Numerical safeguarded use of the implicit restarted Lanczos algorithm for solving nonlinear eigenvalue problems and its monotonicity analysis

In this thesis, we develop an efficient accurate numerical algorithm for evaluating a few of the smallest eigenvalues and their corresponding eigenvectors for large scale nonlinear eigenproblems. The entries of the matrices in these problems are transcendental functions approximated well by rational functions. This algorithm is based upon the Implicit Restarted Lanczos method for solving the linear eigenvalue sub-problems that arise in conjunction with a new zero-finding technique that uses rational function interpolation to approximate the generalized eigenvalues. We have tested this technique on high performance computers and we present some numerical experiments that demonstrate the efficiency and the accuracy of this procedure. Our monotonicity analysis theory shows that the parameterized eigenvalue curves (monotone increasing) are much better behaved than the parameterized determinant curves that have erratic behavior. Our numerical and monotonicity analyses are sufficiently general that they hold for any problem having monotone increasing generalized eigenvalues. This type of problem is associated with the mixed finite element formulation that involves a frequency independent stiffness and frequency dependent mass matrices