3.2.3.1. BandGeneral System

This command is used to construct a BandGeneral linear system of equation object. As the name implies, this class is used for matrix systems which have a banded profile. The matrix is stored as shown below in a 1dimensional array of size equal to the bandwidth times the number of unknowns. When a solution is required, the Lapack routines DGBSV and SGBTRS are used. The following command is used to construct such a system:

system BandGeneral

An n×n matrix A=(ai,j) is a band matrix if all matrix elements are zero outside a diagonally bordered band whose range is determined by constants k1 and k2:

\(a_{i,j}=0 \quad\mbox{if}\quad j<i-k_1 \quad\mbox{ or }\quad j>i+k_2; \quad k_1, k_2 \ge 0.\ \) The quantities k1 and k2 are the left and right half-bandwidth, respectively. The bandwidth of the matrix is \(k1 + k2 + 1\) and only the entries in the band are stored; the rest being implicitly zero.

For example, 6-by-6 a matrix with bandwidth 3:

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

is stored as the 6-by-3 matrix

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

Example

The following example shows how to construct a ProfileSPD system

  1. Tcl Code

system BandGeneral

  1. Python Code

system('BandGeneral')

Code Developed by: fmk