  In this paper an approach for finding a sparse incomplete Cholesky factor
through an incomplete orthogonal factorization with Givens rotations is
discussed and applied to Gaussian Markov random fields (GMRFs). The incomplete
Cholesky factor obtained from the incomplete orthogonal factorization is
usually sparser than the commonly used Cholesky factor obtained through the
standard Cholesky factorization. On the computational side, this approach can
provide a sparser Cholesky factor, which gives a computationally more efficient
representation of GMRFs. On the theoretical side, this approach is stable and
robust and always returns a sparse Cholesky factor. Since this approach applies
both to square matrices and to rectangle matrices, it works well not only on
precision matrices for GMRFs but also when the GMRFs are conditioned on a
subset of the variables or on observed data. Some common structures for
precision matrices are tested in order to illustrate the usefulness of the
approach. One drawback to this approach is that the incomplete orthogonal
factorization is usually slower than the standard Cholesky factorization
implemented in standard libraries and currently it can be slower to build the
sparse Cholesky factor.
