A Matrix Decomposition Method for Orthotropic Elasticity Problems

Chen, Hsin-Chu; Sameh, Ahmed H.

doi:10.1137/0610004

Cited by 39 publications

(14 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since z ∈ C n is J -symmetric (J -skew symmetric) if and only if J z = z (J z = −z), our definition is an extension of Andrew's [1] definition of symmetric and skew symmetric vectors. Other authors (see, e.g., [5,10,12,14]) have made similar extensions; however, these papers deal mainly with the case where R is real and symmetric. We do not assume this in general.…”

Section: Introductionmentioning

confidence: 93%

See 1 more Smart Citation

Characterization and properties of matrices with generalized symmetry or skew symmetry

Trench

2004

Linear Algebra and its Applications

View full text Add to dashboard Cite

Let R ∈ C n×n be a nontrivial involution; i.e., R = R −1 / = ±I . We say that A ∈ C n×n is R-symmetric (R-skew symmetric) if RAR = A (RAR = −A).There are positive integers r and s with r + s = n and matrices P ∈ C n×r and Q ∈ C n×s such that P * P = I r , Q * Q = I s , RP = P , and RQ = −Q. We give an explicit representation of an arbitrary R-symmetric matrix A in terms of P and Q, and show that solving Az = w and the eigenvalue problem for A reduce to the corresponding problems for matrices A P P ∈ C r×r and A QQ ∈ C s×s . We also express A −1 in terms of A −1 P P and A −1 QQ . Under the additional assumption that R * = R, we show that Moore-Penrose inversion and singular value decomposition of A reduce to the corresponding problems for A P P and A QQ . We give similar results for R-skew symmetric matrices. These results are known for the case where R = J = (δ i,n−j +1 ) n i,j =1 ; however, our proofs are simpler even in this case. We say that A ∈ C n×n is R-conjugate if RAR =Ā where R ∈ R n×n and R = R −1 / = ±I . In this case (A) is R-symmetric and (A) is R-skew symmetric, so our results provide explicit representations for R-conjugate matrices in terms of P and Q, which are now in R n×r and R n×s respectively. We show that solving Az = w, inverting A, and the eigenvalue problem for A reduce to the corresponding problems for a related matrix S ∈ R n×n . If R T = R this is also true for Moore-Penrose inversion and singular value decomposition of A.

show abstract

Section: Introductionmentioning

confidence: 93%

“…Since matrices with other types of R-symmetry and R-skew symmetry are now occurring in applications [5,10], it seems worthwhile to consider R-symmetric and R-skew symmetric matrices from a general point of view. We undertake such a study in this paper.…”

Section: Introductionmentioning

confidence: 99%

Characterization and properties of matrices with generalized symmetry or skew symmetry

Trench

2004

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…We continue along the same line of research of [13] and our ideas stem from the following two observations: 1) By an eigenvalue decomposition of P, the constrained problem can be equivalently transformed to two constrained problems in the similar forms as (1.1). Then the general solutions of the original constrained problems, together with the existence conditions, can be represented by the eigenvectors of P; (2) Since the orthogonal projections onto the invariant eigen-subspaces of P can be easily represented in terms of P, the involved eigenvectors can be released and eigenvector-free formulas of the existence conditions and general solution will be given. And,we also consider the optimal approximate solution to the given matrix E from the constrained solution set of problem (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…The reflexive and anti-reflexive matrices possess special properties and have wide applications in engineering and scientific computations [1,2]. So seeking the solutions to matrix equation or matrix equations with these constraints maybe is interesting [6][7][8][9][10]13].…”

Section: Introductionmentioning

confidence: 99%

Eigenvector-free solutions to AX=B with PX=XP and XH=sX constraints

Qiu

Wang

2011

Applied Mathematics and Computation

View full text Add to dashboard Cite

“…Adaptations of this include attempts to accelerate convergence by decreasing the number of iterations required for convergence (Douglas & Miranker, 1987;Hackbusch, 1987;Tuminaro & Chan, 1987;Chen & Sameh, 1989;Tuminaro, 1989;and others). Making the proper choices of the operators (interpolation and restriction, for example) is critical to effective decomposition of the problem.…”

mentioning

confidence: 99%

Massively parallel fast elliptic equation solver for three dimensional hydrodynamics and relativity

Sholl¹,

Wilson²,

Mathews³

et al. 1995

View full text Add to dashboard Cite

Lawrence Livermore Natio nul Labo rato ry G r a n t J. M a t h e w s Physics and Space Technology Program/ V-DivisionLawrence Livermore National Laboratory John €3. Avila Mathematics and Computer Science DepartmentSun Jose State University A B S T R A C TThrough the work proposed in this document we expect to advance the forefront of large scale computational efforts on massively parallel distributed-memory multiprocessors. We will develop tools for effective conversion to a parallel implementation of sequential numerical methods used to solve large systems of partial differential equations. The research supported by this work will involve conversion of a program which does state of the art modeling of multi-dimensional hydrodynamics, general relativity and particle transport in energetic astrophysical environments. The proposed parallel algorithm development, particularly the study and development of fast elliptic equation solvers, could significantly benefit this program and other applications involving solutions to systems of differential equations.We shall (1) develop a data communication manager for distributed memory computers as an aid in program conversions to a parallel environment and implement it in the three dimensional relativistic hydrodynamics program discussed below; (2) develop a concurrent system/concurrent subgrid multigrid method. Currently, five systems are approximated sequentially using multigrid successive overrelaxation. Results from an iteration cycle of one multigrid system are used in following multigrid systems iterations. We shall develop a multigrid algorithm for simultaneous computation of the sets of equations. In addition, we shall implement a method for concurrent processing of the subgrids in each of the multigrid computations. The conditions for convergence of the method will be examined. We'll compare this technique to other parallel multigrid techniques, such as distributed data/sequential subgrids and the ParalIel Superconvergent Multigrid (PSMG) of Frederickson and McBryan. We expect the results of these studies to offer insight and tools both for the selection of new algorithms as well as for conversion of existing large codes for massively parallel architectures. We shall implement the programs on LLNL's Meiko-CS2 which is a distributed memory multiprocessor with 256 nodes, each node having two vector processors and its own communications processor.The goal of this work is to advance the understanding of the fundamental concepts for the solution of systems of differential equations in a massively parallel environment.This solution process is key to DOE research and development programs. This work would contribute to the knowledge of the use of massively parallel distributed memory multigrid algorithms and several areas of current interest in theoretical astrophysics. DISCLAIMER

show abstract

A Matrix Decomposition Method for Orthotropic Elasticity Problems

Cited by 39 publications

References 6 publications

Characterization and properties of matrices with generalized symmetry or skew symmetry

Characterization and properties of matrices with generalized symmetry or skew symmetry

Eigenvector-free solutions to AX=B with PX=XP and XH=sX constraints

Massively parallel fast elliptic equation solver for three dimensional hydrodynamics and relativity

Contact Info

Product

Resources

About