# Understanding the concept of Cholesky decomposition part1 | by Monodeep | Jul 2022

**Calculation of NMR shields at the CASSCF level using gauge atomic orbitals and Cholesky decomposition (****arXiv****)**

**Author : **Tommaso Nottoli, Sophie Burger, Stella Stopkowicz, Jürgen Gauss, Philippe Lipparini

**Summary : **We present an implementation of coupled-perturbed complete active space self-consistent field theory (CP-CASSCF) for calculating nuclear magnetic resonance chemical shifts using gauge atomic orbitals and integrals. two-electron Cholesky decomposition. The CP-CASSCF equations are solved using a direct algorithm where the magnetic Hessian matrix-vector product is expressed in terms of subscript-transformed quantities. Numerical tests on systems with up to about 1300 basis functions provide information regarding both the computational efficiency and the limitations of our implementation.

**2. Cholesky decomposition of two-electron integrals in quantum chemical calculations with perturbative or finite magnetic fields using gauge atomic orbitals (****arXiv****)**

**Author : **Jurgen Gauss, Simon Blaschke, Sophie Burger, Tommaso Nottoli, Philippe Lipparini, Stella Stopkowicz

**Summary : **A rigorous analysis is performed regarding the use of the Cholesky decomposition (CD) of two-electron integrals in the case of quantum chemical calculations with finite or perturbative magnetic fields and gauge atomic orbitals. In particular, we study how permutational symmetry can be taken into account in such calculations and how this symmetry can be exploited to reduce computational requirements. A modified CD procedure is suggested for the case of finite fields which roughly halves the memory demands for storing Cholesky vectors. The resulting symmetry of the Cholesky vectors also saves computational costs. For the two-electron derivative integrals in the case of a perturbative magnetic field, we derive the CD expressions by means of a first-order Taylor expansion of the corresponding finite magnetic field formulas with the no-field case as the reference point. The perturbed Cholesky vectors turn out to be antisymmetric (as already proposed by Burger et al. (J. Chem. Phys., 155, 074105 (2021))) and the corresponding expressions allow significant savings in the required integral evaluations (of a factor of about four) as well as in the actual construction of the Cholesky vectors (using a two-step procedure similar to that presented by Folkestad et al. (J. Chem. Phys., 150, 194112 (2019) ) and Zhang et al (J. Phys. Chem. A, 125, 4258–4265 (2021))). Numerical examples with cases involving several hundred basis functions verify our suggestions regarding CD in the case of finite and perturbative magnetic fields.

**3. Estimates of the determinants of the core matrix from the arrested Cholesky decomposition (****arXiv****)**

**Author :** Simon Bartel, Wouter Boomsma, Jes Frellsen, Damien Garreau

**Summary : **Algorithms involving Gaussian processes or determinant point processes usually require computing the determinant of a kernel matrix. Frequently, the latter is calculated from the Cholesky decomposition, an algorithm of cubic complexity in the size of the matrix. We show that, under light assumptions, it is possible to estimate the determinant from a single sub-matrix, with a probabilistic guarantee on the relative error. We present an augmentation of the Cholesky decomposition which stops under certain conditions before processing the whole matrix. Experiences show that this can save considerable time while having an overload of less than 5% when not stopped early. More generally, we present a probabilistic stopping strategy for the approximation of a sum of known length where the additions are revealed sequentially. We do not assume independence between addends, only that they are bounded from below and decrease the conditional expectation.

**4. Hierarchical sparse Cholesky decomposition with applications to high-dimensional spatio-temporal filtering(****arXiv****)**

**Author :** Marcin Jurek, Matthias Katzfuss

**Summary : **Spatial statistics often involve the Cholesky decomposition of covariance matrices. To ensure scalability at high dimensions, several recent approximations have assumed a sparse Cholesky factor of the precision matrix. We propose a hierarchical Vecchia approximation, whose conditional independence assumptions imply a parsimony in the Cholesky factors of the precision and the covariance matrix. This remarkable property is crucial for high-dimensional spatio-temporal filtering applications. We present a fast and simple algorithm to compute our hierarchical Vecchia approximation, and we provide extensions to nonlinear data assimilation with non-Gaussian data based on the Laplace approximation. In several numerical comparisons, including a filtering analysis of satellite data, our methods significantly outperformed alternative approaches.