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Re: FindRoot / Jacobian

• To: mathgroup at smc.vnet.net
• Subject: [mg84340] Re: FindRoot / Jacobian
• From: sigmundv at gmail.com
• Date: Wed, 19 Dec 2007 23:57:30 -0500 (EST)
• References: <fk73ub$6el$1@smc.vnet.net> <fk87t2$4f6$1@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...
>
> > 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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