27.1: Banded Matrices
- Page ID
- 55704
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A class of sparse matrices that often arise in engineering practice - especially in continuum mechanics - is the banded matrix. An example of banded matrix is shown in Figure 27.1. As the figure shows, the nonzero entries of a banded matrix is confined to within \(m_{\mathrm{b}}\) entries of the main diagonal. More precisely, \[A_{i j}=0, \quad \text { for } \quad j>i+m_{\mathrm{b}} \text { or } j<i-m_{\mathrm{b}},\] and \(A\) may take on any value within the band (including zero). The variable \(m_{\mathrm{b}}\) is referred to as the bandwidth. Note that the number of nonzero entries in a \(n \times n\) banded matrix with a bandwidth \(m_{\mathrm{b}}\) is less than \(n\left(2 m_{\mathrm{b}}+1\right)\).
Let us consider a few different types of banded matrices.
Example 27.1.1 Tridiagonal matrix: \(m_{\mathrm{b}}=1\)
As we have discussed in the previous two chapters, tridiagonal matrices have nonzero entries only along the main diagonal, sub-diagonal, and super-diagonal. Pictorially, a tridiagonal matrix takes the following form:
Clearly the bandwidth of a tridiagonal matrix is \(m_{\mathrm{b}}=1\). A \(n \times n\) tridiagonal matrix arise from, for example, computing the equilibrium displacement of \(n\) masses connected by springs, as we have seen in previous chapters.
Example 27.1.2 Pentadiagonal matrix: \(m_{\mathrm{b}}=2\)
As the name suggests, a pentadiagonal matrix is characterized by having nonzero entries along the main diagonal and the two diagonals above and below it, for the total of five diagonals. Pictorially, a pentadigonal matrix takes the following form:
Example 27.1.3 "Outrigger" matrix
Another important type of banded matrix is a matrix whose zero entries are confined to within the \(m_{\mathrm{b}}\) band of the main diagonal but for which a large number of entries between the main diagonal and the most outer band is zero. We will refer to such a matrix as "outrigger." An example of such a matrix is
In this example, there are five nonzero diagonal bands, but the two outer bands are located far from the middle three bands. The bandwidth of the matrix, \(m_{\mathrm{b}}\), is specified by the location of the outer diagonals. (Note that this is not a pentadiagonal matrix since the nonzero entries are not confined to within \(m_{\mathrm{b}}=2\).) "Outrigger" matrices often arise from finite difference (or finite element) discretization of partial differential equations in two or higher dimensions.