The Deflated Techniques Of Lanczos Algorithm For Solving Large Symmetric Indefinite Linear Systems

This thesis derives three algorithms for solving large indefinite symmetric sparse linear systems on the basis of Lanczos method.The first algorithm is the restarted and deflated Lanczos algorithm. At each beginning, we rich approximate eigenvectors corresponding to a few of smallest eigenvalues in magnitude to the krylov subspace. Numerical result shows that the new method is more efficient in convergence than the standard Lanczos algorithm; The second algorithm generalizes the implicitly restarted Arnoldi(IRA) augmented by Soreesen to the implicitly restarted Lanczos algorithm, which improves the convergence rate of Lanczos algorithm by making good use of the spectral information obtained from the previous process. The last algorithm utilizes deflation strategies to the second algorithm to forming invariant subspace for A, so that the stability can be kept in computing process.We have given theoretical analysis on each new algorithm and done may numerical experiments and comparisons. Theoretical results and numerical experiments have confirmed that the three new algorithms have better practical performance significantly less computational cost and less CPU time.