  Bayesian variable selection regression (BVSR) is able to jointly analyze
genome-wide genetic datasets, but the slow computation via Markov chain Monte
Carlo (MCMC) hampered its wide-spread usage. Here we present a novel iterative
method to solve a special class of linear systems, which can increase the speed
of the BVSR model-fitting tenfold. The iterative method hinges on the complex
factorization of the sum of two matrices and the solution path resides in the
complex domain (instead of the real domain). Compared to the Gauss-Seidel
method, the complex factorization converges almost instantaneously and its
error is several magnitude smaller than that of the Gauss-Seidel method. More
importantly, the error is always within the pre-specified precision while the
Gauss-Seidel method is not. For large problems with thousands of covariates,
the complex factorization is 10 -- 100 times faster than either the
Gauss-Seidel method or the direct method via the Cholesky decomposition. In
BVSR, one needs to repetitively solve large penalized regression systems whose
design matrices only change slightly between adjacent MCMC steps. This slight
change in design matrix enables the adaptation of the iterative complex
factorization method. The computational innovation will facilitate the
wide-spread use of BVSR in reanalyzing genome-wide association datasets.
