Updating URV decompositions in parallel

We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.

Inverting linear combinations of identity and generalized Catalan matrices

Efficient implementation of global computations on a linear array of processors is complicated due to the small communication bandwidth and the large communication diameter of the array. This paper presents efficient parallel techniques for partitioning, movement, and reduction of data on linear arrays. Also, efficient data structures are used to enable fast sequential access of query points within each processor. This combination of serial and parallel techniques is used to derive an optimal parallel algorithm for computing the convex hull of each connected region in an n×n image. The algorithm takes O(n 2/p) time on a linear array with p processors, where 1≤p≤n/log n. This result is processor-time optimal since an optimal sequential algorithm takes O(n 2) to solve the problem. Thus, a linear array with n/log n processors can solve the above problem in O(n log n) time. In comparison, a two dimensional mesh-connected array of processors can solve this problem in O(n) time using n 2 processors. The processor-time product for the mesh is O(n 3), which is not optimal.KeywordsConvex HullExtreme PointLinear ArrayLocal MemoryQuery PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This study develops and tests a MATLAB program for neutralising systems with upper triangular coefficient matrices. Augmented matrices for systems of linear equations are reduced to produce a neutral (identity matrix) left hand side, at which point the right hand side has the solutions for the system. A strategy for minimal error is pursued, where elimination operations must be completed in one step for each number. The strategy was expressed in pseudocode, a flowchart developed, and then the program was coded in MATLAB. The program was tested for four sample augmented matrices of varying sizes. The results demonstrated that the program was 100% accurate. The technique can be applied to any square matrix system, since any square matrix can be expressed as an upper triangular matrix. The automation of matrix operations and achievement of 100% accuracy allows for the use of the program for very large data sets, extending the potential for research with accurate finding.

Parallel sorting with cooperating heaps in a linear array of processors

The authors describe a hardware architecture called Ringtree for 2D geometry generation and processing such as image processing (noise suppression, notch elimination and contour extraction), graphics processing (polygon filing and multiwindowing with nonrectangular windows), and VLSI layout verification (design-rule checking). Ringtree consists of a ring memory which is a special rotating frame buffer, an edge painting tree (EPT), which is a polygon rasterizing hardware, and a linear processor array (LPA). The LPA executes a set of basic operations applicable for bit map data manipulation while the EPT generates the bit map data from a set of scanline commands received from host processor. VLSI implementation issues are also discussed for practical display screen size of 1024*1024 pixels, utilizing the Super Tree concept of J. Poulton et al. (1985) for realizing the whole system using identical VLSI chips. >

This paper shows how a real-time simulator of moving pictures compression algorithms can be rapidly assembled using a basic building block, here called MOVIE (MOdule for Video Experimentation). The internal architecture of the MOVIE VLSI chip can be compared to a small systolic machine made of a 32-bit I/O processor, a reduced linear array of 16-bit computation processors and data video input/output mechanisms. Externally, the chip is provided with four 16-bit bidirectional data ports and three 16-bit bidirectional data video port. Several MOVIE chips can be easily clustered to allow the size of the linear array of computation processors to be increased. The MOVIE chip is fully programmable in a high level language in order to make program developments easier.

Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This second edition of this acclaimed text presents results of both classic and recent matrix analysis using canonical forms as a unifying theme and demonstrates their importance in a variety of applications. This thoroughly revised and updated second edition is a text for a second course on linear algebra and has more than 1,100 problems and exercises, new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices, and much more.

A QR decomposition aided belief propagation (QR‐BP) detector for multiple‐input multiple‐output (MIMO) systems is proposed. Before the iterative message passing process, linear transformation is performed by using the unitary matrix obtained from the QR decomposition of the original MIMO channel matrix. Then, the original MIMO channel matrix is replaced by an upper triangular channel matrix after the linear transformation and the proposed QR‐BP detector operates over the new bipartite graph from the upper triangular channel matrix. Therefore, the number of cycles in the new bipartite graph is significantly decreased when compared with the fully connected bipartite graph from the original MIMO channel matrix. In addition, since the unitary matrix is utilised for the linear transformation, there is no information loss on the likelihoods estimated from noisy observations. Simulation results verify that the proposed QR‐BP detector outperforms existing BP detectors and achieves near‐optimum performance close to the maximum‐likelihood detector in terms of bit error ratio.

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.

In Gaussian elimination and LU factorization we solve a linear system by transforming it to triangular form. These are not the only kind of transformations that can be used for such a task. Matrices with orthonormal columns, called unitary matrices can be used to reduce a square matrix to upper triangular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a QR decomposition and a reduced form which we refer to as a QR factorization. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.

Systolic algorithm for polynomial interpolation and related problems

Lower bounds for two call control problems

The paper presents a parallel architecture for a pel-recursive motion estimation algorithm. It is a linear array of processors, each consisting of an initialization part, a data-routing part and an updating part. The initializing part performs a prediction of the motion vector. The routing parts constitute a routing path along which previous-frame data is routed from processors that store to processors that request such data. A clocked version of the router is presented with some detail. The updating part calculates an update to the predicted motion vector. The architecture proposed is derived in a systematic way and is parameterized w.r.t. certain window sizes. It is thus completely different from the few existing pel-recursive motion estimation architectures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The effects of amplitude and phase errors are determined for symmetric and asymmetric linear predictive array processors. The maximum entropy method (MEM) corresponds to linear prediction in the asymmetric case of a line array of equispaced elements with an end reference. A comparison to conventional beamforming and maximum-likelihood processing shows that they have the same peak location as the symmetric linear predictive array processor. >