Section 3.5: Iterative Solution of Sets of Equations
We have now looked at:
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
- 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.
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.
- 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.
The second situation is the more usual one. There are several possible variants.
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.
- There is set of several nonlinear equations which have to be
- 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
- 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 practice a number of
important chemical engineering problems can be reduced to the
The incidence matrix analysis introduced earlier
enables us to distinguish between the first and second cases,
and is dealt with
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.
Return to Section 3 Index.