3.2.3.2. BandSPD System

This command is used to construct a BandSPD linear system of equation object. As the name implies, this class is used for symmetric positive definite matrix systems which have a banded profile. The matrix is stored as shown below in a 1 dimensional array of size equal to the (bandwidth/2) times the number of unknowns. When a solution is required, the Lapack routines DPBSV and DPBTRS are used. The following command is used to construct such a system:

system BandSPD

An n×n matrix is a symmetric positive definite banded matrix if:

  1. \(A_{i,j}=0 \quad\mbox{if}\quad j<i-k \quad\mbox{ or }\quad j>i+k; \quad k \ge 0.\)

  2. \(A_{i,j} = A_{j,i}\)

  3. \(y^T A y != 0\) for all non-zero vectors y with real entries (\(y \in \mathbb{R}^n\)),

The bandwidth of the matrix is k + k + 1.

For example, a symmetric 6-by-6 matrix with a right bandwidth of 2:

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

is stored as follows:

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

Example

The following example shows how to construct a BandSPD system

  1. Tcl Code

system BandSPD

  1. Python Code

system('BandSPD')

Code developed by: fmk