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A new algorithm to approximate Hermitian matrices by positive semidefiniteHermitian matrices based on modified Cholesky decompositions is presented. Incontrast to existing algorithms, this algorithm allows to specify bounds on thediagonal values of the approximation. It has no significant runtime and memoryoverhead compared to the computation of a classical Cholesky decomposition.Hence it is suitable for large matrices as well as sparse matrices since itpreserves the sparsity pattern of the original matrix. The algorithm tries tominimize the approximation error in the Frobenius norm as well as the conditionnumber of the approximation. Since these two objectives often contradict eachother, it is possible to weight these two objectives by parameters of thealgorithm. In numerical experiments, the algorithm outperforms existingalgorithms regarding these two objectives. A Cholesky decomposition of theapproximation is calculated as a byproduct. This is useful, for example, if acorresponding linear equation should be solved. A fully documented andextensively tested implementation is available. Numerical optimization andstatistics are two fields of application in which the algorithm can be ofparticular interest.

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