ProfileSPD SOE: Difference between revisions
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| style="background:yellow; color:black; width:800px" | '''system | | style="background:yellow; color:black; width:800px" | '''system ProfileSPD''' | ||
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:<math> | :<math> | ||
\begin{bmatrix} | \begin{bmatrix} | ||
A_{11} & A_{12} & A_{13} & 0 & 0 | A_{11} & A_{12} & A_{13} & 0 & 0 \\ | ||
& A_{22} & A_{23} & 0 & | & A_{22} & A_{23} & 0 & A_{25} \\ | ||
& & A_{33} & 0 & 0 | & & A_{33} & 0 & 0 \\ | ||
& & & A_{44} & A_{45 | & & & A_{44} & A_{45} \\ | ||
& sym & & & A_{55 | & sym & & & A_{55} | ||
\end{bmatrix}. | \end{bmatrix}. | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{bmatrix} | \begin{bmatrix} | ||
A_{11} & A_{12} & A_{22} & A_{13} & A_{23} & A_{33} & A_{44} & A_{ | A_{11} & A_{12} & A_{22} & A_{13} & A_{23} & A_{33} & A_{44} & A_{25} & 0 & A_{45} & A_{55} | ||
\end{bmatrix}. | \end{bmatrix}. | ||
</math> | </math> | ||
Revision as of 23:13, 9 March 2010
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This command is used to construct a profileSPDSOE 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. To following command is used to construct such a system:
| system ProfileSPD |
NOTES:
THEORY:
An n×n matrix A=(ai,j ) is a symmmetric postive definite matrix if:
- <math>a_{i,j} = a_{j,i}\,</math>
- <math> y^T A y != 0 \,</math> for all non-zero vectors y with real entries (<math>y \in \mathbb{R}^n</math>).
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:
- <math>
\begin{bmatrix}
A_{11} & A_{12} & A_{13} & 0 & 0 \\
& A_{22} & A_{23} & 0 & A_{25} \\
& & A_{33} & 0 & 0 \\
& & & A_{44} & A_{45} \\
& sym & & & A_{55}
\end{bmatrix}. </math>
The matrix is stored as 1-d arary
- <math>
\begin{bmatrix}
A_{11} & A_{12} & A_{22} & A_{13} & A_{23} & A_{33} & A_{44} & A_{25} & 0 & A_{45} & A_{55}
\end{bmatrix}. </math>
Code Developed by: fmk