ProfileSPD SOE

This command is used to construct a ProfileSPDSOE linear system of equation. 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 

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THEORY:

An n×n matrix A=(a_i,j) is a symmmetric postive definite matrix if:

\[a_{i,j} = a_{j,i}\,\]

\(y^T A y != 0\) 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 structura as shown below:

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

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:

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

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Code Developed by: fmk