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MathGroup Archive 2006

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Re: FindRoot

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70469] Re: FindRoot
  • From: dimmechan at yahoo.com
  • Date: Tue, 17 Oct 2006 02:58:43 -0400 (EDT)
  • References: <20061014184518.29029.qmail@web30201.mail.mud.yahoo.com> <egvaks$rk5$1@smc.vnet.net>

Thanks to Chriss and Andrzej for their responses.

Andrzej Kozlowski wrote:
> O.K, it seems I missed your point, which I now think concerned the
> difference between StepMonitor and EvaluationMonitor (whereas I
> originally thought you were concerned with the difference between
> using Sow-Reap and Print).
>   In general EvaluationMonitor will produce more values than
> StepMonitor, because it records each time the specified numerical
> function is evaluated. Sometimes to make one step in an iterative
> procedure you need to evaluate the function several times at
> different points (the standard example is when you need to compute a
> numerical derivative using finite differences) and all these
> computations will be part of one step (so will not be recorded by
> StepMonitor) but will be recored by EvaluationMonitor.
>
> Andrzej
>
>
> On 15 Oct 2006, at 03:45, dimitris anagnostou wrote:
>
> > Dear Andrzej,
> >
> > I really appreciate your response.
> >
> > By " what exactly the following command gives?" I mean
> > from mathematical point of view what the command
> >
> > Reap[FindRoot[Cos[x] == x, {x, 5}, EvaluationMonitor :> Sow[x]]]
> > gives.
> >
> > For example this gives a list of the steps taken
> >
> > Reap[FindRoot[Cos[x] == x, {x, 5}, StepMonitor :> Sow[x]]]
> >
> >
> > Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> > (tm) Pro*
> >
> > On 14 Oct 2006, at 16:06, dimmechan at yahoo.com wrote:
> >
> > > Hello.
> > >
> > > Let me deal with something elementary but yet confusing for me.
> > >
> > > Consider the equation cos(x)=x.
> > >
> > > FindRoot[Cos[x] == x, {x, 5}]
> > > {x -> 0.7390851332151607}
> > >
> > > This prints the value of x every time a step is taken.
> > >
> > > FindRoot[Cos[x] == x, {x, 5}, StepMonitor :> Print[x]]
> > >
> > > This gives a list of the steps taken.
> > >
> > > Reap[FindRoot[Cos[x] == x, {x, 5}, StepMonitor :> Sow[x]]]
> > > {{x -> 0.7390851332151607}, {{-1., -0.02837830412204312,
> > > 1.0296174587519653, 0.7525886779802748, 0.7391248287000711,
> > > 0.7390851335630761, 0.7390851332151607}}}
> > >
> > > This counts the steps.
> > >
> > > Block[{st = 0}, {FindRoot[Cos[x] == x, {x, 5}, StepMonitor :> st++],
> > > st}]
> > > {{x -> 0.7390851332151607}, 7}
> > >
> > > I want to know what exactly the following command gives.
> > >
> > > Reap[FindRoot[Cos[x] == x, {x, 5}, EvaluationMonitor :> Sow[x]]]
> > > {{x -> 0.7390851332151607}, {{5., -54.99999999999999, -1.,
> > > 8.716216958779569,
> > > -0.02837830412204312, 1.0296174587519653, 0.7525886779802748,
> > > 0.7391248287000711, 0.7390851335630761, 0.7390851332151607}}}
> > >
> > > In versions earlier than 5.0 you could use the following command
> > >
> > > FindRoot[Print[x]; Cos[x] == x, {x, 5}]
> > >
> > > Why it cannot be used now?
> > >
> > > Thanks in advance for any help.
> > >
> >
> >
> > If you want you can still do it in the old way:
> >
> > First evaluate:
> >
> > Developer`SetSystemOptions["EvaluateNumericalFunctionArgument" ->
> > False];
> >
> > and then
> >
> > FindRoot[(Print[x]; Cos[x]) == x, {x, 5}]
> > will work.
> > But note the parentheses! It won't work without them.
> > However, the approach using Sow and Reap is vastly more useful since
> > you can't manipulate data returned by Print statements (well, at
> > least not without changing the value of $Output and some extra
> > programming). I am not sure what kind of answer you expected to your
> > question " what exactly the following command gives". It gives
> > exactly the same list of values (of successive approximations to the
> > root of the equation tried by FindRoot) as your Print method gives,
> > but in a much more convenient form. Does this answer your question?
> >
> > Andrzej Kozlowski
> >
> >
> >
> > All-new Yahoo! Mail - Fire up a more powerful email and get things
> > done faster.


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