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Re: FindRoot / Jacobian

  • To: mathgroup at smc.vnet.net
  • Subject: [mg84389] Re: FindRoot / Jacobian
  • From: "Steve Luttrell" <steve at _removemefirst_luttrell.org.uk>
  • Date: Fri, 21 Dec 2007 03:18:17 -0500 (EST)
  • References: <fk73ub$6el$1@smc.vnet.net> <fk87t2$4f6$1@smc.vnet.net> <fkcst0$1fv$1@smc.vnet.net>

I don't think the problem is to do with precision issues. If you look at the 
plots

Plot3D[solU, {p, 0, 2 \[Pi]}, {q, 0, 2 \[Pi]}]
Plot3D[solV, {p, 0, 2 \[Pi]}, {q, 0, 2 \[Pi]}]

it is clear that there is a continuum of solutions from which

FindRoot[solU == N[u] && solV == N[v], {{p, 0}, {q, 0}}]

picks a single solution, whilst warning you that it is having difficulty 
finding a unique solution by outputting the following message:

"The line search decreased the step size to within tolerance specified by 
AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease 
in the merit function."

I seem to have I overlooked this warning message earlier.

Stephen Luttrell
West Malvern, UK

<sigmundv at gmail.com> wrote in message news:fkcst0$1fv$1 at smc.vnet.net...
> Thank you for the suggestion, Steve. It sure helped me clean up the
> code and get rid of the complex values (which was expected since I
> used complex starting points in FindRoot before).
>
> Do you know how I can increase the precision in FindRoot, without
> increasing the computing time dramatically? If you plot p/.solutions
> and q/.solutions, respectively, using e.g. the ListSurfacePlot3D
> command, you see some peaks. Ideally those peaks should not be there.
> Do you have any clever suggestion on how to get rid of those peaks?
>
> Sigmund Vestergaard
> Kongens Lyngby, Denmark
>
> On Dec 18, 11:37 am, "Steve Luttrell"
> <steve at _removemefirst_luttrell.org.uk> wrote:
>> Here is how you can solve your problem:
>>
>> solU=(E^((-I)*p-(2*ArcTan[Tan[q/2]/Sqrt[3]])/Sqrt[3])*(1+E^((2*I)*p)))/2
>> solV=Sin[p]/E^((2*ArcTan[Tan[q/2]/Sqrt[3]])/Sqrt[3])
>>
>> solutions=Table[FindRoot[solU==N[u]&&solV==N[v],{{p,0},{q,0}}],{u,-\[Pi],\[Pi],(2\[Pi])/70},{v,-\[Pi],\[Pi],(2\[Pi])/70}];
>>
>> You can do a quick visualisation of the solutions with the following
>> graphics:
>>
>> Graphics[{Map[Point,#,{2}],Map[Line,#],Map[Line,Transpose[#]]}&[{p,q}/.solutions//Chop]]
>>
>> Stephen Luttrell
>> West Malvern, UK
>>
>> <sigmu... at gmail.com> wrote in messagenews:fk73ub$6el$1 at smc.vnet.net...
>> > Dear readers,
>>
>> > Consider the following expressions:
>>
>> > solU = (E^((-I)*p - (2*ArcTan[Tan[q/2]/Sqrt[3]])/Sqrt[3])*(1 +
>> > E^((2*I)*p)))/2
>>
>> > solV = Sin[p]/E^((2*ArcTan[Tan[q/2]/Sqrt[3]])/Sqrt[3])
>>
>> > I then want to solve the system of equations (u = solU, v = solV)
>> > numerically on a grid (u,v) \in (-pi,pi). For that purpose I use
>> > FindRoot, and execute the following:
>>
>> > u1 = Pi; du = 2*Pi/70;
>>
>> > uvPts1 = Flatten[Table[{u, v}, {u, u1 + du, u2 - du, du}, {v, v1 +
>> > du,
>> >    v2 - du, dv}], 1];
>>
>> > uvNew = {p, q} /. Table[FindRoot[{solU == uvPts1[[i, 1]], solV ==
>> > uvPts1[[i, 2]]}, {p,
>> >    -2}, {q, 1+I}], {i, Length[uvPts1]}];
>>
>> > This gives me a p-value and a q-value for each point on the grid, no
>> > problem. However, I get the wrong solution at some of the grid points.
>> > Since I get several of the following type of messages, this would be
>> > expected:
>>
>> > FindRoot::jsing:Encountered a singular Jacobian at the point {p,q} = \
>> > {-10.4516+0.784848 \[ImaginaryI],108.092+104.013 \[ImaginaryI]}. Try \
>> > perturbing the initial point(s). >>
>>
>> > Then I would like to know if there is any 'nice' way to handle these
>> > 'singular Jacobian' errors? Does there exist any kind of 'event
>> > handler', as in NDSolve?
>>
>> > Kind regards,
>> > Sigmund Vestergaard
>
> 



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