Showing papers on "Singular value decomposition published in 1986"

Generalized rank annihilation factor analysis

Abstract: : The method of rank annihilation is shown to be a particular case of a more general method for quantitation in bilinear data arrays such as LC/UV, GC/ MS or emission-excitation fluorescence Generalized rank annihilation is introduced as a calibration method that allows for simultaneous quantitative determination of all the analyses of interest in a mixture of unknowns Only one calibration mixture is required The bilinear spectra of both unknown and calibration sample must be obtained Bilinear target factor analysis is introduced as a projection of a target bilinear matrix onto another principal component bilinear matrix space Keywords: Multivariate analysis; Principal component regression (PCR); Two-dimensional data; Singular value decomposition; and Pseudoinverse

Computing the polar decomposition with applications

Abstract: A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice.To add to the known best approximation property of the unitary polar factor, the Hermitian polar factor H of a nonsingular Hermitian matrix A is shown to be a good positive definite approximation to Aand $\frac{1}{2}(A + H)$ is shown to be a best Hermitian positive semi-definite approximation to A. Perturbation bounds for the polar factors are derived.Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix.

Effects of Noise on Modal Parameters Identified by the Eigensystem Realization Algorithm

Abstract: The basic concept of the Eigensystem Realization Algorithm for modal parameter identification and model reduction is extended to minimize the distortion of the identified parameters caused by noise. The mathematical foundation for the properties of accuracy indicators, such as the singular values of the data matrix and modal amplitude coherence, is provided, based on knowledge of the noise characteristics. These indicators quantitatively discriminate noise from system information and are used to reduce the realized system model to a better approximation of the true model. Monte Carlo Simulations are included to support the analytical studies.

Implementation of adaptive array algorithms

Abstract: 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.

Incomplete data problems in X-ray computerized tomography. I. Singular value decomposition of the limited angle transform

Abstract: Etude des problemes intrinseques relatifs aux donnees incompletes dans la tomographie aux rayons X assistee par ordinateur. Decomposition de la transformation de Radon

A Hamiltonian $QR$ Algorithm

Abstract: This paper presents a variant $QR$ algorithm for calculating a Hamiltonian–Schur decomposition [10]. It is defined for Hamiltonian matrices that arise from single input control systems. Numerical stability and Hamiltonian structure are preserved by using unitary symplectic similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of similarity transformations yields a permuted triangular matrix. As the iteration converges, it deflates into problems of lower dimension. Convergence is accelerated by varying a scalar shift. When the Hamiltonian matrix is real, complex arithmetic can be avoided by using an implicit double shift technique. The Hamiltonian-Schur decomposition yields the same invariant subspace information as a Schur decomposition but requires significantly less work and storage for problems of size greater than about 20.

Computing the singular value decompostion of a product of two matrices

Abstract: An algorithm is developed for computing the singular value decomposition of a product of two general matrices without explicitly forming the product. The algorithm is based on an earlier Jacobi-like method due to Kogbetliantz and uses plane rotations applied to the two matrices separately. A triangular variant of the basic algorithm is developed that reduces the amount of work required.

Relationships among eigenshape analysis, Fourier analysis, and analysis of coordinates

Abstract: Eigenshape analysis (a singular value decomposition of a matrix of the tangent angle function φ*(t)) has recently been proposed as an alternative to Fourier analysis for description of outline shapes of organisms. Whenall eigenvectors andall harmonics are retained both approaches represent orthogonal rotations of the same points. Thus distances between pairs of shapes (and any multivariate analyses based on distances) must be the same for both analyses. When true shapes are known to be smooth, dropping higher-order Fourier harmonics results in a desirable smoothing of the digitized outline and a large reduction in computational cost. An alternative method of eigenshape analysis is presented and related to elliptical Fourier analysis and analysis of raw coordinates.

An analysis of the magnetotelluric impedance for three-dimensional conductivity structures

Abstract: The eigenstate analysis of Lanczos, also known as singular value decomposition (SVD), is used to define eight parameters which uniquely describe the magnetotelluric impedance Z. These parameters are independent of a priori assumptions about Z and can be interpreted in terms of three‐dimensional conductivity structures. Through SVD, the impedance is represented by two characteristic states. These states consist of two pairs (E and H) of complex vectors and two corresponding, real, singular values which together describe the extremal properties of Z. The singular values are the maximum and minimum |E|/|H| ratios possible at the observation site and therefore yield the true maximum and minimum apparent resistivities. We use a variation of SVD analysis by incorporating phases in the singular values, which are then called characteristic values. These phases reflect the delay (caused by the earth’s conductivity) of the electric fields relative to their associated magnetic fields. In this analysis of Z, the char...

Parallel QR Decomposition of a rectangular matrix

Abstract: We show that the greedy algorithm introduced in [1] and [5] to perform the parallel QR decomposition of a dense rectangular matrix of sizem×n is optimal. Then we assume thatm/n2 tends to zero asm andn go to infinity, and prove that the complexity of such a decomposition is asymptotically2n, when an unlimited number of processors is available.

Optimal choice of a truncation level for the truncated SVD solution of linear first kind integral equations when data are noisy

Abstract: Given error contaminated discrete data $z_i = \int_0^1 {k(s_i ,t)f(t)dt + \varepsilon _i } $, $i = 1, \cdots ,n$, we apply the truncated singular value decomposition to find an approximate solution to the Fredholm first kind integral equation $(Kf)(x) = \int_0^1 {k(s,t)f(t)dt = g(s)} $. We define an optimal truncation level m and apply generalized cross validation to choose an estimate of this optimal truncation level which depends on the data. Convergence rates for $\|f_{m;n} - f\|$ , where $f_{m;n} $ denotes the approximation obtained from the data with this optimal truncation level, are obtained. A numerical example is included.

Solving Eigenvalue and Singular Value Problems on an Undersized Systolic Array

Abstract: Systolic architectures due to Brent, Luk and Van Loan are today the most promising method for computing the symmetric eigenvalue and singular value decompositions in real time. These systolic arrays, however, are only able to decompose matrices of a given fixed size. Here we present two modified algorithms and a modified array that do not have this disadvantage. The results of a numerical experiment show that a combination of one of our new algorithms and the modified array can decompose matrices of arbitrary size with little or no loss of efficiency.

Application of Singular Value Decomposition to the Design, Analysis, and Control of Industrial Processes

Abstract: Singular Value Decomposition (SVD) is proving to be a very useful tool in modern linear system theory. It is also finding an important role in analysis and design of control systems for those real industrial processes which tend to not be so well behaved as assumed by the theory. This paper will describe how the concepts of SVD analysis can be used to help determine realistic answers to the many practical questions that must be addressed by the control engineer if he or she is going to tackle real multivariable control problems. Such questions as: What is the effect of process design alternatives and operating conditions on the control problem? The paper will also show how SVD can be used to design simple but effective multivariable control systems.

Complexity of parallel QR factorization

Abstract: An optimal algorithm to perform the parallel QR decomposition of a dense matrix of size N is proposed. It is deduced that the complexity of such a decomposition is asymptotically 2N, when an unlimited number of processors is available.

Computation of system balancing transformations

Abstract: An algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization. The algorithm requires no "squaring up." Various algorithmic aspects are discussed in detail. Applications to numerous other closely-related problems are also mentioned. The key idea throughout involves determining a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product.

Architectures for Computing Eigenvalues and SVDs

Abstract: Systolic architectures for eigenvalues and singular values are discussed. One "triangular" processor array and associated algorithms for computing the QR-decomposition, the singular value decomposition, the generalized singular value decomposition and the CS-decomposition are described in detail.

Architectures for a CORDIC SVD processor

Abstract: Architectures for systolic array processor elements for calculating the singular value decomposition (SVD) are proposed. These special purpose VLSI structures incorporate the coordinate rotation (coRDic) algorithms to diagonalize 2X 2 submatrices of a large array. The area-time complexity of the proposed architectures is analyzed along with topics related to a prototype implementation.

An unconstrained partitioned realization for derivative constrained broad-band antenna array processors

Abstract: This paper presents an orthogonal decomposition approach for realizing the derivative constrained broad-band processor as an unconstrained partitioned form. The structure presented is similar to the generalized sidelobe canceller (GSC) structure described in the literature. Furthermore, the paper also examines the components of the unconstrained partitioned structure for the first-order case in detail by using a singular value decomposition (SVD) of the constraint matrix. It is shown for an arbitrary array that for the first-order case, the fixed section is a conventional beamformer.

Approximate realization algorithms for truncated impulse response data

Abstract: The objectives of this paper are: 1) to present a novel derivation of an approximate realization algorithm using singular value decomposition proposed by Kung; 2) to explain the difference between two versions of this algorithm which have been used interchangeably in the literature; and 3) to introduce a new version of this algorithm which exhibits the accuracy of the original algorithm and the theoretical guarantee of stability.

On balancing linear symmetric systems

Abstract: The definition is introduced of a balanced system through the singular value decomposition of the extended reachability and observability matrices. On this basis, two methods for determining the balanced representation are presented, the properties of the balanced systems are derived and the usefulness of the balancing method to system reduction is demonstrated.

Multiprocessor Jacobi algorithms for dense symmetric eigenvalue and singular value decompositions

Abstract: Two parallel algorithms are presented based on Jacobi's method for real symmetric matrices to determine the complete eigensystem of a dense real symmetric matrix and the singular value decomposition of rectangular matrices on a multiprocessor. The intent is to study the advantages of using Jacobi and Jacobi-like schemes over new and existing EISPACK and LINPACK routines on an Alliant FX/8 computer system. For the dense symmetric eigenvalue problem, promising results are shown for small-order matrices. A ''one-sided'' Jacobi-like algorithm which produces the singular value decomposition of a rectangular matrix is shown to provide superior performance for rectangular matrices in which the number of rows is much larger than the number of columns. 17 refs., 9 figs., 5 tabs.