GPU Acceleration of Finding Maximum Eigenvalue of Positive Matrices

Abstract

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.