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## Relative perturbation theory for diagonally dominant matrices

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

Publicado em /10/2014
Português

Relevância na Pesquisa

26.4%

#Accurate computations#Diagonally dominant matrices#Diagonally dominant parts#Eigenvalues#Inverses#Linear systems#Relative perturbation theory#Singular values#Eigenvalues and eigenfunctions#Inverse problems#Matrix algebra

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.

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

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

Publicado em /07/2014
Português

Relevância na Pesquisa

67.37%

#Accurate computations#column diagonal dominance pivoting#diagonally dominant matrices#LDU factorization#rank-revealing decomposition#relative perturbation theory#diagonally dominant parts#Matemáticas

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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36.69%

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/12/2014
Português

Relevância na Pesquisa

26.4%

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.

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## Matrix Factorizations via the Inverse Function Theorem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/08/2014
Português

Relevância na Pesquisa

26.69%

We give proofs of QR factorization, Cholesky's factorization, and LDU
factorization using the inverse function theorem. As a consequence, we obtain
analytic dependence of these matrix factorizations which does not follow
immediately using Gaussian elimination.; Comment: 6 pages

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