band matrix

An important special type of sparse matrices is band matrix, defined as follows.

The following are special cases of band matrices:

Banded matrices, which store only a few diagonals.

Beware, these examples appear to be of banded matrices, that do not have variable band-widths.

An example of packed storage for banded matrix:

Moreover, the storage format appears to be for banded matrices, not skyline matrices as well.

Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse.

Band matrices are usually stored by storing the diagonals in the band; the rest is implicitly zero.

Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero.

He also relates the problem to the representation of sparse linear systems as band matrices with low bandwidth, and to the quadratic assignment problem.

A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.

From a computational point of view, working with band matrices is always preferential to working with similarly dimensioned square matrices.

The inverses of Lehmer matrices are constant tridiagonal matrices, and are thus band matrices.

A band matrix can be likened in complexity to a rectangular matrix whose row dimension is equal to the bandwidth of the band matrix.

One area is sparse matrix/band matrix handling, and general algorithms from this area, such as Cuthill-McKee algorithm, may be applied to find approximate solutions for the graph bandwidth problem.

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.

A very efficient structure for an extreme case of band matrices, the diagonal matrix, is to store just the entries in the main diagonal as a one-dimensional array, so a diagonal nxn matrix requires only n entries.

Matrices with reasonably small upper and lower bandwidth are known as band matrices and often lend themselves to simpler algorithms than general sparse matrices; or one can sometimes apply dense matrix algorithms and gain efficiency simply by looping over a reduced number of indices.

In addition, the effort of coding skyline Cholesky is about same as for Cholesky for banded matrices (available for banded matrices, e.g. in LAPACK; for a prototype skyline code, see ).