The Jacobi-davidson Method With Deflation For Solving Eigenvalue Problems Of Gyroscopic Systems

The Jacobi-Davidson method for solving eigenvalue problems of gyroscopic systems is consideredin this paper. The convergence of the Jacobi-Davidson method for solving quadratic eigenvalueproblems is analyzed. It is shown that the Jacobi-Davidson method has asymptotically quadratic andlinear convergence if the correction equation is solved exactly and inexactly, respectively. Using thestructural properties of gyroscopic systems, a new algorithm for solving the correction equation in theJacobi-Davidson method is presented. Combining the Jacobi-Davidson method for solving eigenvalueproblems of undamped gyroscopic systems with the inverse iteration for solving standard eigenvalueproblems, the inverse iteration is proposed to solve eigenvalue problems of slightly dampedgyroscopic systems. To compute some sucessive eigenvalues of gyroscopic systems, thenon-equivalence low-rank deflation technique for eigenvalue problems of gyroscopic systems isdeveloped based on the ideal of model updating with no spillover, and the Jacobi-Davidson methodwith deflation for computing some eigenvalues of gyroscopic systems is presented. Some numericalresults are included to demonstrate the efficiency of the proposed algorithms.