Off-diagonal Blocks Research Articles

Exponential Convergence Properties of Autocovariance Matrix Inverses and Latent Vector Prediction Coefficients

This paper presents some convergence properties of the inverse of the autocovariance matrix of a vector autoregressive moving average model. Under causality and invertibility conditions, the inverse has bounded on-diagonal blocks and exponentially declining off-diagonal blocks and some elements of the inverse converge to limiting values at exponential rates. These properties lead to similar convergence rates for filter weights and prediction error autocovariances in a latent-variable prediction problem. Uniform convergence rates and the convergence properties of some matrix derivatives are also discussed. These results are useful in developing some distributional properties of estimated autocovariance matrix inverses and estimated latent- variable filter weights.

Light scattering by Prorocentrum micans: a new method and results

Striking light-scattering behavior was observed from a marine dinoflagellate, Prorocentrum micans. Measurements of the angular dependence of the 16 Mueller matrix elements were performed on single cells with a polarization-modulation nephelometer by using a new method for cell immobilization. First the dinoflagellate cells were immobilized in a transparent silica gel containing alcohol, and then a second liquid was diffused into the gel to match the index of refraction of the gel network, thereby producing a transparent support medium that scatters less than one tenth the amount of light scattered by a single cell at 90 degrees . Measurements of scattering by a single cell revealed that all 16 matrix elements were significantly nonzero and different from each other. All matrix elements have an extremely rich, reproducible structure that is highly dependent on cell orientation. The matrix elements symmetrically across the diagonal were not equivalent. Striking features of the measurements are the large peak values of S(13), S(14), and other off-diagonal block elements. We believe that this is the first report of such scattering signals by single, suspended marine microorganisms.

A parallel gaussian elimination method for general linear systems

Based on the folding method of Evans and Hatzopoulos [5] (see also [1]), in [2] we had described a bi-directional Gaussian elimination algorithm for linear systems Ax= dwith a full general coefficient matrix . Extending these ideas in the present paper we present a parallel Gaussian elimination method suitable for a multiprocessor machine. By partitioning the coefficient matrix into r 2blocks each of size n × n (N = rn), we introduce a series of transformations L (k) which at stage of the process simultaneously eliminate the kth column in each off-diagonal block and all the elements below the main diagonal in column kof each diagonal block . The thus transformed system uncouples into rsubsystems each of size n × n with an upper triangular coefficient matrix. The present method has an arithmetical operations count of which is consistent with the count of for the standard Gaussian method; thus, the present method could posibly perform with efficiency 2/(3 - 1/r) if implemented on an r-processor machiner≥2.

Some remarks on the spectra of Hermitian matrices

The well-known Cauchy theorem connects the eigenvalues of a Hermitian matrix to the eigenvalues of a principal submatrix by a sequence of interlacing inequalities. In this note we derive some consequences of the assumption that some of these inequalities become equalities or near-equalities. Our results concern the tininess of the coupling off-diagonal block as well as corresponding subspace estimates.

Grouping efficacy: a quantitative criterion for goodness of block diagonal forms of binary matrices in group technology

SUMMARY Block diagonalization of binary matrices is a primary step in the design of cellular production systems. ‘Grouping efficiency’ which is a weighted average of two measures that consider the voids in the diagonal blocks and exceptional elements in the off-diagonal blocks was the only criterion available to measure the goodness of block diagonal forms. The present work critically analyses this function and brings out its shortcomings, the most severe of them being its low discriminating power. A simple and elegant function has been derived in its place. The new function called grouping efficacy obviates all the defects of the earlier function while retaining the requisite properties. The mathematical properties of the function have been analysed and the function values compared with those of grouping efficiency in the case of well-structured and ill-structured data sets.

Computing the singular value decomposition on a distributed system of vector processors

Jacobi methods for computing the singular value decomposition (SVD) of a matrix are ideally suited for multiprocessor environments due to their inherent parallelism. In this paper we show how a block version of the two-sided Jacobi method can be used to compute the SVD efficiently on a distributed architecture. We compare two variants of this method that differ mainly in the degree to which they diagonalize a given subproblem. The first method is a true block generalization of the scalar scheme in that each off-diagonal block is completely annihilated. The second method is a scalar Jacobi algorithm reorganized in such a manner that it conforms to the block decomposition of the problem. Experiments on the LCAP system at IBM Kingston show that the block Jacobi algorithm performs very well on a distributed system, especially when the processors have vector processing hardware. As an example, we were able to achieve a sustained performance of 159 MFlops on a 960 × 720 SVD problem using eight processors. A suprising outcome of these experiments is that the determining factor for the performance of the algorithm on high-performance architectures is the subproblem solver, not the communication overhead of the algorithm.

Eigenvalues of completions of submatrices

This paper concerns the following problem:Given a submatrix of an n× nmatrix Ztogether with its position in Z, determine the restrictions imposed on the eigenvalues of Zand their elementary divisors by this prescribed part. The terms on which the final results depend are identified as the block similarity invariants. Special cases, which were considered before, are reviewed, and the case of off-diagonal blocks is solved.

SOLVING SPARSE TRIANGULAR LINEAR SYSTEMS ON PARALLEL COMPUTERS

This paper describes and compares three parallel algorithms for solving sparse triangular systems of equations. These methods involve some preprocessing overhead and are primarily of interest in solving many systems with the same coefficient matrix. The first approach is to use a fixed blocksize and form the inverse of the diagonal blocks. The second approach is to use a variable blocksize and reorder the unknowns so that the diagonal blocks are diagonal matrices. The latter technique is called level scheduling because of how it is represented in the adjacency graph, and both row-wise and jagged diagonal storage for the off-diagonal blocks are considered. These techniques are analyzed for general parallel computers and experiments are presented for the eight-processor Alliant FX/8.

Linear response treatment of multipole excitations at finite temperature.

Multipole excitations ${\ensuremath{\lambda}}^{\ensuremath{\pi}}$${=2}^{+}$ and ${3}^{\mathrm{\ensuremath{-}}}$, in $^{116}\mathrm{Sn}$ and $^{208}\mathrm{Pb}$, are calculated within the formalism of the finite-temperature linear response theory. Low- and high-energy strength distributions, for quadrupole and octupole vibrations, are shown as a function of the nuclear temperature. The resulting energy-weighted sum rules, for the associated multipole operators, are found to be weakly varying functions of the temperature T, in the interval 0\ensuremath{\le}T\ensuremath{\le}2 MeV, provided off-diagonal blocks of the response matrix are included in the formalism.

Almost invariance and noninteracting control: a frequency-domain analysis

We solve a number of feedback synthesis problems in the context of noninteracting control or block-diagonal decoupling for finite-dimensional linear time-invariant systems. We consider a plant that, apart from a control input and a measurement output, has a given number of exogenous input vectors and the same number of exogenous output vectors. The decoupling problem studied is to find dynamic compensators from the plant measurement output (which in this paper will be assumed to be the full plant state) to the plant control input in such a way that the following requirements are met: (1) the closed-loop transfer matrix is block-diagonal, (2) the remaining diagonal blocks are stable with respect to an a priori given first stability set, and (3) the clossed-loop system is internaly stable with respect to an a priori given second (in general larger) stability set. In addition, we study the “almost” version of the above problem. In the latter the requirement of exact decoupling is replaced by a requirement of approximate decoupling in the sense that the compensators to be designed should yield off-diagonal blocks in the closed-loop transfer matrix that are arbitrarily small in H ∞-norm. Necessary and sufficient conditions for the existence of such dynamic compensators are formulated in terms of controlled invariant and almost controlled invariant subspaces.

On Kogbetliantz's SVD Algorithm in the Presence of Clusters

We consider matrices with off-diagonal blocks of small norm and derive tight bounds for the approximation of their singular values by those of their diagonal blocks. These results are used to show that triangular matrices with clusters of singular values must possess a principal submatrix of “nearly” diagonal form. From the latter we then derive results pertaining to the quadratic convergence of Kogbetliantz's algorithm for computing the SVD, in the presence of clusters.

Almost non-interacting control by measurement feedback

Consider a linear system Σ that, apart from a control input and a measurement output, has two exogeneous inputs and two exogenous outputs. Controlling such a system by means of a measurement feedback compensator Σ c results in a closed loop system with two inputs and two outputs. Hence, the closed loop transfer matrix can be partitioned as a two by two block matrix. The problem addressed in this paper consists of the following. Given Σ and any positive number ge, is it possible to find Σ c such that the off-diagonal blocks of the closed transfer matrix, in a suitable norm, are smaller than ϵ? For the solvability of this problem necessary and sufficient conditons will be derived.

Approximating Prediction Error Variances for Multiple Trait Sire Evaluations

The coefficient matrix for multiple trait (milk, fat, and protein) mixed model equations may be too large to obtain prediction error variances from inverse elements. The commonly used reciprocals of diagonal elements may not be accurate approximations when sire relationships or multiple traits are included since much information is contained in offdiagonal elements. Approximations incorporating increased information from coefficient matrix were compared with actual prediction error variances for multiple trait evaluations for milk, fat, protein, and dollar value (relationships included) of 229 Ayrshire and 248 Brown Swiss bulls. Six approximations were selection index using number of daughter records, inverses of individual sire diagonal blocks, inverses of group and individual sire blocks, and inverses of all diagonal blocks and offdiagonal blocks associated with individual sires. All approximations underestimated actual prediction error variances, but most, except selection index, were highly correlated (.90 to .99) with actual prediction error variances of sire evaluations for milk yield and product value for contemporary bulls. The approximation incorporating most information from the coefficient matrix is recommended for use on basis of high correlation with and closeness to actual prediction error variances.

Resonances from the complex dilated Hamiltonians in a dilation-adapted basis set with a new stabilization parameter

Explicit consideration of the analytic properties of the solutions to the dilated Hamiltonian is taken into account in the construction of the matrix representation of the latter in an L2 basis. The total dilated matrix is blocked according to division of the basis into ‘‘bound’’ and ‘‘scattering’’ subspaces, which are interacting via the off-diagonal blocks, leading to a coupling maintaining the adequacy of the bound part of the basis throughout the wide range of the dilation angle. The size of the bound subspace, M, becomes a new stabilization parameter; its variation covers the entire range of situations between a real stabilization calculation and the conventional complex-scaling calculation. This construction allows for a systematic analysis of the dilated Hamiltonian, bringing forward the physical interpretation of the configuration interaction while suppressing the disadvantageous effects of the dilation transformation, manifested by poor convergence. The connections to Junker’s complex stabilization method are discussed.

Robust stability theory using both singular value and numerical range data

An approach to robust stability analysis for block structured multiplicative plant perturbations is presented where off-diagonal block uncertainty is described by singular value bounds and diagonal block uncertainty by (mixed) singular value bounds and inclusion conditions on the numerical range.

A note on the structure of space frame stiffness equations for iterative solution storage schemes

The off-diagonal blocks in space frame stiffness equations are observed to have a structure which allows them to be stored in 21 memory locations instead of 36. This allows iterative and semi-iterative (such as conjugate gradient methods) storage scheme algorithms to store the equations more compactly.

On efficient solution of large-scale Newton-Raphson-based flowsheeting problems in limited core

A strategy is presented for solving in limited core the very large sparse linear equation systems that arise in the Newton-Raphson approach to large-scale flowsheeting problems. Such systems may be too large to handle in core all at once and thus must be decomposed. The strategy employs a bordered-block-triangular decomposition and solves the system by block elimination. This prevents loss of sparsity in off-diagonal blocks and facilitiates the efficient use of core and mass storage. Results indicate the strategy to be more effective in this regard than strategies based on a bordered-block-diagonal decomposition.

Standard Errors of Multipliers and Forecasts from Structural Coefficients with Block-Diagonal Covariance Matrix

For some structural econometric models, the contribution of the off-diagonal blocks of the coefficients covariance matrix to the asymptotic standard errors of multipliers and forecasts is empirically evaluated. The reasons suggesting these experiments are briefly discussed. Although the results should not be generalized, it could be useful, in the model building process, to perform the above mentioned computations, even when only the diagonal blocks of the covariance matrix of the coefficients are available.

ChemInform Abstract: MICROWAVE SPECTRUM OF METHYLENE FLUORIDE IN EXCITED VIBRATIONAL STATES

Abstract The rotational spectra of 12CH2F2 in seven of the nine fundamental vibrational states and also in overtone and combination states involving the ν4 mode were observed and assigned. Coriolis interactions between ν3 and ν7, ν2 and ν8, ν3 and ν9, and ν5 and ν7 were analyzed by using approximate expressions for the rotational levels. An effective Hamiltonian with the Coriolis term in the off-diagonal block was applied to stronger interaction between ν3 and ν9. Fermi resonance between ν3 and 2ν4 was found to be negligible. The ground state spectra of 12CH2F2 and of 13CH2F2 were remeasured to improve the accuracy of the rotational and centrifugal distortion constants. The Coriolis coupling constants and the energy differences between two vibrational levels in resonance, which were obtained through an analysis of the satellite spectra, are compared with the results derived from a normal coordinate analysis.

Microwave spectrum of methylene fluoride in excited vibrational states

The rotational spectra of 12CH 2F 2 in seven of the nine fundamental vibrational states and also in overtone and combination states involving the ν 4 mode were observed and assigned. Coriolis interactions between ν 3 and ν 7, ν 2 and ν 8, ν 3 and ν 9, and ν 5 and ν 7 were analyzed by using approximate expressions for the rotational levels. An effective Hamiltonian with the Coriolis term in the off-diagonal block was applied to stronger interaction between ν 3 and ν 9. Fermi resonance between ν 3 and 2 ν 4 was found to be negligible. The ground state spectra of 12CH 2F 2 and of 13CH 2F 2 were remeasured to improve the accuracy of the rotational and centrifugal distortion constants. The Coriolis coupling constants and the energy differences between two vibrational levels in resonance, which were obtained through an analysis of the satellite spectra, are compared with the results derived from a normal coordinate analysis.