opportunity to choose a solution procedure much simpler than would be

expected from an examination of the decomposed problem. Consider the

decomposed problem in Fig. 3-8. At first glance it would appear that in

order to solve for either x or y in f, the variable type not chosen

as the output must be torn. This is not the case. By assigning y to f

and x to g, and by making the index output assignments ii and i2+1, the

incidence matrix in Fig. 3-9 results. For the outputs shown, the first

nine columns represent decision variables. The first block of variables

of output type y can be calculated without tearing any variables of type

x. In fact, the entire set of equations can be solved without tearing

any variables of type x. This phenomenon is called decouplingg."

Decoupling is made possible by a judicious choice of index outputs.

This problem is said to have decoupled in the variable x. Decoupling

of a function type f in its output variable type x occurs when the

function ordering would appear to require that the variable x be torn to

solve functions not of type f, but the variable type x actually does not

need to be torn for that reason.

Theorem 3-5: A variable type x will decouple a function type f, for

which it is the output, if and only if the following conditions exist:

1) A variable index for x must have index output offsets strictly

greater than (or less than) the possible index output offsets for all

other incidences of that variable in any other function type. 2) The

index in 1) must be nested outside the index for function type.

Proof: The proof will be presented for the "greater than" case. The

proof for the "less than" case is analogous.

 If part The proof will be by induction. Assume 1) and 2) hold

 for some set of indexed equations. Let the index in question be called