3.6 Blocking Factors

 The blocking factor defines the size of the Index Display Matrices

for the various function indices. It is desired to perform analyses on

the IDM's and have the results of those analyses be applicable to an

expanded incidence matrix with arbitrary index limits.

 It is possible, by employing different blocking factors, to

analyze an IDM in the same manner twice and reach different results.

For example, the blocking factor can affect the ratio between the number

of tears and the number of rows in a problem. (See Fig. 3-10.) In this

figure, a) and b) are Index Display Matrices representing a tri-diagonal

matrix with first and last columns declared as decisions and omitted

from the figure. For a blocking factor of 2, either of the two possible

output set assignments results in one tear for the two rows. When the

blocking factor is increased from 2 to 3, two of three possible output

set assignments only require one tear for the three rows. The require-

ment that the order of solution be dictated by the IDM (i.e., be top to

bottom or bottom to top) is relaxed for the analysis within the blocks.

When the problem is expanded, the IDM for an index is made up of many

blocks. These blocks must be solved in either ascending or descending

order.

 It is not always the case that the solution procedure derived

 for a block will hold for the expanded IDM. Consider the two expanded

 IDM's in Fig. 3-11, each representing one of the one-tear, blocking

 factor of 3, solution procedures from Fig. 3-10. The first IDM reflects

 a solution procedure which still allows for solution of the equations by

 solving the (1,1) block then the (2,2) block. The second, however, does

 not allow for solving either block before the other. When a solution