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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
).
Find a minimum point of a univariate function.
fun should be a function handle or name. a, b specify a
starting interval. options is a structure specifying additional
options. Currently, fminbnd
recognizes these options:
"FunValCheck"
, "OutputFcn"
, "TolX"
,
"MaxIter"
, "MaxFunEvals"
. For a description of these
options, see optimset.
On exit, the function returns x, the approximate minimum point and fval, the function value thereof.
info is an exit flag that can have these values:
Notes: The search for a minimum is restricted to be in the interval bound by
a and b. If you only have an initial point to begin searching
from 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.
Solve an unconstrained optimization problem defined by the function fcn.
fcn should accept a vector (array) defining the unknown variables, and
return the objective function value, optionally with gradient.
fminunc
attempts to determine a vector x such that
fcn (x)
is a local minimum.
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 options. Currently,
fminunc
recognizes these options:
"FunValCheck"
, "OutputFcn"
, "TolX"
,
"TolFun"
, "MaxIter"
, "MaxFunEvals"
,
"GradObj"
, "FinDiffType"
, "TypicalX"
,
"AutoScaling"
.
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. 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-7
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:
Converged to a solution point. Relative gradient error is less than
specified by TolFun
.
Last relative step size was less than TolX
.
Last relative change in function value was less than TolFun
.
Iteration limit exceeded—either maximum number of algorithm iterations
MaxIter
or maximum number of function evaluations MaxFunEvals
.
Algorithm terminated by OutputFcn
.
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.
Find a value of x which minimizes the function fun.
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:
"TolX"
, "TolFun"
, "MaxFunEvals"
, "MaxIter"
,
"Display"
, "FunValCheck"
, and "OutputFcn"
.
For a description of these options, see optimset
.
Additional inputs for the function fun can be passed as the fourth
and higher arguments. To pass function arguments while using the default
options values, use []
for options.
On exit, the function returns x, the minimum point, and fval, the function value at the minimum.
The third return value exitflag is
if the algorithm converged
(size of the simplex is smaller than options.TolX
AND
the step in the function value between iterations is smaller than
options.TolFun
).
if the maximum number of iterations or the maximum number of function evaluations are exceeded.
if the iteration is stopped by the "OutputFcn"
.
The fourth return value is a structure output with the fields,
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])
The function humps
is a useful function for testing zero and
extrema finding functions.
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.
Previous: Solvers, Up: Nonlinear Equations [Contents][Index]