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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. ==Band matrix== Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: : then the quantities ''k''1 and ''k''2 are called the lower and upper bandwidth, respectively. The bandwidth of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that if . A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small. A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix. ==Applications== In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems. Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Band matrix」の詳細全文を読む スポンサード リンク
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