A solution of the random eigenvalue problem by a dimensional decomposition method

This paper presents a dimensional decomposition method for obtaining probabilistic descriptors of realvalued eigenvalues of positive semi-definite random matrices. The method involves a novel function decomposition allowing lower-variate approximations of eigenvalues, lower-dimensional numerical integration for statistical moments, and Lagrange interpolation facilitating efficient Monte Carlo simulation for probability density functions. Compared with commonly-used perturbation and recently-developed asymptotic methods, no derivatives of eigenvalues are required by the new method developed. Results of numerical examples from structural dynamics indicate that the decomposition method provides excellent estimates of moments and probability densities of eigenvalues for various cases including closely-spaced modes and large statistical variations of input