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## Relative perturbation theory for diagonally dominant matrices

Dailey, Megan; Martínez Dopico, Froilán C.; Ye, Qiang
Fonte: Society for Industrial and Applied Mathematics Publicador: Society for Industrial and Applied Mathematics
Tipo: info:eu-repo/semantics/publishedVersion; info:eu-repo/semantics/article
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In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.; This research was partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542.

## A new perturbation bound for the LDU factorization of diagonally dominant matrices

Dailey, Megan; Martínez Dopico, Froilán C.; Ye, Qiang
Fonte: Society for Industrial and Applied Mathematics Publicador: Society for Industrial and Applied Mathematics
Tipo: info:eu-repo/semantics/publishedVersion; info:eu-repo/semantics/article
Relevância na Pesquisa
67.37%
This work introduces a new perturbation bound for the L factor of the LDU factorization of (row) diagonally dominant matrices computed via the column diagonal dominance pivoting strategy. This strategy yields L and U factors which are always well-conditioned and, so, the LDU factorization is guaranteed to be a rank-revealing decomposition. The new bound together with those for the D and U factors in [F. M. Dopico and P. Koev, Numer. Math., 119 (2011), pp. 337– 371] establish that if diagonally dominant matrices are parameterized via their diagonally dominant parts and off-diagonal entries, then tiny relative componentwise perturbations of these parameters produce tiny relative normwise variations of L and U and tiny relative entrywise variations of D when column diagonal dominance pivoting is used. These results will allow us to prove in a follow-up work that such perturbations also lead to strong perturbation bounds for many other problems involving diagonally dominant matrices.; Research supported in part by Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542.

## Compact symmetric spaces, triangular factorization, and Cayley coordinates

Habermas, Derek
Tipo: Artigo de Revista Científica
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Let U/K represent a connected, compact symmetric space, where theta is an involution of U that fixes K, phi: U/K to U is the geodesic Cartan embedding, and G is the complexification of U. We investigate the intersection of phi(U/K) with the Bruhat decomposition of G corresponding to a theta-stable triangular, or LDU, factorization of the Lie algebra of G. When g in phi(U/K) is generic, the corresponding factorization g=ld(g)u is unique, where l in N^-, d(g) in H, and u in N^+. In this paper we present an explicit formula for d in Cayley coordinates, compute it in several types of symmetric spaces, and use it to identify representatives of the connected components of the generic part of phi(U/K). This formula calculates a moment map for a torus action on the highest dimensional symplectic leaves of the Evens-Lu Poisson structure on U/K.; Comment: 19 pages: Main proof entirely rewritten, sections reorganized, exposition made more precise and concise. To appear in Pacific Journal of Mathematics

## A note on robust preconditioners for monolithic fluid-structure interaction systems of finite element equations

Langer, U.; Yang, H.
Tipo: Artigo de Revista Científica
In this note, we consider preconditioned Krylov subspace methods for discrete fluid-structure interaction problems with a nonlinear hyperelastic material model and covering a large range of flows, e.g, water, blood, and air with highly varying density. Based on the complete $LDU$ factorization of the coupled system matrix, the preconditioner is constructed in form of $\hat{L}\hat{D}\hat{U}$, where $\hat{L}$, $\hat{D}$ and $\hat{U}$ are proper approximations to $L$, $D$ and $U$, respectively. The inverse of the corresponding Schur complement is approximated by applying one cycle of a special class of algebraic multigrid methods to the perturbed fluid sub-problem, that is obtained by modifying corresponding entries in the original fluid matrix with an explicitly constructed approximation of the exact perturbation coming from the sparse matrix-matrix multiplications.