Find minimum of singlevariable function on fixed interval
Finds a minimum for a problem specified by
$$\underset{x}{\mathrm{min}}f(x)\text{suchthat}{x}_{1}x{x}_{2}.$$
x, x_{1}, and x_{2} are scalars and f(x) is a function that returns a scalar.
x = fminbnd(fun,x1,x2)
x = fminbnd(fun,x1,x2,options)
x = fminbnd(problem)
[x,fval] = fminbnd(...)
[x,fval,exitflag] = fminbnd(...)
[x,fval,exitflag,output] = fminbnd(...)
fminbnd
finds the minimum of a function
of one variable within a fixed interval.
x = fminbnd(fun,x1,x2)
returns
a value x
that is a local minimizer of the function
that is described in fun
in the interval x1
< x < x2
. fun
is a function handle;
see Function Handles.
Parameterizing Functions in the MATLAB^{®} Mathematics
documentation, explains how to pass additional parameters to your
objective function fun
.
x = fminbnd(fun,x1,x2,options)
minimizes
with the optimization parameters specified in the structure options
.
You can define these parameters using the optimset
function. fminbnd
uses
these options
structure fields:
 Level of display. 
 Check whether objective function values are valid. 
 Maximum number of function evaluations allowed, a positive
integer. The default is 
 Maximum number of iterations allowed, a positive integer.
The default is 
 Userdefined function that is called at each iteration.
The default is none ( 
 Plots various measures of progress while the algorithm
executes, select from predefined plots or write your own. Pass a function
handle or a cell array of function handles. The default is none (
See Plot Functions in MATLAB Mathematics for more information. 
 Termination tolerance on 
x = fminbnd(problem)
finds the minimum
for problem
, where problem
is
a structure with the following fields:
 Objective function 
 Left endpoint 
 Right endpoint 
 'fminbnd' 
 Options structure created using optimset 
[x,fval] = fminbnd(...)
returns
the value of the objective function computed in fun
at x
.
[x,fval,exitflag] = fminbnd(...)
returns
a value exitflag
that describes the exit condition
of fminbnd
:


 Maximum number of function evaluations or iterations was reached. 
 Algorithm was terminated by the output function. 
 Bounds are inconsistent ( 
[x,fval,exitflag,output] = fminbnd(...)
returns
a structure output
that contains information about
the optimization in the following fields:


 Number of function evaluations 
 Number of iterations 
 Exit message 
fun
is the function to be minimized. fun
accepts
a scalar x
and returns a scalar f
,
the objective function evaluated at x
. The function fun
can
be specified as a function handle for a function file
x = fminbnd(@myfun,x1,x2);
where myfun.m
is a function file such as
function f = myfun(x) f = ... % Compute function value at x.
or as a function handle for an anonymous function:
x = fminbnd(@(x) sin(x*x),x1,x2);
Other arguments are described in the syntax descriptions above.
x = fminbnd(@cos,3,4)
computes π to
a few decimal places and gives a message on termination.
[x,fval,exitflag] = ... fminbnd(@cos,3,4,optimset('TolX',1e12,'Display','off'))
computes π to about 12 decimal places,
suppresses output, returns the function value at x
,
and returns an exitflag
of 1.
The argument fun
can also be a function handle
for an anonymous function. For example, to find the minimum of the
function f(x) = x^{3} – 2x – 5 on the interval (0,2)
,
create an anonymous function f
f = @(x)x.^32*x5;
Then invoke fminbnd
with
x = fminbnd(f, 0, 2)
The result is
x = 0.8165
The value of the function at the minimum is
y = f(x) y = 6.0887
If fun
is parameterized, you can use anonymous
functions to capture the problemdependent parameters. For example,
suppose you want to minimize the objective function myfun
defined
by the following function file:
function f = myfun(x,a) f = (x  a)^2;
Note that myfun
has an extra parameter a
,
so you cannot pass it directly to fminbnd
. To optimize
for a specific value of a
, such as a =
1.5
.
Assign the value to a
.
a = 1.5; % define parameter first
Call fminbnd
with
a oneargument anonymous function that captures that value of a
and
calls myfun
with two arguments:
x = fminbnd(@(x) myfun(x,a),0,1)
The function to be minimized must be continuous. fminbnd
may
only give local solutions.
fminbnd
often exhibits slow convergence
when the solution is on a boundary of the interval.
fminbnd
only handles real variables.
[1] Forsythe, G. E., M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations, PrenticeHall, 1976.
[2] Brent, Richard. P., Algorithms for Minimization without Derivatives, PrenticeHall, Englewood Cliffs, New Jersey, 1973
fminsearch
 fzero
 optimset