Nonlinear
Simultaneous Equations
We’re
going to develop a Matlab function to solve systems of nonlinear simultaneous equations.

We’ll
use the ‘fminsearch’
function to find the intersection of the given
curves or functions with several variables. The method ends when a solution is
found, when a maximum
number of iterations is reached, or when some
specifications of tolerances
are met, according to the default options for
fminsearch.
For
each system of
nonlinear equations and starting

points, we
need to be aware of the solution found, the number of iterations
needed, the
number of function evaluations needed, and the exit flag value. This
helps us
know whether the found solution is good or bad.
It’s important to note
that this minimization
or optimization
function (fminsearch) doesn’t solve simultaneous equations. We have to adapt
the system or formulate
the problem in such a way that we can accomplish this.
In this particular case, we’re going to express the nonlinear system as
a scalar
value, and it’s going to be reduced to its minimum value by fminsearch.
That
value will be the solution to our problem.
Naturally, different
tolerances or starting points deliver
different results. Only one intersection is found for every starting
point,
even though the nonlinear system may have more than one
solution.
If we have this nonlinear
system of simultaneous equations:
We can
express the system in Matlab as a function delivering a scalar:
function y =
nls1(x)
y1 = 2 *
x;
y2 = 4*
x^2 + x;
y3 =
exp(2 * x)  1;
e1 =
abs(y1  y2);
e2 =
abs(y2  y3);
e3 =
abs(y1  y3);
y =
max([e1 e2 e3]);
The
important
part is how we define our goal to be minimized. We want
the
differences of the functions to be as close to zero as possible. If the
diferences
of the functions (or errors between them) are not very close to zero,
then we
don’t have a good solution. We expect y to be zero at the
end (where the curves
intersect). 'fminsearch' will find the value of x that minimizes
the maximum error
between each pair of curves (that's called a minimax approach).
Of course, we can formulate or express the system
in different ways, this is only one idea. This function is called the
‘objective function’ (how we are going to measure the result), in
numerical optimization terms.
Finally,
we can minimize this type of systems with an optimization instruction.
We pass
two parameters: the name of the nonlinear system and the starting point
(we’re
using the default options, which can be modified with ‘optimset’).
fx = 'nls1';
x0 = 5
[x_sol, y_sol, EF, out] = fminsearch(fx, x0)
The
answer is:
x_sol =
0
y_sol
= 0
EF =
1
out =
iterations: 20
funcCount: 40
algorithm: 'NelderMead simplex direct
search'
message: [1x196 char]
This answer means that the intersection of the curves occurs at x_sol = 0, and y_sol = 0.
Now, we
can try a different starting point, to compare solutions:
x0
= 10
[x_sol, y_sol, EF, out] = fminsearch(fx, x0)
The new
answer is:
x_sol
=
0
y_sol
= 0
EF =
1
out =
iterations: 22
funcCount: 44
algorithm: 'NelderMead simplex direct
search'
message: [1x196 char]
Note
that the solution in both cases is the same, but the number of function
evaluations to get to the solution is not the same.
If the curves are very
complex, with many ups and downs, the algorithm
might not get a good answer. You should try then several starting
points.
Another
example. Let’s solve this system:
Our
‘objective function’
is defined:
function y =
nls3(x)
y1 = (x
 4)^2 + 2;
y2 = (x
 3)^2 + 5;
y =
norm(y1  y2, 2);
The function is minimizing the
absolute error between y1
and y2.
Each run finds only one answer. We must try
different starting points to find different intersections. We run
the script like this:
fx
= 'nls3';
x0 = 20
[x_sol,
f, EF, out] = fminsearch(fx, x0)
x0 = 15
[x_sol,
f, EF, out] = fminsearch(fx, x0)
The
two solutions are:
x_sol
=
2.3820
f = 3.5793e005
EF = 1
out =
iterations:
23
funcCount: 47
x_sol =
4.6180
f =
1.0069e004
EF = 1
out =
iterations: 21
funcCount: 43
We find
both solutions, x1
= 2.382 when we start at x0
= 20, and x2
= 4.618 when we
start at x0
= 15.
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