scholarly journals A parallel algorithm for the sparse QR decomposition of a rectangular upper quasi-triangular matrix with ND-type sparsity

2015 ◽  
pp. 566-577
Author(s):  
С.А. Харченко
Keyword(s):  
Parallel Algorithm ◽  
Triangular Matrix ◽  
Parallel Computations ◽  
Qr Decomposition ◽  
Nested Dissection ◽  
Rectangular Matrix ◽  
Computational Mesh ◽  
The Matrix ◽  
Householder Transformations ◽  
Initial Zero

Рассматривается параллельный алгоритм вычисления разреженного $QR$-разложения специальным образом упорядоченной прямоугольной матрицы на основе разреженных блочных преобразований Хаусхолдера. Для построения необходимого упорядочивания можно использовать столбцевое упорядочивание типа вложенных сечений, построенное по структуре матрицы $A^{T}A$, где $A$ - исходная прямоугольная матрица. Для сеточных задач упорядочивание может быть построено на основе известного объемного разбиения расчетной сетки. В качестве базового алгоритма для организации параллельных вычислений используется $QR$-разложение для наборов строк матрицы с дополнением в виде нулевого начального блока. An algorithm for computing the sparse $QR$ decomposition of a specially ordered rectangular matrix is proposed. This decomposition is based on the block sparse Householder transformations. For ordering computations, the nested dissection ordering is used for the matrix $A^{T}A$, where $A$ is the original rectangular matrix. For mesh based problems, the ordering can be constructed starting from an appropriate volume partitioning of the computational mesh. Parallel computations are based on sparse $QR$ decomposition for sets of rows with an additional initial zero block.

scholarly journals Construction of Measurement Matrix Based on Cyclic Direct Product and QR Decomposition for Sensing and Reconstruction of Underwater Echo

2018 ◽  
Vol 8 (12) ◽  
pp. 2510 ◽  
Author(s):  
Tongjing Sun ◽  
Hong Cao ◽  
Philippe Blondel ◽  
Yunfei Guo ◽  
Han Shentu
Keyword(s):  
Direct Product ◽  
Hadamard Matrix ◽  
Triangular Matrix ◽  
Field Measurements ◽  
Qr Decomposition ◽  
Orthogonal Matrix ◽  
Measurement Matrix ◽  
Reconstruction Accuracy ◽  
Weak Signals ◽  
Construction Time

Compressive sensing is a very attractive technique to detect weak signals in a noisy background, and to overcome limitations from traditional Nyquist sampling. A very important part of this approach is the measurement matrix and how it relates to hardware implementation. However, reconstruction accuracy, resistance to noise and construction time are still open challenges. To address these problems, we propose a measurement matrix based on a cyclic direct product and QR decomposition (the product of an orthogonal matrix Q and an upper triangular matrix R). Using the definition and properties of a direct product, a set of high-dimensional orthogonal column vectors is first established by a finite number of cyclic direct product operations on low-dimension orthogonal “seed” vectors, followed by QR decomposition to yield the orthogonal matrix, whose corresponding rows are selected to form the measurement matrix. We demonstrate this approach with simulations and field measurements of a scaled submarine in a freshwater lake, at frequencies of 40 kHz–80 kHz. The results clearly show the advantage of this method in terms of reconstruction accuracy, signal-to-noise ratio (SNR) enhancement, and construction time, by comparison with Gaussian matrix, Bernoulli matrix, partial Hadamard matrix and Toeplitz matrix. In particular, for weak signals with an SNR less than 0 dB, this method still achieves an SNR increase using less data.

scholarly journals Blind Color Image Watermarking Using Fan Beam Transform and QR Decomposition

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 486
Author(s):  
Pranab Kumar Dhar ◽  
Pulak Hazra ◽  
Tetsuya Shimamura
Keyword(s):  
Color Image ◽  
State Of The Art ◽  
Signal To Noise Ratio ◽  
Image Watermarking ◽  
Triangular Matrix ◽  
Structural Similarity ◽  
Qr Decomposition ◽  
Original Image ◽  
Color Image Watermarking ◽  
Multimedia Contents

Digital watermarking has been utilized effectively for copyright protection of multimedia contents. This paper suggests a blind symmetric watermarking algorithm using fan beam transform (FBT) and QR decomposition (QRD) for color images. At first, the original image is transferred from RGB to L*a*b* color model and FBT is applied to b* component. Then the b*component of the original image is split into m × m non-overlapping blocks and QRD is conducted to each block. Watermark data is placed into the selected coefficient of the upper triangular matrix using a new embedding function. Simulation results suggest that the presented algorithm is extremely robust against numerous attacks, and also yields watermarked images with high quality. Furthermore, it represents more excellent performance compared with the recent state-of-the-art algorithms for robustness and imperceptibility. The normalized correlation (NC) of the proposed algorithm varies from 0.8252 to 1, the peak signal-to-noise ratio (PSNR) varies from 54.1854 to 54.1892, and structural similarity (SSIM) varies from 0.9285 to 0.9696, respectively. In contrast, the NC of the recent state-of-the-art algorithms varies from 0.5193 to 1, PSNR varies from 38.5471 to 52.64, and SSIM varies from 0.9311 to 0.9663, respectively.

scholarly journals A 3D Parallel Algorithm for QR Decomposition

2018 ◽  
Author(s):  
Grey Ballard ◽  
James Demmel ◽  
Laura Grigori ◽  
Mathias Jacquelin ◽  
Nicholas Knight
Keyword(s):  
Parallel Algorithm ◽  
Qr Decomposition

SSP Podded Propulsion Motor Control Based on SR-CDKF

2014 ◽  
Vol 986-987 ◽  
pp. 1142-1145
Author(s):  
Wen Long Yao
Keyword(s):  
Qr Decomposition ◽  
Estimation Accuracy ◽  
Square Root ◽  
Rotor Speed ◽  
Closed Loops ◽  
Speed Controller ◽  
Sensorless Vector Control ◽  
Matrix Square Root ◽  
The Matrix ◽  
Vector Control System

In this paper,the rotor speed and the position of the SSP propulsion motor are estimated for building sensorless vector control system with speed and current double closed loops based on square root center difference Kalman filter (SR-CDKF) algorithm. This method makes use of the QR decomposition linear algebra techniques and so on, and it updates the matrix square-root of the state covariance by the Cholesky factor updating. This method can not only get the more steady results but also improve the estimation accuracy of the SSP podded propulsion system. Simulation result shows that the improved CDKF algorithm is not only more accurate but also has higher rate of convergence compared with CDKF speed controller.

On a class of evolution algebras of "chicken" population

2014 ◽  
Vol 25 (08) ◽  
pp. 1450073 ◽  
Author(s):  
A. Dzhumadil'daev ◽  
B. A. Omirov ◽  
U. A. Rozikov
Keyword(s):  
Three Dimensional ◽  
Jordan Form ◽  
Simple Description ◽  
Rectangular Matrix ◽  
Chicken Population ◽  
The Matrix

This paper is devoted to the description of structure of evolution algebras of "chicken" population (EACP). Such an algebra is determined by a rectangular matrix of structural constants. Using the Jordan form of the matrix of structural constants we obtain a simple description of complex EACP. We give the classification of three-dimensional complex EACP. Moreover, some (n + 1)-dimensional EACP are described.

scholarly journals Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

2019 ◽  
Vol 35 ◽  
pp. 116-155
Author(s):  
Biswajit Das ◽  
Shreemayee Bora
Keyword(s):  
Eigenvalue Problem ◽  
Matrix Polynomial ◽  
Vector Spaces ◽  
Backward Error ◽  
Rectangular Matrix ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
Stable Manner ◽  
The Matrix ◽  
Matrix Pencils

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

Search for identical regions in images using invariant moments

2021 ◽  
Vol 26 (jai2021.26(2)) ◽  
pp. 55-62
Author(s):  
Sabelnikov P ◽  
◽  
Sabelnikov Yu ◽  
Keyword(s):  
Integral Representation ◽  
Parallel Computations ◽  
Invariant Moments ◽  
Geometric Moments ◽  
The Matrix ◽  
Rectangular Area ◽  
Characteristic Points ◽  
Speed Up ◽  
Almost All ◽  
Local Image

One of the ways to describe objects on images is to identify some of their characteristic points or points of attention. Areas of neighborhoods of attention points are described by descriptors (lots of signs) in such way that they can be identified and compared. These signs are used to search for identical points in other images. The article investigates and establishes the possibility of searching for arbitrary local image regions by descriptors constructed with using invariant moments. A feature of the proposed method is that the calculation of the invariant moments of local areas is carried out with using the integral representation of the geometric moments of the image. Integral representation is a matrix with the same size as the image. The elements of the matrix is the sums of the geometric moments of individual pixels, which are located above and to the left with respect to the coordinates of this element. The number of matrices depends on the order of the geometric moments. For moments up to the second order (inclusively), there will be six such matrices. Calculation of one of six geometric moments of an arbitrary rectangular area of the image comes down up to 3 operations such as summation or subtraction of elements of the corresponding matrix located in the corners of this area. The invariant moments are calculated on base of six geometric moments. The search is performed by scanning the image coordinate grid with a window of a given size. In this case, the invariant moments and additional parameters are calculated and compared with similar parameters of the neighborhoods of the reference point of different size (taking into account the possible change in the image scale). The best option is selected according to a given condition. Almost all mass operations of the procedures for calculating the parameters of standards and searching of identical points make it possible explicitly perform parallel computations in the SIMD mode. As a result, the integral representation of geometric moments and the possibility of using parallel computations at all stages will significantly speed up the calculations and allow you to get good indicators of the search efficiency for identical points and the speed of work

A PARALLEL FAST MULTIPOLE METHOD FOR THE HELMHOLTZ EQUATION

1995 ◽  
Vol 05 (02) ◽  
pp. 263-274 ◽  
Author(s):  
MARK A. STALZER
Keyword(s):  
Helmholtz Equation ◽  
Parallel Algorithm ◽  
Fast Multipole Method ◽  
Iterative Solvers ◽  
Dense Matrix ◽  
Vector Product ◽  
Fast Multipole ◽  
Multipole Method ◽  
The Matrix ◽  
Matrix Vector

Presented is a parallel algorithm based on the fast multipole method (FMM) for the Helmholtz equation. This variant of the FMM is useful for computing radar cross sections and antenna radiation patterns. The FMM decomposes the impedance matrix into sparse components, reducing the operation count of the matrix-vector multiplication in iterative solvers to O(N3/2) (where N is the number of unknowns). The parallel algorithm divides the problem into groups and assigns the computation involved with each group to a processor node. Careful consideration is given to the communications costs. A time complexity analysis of the algorithm is presented and compared with empirical results from a Paragon XP/S running the lightweight Sandia/University of New Mexico operating system (SUNMOS). For a 90,000 unknown problem running on 60 nodes, the sparse representation fits in memory and the algorithm computes the matrix-vector product in 1.26 seconds. It sustains an aggregate rate of 1.4 Gflop/s. The corresponding dense matrix would occupy over 100 Gbytes and, assuming that I/O is free, would require on the order of 50 seconds to form the matrix-vector product.

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