scholarly journals Efficient Implementation of Gaussian Belief Propagation Solver for Large Sparse Diagonally Dominant Linear Systems

2012 ◽  
Vol 48 (2) ◽  
pp. 471-474 ◽  
Author(s):  
Yousef El-Kurdi ◽  
Warren J. Gross ◽  
Dennis Giannacopoulos
Keyword(s):  
Linear Systems ◽  
Belief Propagation ◽  
Efficient Implementation ◽  
Diagonally Dominant

Some Remarks About the Decoupling Approximation of Damped Linear Systems

2009 ◽  
Author(s):  
Matthias Morzfeld ◽  
Nopdanai Ajavakom ◽  
Fai Ma
Keyword(s):  
Linear Systems ◽  
Approximation Error ◽  
Driving Frequency ◽  
Finite Range ◽  
Diagonal Dominance ◽  
Damping Matrix ◽  
Modal Damping ◽  
Diagonally Dominant ◽  
Decoupling Approximation ◽  
Damped Systems

A common approximation in the analysis of non-classically damped systems is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant. In addition, the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant. It is shown numerically in this paper that, over a finite range, errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off-diagonal elements decreasing continuously in magnitude. These unexpected drifts in errors due to the decoupling approximation can be observed at any driving frequency. Small off-diagonal elements in the modal damping matrix may not be sufficient to ensure small errors due to the decoupling approximation. Error-criteria based solely upon diagonal dominance of the modal damping matrix cannot be accurate.

Accurate solutions of diagonally dominant tridiagonal linear systems

2014 ◽  
Vol 54 (3) ◽  
pp. 711-727 ◽  
Author(s):  
Rong Huang ◽  
Jianzhou Liu ◽  
Li Zhu
Keyword(s):  
Linear Systems ◽  
Diagonally Dominant

scholarly journals Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices

2015 ◽  
Vol 30 ◽  
pp. 843-870 ◽  
Author(s):  
Cheng-yi Zhang ◽  
Dan Ye ◽  
Cong-Lei Zhong ◽  
SHUANGHUA SHUANGHUA
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sufficient Conditions ◽  
Numerical Linear Algebra ◽  
Convergence Results ◽  
Diagonally Dominant Matrices ◽  
Sufficient Conditions For Convergence ◽  
Diagonally Dominant ◽  
Necessary And Sufficient ◽  
Theoretical Results

It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible H−matrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general H−matrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general H−matrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general H−matrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.

scholarly journals Implementing and Evaluating an Heterogeneous, Scalable, Tridiagonal Linear System Solver with OpenCL to Target FPGAs, GPUs, and CPUs

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Hamish J. Macintosh ◽  
Jasmine E. Banks ◽  
Neil A. Kelson
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
High Performance ◽  
Performance Comparison ◽  
Third Party ◽  
Global Memory ◽  
Diagonally Dominant ◽  
The Individual ◽  
Performance Computing ◽  
Tridiagonal Systems

Solving diagonally dominant tridiagonal linear systems is a common problem in scientific high-performance computing (HPC). Furthermore, it is becoming more commonplace for HPC platforms to utilise a heterogeneous combination of computing devices. Whilst it is desirable to design faster implementations of parallel linear system solvers, power consumption concerns are increasing in priority. This work presents the oclspkt routine. The oclspkt routine is a heterogeneous OpenCL implementation of the truncated SPIKE algorithm that can use FPGAs, GPUs, and CPUs to concurrently accelerate the solving of diagonally dominant tridiagonal linear systems. The routine is designed to solve tridiagonal systems of any size and can dynamically allocate optimised workloads to each accelerator in a heterogeneous environment depending on the accelerator’s compute performance. The truncated SPIKE FPGA solver is developed first for optimising OpenCL device kernel performance, global memory bandwidth, and interleaved host to device memory transactions. The FPGA OpenCL kernel code is then refactored and optimised to best exploit the underlying architecture of the CPU and GPU. An optimised TDMA OpenCL kernel is also developed to act as a serial baseline performance comparison for the parallel truncated SPIKE kernel since no FPGA tridiagonal solver capable of solving large tridiagonal systems was available at the time of development. The individual GPU, CPU, and FPGA solvers of the oclspkt routine are 110%, 150%, and 170% faster, respectively, than comparable device-optimised third-party solvers and applicable baselines. Assessing heterogeneous combinations of compute devices, the GPU + FPGA combination is found to have the best compute performance and the FPGA-only configuration is found to have the best overall estimated energy efficiency.

scholarly journals Convergence of GAOR Iterative Method with Strictly Diagonally Dominant Matrices

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Guangbin Wang ◽  
Hao Wen ◽  
Ting Wang
Keyword(s):  
Iterative Method ◽  
Linear Systems ◽  
Numerical Examples ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
Better Than

We discuss the convergence of GAOR method for linear systems with strictly diagonally dominant matrices. Moreover, we show that our results are better than ones of Darvishi and Hessari (2006), Tian et al. (2008) by using three numerical examples.

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