Page 155 -

Minimum of a Function of Two Variables
To find the local minimum of a multivariable function, we use the MATLAB
fmins function. Finding the maximum can be handled by the same tech-
nique as outlined for the one variable case.

Example 5.4
2
2
Find the position of the minimum of the surface f(x, y) = x + y .
Solution:
1. First, make a function file and save it as fname.m.
function f=fname(array)

x=array(1); % x is stored in first element of array
y=array(2); % y is stored in second element of
%array
f=x.^2+y.^2; % function stored in f

2. Graph the contour plot for the surface; and from it, estimate the
coordinates of the minimum:

arrayguess=[.1 .1];
The arrayguess holds the initial guess for both coordinates at
the minimum. That is,
arrayguess=[xguess yguess];

3. The coordinates of the minimum are then obtained by entering the
following commands in the command window:

arraymin=fmins('fname',arrayguess)
fmin=feval('fname',arraymin)

Homework Problem

Pb. 5.24 In this problem we propose to apply the above optimization tech-
niques to the important problem of the optical narrow band transmission fil-
ter. This filter, in very wide use in optics, consists of two parallel semi-
reflective surfaces (i.e., mirrors) with reflection coatings R and R and sepa-
2
1
rated by a distance L. Assuming that the material between the mirrors has an
index of refraction n and that the incoming beam of light has frequency ω and
is making an angle θ with the normal to the semi-reflective surfaces, then the
i
ratio of the transmitted light intensity to the incident intensity is

© 2001 by CRC Press LLC