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MathGroup Archive 2007

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FindRoot can NOT handle mixed real and complex variables

  • To: mathgroup at smc.vnet.net
  • Subject: [mg79899] FindRoot can NOT handle mixed real and complex variables
  • From: AES <siegman at stanford.edu>
  • Date: Wed, 8 Aug 2007 04:53:08 -0400 (EDT)
  • Organization: Stanford University

I'm (re)posting this as an assertion, not a question, hoping to rouse a 
little more interest, since it appears to be a significant weakness in 
FindRoot, and a previous post, rather unusually, brought no satisfactory 
resolution;

The problem is to find the roots of two complex equations

   u * BesselJ[1, u] * BesselK[0, w] == w * BesselK[1, w] * BesselJ[0, u]

   u^2 + w^2 == g

with constraints

   Re[g] == <an input value, DN>    

   Re[w] ==  0     

So that's two complex (or four real) equations; four real numbers in the 
desired output; and at least one solution exists in general for any 
choice of DN and can be found using other methods -- but there appears 
to be NO WAY (no straightforward way, anyway) to find it using FindRoot, 
-- or even to get FindRoot to tackle the basic problem.

Right????????

------

[For testing purposes, a sample starting point close to but not exactly 
one particular solution, would be  DN = -200,  g0 = DN + 0.8 I = -200 + 
0.8 I,  u0 = 2.39 + 0.17 I,  w0 = 0 + 14.34 I  ]


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