# Section 3.5: Iterative Solution of Sets of Equations

We have now looked at:
• how to reorder and rearrange sets of equations which can be solved directly and explicitly, and
• how to solve single equations which cannot be rearranged for explicit solution by an iterative procedure.
Many problems in practice do not fall into the above two categories; indeed there are very few problems which are described by a single equation. There are two other factors which may prevent us from using the two simple techniques.
• We may be able to reorder the set of equations for direct solution, i.e. we can obtain an incidence table which is lower triangular, but we cannot rearrange one or more of the individual equations to make a formula explicitly giving the appropriate variable on the left hand side.
• A reordering of the equations into lower triangular form is not possible.
In fact we have already shown how to deal with the first situation. The individual equation or equations involve only a single unknown, all others having been calculated, and so may be solved using an appropriate iterative method as single equations. This thus presents no new problem.

The second situation is the more usual one. There are several possible variants.

1. There is set of several nonlinear equations which have to be solved simultaneously.
2. The problem involves a set of nonlinear equations but it can be reduced to iteration in one variable, i.e. not all the equations need to be solved simultaneously.
3. There is a set of equations which do have to be solved simultaneously, but they are all linear in the unknowns for which they must be solved.
In the first, most general situation, it is most convenient to use on of a range of packaged procedures. We will not discuss these here.

In practice a number of important chemical engineering problems can be reduced to the second case. The incidence matrix analysis introduced earlier enables us to distinguish between the first and second cases, and is dealt with in section 3.5.1, systems reducible to iteration in a single unknown.

In the third case, because the equations are are linear, it is possible to obtain what is in effect an analytical solution. This is covered in section 3.6, systematic methods for sets of linear equations.

Next - Section 3.6: Systematic Methods for Sets of Linear Equations.