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We describe randomized algorithms for computing the dominant eigenmodes ofthe Generalized Hermitian Eigenvalue Problem (GHEP) $Ax=\lambda Bx$, with $A$Hermitian and $B$ Hermitian and positive definite. The algorithms we describeonly require forming operations $Ax$, $Bx$ and $B^{-1}x$ and avoid formingsquare-roots of $B$ (or operations of the form, $B^{1/2}x$ or $B^{-1/2}x$). Weprovide a convergence analysis and a posteriori error bounds that build uponthe workof~\cite{halko2011finding,liberty2007randomized,martinsson2011randomized}(which have been derived for the case $B=I$). Additionally, we derive some newresults that provide insight into the accuracy of the eigenvalue calculations.The error analysis shows that the randomized algorithm is most accurate whenthe generalized singular values of $B^{-1}A$ decay rapidly. A randomizedalgorithm for the Generalized Singular Value Decomposition (GSVD) is alsoprovided. Finally, we demonstrate the performance of our algorithm on computingthe Karhunen-Lo\`{e}ve expansion, which is a computationally intensive GHEPproblem with rapidly decaying eigenvalues.

Randomized algorithms for Generalized Hermitian Eigenvalue Problems with application to computing Karhunen-Lo\`{e}ve expansion