Partitioning of an Element

Partitioning on an Element is used in a cyclical way to order the indexing of the matrix under development.

In the process of partitioning, data are supplied that partially fill the matrix. Each cycle may involve several parts, and each part involves the same four steps:

• First cycle has only one part, and four steps.
• Second cycle may have as many as three parts, depending on the results of the first cycle.
• Third cycle may have as many as nine parts.
• The nth cycle may have as many as 3^n-1 parts.

Experience suggests that only a few cycles will normally be needed to develop a Reachability Matrix, and the number of parts will seldom be as high as the stated maxima.

<math>\begin{pmatrix}

  1 & 1 & 1 & 1 \\
  0 & 1 & 0 & 1 \\
  0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 1
\end{pmatrix}</math> which includes a diagonal of ones since each number divides itself.