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. The bandwidth is defined as one more than twice the half bandwidth or one more than the sum of the left and right bandwidths. In banded storage, all entries within the band are stored. In 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.[1]
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 Cholesky[2] is about same as for Cholesky for banded matrices (available, e.g. in ScaLAPACK).
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 algoritm. The basic and one of the earliest algorithm to do that is RCM (Reverse Cuthill-McKee algorithm). Many more reordering algorithms are implemented in contemporary packages, such as MATLAB.
However, skyline storage is not as suitable for very large systems (many millions of equations) because is is not easily adapted for massively parallel computing, and general sparse methods,[3] which store only nonzero entries of the matrix, have become more efficient for very large problems.
References
- ^ 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, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, SIAM, 1994
- ^ George, 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book contains also the description and source code of simple routines that are still useful even if they have been long superseded.
- ^ Iain S Duff, Albert M. Erisman, and John K. Reid, Direct methods for sparse matrices, Oxford University Press, 1986 ISBN:0-198-53408-6