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MathGroup Archive 1996

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Re: FindRoot termination criteria

  • To: mathgroup at smc.vnet.net
  • Subject: [mg5561] Re: FindRoot termination criteria
  • From: Robert Knapp <rknapp at sover.net>
  • Date: Thu, 19 Dec 1996 01:02:47 -0500
  • Organization: Wolfram Research
  • Sender: owner-wri-mathgroup at wolfram.com

Sysabel wrote:
> 
> Could someone tell me what criteria is used in FindRoot to terminate
> the search?
> I'm running FindMinimum on
> a function that does not have local minima yet it terminates.  I've
> tried upping the MaxIterations and the AccuracyGoal but the results
> don't seem to improve.
> 
> For example :
> numerical_solution = FindRoot[f[x]==0, {x, {0,1}},
> AccuracyGoal->0.001];
> 
> If exact solution is name exact_solution,
> 
> is Findroot exit the search when :
>         abs(numerical_solution - exact_solution) < 0.001 ???
> or when :
>         abs[f[numerical_solution ] - f[exact_solution]] < 0.001 ???
> 
> or others ?
> 
> Thanks in advance,
> Eric

It is hard to tell from your message whether you are wanting to use 

FindRoot -- for numreically finding a solution to an equation..or
FindMinimum -- for numerically minimizing a function.

The criteria for both are based on the objective function.

For FindRoot, at the exact solution, the objective function is 0, so
the stopping criterion is that 

Abs[f[numerical_solution]] <= 10^(-AccuracyGoal)

For FindMinimum, estimates are used so that

Abs[f[numerical_solution]-f[exact_solution]] <= 10^(-AccuracyGoal)

and (if f[exact_solution] is non zero)

Abs[(f[numerical_solution]-f[exact_solution])/f[exact_solution]] <=
                                                10^(-PrecisionGoal)

FindRoot does not have a PrecisionGoal option because relative error at
zero makes no sense.

Rob Knapp
WRI


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