Purpose:
To develop efficient methods for approximating eigenvalues and
eigenvectors of a symmetric matrix A that is block tridiagonal,
banded, or the magnitudes of its elements rapidly decrease when moving away from
the diagonal. More specifically, given a relative error
parameter  ,
the objective is to develop and implement
efficient methods for finding approximations to all or a subset of the
eigenvalues 
and eigenvectors 
of the matrix A such that:
with
 ,
where the diagonal matrix 
contains the approximations 
to the eigenvalues
of A and the column vectors
 of
 are the approximations
to the eigenvectors
of A.
the eigenvectors are numerically orthogonal, and
their computational complexity is reduced significantly (by
one or two orders of magnitude) compared to computing "exact"
(except for rounding errors) eigenvalues and eigenvectors.
Research Leader:
Robert C. Ward
Collaborators:
Yihua Bai, Indiana State University
Wilfried Gansterer, University of Vienna
Rick Muller, Sandia National Laboratories, Albuquerque
George Fann, Oak Ridge National Laboratory
Bill Goddard, California Institute of Technology
Robert Harrison, Oak Ridge National Laboratory
R. J. Hinde, University of Tennessee
Guoping Zhang, Indiana State University
Publications:
Journal Publications:
Bai, Yihua and Robert C. Ward: Parallel Block Tridiagonalization of Real Symmetric Matrices,
J Parallel Distrib. Comput. 68 (2008) pp. 703-715.
Bai, Yihua and Robert C. Ward: A Parallel Symmetric Block-Tridiagonal Divide-and-Conquer Algorithm,
ACM Trans. Math. Software 33 (2007) Article 25.
Bai, Yihua, Wilfried N. Gansterer, Robert C. Ward: Block-Tridiagonalization of "Effectively"
Sparse Symmetric Matrices, ACM Trans. Math Software 30 (2004) pp. 326-352.
Gansterer, Wilfried N., Yihua Bai, Robert M. Day and Robert C. Ward: A Framework for Approximating
Eigenpairs of Symmetric Block Tridiagonal Matrices, IEEE Computing in Science & Engineering 6
(2004) pp. 50-59.
Gansterer, Wilfried N., Robert C. Ward, Richard P. Muller and William A. Goddard, III: Computing
Approximate Eigenpairs of Symmetric Block Tridiagonal Matrices, SIAM J. Sci. Comput. 25 (2003)
pp. 65-85.
Gansterer, Wilfried N., Robert C. Ward and Richard P. Muller: An Extension of the Divide-and-Conquer
Method for a Class of Symmetric Block-Tridiagonal Eigenproblems, ACM Trans. Math. Software 28
(2002) pp. 45-58.
Others:
Ward, Robert C. and Yihua Bai:
Performance of Parallel Eigensolvers on Electronic
Structure Calculations II, Technical Report UT-CS-06-572,
University of Tennessee, Knoxville, TN, September 2006.
Yihua Bai: High Performance Parallel Approximate Eigensolver for
Real Symmetric Matrices, PhD Dissertation, University of Tennessee, Knoxville, December 2005.
Robert C. Ward, Yihua Bai and Justin Pratt:
Performance of Parallel Eigensolvers on Electronic
Structure Calculations, Technical Report UT-CS-05-560,
University of Tennessee, Knoxville, TN, February 2005.
R. M. Day:
A Coarse-Grain Parallel Implementation of the Block Tridiagonal
Divide and Conquer Algorithm for Symmetric Eigenproblems,
Master Thesis, University of Tennessee, Knoxville, May 2003
Bai, Y., R. M. Day, W. N. Gansterer and R. C. Ward:
New Algorithmic Tools For Electronic Structure Computations,
Proceedings of the Fourth IMACS Symposium on Mathematical Modelling, Volume 2 (CD-ROM), Vienna University of Technology (February 2003)
Partial Research Sponsors:
The Academic Strategic Alliances Program of the DOE
Accelerated Strategic Computing Initiative (ASCI/ASAP) under subcontract
to the California Institute of Technology.
Computing and Computational Sciences Directorate, Oak Ridge National Laboratory
Science Alliance Program, University of Tennessee
Contact:
Dr. Robert C. Ward, ward@eecs.utk.edu
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