Vacuum stability and the Cholesky decomposition

We discuss how the Cholesky decomposition may be used to ascertain whether a critical point of the field theory scalar potential provides a stable vacuum configuration. We then use this method to derive the stability conditions in a specific example.Comment: 7 page

Reduced scaling in electronic structure calculations using Cholesky decompositions

We demonstrate that substantial computational savings are attainable in electronic structure calculations using a Cholesky decomposition of the two-electron integral matrix. In most cases, the computational effort involved calculating the Cholesky decomposition is less than the construction of one Fock matrix using a direct O(N2) [email protected]

On the Cholesky Decomposition for electron propagator methods: General aspects and application on C60

To treat the electronic structure of large molecules by electron propagator methods we developed a parallel computer program called P-RICD$\Sigma$. The program exploits the sparsity of the two-electron integral matrix by using Cholesky decomposition techniques. The advantage of these techniques is that the error introduced is controlled only by one parameter which can be chosen as small as needed. We verify the tolerance of electron propagator methods to the Cholesky decomposition threshold and demonstrate the power of the P-RICD$\Sigma$ program for a representative example (C60). All decomposition schemes addressed in the literature are investigated. Even with moderate thresholds the maximal error encountered in the calculated electron affinities and ionization potentials amount to a few meV only, and the error becomes negligible for small thresholds.Comment: 30 pages, 6 figures submitted to J.Chem. Phy

A generalized non-square Cholesky decomposition algorithm with applications to finance

In several applications there is the need to compute a Cholesky decomposition of a given symmetric matrix. The usual Cholesky decomposition algorithm will fail if the given matrix is semi-positive, although such a decomposition exists. To overcome this problem there exists a LDLT decomposition for semi-positive matrices. In the case that the given symmetric matrix is not semi-positive, no Cholesky decomposition exists. In such a situation one aims to approximate this matrix by a (semi-)positive one and computes the Cholesky decomposition of the approximation. From the context of numerical optimization there exist algorithms by Gill, Murray and Wright and a refinement by Eskow and Schnabel. Both methods basicly return a Cholesky decomposition of a positive approximation of an indefinite input matrix. In this paper we extend the LDLT algorithm such that it coincides for a semi-definite input with the LDLT decomposition and for indefinite input it gives the decomposition of a semi-positive approximation. In contrast to the algorithms mentioned before, for indefinite input matrices our algorithm gives a decomposition, which has a lower rank. This gives the important opportunity to introduce a dimension reduction, if possible, and we will show that this algorithm can save computation time in several applications in finance, especially for risk management

Size-intensive decomposition of orbital energy denominators

We introduce an alternative to Almlöf and Häser’s Laplace transform decomposition of orbital energy denominators used in obtaining reduced scaling algorithms in perturbation theory based methods. The new decomposition is based on the Cholesky decomposition of positive semidefinite matrices. We show that orbital denominators have a particular short and size-intensive Cholesky decomposition. The main advantage in using the Cholesky decomposition, besides the shorter expansion, is the systematic improvement of the results without the penalties encountered in the Laplace transform decomposition when changing the number of integration points in order to control the convergence. Applications will focus on the coupled-cluster singles and doubles model including connected triples corrections [CCSD(T)], and several numerical examples are discussed.Alfredo.Sá[email protected]