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MathGroup Archive 2003

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RE[2]: ugly Plot Eigenvalues ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg39731] RE[2]: [mg39670] ugly Plot Eigenvalues ?
  • From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
  • Date: Mon, 3 Mar 2003 23:50:52 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com


>-----Original Message-----
>From: Wolf, Hartmut 
To: mathgroup at smc.vnet.net
>Sent: Monday, March 03, 2003 10:03 AM
>To: 'Pavel Pokorny'; mathgroup at smc.vnet.net
>Subject: [mg39731] RE: [mg39670] ugly Plot Eigenvalues ?
>
>
>
>>-----Original Message-----
>>From: Pavel Pokorny [mailto:Pavel.Pokorny at vscht.cz]
To: mathgroup at smc.vnet.net
>>Sent: Friday, February 28, 2003 10:46 AM
>>To: mathgroup at smc.vnet.net
>>Subject: [mg39731] [mg39670] ugly Plot Eigenvalues ?
>>
>>
>>    Dear Mathematica friends,
>>
>>How can I plot Re Eigenvalues as a function of a parameter
>>without unwanted ugly connections between branches?
>>In the example (computed by Mathematica 4.2 for HP-UX PA-RISC)
>>
>>   m1={{8., 8, 7}, {0, 5, 5}, {8, 0, 0}}
>>   m2={{6., 0, 6}, {10, 8, 3}, {5, 8, 6}}
>>   Plot[Evaluate[Re[Eigenvalues[m1+w*m2]]], {w,-3,3}];
>>
>>for w approximately -0.4 there is an unwanted vertical bar.
>>This is related to "ordering" of solutions to equation.
>>
>>Is there a simple nice solution for this and related examples?
>>(taking exact numbers in the example above avoids the unwanted 
>>connection,
>>but this is not a good solution for a general machine 
>>precission problem)
>>
>>-- 
>>Pavel Pokorny
>>Math Dept, Prague Institute of Chemical Technology
>>http://staffold.vscht.cz/mat/Pavel.Pokorny
>>
>
>A suggestion:
>
>ev = Re[Eigenvalues[m1 + w*m2]];
>
>evs := Sort[ev]
>
>Plot[{evs[[1]], evs[[2]], evs[[3]]}, {w, -3, 3}]
>
>Of course a problem of reference still remains.
>
>
>--
>Hartmut Wolf
>

As I said, a problem of reference. The idea is to seperate out regions of w
were the eigenvalues (here their real parts) are smooth functions of w, and
then combine to globally smooth solutions. In this case the ordering of the
real part itself cannot be used, since we have a degenerate case for most w
for that, and numerical instabilities creap in. But the the imaginary part
does better:


Clear[ev, evs]

ev = Eigenvalues[m1 + w*m2];   (* this is better usable *)

Block[{f = Composition[Ordering, Chop, Im]}, 
  Plot[{f[ev][[1]], f[ev][[2]], f[ev][[3]]}, {w, -3, 3}, 
    PlotStyle -> {Hue[0], Hue[1/3], Hue[2/3]}]]


with that, we find

Block[{f = Composition[Ordering, Chop, Im], g = Re}, 
  Plot[{g[ev[[Min[f[ev][[1]] + f[ev][[2]], 4] - 1]]], 
      g[ev[[Min[f[ev][[1]], f[ev][[2]]]]]], g[ev[[f[ev][[3]]]]]}, {w, -3,
3}, 
    PlotStyle -> {Hue[0], Hue[1/3], Hue[2/3]}, AspectRatio -> 1]]

giving a smooth numbering (or coloring) of the eigenvalues.


This of course is pure (or poor) handwork; a challenge would be to
automatically find the continuous paths when crossing the branch cuts of the
functions involved as w varies.

--
Hartmut Wolf





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