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It is of critical importance to efficiently and accurately predict global resonances of a 3-D integrated circuit system that involves arbitrarily shaped lossy conductors and inhomogeneous materials. A quadratic eigenvalue solver of linear complexity and electromagnetic accuracy is developed in this paper to fulfill this task. Without sacrificing accuracy, the proposed eigenvalue solver has shown a clear advantage over state-of-the-art eigenvalue solvers in fast CPU time. It successfully solves a quadratic eigenvalue problem of over 2.5 million unknowns associated with a large-scale 3-D on-chip circuit embedded in inhomogeneous materials in 40 min on a single 3 GHz 8222SE AMD Opteron processor.