Parallel Computing Research On The Matrix Eigenproblem

Matrix eigenproblem plays an important part of numerical calculation. In addition, it is also an active research subject of computer science and numerical algebra. With the development of computer science and the advent of parallel computing, matrix eigenproblem have become one of the chief tasks of the large-scale computers. It is impossible to define large-scale eigenvalue and eigenvector with uniprocessor because it is a time-consuming process. Thanks to parallel computing, the increasing capacity of computer power and storage makes the effective computing possible.Not only can matrix eigenproblem directly solve nonlinear programming, ordinary differential equation as well as various mathematical calculations, but also they play an important role in structural mechanics, structural design, computational physics and quantum mechanics. At present, the applications of matrix eigenproblem are mostly from the solution of equation of mathematical physics, difference equation and Markovprocess. Therefore it is of great practical significance to study efficient and practicable parallel algorithm used to solve matrix eigenproblem.Based on the correlation studies at home and abroad, the paper comprises the main research work and innovation as follows:Firstly, the paper analyses the current situation and significance of the parallel computing on matrix eigenproblem and makes a summary of the features and development prospects of the parallel computing at home and abroad. In addition, it makes a comprehensive study of parallel environment, and focuses on syntactic model, access and storage model of parallel machine and parallel programming environment. And this paper briefly introduces hardware and software systems of high performance parallel computer drawing 4000L.Secondly, in terms of numerical methodes, this paper explores into the matrix calculation methods and strategies involved in solving matrix eigenproblem and realizes two parallel algorithms of matrix multiplication in parallel machine.Then it makes a comparison and analysis between serial and parallel, summarizes the features of various algorithms and gives the research direction of every algorithm. And it lays down a solid foundation for the following research work.Thirdly, this paper emphasizes the maximum absolute value of the general matrix eigenvalue by the power method and solves the matrix of all eigenvalues of QR methods. Also it makes an intensive study of traditional Jacobi method, realizing the parallel algorithms of Jacobi method in parallel machine.One-side Jacobi transformation has an advantage of actualizing column transformation of matrix only. Because of that, combining the properties of symmetry and tridiagonal, this thesis puts forward a one-side Jacobi parallel algorithm for symmetric tridiagonal matrix eigenproblem. The algorithm subdivides tridiagonal matrix by columns, every processor only needs to actualize one-side orthogonal transformation of three contiguous columns in local storage, and then it can carry out one-side orthogonal transformation between processor only after transmitting two prior columns of data. The algorithm has high efficiency by theoretical analysis and experiments.Finally, the paper gives a summary of the research and also puts forward some questions to be further studied.