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GPU Acceleration of Finding Maximum Eigenvalue of Positive Matrices

机译：GPU加速找到正矩阵的最大特征值

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Matrix eigenvalue theory has become an important analysis tool in scientific computing. Sometimes, people do not need to find all eigenvalues but only the maximum eigenvalue. Existing algorithms of finding the maximum eigenvalue of matrices are implemented sequentially. With the increasing of the orders of matrices, the workload of calculation is getting heavier. Therefore, traditional sequential methods are unable to meet the need of fast calculation for large matrices. This paper proposes a parallel algorithm named PA-ST to find the maximum eigenvalue of positive matrices by using similarity transformation which is implemented by CUDA (Computer Unified Device Architecture) on GPU (Graphic Process Unit). To the best of our knowledge, this is the first CUDA based parallel algorithm of calculating maximum eigenvalue of matrices. In order to improve the performance, optimization techniques are applied in this paper such as using the shared memory rather than the global memory to improve the speed of computation, avoiding bank conflicts by setting the span index, satisfying the principle of coalesced memory access, and by using single-precision floating-point arithmetic and the pinned memory to reduce the copy operation and obtain higher data transfer bandwidth between the host and the GPU device. The experimental results show that the similarity transformation technique can significantly shorten the running time compared to the sequential algorithm and the speedup ratio is nearly stable when the number of iterations increases. As the matrix order increases, the running time of the sequential algorithm and PA-ST increases correspondingly. Experiments also show that the speedup ratio of the PA-ST is between 2.85 and 35.028.
机译：矩阵特征值理论已成为科学计算中的重要分析工具。有时，人们不需要找到所有特征值，而是只有最大的特征值。依次实现找到矩阵最大特征值的现有算法。随着矩阵秩序的增加，计算的工作量正在变得越来越重。因此，传统的顺序方法无法满足大矩阵快速计算的需要。本文提出了一种名为PA-ST的并行算法，通过使用CUDA（计算机统一设备架构）在GPU（图形处理单元）实现的相似性转换来找到正矩阵的最大特征值。据我们所知，这是第一个基于CUDA的并行算法计算了矩阵最大特征值。为了提高性能，在本文中应用优化技术，例如使用共享内存而不是全局内存来提高计算速度，通过设置跨度指数来避免银行冲突，满足聚结的内存访问的原理，以及通过使用单精度浮点算术和固定存储器来减少复制操作并在主机和GPU设备之间获得更高的数据传输带宽。实验结果表明，与顺序算法相比，相似性变换技术可以显着缩短运行时间，并且当迭代的数量增加时，加速比几乎稳定。随着矩阵顺序增加，顺序算法和PA-ST的运行时间相应地增加。实验还表明，PA-ST的加速比在2.85和35.028之间。

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