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FindMinimum
FindMinimum[
f
,
x
,
] searches for a local minimum in f, starting from the point x
=
 .
FindMinimum returns a list of the form 
,
x
->
 , where  is the minimum value of f found, and  is the value of x for which it is found.
FindMinimum[
f
,
x
,
,
] searches for a local minimum in f using  and  as the first two values of x. This form must be used if symbolic derivatives of f cannot be found.
FindMinimum[
f
,
x
,
xstart
,
xmin
,
xmax
] searches for a local minimum, stopping the search if x ever gets outside the range xmin to xmax.
FindMinimum[
f
,
x
,
,
y
,
, ... ] searches for a local minimum in a function of several variables.
FindMinimum has attribute HoldAll.
FindMinimum works by following the path of steepest descent from each point that it reaches. The minima it finds are local, but not necessarily global, ones.
The following options can be given:
The default settings for AccuracyGoal and PrecisionGoal are 10 digits less than WorkingPrecision.
FindMinimum continues until either of the goals specified by AccuracyGoal or PrecisionGoal is achieved.
See the Mathematica book: Section 1.6.5, Section 3.9.8.
See also Implementation NotesA.9.44.23MainBookLinkOldButtonDataA.9.44.23.
See also: ConstrainedMin, LinearProgramming, D, Fit.
Related package: Statistics`NonlinearFit`.
Further Examples
This finds the value of x which minimizes  , starting from x = 2.
In[1]:= 
Out[1]= 
Here is a function with many local minima.
Evaluate the cell to see the graphic.
In[2]:= 
FindMinimum finds the local minimum closest to x = 1. This is not the global minimum for the function.
In[3]:= 
Out[3]= 
This finds the local minimum of a function of two variables. As in FindRoot, it is a good idea to choose starting values that are not too special.
In[4]:= 
Out[4]= 
This finds a minimum, explicitly specifying a gradient to use.
In[5]:= 
Out[5]= 
Using the default setting, a different local minimum is found.
In[6]:= 
Out[6]= 
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