3.2.3.3. ProfileSPD System

This command is used to construct a ProfileSPD linear system of equation object. As the name implies, this class is used for symmetric positive definite matrix systems. The matrix is stored as shown below in a 1 dimensional array with only those values below the first non-zero row in any column being stored. This is sometimes also referred to as a skyline storage scheme. The following command is used to construct such a system:

system ProfileSPD

An n×n matrix A= is a symmetric positive definite matrix if:

  1. \(a_{i,j} = a_{j,i}\)

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

In the skyline or profile storage scheme only the entries below the first no-zero row entry in any column are stored if storing by rows: The reason for this is that as no reordering of the rows is required in gaussian eleimination because the matrix is SPD, no non-zero entries will ocur in the elimination process outside the area stored.

For example, a symmetric 6-by-6 matrix with a structure as shown below:

\[\begin{split}\begin{bmatrix} A_{11} & A_{12} & 0 & 0 & 0 \\ & A_{22} & A_{23} & 0 & A_{25} \\ & & A_{33} & 0 & 0 \\ & & & A_{44} & A_{45} \\ & sym & & & A_{55} \end{bmatrix}\end{split}\]

The matrix is stored as 1-d array

\[\begin{bmatrix} A_{11} & A_{12} & A_{22} & A_{23} & A_{33} & A_{44} & A_{25} & 0 & A_{45} & A_{55} \end{bmatrix}\]

with a further array containing indices of diagonal elements:

\[\begin{bmatrix} 1 & 3 & 5 & 6 & 10 \end{bmatrix}\]

Example

The following example shows how to construct a ProfileSPD system

  1. Tcl Code

system ProfileSPD

  1. Python Code

system('ProfileSPD')

Code Developed by: fmk