Refined Jacobi-Davidson Algorithms For Large Unsymmetric Matrix Eigenproblems

Large unsymmetric eigenproblem arise in many applications in scientific and engineering computing. Theoretical analysis and numerical experiments show that Ritz vectors obtained by the classical orthogonal projection methods may not converge to the desired eigenvectors even if the corresponding Ritz values converge. In order to circumvent this flaw, Jia suggested to use the refined vectors instead of the Ritz vectors to approximate the desired eigenvectors. The resulting refined projection methods correct the deficiency of the possible nonconvergence of eigenvectors and converge faster.It is important to combine the refined projection principle with many other techniques. More robust and reliable algorithms can be derived from the combination. This thesis consists of four parts.In Chapter One we give the background of large unsymmetric matrix eigenproblems and basic numerical algorithms for solving them. We review the state of the art of this subject. Finally, we describe the work of this thesis.In Chapter Two we make two improvements for the Jacobi-Davidson method combining with the refined projection principle. One is use a refined approximate eigenvector to get a new correction linear system, in which the resulting projection subspace can contain more information on desired eigenvectors. Another is change the way of restarting, in which the refined vectors are used to form a basis of the new subspace. In such a way, More information on the desired eigenvectors obtained during the previous cycle can be retained. Several numerical examples are reported to compare the new refined JD method with the JD method. Numerical results confirm efficency of the new algorithm. We also consider some properties on the solution of the correction linear system.In Chapter Three we investigate a refined harmonic Rayleigh-Ritz method that is developed by combining the harmonic JD method and the refined projection principle. New correction equation has been supposed to accelerate the convergence. We investigate the restarting issue and propose a new refined harmonic JD algorithm with thick restart. Numerical examples are reported that compare the new algorithm with the harmonic JD algorithm and refined JD algorithm.IIJacobi-DavidsonIn Chapter Four we investigate the problem of computing the singular value decomposition (SVD). We improve the approximate singular vector using the refined projection principle and use it to improve the correction equation of JDSVD. Numerical results confirm the efficency of the the refined JDSVD.