(i.e., the order of nesting) in terms of do-loops makes the transition

to computer implemented solution procedures in FORTRAN using do-loops

straightforward. The advantages of using the do-loop in a solution

procedure are primarily the compactness of code and the speed of

execution. Indeed, most solution procedures implemented in order

to solve indexed equations use do-loops or code similar to do-loops.

 In order to fully realize the consequences of the nesting of the

indices and the effect of the Function-Variable Incidence Matrix and

Index Display Matrices on the actual incidence matrix, examine Fig. 2-3.

This is the expanded incidence matrix for the problem in Fig. 2-2.

First look at Fig. 2-3 a), the entire incidence matrix. The matrix is

partitioned into four submatrices, the divisions being chosen along the

boundaries between function types and between variable types. These are

the outermost nested "indices" and hence the ones examined first. The

structure of the partitioned matrix is that of a 2x2, non-zero elements

in all positions. That is the same as the structure of the FVIM, which

has its rows and columns labeled by function and variable type. The

next matrix, actually the (1,1) partition of Fig. 2-3 a), is itself

partitioned along boundaries between the indices nested next innermost

(corresponding to i2). Figure 2-2 d) exhibits the s;me structure as

Fig. 2-3 b) (in a blockwise nature). This is what might be expected

since Fig. 2-2 d) is the IDM for j2, which is a subscript of x in f.

The pattern now becomes clear and as would be expected, Fig. 2-3 c)

exhibits: the same structure as Fig. 2-2 b). Figures 2-4 a)-e) are

representations of 2-3 a) achieved by varying the index orderings among

the five other possible orderings. The type of behavior described above

is evident in all of these incidence matrices also. Knowledge of this