Block Jacobi-Davidson Method For Large Eigenproblems

With the conventional projection algorithm problems can arise, if the desired eigenvalues are multiple or if they are very close together: the ill-conditioned may depress the efficiency and reliability of the method. There are many methods to solve it and the "block" method is a efficient ones.Jacobi-Davidson method is very efficient for computing the extreme eigenpairs of large-scale matrices, but the system matrix of the correction equation may become ill-conditioned if it encounters the above situation. Therefore, the block version of the algorithm presented in this pape tries to fix these issues. The idea's the following: The block algorithm is parametrised by an additional parameter: the block size p. Now p Ritz pairs are computed in each JD iteration, instead of only one. For each of these p Ritz pairs a search direction is computed from the correction equation. The projections in the correction equation are adjusted in such a way, that they keep the solutions orthogonal to all Ritz vectors. These modification prevent the system matrix from becoming ill-conditioned and nearly singular.By using the refining strategy and deflation technique, furthermore, the paper improves the new method and proposes the refined block Jacobi-Davidson method and the refined block Jacobi-Davidson method with deflation. The first one improves the convergence qualities of the approximation eigenvectors and the last avoid the potential repetition work. Both the new methods improve the efficiency and reliability of block Jacobi-Davidson method greatly in theory .