First it is necessary to introduce the concept of index ordering.

In naming the rows of an incidence matrix for a set of indexed equations

it is desirable to have an orderly way to "step through" the various

index values. The reasons for this are first, to insure that each

function is represented and that no functions are included more than

once,and second, to provide a logical means of stepping through the

various functions which is easily adaptable to an iterative solution

procedure. The method adopted is similar to the nesting of FORTRAN

"do-loops." There is one important deviation, however, in that the

function type is also treated as an index. The reasons for this modifi-

cation become clear upon examination of the problem in Fig. 2-2. If we

consider the three indices (function type, ii, and i2) as do-loop

indices, obviously there are six ways in which these indices can be

nested. This gives rise to six different structures for the indidence

matrix. Here we have assumed that the ordering of the functions which

would be dictated by the do-loop would be the order of the functions

occurring in sequential rows. Also, the variables would be ordered

similarly, i.e., each variable index would have the same nested position

as the function index upon which it depends and the variable type would

be assumed to depend upon function type. While it is obvious that the

index I precedes the index 2 in a normal ordering, there is no such

normal ordering when considering the function or variable type. It is

not known whether f precedes g or whether x precedes y and at this point

it does not really matter and the functions and variables will be put

into an arbitrary order. Later the actual order will be dictated by

precedence ordering considerations, similar to those applied to non-

indexed equations and variables. The description of the index orderings