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Re: problems with NMinimize

On 4/22/10 at 6:43 AM, ngb at (Neil Broderick) wrote:

>Hi, I am trying to use NMinimize to find the solutions to various
>numerical equations and I keep getting error messages concerning
>non-numerical values. For example consider the following:

>In[2]:= NMinimize[Abs[NIntegrate[Sin[x], {x, -a, b}]]^2, {a, b}]

<error messages snipped>

Even if this did work as you expected with no error messages,
this would be a very inefficient way to solve the problem.
NMinimize works by repeatedly evaluating the expression to be
minimized. That means the numerical integration problem is
repeatedly solved. In this case, there is a symbolic result for
the integral. So doing:

In[4]:= int = Integrate[Sin[x], {x, -a, b}]

Out[4]= cos(a)-cos(b)

And this clearly is zero for Abs[a]==Abs[b]. So, any pair of
numbers satisfying this last will be a minimum for the square of
the absolute value for the integral.

But if you didn't want to find the minimum by inspection, then doing

In[5]:= NMinimize[Abs[int], {a, b}]

Out[5]= {4.80727*10^-14,{a->0.0961182,b->0.0961182}}

returns one of many possible solutions. Note, I didn't bother
with squaring the absolute value since minimizing the absolute
value is the same as minimizing the square of the absolute value.

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