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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.

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:

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. See Iterative Display in MATLAB Mathematics for more information.

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 Functions in MATLAB Mathematics for more information.

PlotFcns

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 ([]).

  • @optimplotx plots the current point

  • @optimplotfval plots the function value

  • @optimplotfunccount plots the function count

See Plot Functions in MATLAB Mathematics 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 in the following fields:

algorithm

Algorithm used

funcCount

Number of function evaluations

iterations

Number of iterations

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 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.

Examples

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',1e-12,'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) = x3 – 2x – 5 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 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 fminbind. To optimize for a specific value of a, such as a = 1.5.

  1. Assign the value to a.

    a = 1.5; % define parameter first
  2. 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)

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.

More About

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Algorithms

fminbnd is a function file. Its 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 a point x in the interior of the interval (x1,x2) that is close to the minimizer. In this case, the distance of x from the minimizer is no more than 2*(TolX + 3*abs(x)*sqrt(eps)). See [1] or [2] for details about the algorithm.

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

See Also

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