Algorithms for solving narrowly-banded linear systems on parallel computers by direct methods

Thesis/Dissertation · Sun Jan 01 04:00:00 EST 1989

OSTI ID:6037034

Cleary, A J

This research concerns the development, implementation, and comparison of direct methods for solving linear systems A{sub {beta}}x = b on parallel computers, where A{sub {beta}} is a banded symmetric positive definite matrix with bandwidth {beta}. His basic method of solution will use the Choleski decomposition. The first class of algorithms factors A{sub {beta}} without reordering the matrix. The author will present column oriented algorithms with storage by columns and rows and give results from the Flex/32 that show that these algorithms are efficient on problems with large bandwidths. He also considers a root-free version of the row algorithm. In the second class of algorithms, a Choleski factorization is performed on a reordered matrix A = PA{sub {beta}}P{sup T}, where P is a permutation matrix. He notes that the factorization of A results in fill-in that is responsible for a four-fold increase in the operation count of the factorization. He then presents theoretical results quantifying the tradeoff between parallelism and fill in the factorization of all possible A. From this result he predicts the existence of a two-way Choleski algorithm; he presents and analyzes this algorithm, and present computational results that show that it consistently outperforms the previous algorithms. Next, he considers optimal orderings. He presents theory that shows that an optimal ordering for any banded matrix and computer system can always be found that has a certain simple form. This theorem is then applied to some example situations to produce optimal orderings for them.

• Research Organization: Virginia Univ., Charlottesville, VA (USA). Dept. of Biology
• OSTI ID: 6037034
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
CHEMICAL REACTIONS
COMPUTER CALCULATIONS
DATA
DECOMPOSITION
EQUATIONS
FACTORIZATION
INFORMATION
LINEAR PROGRAMMING
MATHEMATICAL LOGIC
MATRIX ELEMENTS
NUMERICAL DATA
OPTIMIZATION
PARALLEL PROCESSING
PREDICTION EQUATIONS
PROGRAMMING
THEORETICAL DATA