scholarly journals Acceleration of Hessenberg reduction for nonsymmetric matrix.

2021 ◽  
Author(s):  
Hesamaldin Nekouei
Keyword(s):  
Hybrid Algorithm ◽  
Eigenvalue Problems ◽  
Symmetric Matrices ◽  
Real Matrix ◽  
Great Efficiency ◽  
Matrix Size ◽  
Hessenberg Form ◽  
Large Matrix ◽  
Speed Up ◽  
Nonsymmetric Matrix

The worth of finding a general solution for nonsymmetric eigenvalue problems is specified in many areas of engineering and science computations, such as reducing noise to have a quiet ride in automotive industrial engineering or calculating the natural frequency of a bridge in civil engineering. The main objective of this thesis is to design a hybrid algorithm (based on CPU-GPU) in order to reduce general non-symmetric matrices to Hessenberg form. A new blocks method is used to achieve great efficiency in solving eigenvalue problems and to reduce the execution time compared with the most recent related works. The GPU part of proposed algorithm is thread based with asynchrony structure (based on FFT techniques) that is able to maximize the memory usage in GPU. On a system with an Intel Core i5 CPU and NVIDA GeForce GT 635M GPU, this approach achieved 239.74 times speed up over the CPU-only case when computing the Hessenberg form of a 256 * 256 real matrix. Minimum matrix order (n), which the proposed algorithm supports, is sixteen. Therefore, supporting this matrix size is led to have the large matrix order range.

scholarly journals Acceleration of Hessenberg reduction for nonsymmetric matrix.

2021 ◽  
Author(s):  
Hesamaldin Nekouei
Keyword(s):  
Hybrid Algorithm ◽  
Eigenvalue Problems ◽  
Symmetric Matrices ◽  
Real Matrix ◽  
Great Efficiency ◽  
Matrix Size ◽  
Hessenberg Form ◽  
Large Matrix ◽  
Speed Up ◽  
Nonsymmetric Matrix

The worth of finding a general solution for nonsymmetric eigenvalue problems is specified in many areas of engineering and science computations, such as reducing noise to have a quiet ride in automotive industrial engineering or calculating the natural frequency of a bridge in civil engineering. The main objective of this thesis is to design a hybrid algorithm (based on CPU-GPU) in order to reduce general non-symmetric matrices to Hessenberg form. A new blocks method is used to achieve great efficiency in solving eigenvalue problems and to reduce the execution time compared with the most recent related works. The GPU part of proposed algorithm is thread based with asynchrony structure (based on FFT techniques) that is able to maximize the memory usage in GPU. On a system with an Intel Core i5 CPU and NVIDA GeForce GT 635M GPU, this approach achieved 239.74 times speed up over the CPU-only case when computing the Hessenberg form of a 256 * 256 real matrix. Minimum matrix order (n), which the proposed algorithm supports, is sixteen. Therefore, supporting this matrix size is led to have the large matrix order range.

scholarly journals A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 100 ◽  
Author(s):  
Luca Bergamaschi
Keyword(s):  
Linear Systems ◽  
Eigenvalue Problems ◽  
Real Life ◽  
Low Rank ◽  
Preconditioned Conjugate Gradient ◽  
Krylov Methods ◽  
Rank Matrix ◽  
Recent Developments ◽  
Speed Up ◽  
Low Rank Matrix

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems A k x k = b k , k = 1 , … arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

scholarly journals Adaptive Cat Swarm Optimization Algorithm and Its Applications in Vehicle Routing Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Xiao-Fang Ji ◽  
Jeng-Shyang Pan ◽  
Shu-Chuan Chu ◽  
Pei Hu ◽  
Qing-Wei Chai ◽  
...  
Keyword(s):  
Vehicle Routing ◽  
Hybrid Algorithm ◽  
Simulation Experiments ◽  
Swarm Optimization ◽  
Routing Problem ◽  
Cat Swarm Optimization ◽  
Fixed Dimension ◽  
Speed Up ◽  
Effectiveness And Efficiency ◽  
Experimental Findings

This paper proposes a novel hybrid algorithm named Adaptive Cat Swarm Optimization (ACSO). It combines the benefits of two swarm intelligence algorithms, CSO and APSO, and presents better search results. Firstly, some strategies are implemented to improve the performance of the proposed hybrid algorithm. The tracing radius of the cat group is limited, and the random number parameter r is adaptive adjusted. In addition, a scaling factor update method, called a memory factor y, is introduced into the proposed algorithm. They can be learnt very well so as to jump out of local optimums and speed up the global convergence. Secondly, by comparing the proposed algorithm with PSO, APSO, and CSO, 23 benchmark functions are verified by simulation experiments, which consists of unimodal, multimodal, and fixed-dimension multimodal. The results show the effectiveness and efficiency of the innovative hybrid algorithm. Lastly, the proposed ACSO is utilized to solve the Vehicle Routing Problem (VRP). Experimental findings also reveal the practicability of the ACSO through a comparison with certain existing methods.

scholarly journals Parallel Implementations of RCM Algorithm for Bandwidth Reduction of Sparse Matrices

2018 ◽  
Vol 18 (3) ◽  
pp. 449
Author(s):  
Thiago Nascimento Rodrigues ◽  
Maria Claudia Silva Boeres ◽  
Lucia Catabriga
Keyword(s):  
Linear Systems ◽  
Sparse Matrices ◽  
Symmetric Matrices ◽  
Systems Of Equations ◽  
Original Algorithm ◽  
Bandwidth Reduction ◽  
Parallel Implementations ◽  
Speed Up ◽  
Linear Systems Of Equations

The Reverse Cuthill-McKee (RCM) algorithm is a well-known heuristicfor reordering sparse matrices. It is typically used to speed up the computation ofsparse linear systems of equations. This paper describes two parallel approachesfor the RCM algorithm as well as an optimized version of each one based on someproposed enhancements. The first one exploits a strategy for reducing lazy threads,while the second one makes use of a static bucket array as the main data structureand suppress some steps performed by the original algorithm. These related changesled to outstanding reordering time results and significant bandwidth reductions.The performance of two algorithms is compared with the respective implementationmade available by Boost library. The OpenMP framework is used for supportingthe parallelism and both versions of the algorithm are tested with large sparse andstructural symmetric matrices.

Convergence and Preconditioning of Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems

2020 ◽  
Vol 20 (2) ◽  
pp. 343-359
Author(s):  
Rayan Nasser ◽  
Miloud Sadkane
Keyword(s):  
Eigenvalue Problems ◽  
Matrix Pencil ◽  
Linear Rate ◽  
Subspace Iteration ◽  
Matrix Anal ◽  
Small Constant ◽  
Large Matrix ◽  
Generalized Eigenvalue Problems ◽  
The One ◽  
Inner Iteration

AbstractThis paper focuses on the inner iteration that arises in inexact inverse subspace iteration for computing a small deflating subspace of a large matrix pencil. First, it is shown that the method achieves linear rate of convergence if the inner iteration is performed with increasing accuracy. Then, as inner iteration, block-GMRES is used with preconditioners generalizing the one by Robbé, Sadkane and Spence [Inexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl. 31 2009, 1, 92–113]. It is shown that the preconditioners help to maintain the number of iterations needed by block-GMRES to approximately a small constant. The efficiency of the preconditioners is illustrated by numerical examples.

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