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MathGroup Archive 1995

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Re: Interactive input? and sorting FindRoot list

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg727] Re: Interactive input? and sorting FindRoot list
  • From: rubin at msu.edu (Paul A. Rubin)
  • Date: 11 Apr 1995 21:18:55 GMT
  • Organization: Michigan State University

In article <3m4ogd$e47 at news0.cybernetics.net>,
   REECE_D <REECE+_D%A1%Electromagnetic_Sciences at mcimail.com> wrote:
->
->   Two questions:
->   
->   1)  Is there a way to have a module execute up to a point, ask the 
user 
->for input and then continue?

Yes.  Input["prompt"] does it.

->   2)  How does one take a list of roots returned from
->   Table[FindRoot[eq, i], {i,x0,x1,di}] and return only the unique roots? 
->
->The problem I am encountering is that FindRoot will return two or more 
->versions of the same root so Union sees them as distinct while in reality 
->they are approximations to the same root.  I've tried N[] and 
->SetPrecision[] with no luck.
->   

You can specify your own comparison function for the option SameTest in 
Union.  For example:

In[1]:=  s = Table[ FindRoot[ x^3 - 2x == 0, {x, i} ], {i, -3, 3} ]
Out[1]=  {{x -> -1.41421}, {x -> -1.41421}, {x -> -1.41421},
          {x -> 0.}, {x -> 1.41421}, {x -> 1.41421},  {x -> 1.41421}}
(*note repeated roots*)
In[2]:=  stest[ {Rule[ x_, n_ ]}, {Rule[ y_, m_ ]} ] :=  x === y && n == m
In[3]:=  Union[ s, SameTest -> stest ]
Out[3]=  {{x -> -1.41421}, {x -> 0.}, {x -> 1.41421}} (*repetitions gone*)

->   I am trying to write a module which will calculate a characteristic 
->equation, plot it, prompt the user for approximate distance between roots 
->and then return the first n roots.  I could eliminate the plot and prompt 
->if I knew of a robust method to calculate the first n roots of the 
->characteristic equation.

I take it from context that (a) you are only interested in real roots and 
(b) Solve[] isn't doing the job for you.  If you have a priori knowledge of 
a finite interval containing all roots, you could eliminate the plot/prompt 
cycle by doing a grid search of the interval until you find either a sign 
change (indicating a root of odd multiplicity) or a function value "near" 
zero (indicating a root, possibly of even multiplicity), then applying 
FindRoot to that interval to nail a root, then dividing x - root out of the 
characteristic polynomial (reducing the degree by one), and repeating until 
either the degree gets down to 4 (Solve does quartics exactly, I think) or 
you fail to find a sign change before hitting some minimal mesh size.  
(BTW, in the mesh search, I would suggest progressively bisecting all 
subintervals rather than doing a left-to-right search on the final mesh; I 
suspect successive bisection would be quicker, although I wouldn't swear to 
it.)  This is not fool-proof:  if two roots are very close together, or 
worse yet you have a repeated root with even multiplicity, you could miss 
them (it?), and a function value "near" 0 does not guarantee a root nearby.

Paul

**************************************************************************
* Paul A. Rubin                                  Phone: (517) 432-3509   *
* Department of Management                       Fax:   (517) 432-1111   *
* Eli Broad Graduate School of Management        Net:   RUBIN at MSU.EDU    *
* Michigan State University                                              *
* East Lansing, MI  48824-1122  (USA)                                    *
**************************************************************************
Mathematicians are like Frenchmen:  whenever you say something to them,
they translate it into their own language, and at once it is something
entirely different.                                    J. W. v. GOETHE


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