Band matrix

In numerical mathematics, a band matrix is a matrix in which, in addition to the main diagonal, only a certain number of secondary diagonals has non-zero elements. If only one lower and one upper secondary diagonal are not equal to zero, one speaks of tridiagonal matrices . These matrices are thinly populated matrices with a special structure. Band matrices often arise during the discretization of differential equations .

description

Let with , the matrix A is a band matrix of the bandwidth if the following applies to its elements : ${\displaystyle p,q\in \mathbb {N} }$ ${\displaystyle p,q\geq 0}$ ${\displaystyle l=p+q+1}$ ${\displaystyle a_{ij}}$

${\displaystyle a_{ij}=0}$for or ${\displaystyle j+p<i}$ ${\displaystyle i+q<j.}$

In addition to the main diagonal, only p lower and q upper secondary diagonals are occupied.

${\displaystyle \left({\begin{matrix}a_{11}&\ldots &a_{1(q+1)}&0&\ldots &\ldots &\ldots &0\\\vdots &\ddots &&\ddots &\ddots &&&\vdots \\a_{(p+1)1}&&\ddots &&\ddots &\ddots &&\vdots \\0&\ddots &&\ddots &&\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &&\ddots &&\ddots &0\\\vdots &&\ddots &\ddots &&\ddots &&a_{(n-q)n}\\\vdots &&&\ddots &\ddots &&\ddots &\vdots \\0&\ldots &\ldots &\ldots &0&a_{n(n-p)}&\ldots &a_{nn}\end{matrix}}\right)}$

properties

For positively definite band matrices, the band structure is preserved in the Cholesky decomposition . If column pivoting is used for the solution, this also applies to the LR decomposition of a regular band matrix, only the number of diagonals increases slightly. The cost of the calculation is reduced to . ${\displaystyle O(n)}$

Web links

• LP - tape matrices (definition, sentences, evidence, pseudo-code for LU decomposition)