times the number of functions in each i, block, over all such blocks.

 Now suppose that the order of the indices is ili2. The solution

procedure would converge the first variable in the first i, block

simultaneously with the first variable in each ii block. This is

equivalent to converging all variables in the first i2 block. The

number of function evaluations would be

 N2 = kiM1M2 3-6

since all functions must be evaluated. Equation 3-6 represents only the

number of function evaluations required to converge the first variable

in each iI block. The total number of function evaluations required

would be
 M2
 N = MM, I k. 3-7
 T i 2
 i2=1

N is greater than N1 by a factor of M2. Since the IDM for i2 does not

fully precedence order, M2 must be greater than one. The theorem is

proved.

 This theorem indicates the desirability of nesting indices whose

IDM's fully precedence order outside those whose IDM's do not. A

problem arises when a function index is to be nested outside the index

for function type.

 Consider the incidence matrices in Fig. 3-6 a) and b). Any of the

four partitions in the incidence matrix a) themselves have legal output

set assignments. In the incidence matrix b), however, the (1,0) and

(2,1) blocks do not have legal output set assignments. This condition

is caused by ii being nested outside i The function g contains only

variable index j2, and hence no variables with index value equal to

zero. The result is that if the index output is to be assigned inde-

pendent of function type it must be chosen from a special IDM. This IDM