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MathGroup Archive 2006

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Re: Eliminate Complex Roots

  • To: mathgroup at smc.vnet.net
  • Subject: [mg64361] Re: [mg64348] Eliminate Complex Roots
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Tue, 14 Feb 2006 01:31:36 -0500 (EST)
  • Reply-to: hanlonr at cox.net
  • Sender: owner-wri-mathgroup at wolfram.com

Reduce[a^3+10*a^2-15*a+b==0,a,Reals]

(b < (10/27)*(-335 - 29*Sqrt[145]) && a == Root[#1^3 + 10*#1^2 - 15*#1 
+ b & , 1]) || 
  (b == (10/27)*(-335 - 29*Sqrt[145]) && 
   (a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 1] || 
    a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 3])) || 
  ((10/27)*(-335 - 29*Sqrt[145]) < b < (10/27)*(-335 + 29*Sqrt[145]) && 
   (a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 1] || 
    a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 2] || 
    a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 3])) || 
  (b == (10/27)*(-335 + 29*Sqrt[145]) && 
   (a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 1] || 
    a == Root[#1^3 + 10*#1^2 - 15*#1 + b & , 2])) || 
  (b > (10/27)*(-335 + 29*Sqrt[145]) && a == Root[#1^3 + 10*#1^2 - 15*#1 
+ b & , 1])

Reduce[{a^3+10*a^2-15*a+b\[Equal]0,b==(10/27)*(-335-29*Sqrt[145])},a]

b == (10/27)*(-335 - 29*Sqrt[145]) && (a == (1/3)*(-10 - Sqrt[145]) || 
   a == (2/3)*(-5 + Sqrt[145]))

%//N

b == -253.4097195499913 && (a == -7.347198192930765 || a == 
4.6943963858615305)

Plot[Root[#1^3+10*#1^2-15*#1+b&,1],{b,-500,25}];


Bob Hanlon

> 
> From: bghiggins at ucdavis.edu
To: mathgroup at smc.vnet.net
> Subject: [mg64361] [mg64348] Eliminate Complex Roots
> 
> Hi All,
> 
> Suppose I have the following equation and I want to plot the real roots
> using Plot as a function of the parameter b
> 
> a^3 + 10*a^2 - 15*a + b == 0
> 
> For the problem I am interested I have a set of algebraic equations of
> at least order 3, but this is simpler enough to explain what the issue
> is.
> 
> Now if I want just the Real root I can do the following:
> 
> 
> Sol2[b_] := DeleteCases[Solve[a^3 + 10.*a^2 - 15*a + b ==
>      0, a], {___, x_ -> Complex[y_, z_], ___}]
> 
> and evaluating
> 
> Sol2[20]
> 
> {{a -> -11.46104034060055}}
> 
> Gives the desired result. So far so good. Now suppose I want to plot
> the solutions asa function of b. My immediate thought was to try
> 
> Plot[Evaluate[a /. Sol2[b]], {b, -10, 30}, PlotStyle -> {Red, Blue,
> Magenta}]
> 
> This does give the desired result, but gives error messages when it
> tries to plot a complex quantity. That is the DeleteCases does not seem
> to work within Plot, even though I wrapped everything with evaluate.
> 
> Any suggestions?
> 
> Thanks much
> 
> Brian
> 
> 


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