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:

.. function:: system BandSPD

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

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

2. :math:`A_{i,j} = A_{j,i}`

3. :math:`y^T A y != 0` for all non-zero vectors y with real entries (:math:`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:

.. math::
   \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}

is stored as follows:

.. math::
   \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}

.. admonition:: Example 

   The following example shows how to construct a BandSPD system

   1. **Tcl Code**

   .. code-block:: tcl

      system BandSPD

   2. **Python Code**

   .. code-block:: python

      system('BandSPD')

Code developed by: |fmk|