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How are the complex roots ordered in the Root[] function ?

Bob Ager
Posted 10 years ago

I looked at the documentation on the Root[] function, but could not find the definition on how the complex roots are ordered. Say I have a polynomial with complex coefficients and its root are: 3+i, 2-i and -3-i. Which one of these is 1, 2, 3 as returned by the Root[] function ?
POSTED BY: Bob Ager
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Does the following sort of experiment suggest an answer?

    p = -5 + 2 z + 3 z^2 - 9 z^3 - z^4 + 20 z^5 - 9 z^6;
    soln = Solve[p == 0, z]
(* output is list of rules with Root objects *)
    N/@soln
{{z -> 0.953232}, {z -> 1.9405}, 
  {z -> -0.668983 - 0.331246 I}, {z -> -0.668983 + 0.331246 I}, {z -> 0.333231 - 0.654154 I}, {z -> 0.333231 + 0.654154 I}}
POSTED BY: Murray Eisenberg

Can you provide a better example? For the cubic 

     p = (z - (3 + I)) (z - (2 - I)) (z - (-3 - I))

Mathematica gives:

    Solve[p == 0, z]
{{z -> -3 - I}, {z -> 2 - I}, {z -> 3 + I}}

so no Root objects at all.
POSTED BY: Murray Eisenberg
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