For The Gpu Parallel Algorithm Of Matrix Eigenvalues Of Research

Solving the eigenvalues of matrices is an open problem which is often related to scientific computation. With the order of matrices increasing continually, traditional sequential algorithms are unable to meet the needs for the calculation of time. Although people can use cluster systems in a short period of time to solve the eigenvalues of large-scale matrices, it will bring an increase in equipment costs and power consumption.In order to solve the problems above, this paper proposes a parallel algorithm named Hybrid Algorithm (for short HA) which is implemented by CUDA (Computer Unified Device Architecture) on GPU(Graphic Process Unit) to solve the eigenvalues of symmetric matrices. This parallel algorithm is based on Jacobi iteration method. HA algorithm is a combination of the proposed two parallel Jacobi iteration algorithm PA-1and PA-2. The main ideas of PA-1and PA-2are parallel implementation on finding the element who has the maximum absolute value in the non-diagonal elements of symmetric matrix and computing the elements of transformed matrix this two step of sequential Jacobi algorithm. PA-1algorithm is different from PA-2algorithm only in how to find the one from non-diagonal elements whose absolute value is the maximum. The experimental results show that the HA can save more running time than traditional sequential algorithms and the speedup ratio is9.85～13.71. As the size of matrices increase, the general trend of speedup ratio increases correspondingly. Moreover, as the number of iterations increases, the speedup ratio is very stable. Therefore, the computing time of traditional sequential algorithms to solve the eigenvalues of matrices has been reduced significantly.This paper also proposes a parallel QR algorithm named PA-QR which is implemented by CUDA on GPU to solve the eigenvalues of general matrices. But before using the QR method, we need change the general matrices to upper H (Hessenberg) matrices. We proposes a parallel algorithm named PA-H firstly which can change the general matrices to upper H matrices. The main idea of the PA-H algorithm is parallel implementation on finding the element who has the maximum absolute value from column k-1below the k-1element and line exchange, column exchange and elimination transform to the rows and columns. The main idea of the PA-QR algorithm is parallel implementation on four cycles which are a heavy cycle. The experimental results show that the speedup ratio of PA-H is1.79～7.81, while the speedup ratio of PA-QR is3.24-118.9. As the size of matrices increase, the speedup ratio increases correspondingly. Moreover, as the number of iterations increases, the speedup ratio is very stable.