Electromagnetic analysis has become increasingly important in the design of integrated circuits (IC) due to process scaling and frequency scaling. Many problems arising from the electromagnetics-based IC analysis can be formulated as a complex-valued generalized eigenvalue problem A x=λBx. State-of-the-art solutions of the problem require O(kN²+k²N) operations, with N being the number of unknowns and k the number of dominant eigenvalues. No linear complexity solution has been reported so far.

In this work, we first developed an eigenvalue solver of O(k ²N) complexity for the 2.5-D finite-element based full-wave analysis of large-scale on-chip interconnect structures. We reduce the original system and solve the reduced system in a linear complexity. From the solution of the reduced system, we recover the solution of other unknowns also in a linear complexity. Eigenvalue clustering and a fast system reduction algorithm are developed to expedite the process.

The computational complexity of the aforementioned eigenvalue solver is further reduced from O(k²N) to O( N). It is accomplished by eliminating the dependence of the complexity on the number of eigenvalues. We develop an equivalent eigenvalue problem, the solution of which is local. The original large problem is then decomposed into L small problems, each of which has a constant number of dominant eigenvalues. As a result, the computational complexity is reduced from O(k²N) to O(N).

This work is then extended to general 3-D problems. The finite-element based solution of general 3-D problems involving both dielectrics and non-ideal conductors is formulated as a quadratic eigenvalue problem. Orthogonal prism vector basis functions are utilized to construct a linear-complexity solution of the resultant system matrix. The local resonances associated with the orthogonal vector bases are identified and removed. An optimal scaling is employed to accurately solve the ill-conditioned quadratic eigenvalue problem. This O( N) 3-D eigenvalue solver has been applied to extract the resonance frequencies as well as perform modal analysis of 3-D high-speed integrated circuits. It has demonstrated clear advantages over state-of-the-art eigenvalue solvers with fast CPU time and without sacrificing accuracy.

Finally, for applications that only require eigenvalues in a specific range, we cluster the eigenvalues in a unit circle by conformal mapping. We then employ a shift-inverting technique to shift the eigenvalues of the original system to the specific range of interest. The resultant new generalized eigenvalue problem is then solved in linear complexity by developing an iterative solution that has a constant number of iterations.