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Often it is useful to find the minimum value of a function rather than just
the zeroes where it crosses the x-axis. `fminbnd`

is designed for the
simpler, but very common, case of a univariate function where the interval
to search is bounded. For unbounded minimization of a function with
potentially many variables use `fminunc`

or `fminsearch`

. The two
functions use different internal algorithms and some knowledge of the objective
function is required. For functions which can be differentiated,
`fminunc`

is appropriate. For functions with discontinuities, or for
which a gradient search would fail, use `fminsearch`

.
See Optimization, for minimization with the presence of constraint
functions. Note that searches can be made for maxima by simply inverting the
objective function
(`Fto_max = -Fto_min`

).

- :
`x`=**fminbnd***(*¶`fun`,`a`,`b`) - :
`x`=**fminbnd***(*¶`fun`,`a`,`b`,`options`) - :
*[*`x`,`fval`,`info`,`output`] =**fminbnd***(…)*¶ Find a minimum point of a univariate function.

`fun`is a function handle, inline function, or string containing the name of the function to evaluate.The starting interval is specified by

`a`(left boundary) and`b`(right boundary). The endpoints must be finite.`options`is a structure specifying additional parameters which control the algorithm. Currently,`fminbnd`

recognizes these options:`"Display"`

,`"FunValCheck"`

,`"MaxFunEvals"`

,`"MaxIter"`

,`"OutputFcn"`

,`"TolX"`

.`"MaxFunEvals"`

proscribes the maximum number of function evaluations before optimization is halted. The default value is 500. The value must be a positive integer.`"MaxIter"`

proscribes the maximum number of algorithm iterations before optimization is halted. The default value is 500. The value must be a positive integer.`"TolX"`

specifies the termination tolerance for the solution`x`. The default is`1e-4`

.For a description of the other options, see

`optimset`

. To initialize an options structure with default values for`fminbnd`

use`options = optimset ("fminbnd")`

.On exit, the function returns

`x`, the approximate minimum point, and`fval`, the function evaluated`x`.The third output

`info`reports whether the algorithm succeeded and may take one of the following values:- 1 The algorithm converged to a solution.
- 0
Iteration limit (either
`MaxIter`

or`MaxFunEvals`

) exceeded. - -1
The algorithm was terminated by a user
`OutputFcn`

.

Programming Notes: The search for a minimum is restricted to be in the finite interval bound by

`a`and`b`. If you have only one initial point to begin searching from then you will need to use an unconstrained minimization algorithm such as`fminunc`

or`fminsearch`

.`fminbnd`

internally uses a Golden Section search strategy.**See also:**fzero, fminunc, fminsearch, optimset.

- :
**fminunc***(*¶`fcn`,`x0`) - :
**fminunc***(*¶`fcn`,`x0`,`options`) - :
*[*`x`,`fval`,`info`,`output`,`grad`,`hess`] =**fminunc***(*¶`fcn`, …) Solve an unconstrained optimization problem defined by the function

`fcn`.`fminunc`

attempts to determine a vector`x`such that

is a local minimum.`fcn`(`x`)`fun`is a function handle, inline function, or string containing the name of the function to evaluate.`fcn`should accept a vector (array) defining the unknown variables, and return the objective function value, optionally with gradient.`x0`determines a starting guess. The shape of`x0`is preserved in all calls to`fcn`, but otherwise is treated as a column vector.`options`is a structure specifying additional parameters which control the algorithm. Currently,`fminunc`

recognizes these options:`"AutoScaling"`

,`"FinDiffType"`

,`"FunValCheck"`

,`"GradObj"`

,`"MaxFunEvals"`

,`"MaxIter"`

,`"OutputFcn"`

,`"TolFun"`

,`"TolX"`

,`"TypicalX"`

.If

`"AutoScaling"`

is`"on"`

, the variables will be automatically scaled according to the column norms of the (estimated) Jacobian. As a result,`"TolFun"`

becomes scaling-independent. By default, this option is`"off"`

because it may sometimes deliver unexpected (though mathematically correct) results.If

`"GradObj"`

is`"on"`

, it specifies that`fcn`—when called with two output arguments—also returns the Jacobian matrix of partial first derivatives at the requested point.`"MaxFunEvals"`

proscribes the maximum number of function evaluations before optimization is halted. The default value is`100 * number_of_variables`

, i.e.,`100 * length (`

. The value must be a positive integer.`x0`)`"MaxIter"`

proscribes the maximum number of algorithm iterations before optimization is halted. The default value is 400. The value must be a positive integer.`"TolX"`

specifies the termination tolerance for the unknown variables`x`, while`"TolFun"`

is a tolerance for the objective function value`fval`. The default is`1e-6`

for both options.For a description of the other options, see

`optimset`

.On return,

`x`is the location of the minimum and`fval`contains the value of the objective function at`x`.`info`may be one of the following values:- 1
Converged to a solution point. Relative gradient error is less than specified by

`TolFun`

.- 2
Last relative step size was less than

`TolX`

.- 3
Last relative change in function value was less than

`TolFun`

.- 0
Iteration limit exceeded—either maximum number of algorithm iterations

`MaxIter`

or maximum number of function evaluations`MaxFunEvals`

.- -1
Algorithm terminated by

`OutputFcn`

.- -3
The trust region radius became excessively small.

Optionally,

`fminunc`

can return a structure with convergence statistics (`output`), the output gradient (`grad`) at the solution`x`, and approximate Hessian (`hess`) at the solution`x`.Application Notes: If the objective function is a single nonlinear equation of one variable then using

`fminbnd`

is usually a better choice.The algorithm used by

`fminunc`

is a gradient search which depends on the objective function being differentiable. If the function has discontinuities it may be better to use a derivative-free algorithm such as`fminsearch`

.**See also:**fminbnd, fminsearch, optimset.

- :
`x`=**fminsearch***(*¶`fun`,`x0`) - :
`x`=**fminsearch***(*¶`fun`,`x0`,`options`) - :
`x`=**fminsearch***(*¶`problem`) - :
*[*`x`,`fval`,`exitflag`,`output`] =**fminsearch***(…)*¶ -
Find a value of

`x`which minimizes the multi-variable function`fun`.`fun`is a function handle, inline function, or string containing the name of the function to evaluate.The search begins at the point

`x0`and iterates using the Nelder & Mead Simplex algorithm (a derivative-free method). This algorithm is better-suited to functions which have discontinuities or for which a gradient-based search such as`fminunc`

fails.Options for the search are provided in the parameter

`options`using the function`optimset`

. Currently,`fminsearch`

accepts the options:`"Display"`

,`"FunValCheck"`

,`"MaxFunEvals"`

,`"MaxIter"`

,`"OutputFcn"`

,`"TolFun"`

,`"TolX"`

.`"MaxFunEvals"`

proscribes the maximum number of function evaluations before optimization is halted. The default value is`200 * number_of_variables`

, i.e.,`200 * length (`

. The value must be a positive integer.`x0`)`"MaxIter"`

proscribes the maximum number of algorithm iterations before optimization is halted. The default value is`200 * number_of_variables`

, i.e.,`200 * length (`

. The value must be a positive integer.`x0`)For a description of the other options, see

`optimset`

. To initialize an options structure with default values for`fminsearch`

use`options = optimset ("fminsearch")`

.`fminsearch`

may also be called with a single structure argument with the following fields:`objective`

The objective function.

`x0`

The initial point.

`solver`

Must be set to

`"fminsearch"`

.`options`

A structure returned from

`optimset`

or an empty matrix to indicate that defaults should be used.

The field

`options`

is optional. All others are required.On exit, the function returns

`x`, the minimum point, and`fval`, the function value at the minimum.The third output

`exitflag`reports whether the algorithm succeeded and may take one of the following values:- 1
if the algorithm converged (size of the simplex is smaller than

`TolX`

**AND**the step in function value between iterations is smaller than`TolFun`

).- 0
if the maximum number of iterations or the maximum number of function evaluations are exceeded.

- -1
if the iteration is stopped by the

`"OutputFcn"`

.

The fourth output is a structure

`output`containing runtime about the algorithm. Fields in the structure are`funcCount`

containing the number of function calls to`fun`,`iterations`

containing the number of iteration steps,`algorithm`

with the name of the search algorithm (always:`"Nelder-Mead simplex direct search"`

), and`message`

with the exit message.Example:

fminsearch (@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])

Note: If you need to find the minimum of a single variable function it is probably better to use

`fminbnd`

.

The function `humps`

is a useful function for testing zero and
extrema finding functions.

- :
`y`=**humps***(*¶`x`) - :
*[*`x`,`y`] =**humps***(*¶`x`) Evaluate a function with multiple minima, maxima, and zero crossings.

The output

`y`is the evaluation of the rational function:1200*

`x`^4 - 2880*`x`^3 + 2036*`x`^2 - 348*`x`- 88`y`= - --------------------------------------------- 200*`x`^4 - 480*`x`^3 + 406*`x`^2 - 138*`x`+ 17`x`may be a scalar, vector or array. If`x`is omitted, the default range [0:0.05:1] is used.When called with two output arguments, [

`x`,`y`],`x`will contain the input values, and`y`will contain the output from`humps`

.Programming Notes:

`humps`

has two local maxima located near`x`= 0.300 and 0.893, a local minimum near`x`= 0.637, and zeros near`x`= -0.132 and 1.300.`humps`

is a useful function for testing algorithms which find zeros or local minima and maxima.Try

`demo humps`

to see a plot of the`humps`

function.**See also:**fzero, fminbnd, fminunc, fminsearch.

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