2.3 Index Display Matrices

 The method used until now to describe variable indices, i.e.,

lists and ranges, is not well suited for a structural analysis of the

relationship between function indices and variable indices. Since many

existing algorithms are either designed to treat incidence matrices, or

can be easily modified to do so, it is desirable to somehow convert the

lists and ranges to incidence matrix representation. To do this the

rows of the incidence matrix are made to correspond to the values of the

function index and the columns are made to correspond to the values of

the variable index. An element a.. is zero unless the variable index
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can have the value corresponding to the column j when the function index

is equal to the value corresponding to the row i. The resulting matrix

is called an "Index Display Matrix" (IDM).

 The number of rows in an Index Display Matrix is determined by the

range for the function index upon which it depends. Thus,.for the

function index ii and the variable index j, which depends on it defined

as follows:

 ii: L = 1 ji(ii): iu-1

 U = 20 ii

 A = 1 il+i

the Index Display Matrix (with zero elements blank and non-zero elements

denoted by 'x') would be that of Fig. 2-1. This representation of the

IDM occupies considerable space and is certainly larger than is neces-

sary to convey the structural pattern either to a person studying it or

to an algorithm analyzing it. In addition, as the range of the function

index changes, so does the Index Display Matrix. In order to reduce the

size of Index Display Matrices to something small enough to allow