Band matrix

Inmathematics,particularlymatrix theory,aband matrixorbanded matrixis asparse matrixwhose non-zero entries are confined to a diagonalband,comprising themain diagonaland zero or more diagonals on either side.

Band matrix

Bandwidth

Formally, consider ann×nmatrixA=(a_i,j). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constantsk₁andk₂:

$a_{i,j}=0\quad {\mbox{if}}\quad j<i-k_{1}\quad {\mbox{ or }}\quad j>i+k_{2};\quad k_{1},k_{2}\geq 0.\,$

then the quantitiesk₁andk₂are called thelower bandwidthandupper bandwidth,respectively.Thebandwidthof the matrix is the maximum ofk₁andk₂;in other words, it is the numberksuch that$a_{i,j}=0$if$|i-j|>k$.

Examples

• A band matrix withk₁=k₂= 0 is adiagonal matrix,with bandwidth 0.
• A band matrix withk₁=k₂= 1 is atridiagonal matrix,with bandwidth 1.
• Fork₁=k₂= 2 one has a pentadiagonal matrix and so on.
• Triangular matrices
  • Fork₁= 0,k₂=n−1, one obtains the definition of an uppertriangular matrix
  • similarly, fork₁=n−1,k₂= 0 one obtains a lower triangular matrix.
• Upper and lowerHessenberg matrices
• Toeplitz matriceswhen bandwidth is limited.
• Block diagonal matrices
• Shift matricesandshear matrices
• Matrices inJordan normal form
• Askyline matrix,also called "variable band matrix" – a generalization of band matrix
• The inverses ofLehmer matricesare constant tridiagonal matrices, and are thus band matrices.

Applications

Innumerical analysis,matrices fromfinite elementorfinite differenceproblems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the banded property 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.

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 thesquare rootof the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applyingGaussian elimination(or equivalently anLU decomposition) to such a matrix results in the band being filled in by many non-zero elements.

Band storage

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

For example, atridiagonal matrixhas bandwidth 1. The 6-by-6 matrix

${\begin{bmatrix}B_{11}&B_{12}&0&\cdots &\cdots &0\\B_{21}&B_{22}&B_{23}&\ddots &\ddots &\vdots \\0&B_{32}&B_{33}&B_{34}&\ddots &\vdots \\\vdots &\ddots &B_{43}&B_{44}&B_{45}&0\\\vdots &\ddots &\ddots &B_{54}&B_{55}&B_{56}\\0&\cdots &\cdots &0&B_{65}&B_{66}\end{bmatrix}}$

is stored as the 6-by-3 matrix

${\begin{bmatrix}0&B_{11}&B_{12}\\B_{21}&B_{22}&B_{23}\\B_{32}&B_{33}&B_{34}\\B_{43}&B_{44}&B_{45}\\B_{54}&B_{55}&B_{56}\\B_{65}&B_{66}&0\end{bmatrix}}.$

A further saving is possible when the matrix is symmetric. For example, consider a symmetric 6-by-6 matrix with an upper bandwidth of 2:

${\begin{bmatrix}A_{11}&A_{12}&A_{13}&0&\cdots &0\\&A_{22}&A_{23}&A_{24}&\ddots &\vdots \\&&A_{33}&A_{34}&A_{35}&0\\&&&A_{44}&A_{45}&A_{46}\\&sym&&&A_{55}&A_{56}\\&&&&&A_{66}\end{bmatrix}}.$

This matrix is stored as the 6-by-3 matrix:

${\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{22}&A_{23}&A_{24}\\A_{33}&A_{34}&A_{35}\\A_{44}&A_{45}&A_{46}\\A_{55}&A_{56}&0\\A_{66}&0&0\end{bmatrix}}.$

Band form of sparse matrices

From a computational point of view, working with band matrices is always preferential to working with similarly dimensionedsquare 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. Thus the work involved in performing operations such as multiplication falls significantly, often leading to huge savings in terms of calculation time andcomplexity.

As sparse matrices lend themselves to more efficient computation than dense matrices, as well as in more efficient utilization of computer storage, there has been much research focused on finding ways to minimise the bandwidth (or directly minimise the fill-in) by applying permutations to the matrix, or other such equivalence or similarity transformations.

TheCuthill–McKee algorithmcan be used to reduce the bandwidth of a sparsesymmetric matrix.There are, however, matrices for which thereverse Cuthill–McKee algorithmperforms better. There are many other methods in use.

The problem of finding a representation of a matrix with minimal bandwidth by means of permutations of rows and columns isNP-hard.

See also

• Diagonal matrix
• Graph bandwidth

References

• Atkinson, Kendall E. (1989),An Introduction to Numerical Analysis,John Wiley & Sons,ISBN0-471-62489-6.
• Davis, Timothy A. (2006),Direct Methods for Sparse Linear Systems,Society for Industrial and Applied Mathematics,ISBN978-0-898716-13-9.
• Feige, Uriel (2000), "Coping with the NP-Hardness of the Graph Bandwidth Problem",Algorithm Theory - SWAT 2000,Lecture Notes in Computer Science, vol. 1851, pp. 129–145,doi:10.1007/3-540-44985-X_2.
• Golub, Gene H.;Van Loan, Charles F.(1996),Matrix Computations(3rd ed.), Baltimore: Johns Hopkins,ISBN978-0-8018-5414-9.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),"Section 2.4",Numerical Recipes: The Art of Scientific Computing(3rd ed.), New York: Cambridge University Press,ISBN978-0-521-88068-8.

External links

• Information pertaining to LAPACK and band matrices
• A tutorial on banded matrices and other sparse matrix formats