16 research outputs found
Backward Error Analysis of Factorization Algorithms for Symmetric and Symmetric Triadic Matrices
We consider the factorization of a symmetric matrix where is
unit lower triangular and is block diagonal with diagonal blocks of
order or . This is a generalization of the Cholesky factorization,
and pivoting is incorporated for stability. However, the reliability of
the Bunch-Kaufman pivoting strategy and Bunch's pivoting method for
symmetric tridiagonal matrices could be questioned, because they may
result in unbounded . In this paper, we give a condition under which
factorization will run to completion in inexact arithmetic with
inertia preserved. In addition, we present a new proof of the
componentwise backward stability of the factorization using the inner
product formulation, giving a slight improvement of the bounds in Higham's
proofs, which relied on the outer product formulation and normwise
analysis.
We also analyze the stability of rank estimation of symmetric indefinite
matrices by factorization incorporated with the Bunch-Parlett
pivoting strategy, generalizing results of Higham for the symmetric
semidefinite case.
We call a matrix triadic if it has no more than two non-zero off-diagonal
elements in any column. A symmetric tridiagonal matrix is a special case.
In this paper, we display the improvement in stability bounds when the
matrix is triadic
Characterizations of inverse M-matrices with special zero patterns
AbstractIn this paper, we provide some characterizations of inverse M-matrices with special zero patterns. In particular, we give necessary and sufficient conditions for k-diagonal matrices and symmetric k-diagonal matrices to be inverse M-matrices. In addition, results for triadic matrices, tridiagonal matrices and symmetric 5-diagonal matrices are presented as corollaries
MRRR-based Eigensolvers for Multi-core Processors and Supercomputers
The real symmetric tridiagonal eigenproblem is of outstanding importance in
numerical computations; it arises frequently as part of eigensolvers for
standard and generalized dense Hermitian eigenproblems that are based on a
reduction to tridiagonal form. For its solution, the algorithm of Multiple
Relatively Robust Representations (MRRR or MR3 in short) - introduced in the
late 1990s - is among the fastest methods. To compute k eigenpairs of a real
n-by-n tridiagonal T, MRRR only requires O(kn) arithmetic operations; in
contrast, all the other practical methods require O(k^2 n) or O(n^3) operations
in the worst case. This thesis centers around the performance and accuracy of
MRRR.Comment: PhD thesi
ICASE
This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in the areas of (1) applied and numerical mathematics, including numerical analysis and algorithm development; (2) theoretical and computational research in fluid mechanics in selected areas of interest, including acoustics and combustion; (3) experimental research in transition and turbulence and aerodynamics involving Langley facilities and scientists; and (4) computer science
Multidimensional Wavelets and Computer Vision
This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing
Model Order Reduction
An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science