Matrice symétrique

vignette|Matrice 5x5 symétrique. Les coefficients égaux sont représentés par la même couleur. En algèbre linéaire et multilinéaire, une matrice symétrique est une matrice carrée qui est égale à sa propre transposée, c'est-à-dire telle que a = a pour tous i et j compris entre 1 et n, où les a sont les coefficients de la matrice et n est son ordre. Exemples Les coefficients d'une matrice symétrique sont symétriques par rapport à la diagonale principale (du coin en haut à gauche jusqu'à celui en bas à droite). La matrice suivante est donc symétrique : :\begin{pmatrix} 2 & 4 & 6\ 4 & 0 & 10\ 6 & 10 & 12\end{pmatrix} Toute matrice diagonale est symétrique. Propriétés

Une matrice représentant une forme bilinéaire est symétrique si et seulement si cette dernière est symétrique.
L'ensemble des matrices symétriques d'ordre n à coefficients dans un corps commutatif est un sous-espace vectoriel de dimension n(n+1)/2 de l'espace vectoriel des matrices car

Fast hierarchical solvers for symmetric eigenvalue problems

Ana Susnjara

In this thesis we address the computation of a spectral decomposition for symmetric banded matrices. In light of dealing with large-scale matrices, where classical dense linear algebra routines are not applicable, it is essential to design alternative techniques that take advantage of data properties. Our approach is based upon exploiting the underlying hierarchical low-rank structure in the intermediate results. Indeed, we study the computation in the hierarchically off-diagonal low rank (HODLR) format of two crucial tools: QR decomposition and spectral projectors, in order to devise a fast spectral divide-and-conquer method. In the first part we propose a new method for computing a QR decomposition of a HODLR matrix, where the factor R is returned in the HODLR format, while Q is given in a compact WY representation. The new algorithm enjoys linear-polylogarithmic complexity and is norm-wise accurate. Moreover, it maintains the orthogonality of the Q factor. The approximate computation of spectral projectors is addressed in the second part. This problem has raised some interest in the context of linear scaling electronic structure methods. There the presence of small spectral gaps brings difficulties to existing algorithms based on approximate sparsity. We propose a fast method based on a variant of the QDWH algorithm, and exploit that QDWH applied to a banded input generates a sequence of matrices that can be efficiently represented in the HODLR format. From the theoretical side, we provide an analysis of the structure preservation in the final outcome. More specifically, we derive a priori decay bounds on the singular values in the off-diagonal blocks of spectral projectors. Consequently, this shows that our method, based on data-sparsity, brings benefits in terms of memory requirements in comparison to approximate sparsity approaches, because of its logarithmic dependence on the spectral gap. Numerical experiments conducted on tridiagonal and banded matrices demonstrate that the proposed algorithm is robust with respect to the spectral gap and exhibits linear-polylogarithmic complexity. Furthermore, it renders very accurate approximations to the spectral projectors even for very large matrices. The last part of this thesis is concerned with developing a fast spectral divide-and-conquer method in the HODLR format. The idea behind this technique is to recursively split the spectrum, using invariant subspaces associated with its subsets. This allows to obtain a complete spectral decomposition by solving the smaller-sized problems. Following Nakatsukasa and Higham, we combine our method for the fast computation of spectral projectors with a novel technique for finding a basis for the range of such a HODLR matrix. The latter strongly relies on properties of spectral projectors, and it is analyzed theoretically. Numerical results confirm that the method is applicable for large-scale matrices, and exhibits linear-polylogarithmic complexity.

EPFL2018

Sur quelques équations aux dérivées partielles en lien avec le lemme de Poincaré

David Valentin Strütt

In this thesis, we study two distinct problems. The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system. The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

EPFL2018

Fast hierarchical solvers for symmetric eigenvalue problems

Ana Susnjara

EPFL2018

CM–fields and skew–symmetric matrices

Fast hierarchical solvers for symmetric eigenvalue problems

Sur quelques équations aux dérivées partielles en lien avec le lemme de Poincaré

Fast hierarchical solvers for symmetric eigenvalue problems

Sur quelques équations aux dérivées partielles en lien avec le lemme de Poincaré

CM–fields and skew–symmetric matrices