27.1: Banded Matrices
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- 55704
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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.