Approximation of Hermitian Matrices by Positive Semidefinite Matrices using Modified Cholesky Decompositions

A new algorithm to approximate Hermitian matrices by positive semidefinite matrices based on modified Cholesky decompositions is presented. The approximation error and the condition number of the approximation can be controlled by parameters of the algorithm. The algorithm tries to minimize the approximation error in the Frobenius norm. It has no significant runtime and memory overhead compared to the computation of an unmodified Cholesky decomposition. Sparsity and positive diagonal entries can be preserved. A Cholesky decomposition of the approximation matrix is calculated as a byproduct.