Some Refined Algorithms For Large Unsymmetric Matrix Eigenproblems

This thesis presents some refined algorithms for large unsymmetric matrix eigenproblems and investigates their convergence and restart. It consists of five parts.Chapter one gives 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.Chapter two investigates the restarted refined Arnoldi method for solving large matrix eigenproblems. Two improvements have been made for a restarted subspace . One is that using an ingenious linear combination of the refined Ritz vectors forms an initial vector and then generates a new Krylov subspace. Another is that retaining the refined Ritz vector in the new subspace,called augmented Krylov subspace. This way retains useful information and makes the resulting algorithm converge faster. Several numerical examples are reported that compares the new algorithm with the implicitly restarted Arnoldi algorithm (IRA) and the implicitly restarted refined Arnoldi algorithm (IRRA) . Numerical results confirm efficiency of the new algorithm .In Chapter three,we first present a priori theoretical error bounds for eigenelements by the harmonic Rayleigh-Ritz method. Second,we propose and discuss the refined harmonic Rayleigh-Ritz method. We give the error bound for the refined harmonic Ritz vector and establish the relation between the refined harmonic Ritz vector and the harmonic Ritz vector. Third,we discuss the refined harmonic Arnoldi method in Krylov subspace,give a priori residual error bounds for the refined eigenpairs. Fourth,the restarted refined harmonic Arnoldi method is considered and a refined harmonic Arnoldi algorithm in the augmented Krylov subspace is developed. Finally,numerical examples are reported that compare the new algorithm with the .implicitly restarted harmonic Arnoldi algorithm (IRHA) and the implicitly restarted refined harmonic Arnoldi algorithm (IRRHA) . Numerical results confirm efficiency of the new algorithm .Chapter four investigates how to compute the Ritz values i(i = 1,2,...,l) hi the space spanned by the refined Ritz vectors i(i = 1,2,...,l) and use them to approximate the desired eigenvalues. For a Krylov subspase,a priori theoretical error bounds between the and the Ritz value A;are established and the residual relations between the new approximate eigenpairs and are given. Finally,we give a new algorithm that computes the Ritz values in the augmentedKrylov and carry on the numerical examples. Numerical results confirm efficiency of the new algorithm .In Chapter five,we establish three results. First,we prove an reverse order implicit Q-theorem:once the last column of V is given,then V and G are also uniquely determined. Second,we prove that for a Krylov subspase two formulations of the Arnoldi are equivalent and in one to one correspondence. Finally,using the equivalence relation and the reverse order implicit Q-theorem,we prove that for the Krylov subspace,if the last vector of the vector sequence generated by the Arnoldi process is given,then the vector sequence and resulting Hessenberg matrix are uniquely determined.