  A new algorithm to approximate Hermitian matrices by positive semidefinite
Hermitian matrices based on modified Cholesky decompositions is presented. In
contrast to existing algorithms, this algorithm allows to specify bounds on the
diagonal values of the approximation. It has no significant runtime and memory
overhead compared to the computation of a classical Cholesky decomposition.
Hence it is suitable for large matrices as well as sparse matrices since it
preserves the sparsity pattern of the original matrix. The algorithm tries to
minimize the approximation error in the Frobenius norm as well as the condition
number of the approximation. Since these two objectives often contradict each
other, it is possible to weight these two objectives by parameters of the
algorithm. In numerical experiments, the algorithm outperforms existing
algorithms regarding these two objectives. A Cholesky decomposition of the
approximation is calculated as a byproduct. This is useful, for example, if a
corresponding linear equation should be solved. A fully documented and
extensively tested implementation is available. Numerical optimization and
statistics are two fields of application in which the algorithm can be of
particular interest.
