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I was asked about various storage of matrices; in particular, about band storage and other variations. I know it is something related to storage of sparse matrices in an efficient way. But, I have no clear idea about the details.
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In mathematics, particularly matrix theory, a band matrix or banded 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.
Bandwidth of a matrix A is defined as the smallest non-negative integer K such that A(i,j) = 0 for |i - j| > K. For example, a matrix with all zeros will have its bandwith equal to zero. Similarly bandwith of diagonal matrix will also be zero.
Band interleaved by line (BIL), band interleaved by pixel (BIP), and band sequential (BSQ) are three common methods of organizing image data for multiband images. BIL, BIP, and BSQ are not in themselves image formats but are schemes for storing the actual pixel values of an image in a file.
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main diagonal).
Simply put, a matrix is banded if all the non-zero elements are close to the diagonal. More formally, consider a matrix A whose elements are aij. The matrix is said to be banded with bandwith m if aij = 0 for all i, j such that abs(i-j)>=m.
A banded matrix is a sparse matrix of a very special form. Band structure is very easy to understand and operate on. Storage is efficient, and very efficient algorithms exist for banded matrices. More so than for more general sparse matrices.
Band storage takes advantage of the structure of the matrices by only storing the elements that may be non-zero.
Read more about this here: http://en.wikipedia.org/wiki/Band_matrix
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