The Improved Block Jacobi-Davidson Method For Solving Large Symmetric Eigenvalue Problems

Block Jacobi-Davidson method (BJD) is very efficient for computing the multiple or clustered eigenpairs of the symmetric eigenproblems. Several eigenpairs can be computed simultaneously by the method. However, some convergent Ritz pairs generated by the iteration will still participate in the follow-up iteration. The overall convergence rate of the method is reduced. In addition, its computational cost is very expensive when the method is applied to find interior eigenvalues.In order to improve the overall convergence of the block Jacobi-Davidson method, the dynamic deflation technique is used to the method and the dynamic deflated block Jacobi-Davidson method (DBJD) is presented. In order to compute interior eigenvalues, combining the harmonic Rayleigh-Ritz procedure with the block Jacobi-Davidson method, the harmonic block Jacobi-Davidson method (HBJD) is proposed. Finally, applying the dynamic deflation technique to the harmonic block Jacobi-Davidson method, the dynamic deflated harmonic block Jacobi-Davidson method (DHBJD) is given.Efficiently solving the correction equation is crucial for the block Jacobi-Davidson method. The correction equation in BJD is transformed into the classic saddle point system. The preconditioning techniques for the system are analyzed.Numerical experiments show that the dynamic deflated block Jacobi-Davidson method is more efficient than the block Jacobi-Davidson method, moreover, the HBJD and the DHBJD is very efficient for computing the multiple or clustered interior eigenpairs of the symmetric eigenproblems.