Section 3.3: Iterative Solution of Single Algebraic Equations
Introduction
To solve a single equation analytically it must be possible to rewrite it
as a formula with its unknown on the left hand side.
As already noted, this is in general possible only if either:
- The equation is linear in the unknown, i.e. the unknown appears only raised to the first
power, and does not appear as the argument of a function such as log, sin or exp.
- The equation is nonlinear in the unknown, but the unknown appears only once. (This also formally requires that any nonlinear function which appears
has an inverse, but this is true for all normal functions.)
A very few other special equations, such quadratics and cubics, have
analytical solutions. These do not, however, generally appear in
physical systems and so they are not treated here.
In practice there are also very few interesting problem in chemical engineering
which can conveniently be turned into just one equation.
However, we will later see how the methods used for a single equation
can often be extended to more realistic problems described by quite large
numbers of equations by analysing the structure of the set of
equations. This uses essentially the same technique we have just used to
solve certain sets of equations `directly' by rearrangement.
Rather than looking for trivial chemical engineering problems with
single equations we shall deal with entirely abstract examples.
Consider our single equation to involve a single unknown x,
and may be written as:
f( x ) = 0
Here f implies any functional form which can be expressed
using any of the algebraic operators with which we are already familiar.
We will also assume that our equations have a single solution at one
value of x. In principle we know that this is not the
case, e.g. a quadratic has two solutions, a cubic has three, etc. In
practice, however, it is unusual for an engineering problem to have
multiple meaningful solutions. There may be
additional solutions which are mathematically valid, but they usually
have values which are physically unacceptable, e.g. at mole fractions
greater than one, or negative Kelvin temperatures. (Note: exceptions
do occur, but they are outside the scope of this course.)
Since the methods
used to find the solution normally involve searching within predefined
bounds, these physically unrealistic solutions may often be excluded.
Methods
The notes below describe three methods used to solve single algebraic equations:
Single Equation Solvers
Most spreadsheet systems now provide a solve or `goal seek'
facility which can be programmed to adjust a cell, containing
the value of the unknown (e.g. x) until another cell
containing a formula involving the unknown (e.g. f( x ))
is reduced to zero. The methods used in different spreadsheets
vary in both power and sophistication. Try some of the examples below.
A WWW based solver using bisection and written in JavaScript
is available
in the modelling lab. It is worth using this on a few examples to watch the progress towards
the solution.
Example Equations
- x - 3 = 0
A linear equation is actually a special type of algebraic equation.
- x^{2} - x -12 = 0
There are two solutions, one positive and one negative. Run
the solver with different initial bounds (e.g. 0 to 12 and -12 to 0)
and find both solutions separately. What happens if a set of bounds
encompassing both solutions is given?
- x + log (x) = 0
The solution lies between 0 and 1.0 but log(0) of course cannot
be calculated.
- x² - e^{-x} = 0 ;
solution is between -1 and +1
- 1 - exp(-x) - exp(-2x) = 0
;
solution is between -1 and +1
- 1 - (p_{1} + p_{2}) = 0
where p_{1} = exp(11(1-300/x))
and p_{2} = exp(11(1-250/x))
Solution is between 100 and 400.
Next - Section 3.4: Iterative Solution
of Several Equations using Single Equation Methods
Return to Section 3 Index