Re: FindRoots?

*To*: mathgroup at smc.vnet.net*Subject*: [mg112136] Re: FindRoots?*From*: Andrzej Kozlowski <akozlowski at gmail.com>*Date*: Wed, 1 Sep 2010 06:28:41 -0400 (EDT)

On 31 Aug 2010, at 10:17, David Park wrote: > True, Sin[1/x] will eventually break any root finding algorithm because we > will always run out of time or space to store the answers if we push too > close to zero. > > However, this does provide a good example to show how well Ted's algorithm > works. In this case the exact roots are 1/(n Pi) where n is a positive > interger that runs between limits that span the region we are searching. > > So, let's try some cases numerically. To search for roots between 0.01 and > 0.1 n runs from 4 to 21 and there are 28 roots. Here are the results from > RootSearch. > > Needs["Ersek`RootSearch`"] > > RootSearch[Sin[1/x] == 0, {x, 0.01, 0.1}] > Length[%] > > {{x -> 0.0102681}, {x -> 0.0106103}, {x -> 0.0109762}, {x -> > 0.0113682}, {x -> 0.0117893}, {x -> 0.0122427}, {x -> > 0.0127324}, {x -> 0.0132629}, {x -> 0.0138396}, {x -> > 0.0144686}, {x -> 0.0151576}, {x -> 0.0159155}, {x -> > 0.0167532}, {x -> 0.0176839}, {x -> 0.0187241}, {x -> > 0.0198944}, {x -> 0.0212207}, {x -> 0.0227364}, {x -> > 0.0244854}, {x -> 0.0265258}, {x -> 0.0289373}, {x -> > 0.031831}, {x -> 0.0353678}, {x -> 0.0397887}, {x -> > 0.0454728}, {x -> 0.0530516}, {x -> 0.063662}, {x -> 0.0795775}} > > 28 > > For the domain from 0.001 to 0.01 n runs from 32 to 318 and there are 287 > roots. > > RootSearch[Sin[1/x] == 0, {x, 0.001, 0.01}, InitialPrecision :> 40, > InitialSamples -> 5000, MaxBrentSteps -> 1000, > MaxSecantSteps -> 1000, PrecisionGoal :> 10]; > Length[%] > > 287 > > For the domain from 0.0001 to 0.001 n runs from 319 to 3183 and there are > 2865 roots. RootSearch took about 20 minutes on this. (I'm not certain if I > picked optimum values for the parameters to get the best efficiency.) > > results3 = > RootSearch[Sin[1/x] == 0, {x, 0.0001, 0.001}, > InitialPrecision :> 40, InitialSamples -> 30000, > MaxBrentSteps -> 1000, MaxSecantSteps -> 1000, PrecisionGoal :> 10]; > Length[%] > > 2865 > > I would consider this pretty good performance. > Perhaps. But note that: sols = N[Reduce[Sin[1/x] == 0 && 10^-4 < x < 10^-3, x], 10]; // Timing {2.88873,Null} Length[List @@ sols] 2865 So what do you need RootSearch for? Andrzej Kozlowski