Skyline matrix

Skyline matrix

A skyline matrix, or a variable band matrix, is a form of a sparse matrix storage format for a square, banded (and typically symmetric) matrix that reduces the storage requirement of a matrix more than banded storage. In banded storage, all entries within a fixed distance from the diagonal (called half-bandwidth) are stored. In column oriented skyline storage, only the entries from the first nonzero entry to the last nonzero entry in each column are stored. There is also row oriented skyline storage, and, for symmetric matrices, only one triangle is usually stored. [ [http://www.netlib.org/linalg/html_templates/node96.html Skyline Storage (SKS)] , in R. Barrett, M. Berry,T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. Van der Vorst, [http://www.netlib.org/linalg/html_templates/report.html "Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition" (online only)] , SIAM, 1994. [http://www.ec-securehost.com/SIAM/ot43.html 1st edition 1987/1993 hardcopy ] ISBN 978-0-898713-28-2 ISBN ISBN 0-89871-328-5] Skyline storage has become very popular in the finite element codes for structural mechanics, because the skyline is preserved by Cholesky decomposition (a method of solving systems of linear equations with a symmetric, positive-definite matrix; all fill-in falls within the skyline), and systems of equations from finite elements have a relatively small skyline. In addition, the effort of coding skyline CholeskyGeorge, Alan and Liu, Joseph W. H., "Computer solution of large sparse positive definite systems", Prentice-Hall Inc., 1981. ISBN 0131652745 This book contains also the description and source code of simple routines that are still useful even if they have been long superseded.] is about same as for Cholesky for banded matrices (available for banded matrices, e.g. in LAPACK; for a prototype skyline code, see ).

Before storing a matrix in skyline format, the rows and columns are typically renumbered to reduce the size of the skyline (the number of nonzero entries stored) and to decrease the number of operations in the skyline Cholesky algorithm. The same heuristic renumbering algorithm that reduce the bandwidth are also used to reduce the skyline. The basic and one of the earliest algorithms to do that is [http://people.scs.fsu.edu/~burkardt/f_src/rcm/rcm.html RCM] (Reverse Cuthill-McKee algorithm).

However, skyline storage is not as popular for very large systems (many millions of equations) because skyline Cholesky is not so easily adapted for massively parallel computing, and general sparse methods, [Iain S Duff, Albert M. Erisman, and John K. Reid, "Direct methods for sparse matrices", Oxford University Press, 1986 ISBN 0-198-53408-6] which store only the nonzero entries of the matrix, become more efficient for very large problems due to much less fill-in.

See also

*Sparse matrix
*Banded matrix
*Frontal solver

References


Wikimedia Foundation. 2010.

Нужен реферат?

Look at other dictionaries:

  • Skyline (disambiguation) — Skyline may refer to:* Skyline, the artificial horizon that a city s overall structure createsPlaces* Skyline, Alabama * Skyline, Kentucky * Skyline, Minnesota * Skyline Boulevard, a scenic highway in California * Skyline Caverns, Virginia *… …   Wikipedia

  • Matrix representation — This article is about the layout of matrices in the memory of computers. For the representation of groups and algebras by matrices in linear algebra, see representation theory. Matrix representation is a method used by a computer language to… …   Wikipedia

  • Sparse matrix — A sparse matrix obtained when solving a finite element problem in two dimensions. The non zero elements are shown in black. In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros (Stoer Bulirsch 2002,… …   Wikipedia

  • Band matrix — In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. Contents 1 Matrix bandwidth 2… …   Wikipedia

  • Vom Bordstein bis zur Skyline — Studioalbum von Bushido …   Deutsch Wikipedia

  • The Matrix — For the series, see The Matrix (franchise). For other uses, see Matrix. The Matrix Theatrical release poster Directed by Andy Wachowski Larry Wachowski …   Wikipedia

  • Mega City (The Matrix) — Mega City is an enormous virtual megacity in which the inhabitants of the Matrix live their lives in the Matrix series. The City is a conglomeration of many cities, fused into one large city with a gigantic downtown and an impressive skyline.… …   Wikipedia

  • List of matrices — This page lists some important classes of matrices used in mathematics, science and engineering: Matrices in mathematics*(0,1) matrix a matrix with all elements either 0 or 1. Also called a binary matrix . *Adjugate matrix * Alternant matrix a… …   Wikipedia

  • List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

  • Список матриц — Структура матрицы Здесь собраны наиболее важные классы матриц, используемые в математике, науке (в целом) и прикладной науке (в частности). Под матрицей понимается прямоугольный массив чисел …   Википедия

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”