MATLAB Function Reference

fminbnd - Find minimum of single-variable function on fixed interval

Syntax

x = fminbnd(fun,x1,x2) x = fminbnd(fun,x1,x2,options) [x,fval] = fminbnd(...) [x,fval,exitflag] = fminbnd(...) [x,fval,exitflag,output] = fminbnd(...)

Description

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 in the MATLAB^® Programming documentation for more information.

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:

`Display`	Level of display. `'off'` displays no output; `'iter'` displays output at each iteration; `'final'` displays just the final output; `'notify'` (default) displays output only if the function does not converge.
`FunValCheck`	Check whether objective function values are valid. `'on'` displays an error when the objective function returns a value that is `complex` or `NaN`. `'off'` displays no error.
`MaxFunEvals`	Maximum number of function evaluations allowed.
`MaxIter`	Maximum number of iterations allowed.
`OutputFcn`	User-defined function that is called at each iteration. See Output Function in the Optimization Toolbox for more information.
`PlotFcns`	User-defined plot function that is called at each iteration. See Plot Functions in the Optimization Toolbox for more information.
`TolX`	Termination tolerance on `x`.

[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:

`1`	`fminbnd` converged to a solution `x` based on `options.TolX.`
`0`	Maximum number of function evaluations or iterations was reached.
`-1`	Algorithm was terminated by the output function.
`-2`	Bounds are inconsistent (`x1 > x2`).

[x,fval,exitflag,output] = fminbnd(...) returns a structure output that contains information about the optimization:

`output.algorithm`	Algorithm used
`output.funcCount`	Number of function evaluations
`output.iterations`	Number of iterations
`output.message`	Exit message

Arguments

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 an M-file function

x = fminbnd(@myfun,x1,x2);

where myfun.m is an M-file function 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.

Examples

x = fminbnd(@cos,3,4) computes to a few decimal places and gives a message on termination.

[x,fval,exitflag] = ... 
   fminbnd(@cos,3,4,optimset('TolX',1e-12,'Display','off'))

computes 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 on the interval (0,2), create an anonymous function f

f = @(x)x.^3-2*x-5;

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 problem-dependent parameters. For example, suppose you want to minimize the objective function myfun defined by the following M-file function.

function f = myfun(x,a)
f = (x - a)^2;

Note that myfun has an extra parameter a, so you cannot pass it directly to fminbind. 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 one-argument anonymous function that captures that value of a and calls myfun with two arguments:
```
x = fminbnd(@(x) myfun(x,a),0,1)
```

Algorithm

fminbnd is an M-file. The algorithm is based on golden section search and parabolic interpolation. Unless the left endpoint x₁ is very close to the right endpoint x₂, fminbnd never evaluates fun at the endpoints, so fun need only be defined for x in the interval x₁ < x < x₂. If the minimum actually occurs at x₁ or x₂, fminbnd returns an interior point at a distance of no more than 2*TolX from x₁ or x₂, where TolX is the termination tolerance. See [1] or [2] for details about the algorithm.

Limitations

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.

References

[1] Forsythe, G. E., M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, 1976.

[2] Brent, Richard. P., Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, New Jersey, 1973

flow

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