LU Decomposition Of Matrix Research Articles

Performance evaluation of LU matrix decomposition using the SYCL standard

The object of this study is the performance of the SYCL standard tools when solving the LU matrix decomposition problem. SYCL is a fairly new technology for parallel computing in heterogeneous systems, so the topic of evaluating the performance of the standard on specific tasks in the field of parallel computing is relevant. In the study, the algorithm of parallelized LU decomposition of a square matrix was implemented by means of the SYCL standard and standard C++, and an experiment was conducted to test the implementation in a heterogeneous system with several types of processors. During testing, the program received square matrices of various dimensions as input, and the output was the execution time of the LU schedule on the selected processor. The obtained results, presented in the form of tabular and graphic data, show the advantage of the implementation of the SYCL standard over ordinary C++ by more than 2 times when using a graphics processor. It was experimentally shown that the implementation on SYCL is almost not inferior in speed to the implementation on ordinary C++ when executed on a central processor. Such results are caused both by the high possibility of parallelizing the LU schedule algorithm itself, and by the great work of the developers of the standard on its optimization. The obtained results indicate the possibility of speeding up the solution of the LU decomposition of the matrix and similar algorithms by means of SYCL when using heterogeneous systems with processors optimized for data parallelism. The results of the study can be used in justifying the choice of technology for solving LU matrix decomposition problems or problems with a similar parallelization scheme.

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Design and implementation of dual-core MIPS processor for LU decomposition based on FPGA

Many systems like the control systems and in communication systems, there is usually a demand for matrix inversion solution. This solution requires many operations, which makes it not possible or very hard to meet the needs for real-time constraints. Methods were exists to solve this kind of problems, one of these methods by using the LU decomposition of matrix which is a good alternative to matrix inversion. The LU matrices are two matrices, the L matrix, which is a lower triangular matrix, and the U matrix, which is an upper triangular matrix. In this paper, a design of dual-core processor is used as the hardware of the work and certain software was written to enable the two cores of the dual-core processor to work simultaneously in computing the value of the L matrix and U matrix. The result of this work are compared with other works that using single-core processor, and the results found that the time required in the cores of the dual-core is more less than using single-core. The designed dual-core processor is invoked using the VHDL language.

Real-Time HIL Emulation of Faulted Electric Machines Based on Nonlinear MEC Model

In electric machine drive systems, hardware-in-the-loop (HIL) emulation provides accurate testing of actual control system prototypes and protection devices interfaced with the electric machine model on a real-time simulator in a non-destructive environment particularly when faults are studied. A compromise between the model accuracy and computational burden makes the magnetic equivalent circuit (MEC) model ideal for real-time simulation of electric machines. However, satisfying the timing constraints of real-time simulation to accommodate internal machine faults is still challenging due to the nonlinearity and rotation of electric machines. In this paper, the transmission line modeling (TLM) method is utilized to keep the MEC coefficient matrix unchanged during nonlinear iterations. Afterward, for the first time, the entire potential of the TLM method for pre-calculation is exploited by proposing an efficient matrix re-ordering combined with the left-looking Gilbert–Peierls algorithm to minimize the computational burden of the sparse MEC matrix LU decomposition required in each time-step due to rotation. Furthermore, the massive hardware architecture of the field programmable gate array is used as the platform for implementation to fully exploit parallelism. With the proposed MEC-based real-time TLM method, the minimum time-step as low as 500 $\mu \text{s}$ can be achieved and the results validation with two-dimensional finite element model (FEM) of the commercial Jmag-Designer software shows the accuracy and efficiency of the proposed methodology.

Parallel real-world LU decomposition: Gauss vs. Crout algorithm

Abstract This paper presents numerical experiments with assorted versions of parallel LU matrix decomposition algorithms (Gauss and Crout algorithm). The tests have been carried out on the hardware platform with fourcore Skylake processor featuring hyperthreading technology doubling virtually core number. Parallelization algorithms have been implemented with the aid of classic POSIX threads library. Experiments have shown that basic 4-thread acceleration of all parallel implementations is almost equal to the number of threads/processors. Both algorithms are worth considering in real-world applications (Florida University collection). Gauss algorithm is a better performer, with respect to timing, in the case of matrices with lower density of nonzeros, as opposed to higher density matrices. The latter are processed more efficiently with the aid of Crout algorithm implementation.

A novel key pre-distribution approach for high security and efficiency using LU-decomposition of matrix

Wide-spread deployment of wireless sensor network (WSN) is emerging, which allows an economical solution for numerous applications. For the security of communication network, asymmetric cryptography is usually employed. This approach is, however, often unsuitable to WSN due to limited computational and energy power. Among various approaches the random key pre-distribution approach is regarded as the most effective one for WSN. Here one shortcoming is that a common key is not guaranteed to be found between any two communicating nodes. This paper proposes a new key pre-distribution scheme solving this problem by obtaining the keys with LU-decomposition of a symmetric matrix. Mathematical analysis and computer simulation reveal that the proposed scheme significantly reduces the overhead required for secure connectivity and energy efficiency of the sensor nodes compared to the existing approaches. The superiority of the proposed scheme becomes more substantial when the memory size of the sensor nodes is small.

A novel key pre-distribution approach for high security and efficiency using LU-decomposition of matrix

A novel key pre-distribution approach for high security and efficiency using LU-decomposition of matrix

Application of combined Newton–Raphson method to large load flow models

Load flow is the most used calculations in power system operation planning. Renewable resources have caused that system operator has to do power flow analysis and/or contingency analysis as fast as possible in order to predict next step in power system control. LU Decomposition of Jacobian matrix remains the most computationally expensive task during Newton–Raphson iterative method. Computational time appears to be critical issue when load flow calculation is performed on large power system load flow models. In this case, Jacobian matrix LU decomposition should not be performed in iterations in which convergence rate is not violated but performed in iterations in which convergence rate drop below specified level. In other words, modified Newton–Raphson Method which eliminates the repeated Jacobian matrix LU decomposition and generic Newton–Raphson method are combined depending on convergence rate. The paper presents application of proposed combined Newton–Raphson method which is based on convergence rate control. Comparison of combined, Shamanskii, generic, and modified Newton–Raphson methods is carried out taking into consideration computational time and number of iterations required to achieve convergence of load flow models of various dimensions.

A High Performance and Memory Efficient LU Decomposer on FPGAs

LU decomposition for dense matrices is an important linear algebra kernel that is widely used in both scientific and engineering applications. To efficiently perform large matrix LU decomposition on FPGAs with limited local memory, a block LU decomposition algorithm on FPGAs applicable to arbitrary matrix size is proposed. Our algorithm applies a series of transformations, including loop blocking and space-time mapping, onto sequential nonblocking LU decomposition. We also introduce a high performance and memory efficient hardware architecture, which mainly consists of a linear array of processing elements (PEs), to implement our block LU decomposition algorithm. Our design can achieve optimum performance under various hardware resource constraints. Furthermore, our algorithm and design can be easily extended to the multi-FPGA platform by using a block-cyclic data distribution and inter-FPGA communication scheme. A total of 36 PEs can be integrated into a Xilinx Virtex-5 XC5VLX330 FPGA on our self-designed PCI-Express card, reaching a sustained performance of 8.50 GFLOPS at 133 MHz for a matrix size of 16,384, which outperforms several general-purpose processors. For a Xilinx Virtex-6 XC6VLX760, a newer FPGA, we predict that a total of 180 PEs can be integrated, reaching 70.66 GFLOPS at 200 MHz. Compared to the previous work, our design can integrate twice the number of PEs into the same FPGA and has significantly higher performance.

A scalable and robust key pre-distribution scheme with network coding for sensor data storage

In this paper, we propose a scalable and robust key pre-distribution scheme based on network coding and matrix decomposition for distributed sensor data storage. As a promising information dissemination technique, network coding is suitable for the key information exchange due to the reduced number of transmissions and the enhanced robustness. Matrix decomposition can help to exchange key information without any central nodes and location knowledge. Our scheme uses LU matrix decomposition to decompose a shared key into two vectors R and C. The vector R is held privately by the sensor, while the other vector C is protected and disseminated using network coding. The combination of network coding and matrix decomposition enhances the scalability, improves the communication performance, and increases the robustness for sensor data storage networks. Extensive theoretical analysis and simulative results both demonstrate the efficacy and efficiency of the proposed key pre-distribution scheme in distributed sensor data storage.

On the [formula omitted] forms of the truncated LU decomposition

Matrix LU decomposition has six i j k forms. Different forms have different computational complexities and storage requirements, particularly on vector and parallel computers. Other factors governing the choice of a particular form are considered. For treating Fredholm integral equations of the first kind, the truncated LU decomposition of the resulting system matrix is recommended. Required modifications to selected known i j k forms are presented.

Shape determination of attached fluctuating phospholipid vesicles

A method to determine the shapes of a thermally fluctuating phospholipid vesicle is described. The vesicle that was attached at one point to a glass plate was submitted to a flow. It was observed by an inverted optical microscope using phase contrast technique. A large number of consecutive video images were digitised at different flow velocities and stored on computer disk. They were off-line binarised using the global average value as a threshold. The vesicle outlines were determined by a contour following procedure based on the maze walking algorithm. They were described in terms of Fourier coefficients, which were calculated by the least square fitting of the contour points using normal equations and LU matrix decomposition. This method, which was the basis for the determination of the membrane bending modulus, was thoroughly tested and it proved to be robust and efficient.

A parallel solution to linear systems

Streaming SIMD Extensions (SSE) is a unique feature embedded in the Pentium III and IV classes of microprocessors. By fully exploiting SSE, parallel algorithms can be implemented on a standard personal computer and a theoretical speedup of four can be achieved. In this paper, we demonstrate the implementation of a parallel LU matrix decomposition algorithm for solving linear systems with SSE and discuss advantages and disadvantages of this approach based on our experimental study.

Efficient parallel algorithm for dense matrix LU decomposition with pivoting on hypercubes

LU decomposition is intensively used in various scientific and engineering computations. A parallel algorithm for dense matrix LU decomposition with pivoting on hypercubes is presented. Using n processors, the presented algorithm can finish LU decomposition of an n × n matrix in O( n 2 3 + O(n√n log 2 n)) steps, including computations as well as communications, and its efficiency is 1 asymptotically when n becomes large. The algorithm employs row- column- as well as block-parallelisms interchangeably so that the n processors are used efficiently in the whole computation process. Using the rich connectivity, all the data alignment requirements can be realized in O(log 2 n) steps. The algorithm proposed here not only is suitable for systems with small numbers of processors, but also is suitable for systems with large numbers of processors.

Tuning LAPACK codes on hierarchical memory machines

As important as the advance of computer technology is the development of suitable software for the exploitation of computer capacity. In this sense LAPACK (Andersen et al., LAPACK User's Guide, release 1.0, SIAM, Philadelphia, 1992) appears as the most efficient library in the dense linear algebra field, obtaining good results on vector and parallel computers, and in general, on hierarchical memory machines. However, some deficiencies in the general matrix LU decomposition were detected in DGETRF subroutine. On hierarchical memory machines not only the cache and TLB faults have to be minimized, but also the page faults, which cause excessive I/O operations. The blocking strategy used by DGETRF (and LAPACK in general) makes good use of the cache memory, but does not seem to be enough to avoid unnecessary I/O operations. Therefore, DGETRF does not provide satisfactory run times for large dimension matrices. In this paper a new code using a double blocking strategy will be described, which attains better run times than DGETRF.

Minimum cycle fast implementation of two-dimensional digital filters based on the LU decomposition

A new architecture for two-dimensional filters is presented, which is based on the LU matrix decomposition and is characterized by minimum cycle time. The clock cycle time in the proposed structure is determined by one multiplication and three additions, which are the maximum number of arithmetic operations in the critical path, independently of the filter's order. This realization is characterized by all known advantages of decomposition structures (modularity, parallelism, reconfigurability) and may be used in real-time image processing applications.

Numerical perturbation method for approximate solution of poisson's equation on a moderately deforming grid

AbstractIn problems such as the computation of incompressible flows with moving boundaries, it may be necessary to solve Poisson's equation on a large sequence of related grids. In this paper the LU decomposition of the matrix A0 representing Poisson's equation discretized on one grid is used to efficiently obtain an approximate solution on a perturbation of that grid. Instead of doing an LU decomposition of the new matrix A, the RHS is perturbed by a Taylor expansion of A−1 about A0. Each term in the resulting series requires one ‘backsolve’ using the original LU.Tests using Laplace's equation on a square/rectangle deformation look promising; three and seven correction terms for deformations of 20% and 40% respectively yielded better than 1% accuracy.As another test, Poisson's equation was solved in an ellipse (fully developed flow in a duct) of aspect ratio 2/3 by perturbing about a circle; one correction term yielded better than 1% accuracy.Envisioned applications other than the computation of unsteady incompressible flow include: three‐dimensional parabolic problems in tubes of varying cross‐section, use of ‘elimination’ techniques other than LU decomposition, and the solution of PDEs other than Poisson's equation.

Filter realisation by passive network simulation

General simulation methods of passive networks are investigated. A new simulation approach based on LU matrix decomposition is presented. This approach can be used to design active-RC, switched-capacitor and digital networks. The resulting system demonstrates very low sensitivities with respect to component values. New digital realisations, in particular, reveal structures of low complexity, high speed of operation and low cost of implementation.

The LUD approach to switched-capacitor filter design

A new design method for switched-capacitor filters (SCF) is presented. It is based upon an LU matrix decomposition technique and has the distinct advantage of producing SC filter realizations containing no delay free loops. These are formed traditionally by capacitors and opamps in leapfrog realizations. It is demonstrated that this feature should render reduced dependence of the filter response to nonideal effects such as finite amplifier GB and switch resistance. Results from realistic leapfrog and LUD SC filter realizations confirm this.