Re: FindRoot repeatedly evaluating function

• To: mathgroup at smc.vnet.net
• Subject: [mg121107] Re: FindRoot repeatedly evaluating function
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Sun, 28 Aug 2011 04:06:23 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201108251105.HAA24905@smc.vnet.net>

```Storing the result that way is called "memoization" or sometimes "dynamic
programming". (I don't know why it's not called "memorization", since
that's exactly what it is.)

If it makes things faster, do it. If it doesn't, don't.

I don't know why it would call ONLY twice near the same value if you do
that but three times if you don't; possibly because two of the values are
the same to machine precision. That may lead to a less-exact estimate of
the derivative.

Bobby

On Fri, 26 Aug 2011 04:00:14 -0500, Simon Pearce
<Simon.Pearce at nottingham.ac.uk> wrote:

> Hi Bobby,
>
> Thanks for your quick response. Why then if I store the result using
> f[a_]:=f[a] does it only do it twice? The first two results are being
> done at the same value (at least to machine precision I assume), with a
> third being done very slightly away.
>
> My actual problem is not amenable to symbolic solving.
>
> Thanks,
> Simon
>
> -----Original Message-----
> From: DrMajorBob [mailto:btreat1 at austin.rr.com]
> Sent: 25 August 2011 18:10
> To: mathgroup at smc.vnet.net; Simon Pearce
> Subject: Re: FindRoot repeatedly evaluating function
>
> FindRoot makes several evaluations near each guess to estimate the
> derivative numerically, and uses that to compute another guess. There's
> nothing odd or surprising about it.
>
> To avoid all that (where possible), use symbolic solvers instead:
>
> Solve[x[10] == 0 /.
>    First@DSolve[{x''[S] - x'[S] + x[S] == 0, x[0] == 1, x'[0] == a}, x,
>       S], a]
>
> {{a -> (Csc[5 Sqrt[3]] (-3 Cos[5 Sqrt[3]] + Sqrt[3] Sin[5 Sqrt[3]]))/(
>     2 Sqrt[3])}}
>
> Bobby
>
> On Thu, 25 Aug 2011 06:05:19 -0500, Simon Pearce
> <Simon.Pearce at nottingham.ac.uk> wrote:
>
>> Hi Mathgroup,
>>
>> When I use FindRoot[f[y],y] I find that the inner function f is
>> evaluated 3 or 4 times at each value of y (or at least very similar
>> values), even if y is far from the root. This has obvious implications
>> to the speed of my code.
>> Can anyone explain why this is the case, and tell me any way to stop it
>> from repeatedly evaluating f? If I use f[a]:=f[a]=... then it uses the
>> stored result, but I don't want to store thousands of such real valued
>> expressions.
>>
>> The following simple code shows the essence of the problem, using Print
>> to show where the function is evaluated and its value there.
>>
>> f[a_?NumericQ]:=Module[{sol},
>>   sol=NDSolve[{x''[S]-x'[S]+x[S]==0,x[0]==1,x'[0]==a},x,{S,0,10}][[1]];
>>   Print[{a,x[10]/.sol}]; x[10]/.sol ]
>> FindRoot[f[y],{y,6}]
>>
>> Thanks,
>> Simon Pearce
>> Postdoctoral Researcher
>> The Centre for Plant Integrative Biology
>> School of Biosciences
>> University of Nottingham
>>
>>
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>
>

--
DrMajorBob at yahoo.com

```

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