from there being more than one list element for the variable index list.

Theorem 3-2: For a general solution procedure, any index decisions must

be declared as offsets from the function index upper and lower limits.

Proof: Since for a general solution procedure the index range is arbi-

trary, the only function index values which can be guaranteed to occur

are L. and U.. Thus the only variable index values which can be
 7 1
guaranteed to exist are offsets from L. and U.. (Remember that an
 1 1

offset can be equal to zero.) The index decisions must therefore be

chosen as offsets from L. and U.. Theorem 3-2 is proved.
 1 2
 To see the effect of index decisions on a problem recall the

example problem at the beginning of this chapter. For j, the value zero

(actually L. -1) was an index decision. All variables with index j,
 11
equal to zero were decision variables. Similarly, all variables with

index j2 equal to one were decision variables.


3.4 Index Imbedding

 As was seen in Chapter 2, the order of nesting chosen for the

 indices has a profound effect on the appearance of the expanded

 incidence matrix. Also, as in the case of the example problem at the

 beginning of this chapter, it will affect the efficiency of the solution

 procedure.

 First, index ordering (the order of nesting) can affect the

 number of function evaluations necessary to reach convergence. To see

 this, consider the incidence matrices c) and d) in Fig. 3-5, which are

 generated from the Index Display Matrices a) and b) with different index

 orderings. Suppose that these incidence matrices represent a set of

 equations to be solved for the outputs indicated by the circles. To