A QR-like Algorithm And A Symplectic Lanczos Algorithm For Hamiltonian Matrix Eigenproblems And The Error Analysis Of Symplectic Lanczos Algorithm

In this paper, several algorithms for computing the eigenvalues and eigenvectors of a matrix are presented, mainly including an efficient and stable structure preserving algorithm, which is a variant of the QR-like algorithm for computing the eigenvalues and stable invariant subspaces of a Hamiltonian matrix and a modified symplectic Lanczos method for computing the desired eigenvalues and their corresponding eigenvectors of large sparse Hamiltonian matrix. Moreover, a rounding error analysis of the symplectic Lanczos method is given for the Hamiltonian eigenvalue problem. Our work involves the theories of numerical analysis, matrix computation, abstract algebra, control theory and algebraic Riccati equation. Together with them, we mainly do the following works.1. This thesis proposes a technique to improve the instability of symplectic QR-like algorithms (SR algorithms) and discusses some properties of special symplectic Householder transformations and symplectic Givens transformations used in the technique. Finally, the thesis gives a strategy of how to choose these special transformations. Based on this technique one can present a modified symplectic QR-like algorithm.2. An efficient and stable structure-preserving algorithm, which is a variant of the QR like (SR) algorithm due to Bunse-Gerstner and Mehrmann, is presented for computing the eigenvalues and stable invariant subspaces of a Hamiltonian matrix. In the algorithm two strategies are employed, one of which is called dis-unstabilization technique and the other is preprocessing technique, Together with them, a so-called ratio-reduction equation and a backtrack technique are introduced to avoid the instability and breakdown in the original algorithm. It is shown that the new algorithm can overcome the instability and breakdown at low cost. Numerical results have demonstrated that the algorithm is stable and can compute the eigenvalues to very high accuracy.3. There was a QR-like algorithm1'61 for the solution of real algebraic Riccati equation or computing the stable invariant subspace associated with the n eigenvalues with negative real part of the corresponding real Hamiltonian matrix. In the QR-like algorithm, to preserve the Hamiltonian structure,symplectic similarity transformations were employed. Especially a random symplectic matrix with the form of S = I2n -wwT J was used when the algorithm was unstable, so the properties of the symplectic matrix are worth to studying. In this thesis, several important properties of 5 are presented and proved. Especially, the property of the condition number of the random symplectic matrix should be the theoretic basis of constructing the symplectic matrix.4. A rounding error analysis of the symplectic Lanczos method is given for the Hamiltonian eigenvalue problem. It is applicable when no break down occurs and shows that the restriction of preserving the Hamiltonian structure does not destroy the characteristic feature of nonsymmetric Lanczos processes. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. As be expected, it follows that (under certain assumptions) the computed J-orthogonal Lanczos vectors loose J-orthogonality when some Ritz values begin to converge.