  Algorithms are presented for evaluating gradients and Hessians of logarithmic
barrier functions for two types of convex cones: the cone of positive
semidefinite matrices with a given sparsity pattern, and its dual cone, the
cone of sparse matrices with the same pattern that have a positive semidefinite
completion. Efficient large-scale algorithms for evaluating these barriers and
their derivatives are important in interior-point methods for nonsymmetric
conic formulations of sparse semidefinite programs. The algorithms are based on
the multifrontal method for sparse Cholesky factorization.
