37,397 research outputs found
Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors
We introduce a method for obtaining new classes of free divisors from
representations of connected linear algebraic groups where
, with having an open orbit. We give sufficient conditions
that the complement of this open orbit, the "exceptional orbit variety", is a
free divisor (or a slightly weaker free* divisor) for "block representations"
of both solvable groups and extensions of reductive groups by them. These are
representations for which the matrix defined from a basis of associated
"representation vector fields" on has block triangular form, with blocks
satisfying certain nonsingularity conditions.
For towers of Lie groups and representations this yields a tower of free
divisors, successively obtained by adjoining varieties of singular matrices.
This applies to solvable groups which give classical Cholesky-type
factorization, and a modified form of it, on spaces of symmetric,
skew-symmetric or general matrices. For skew-symmetric matrices, it further
extends to representations of nonlinear infinite dimensional solvable Lie
algebras.Comment: 50 pages. Many changes from v1 in response to a thorough review,
mostly concentrated in sections 2, 3, and 4. To appear in Annales de
l'Institut Fourie
On fixed-point, Krylov, and block preconditioners for nonsymmetric problems
The solution of matrices with block structure arises in numerous
areas of computational mathematics, such as PDE discretizations based on
mixed-finite element methods, constrained optimization problems, or the
implicit or steady state treatment of any system of PDEs with multiple
dependent variables. Often, these systems are solved iteratively using Krylov
methods and some form of block preconditioner. Under the assumption that one
diagonal block is inverted exactly, this paper proves a direct equivalence
between convergence of block preconditioned Krylov or fixed-point
iterations to a given tolerance, with convergence of the underlying
preconditioned Schur-complement problem. In particular, results indicate that
an effective Schur-complement preconditioner is a necessary and sufficient
condition for rapid convergence of block-preconditioned GMRES, for
arbitrary relative-residual stopping tolerances. A number of corollaries and
related results give new insight into block preconditioning, such as the fact
that approximate block-LDU or symmetric block-triangular preconditioners offer
minimal reduction in iteration over block-triangular preconditioners, despite
the additional computational cost. Theoretical results are verified numerically
on a nonsymmetric steady linearized Navier-Stokes discretization, which also
demonstrate that theory based on the assumption of an exact inverse of one
diagonal block extends well to the more practical setting of inexact inverses.Comment: Accepted to SIMA
Symmetric indefinite triangular factorization revealing the rank profile matrix
We present a novel recursive algorithm for reducing a symmetric matrix to a
triangular factorization which reveals the rank profile matrix. That is, the
algorithm computes a factorization where is a permutation matrix,
is lower triangular with a unit diagonal and is
symmetric block diagonal with and antidiagonal
blocks. The novel algorithm requires arithmetic
operations. Furthermore, experimental results demonstrate that our algorithm
can even be slightly more than twice as fast as the state of the art
unsymmetric Gaussian elimination in most cases, that is it achieves
approximately the same computational speed. By adapting the pivoting strategy
developed in the unsymmetric case, we show how to recover the rank profile
matrix from the permutation matrix and the support of the block-diagonal
matrix. There is an obstruction in characteristic for revealing the rank
profile matrix which requires to relax the shape of the block diagonal by
allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient.
This relaxed decomposition can then be transformed into a standard
decomposition at a
negligible cost
Functions preserving nonnegativity of matrices
The main goal of this work is to determine which entire functions preserve
nonnegativity of matrices of a fixed order -- i.e., to characterize entire
functions with the property that is entrywise nonnegative for every
entrywise nonnegative matrix of size . Towards this goal, we
present a complete characterization of functions preserving nonnegativity of
(block) upper-triangular matrices and those preserving nonnegativity of
circulant matrices. We also derive necessary conditions and sufficient
conditions for entire functions that preserve nonnegativity of symmetric
matrices. We also show that some of these latter conditions characterize the
even or odd functions that preserve nonnegativity of symmetric matrices.Comment: 20 pages; expanded and corrected to reflect referees' remarks; to
appear in SIAM J. Matrix Anal. App
Minimizing Communication for Eigenproblems and the Singular Value Decomposition
Algorithms have two costs: arithmetic and communication. The latter
represents the cost of moving data, either between levels of a memory
hierarchy, or between processors over a network. Communication often dominates
arithmetic and represents a rapidly increasing proportion of the total cost, so
we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds
were presented on the amount of communication required for essentially all
-like algorithms for linear algebra, including eigenvalue problems and
the SVD. Conventional algorithms, including those currently implemented in
(Sca)LAPACK, perform asymptotically more communication than these lower bounds
require. In this paper we present parallel and sequential eigenvalue algorithms
(for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms
that do attain these lower bounds, and analyze their convergence and
communication costs.Comment: 43 pages, 11 figure
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