations can be applied to a(t) (or A) to change V, potentially reducing its rank In processing multichannel outputs from a linear receiving aperture, such a transformation can reduce the rank to the number of plane wave sources present, irrespective of the number of frequencies that each source emits

Decomposition of Cohen's matrix R into simpler color invariants.

Abstract: Matrix R of linear-group tristimulus invariants, presented by Cohen and Friden (1976) and Cohen and Kappauf (1982), was decomposed into functions of simpler tristimulus volume ratios (also linear-group invariants). Such volume ratios were applied to the dual problems of color constancy and illuminant-invariant object-color recognition by an artificial trichromatic photosensor. The proof of the decomposition theorem relies on a lemma used in previous work elucidating the conditions for illuminant-invariance of clockwise/counterclockwise ordering of triads of object colors in chromaticity space.

Cascade decomposition and realization of multivariable systems via block-pole and block-zero placement

Abstract: A technique is presented for state-feedback and state-feedforward block decomposition of a class of multivariable systems into a block-cascaded form structure having the assigned block poles and block zeros. A state-space technique is also presented for minimal realization of a multivariable compensator, represented by a voltage transfer function matrix, using the decoupled RC cascaded networks and summers.