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Re: How accurate is the solution for high degree algebraic equation?

• To: mathgroup at smc.vnet.net
• Subject: [mg128499] Re: How accurate is the solution for high degree algebraic equation?
• From: Sseziwa Mukasa <mukasa at gmail.com>
• Date: Thu, 25 Oct 2012 01:43:14 -0400 (EDT)
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For a polynomial of such large degree it's unlikely to can use machine precision.  Mathematica can solve this in infinite precision quickly:

(Debug) In[12]:= d = 54; f = (-z - 1)^d - (-z^d - 1);
sol = Solve[f == 0, z];
a = z /. sol;
(Debug) In[17]:= FullSimplify[f /. z -> a[[1]]]
(Debug) Out[17]= 0

You can see numerical approximations of the solutions using N:

(Debug) In[20]:= N[a[[1]]]
N[a[[1]], 30]
(Debug) Out[20]= -0.5 - 17.1839 I
(Debug) Out[21]= -0.5000000000000000000000000000 - 17.1838854436050918792404513804 I

On Oct 24, 2012, at 3:32 AM, Alexandra wrote:

> I wanted to know all the solutions of f = (-z - 1)^d - (-z^d - 1)==0, where  d=54.
> I did the following:
>
> d = 54; f = (-z - 1)^d - (-z^d - 1);
> sol = NSolve[f == 0,z];
> a = z /. sol;
>
> So    a    is a set of solutions.
>
> If I  compute 
> f /. z -> a[[50]] // N
> It returns a number very close to zero.  This is natural. 
>
> But if I compute 
> f /. (z -> a[[1]]) // N
>
> Then
> Mathematica returns 
> 12.0047 + 14.7528 I
>
> I cannot say a[[1]] is a solution of f=0.
>
> Many other elements in the solution set    a    does not seem to satisfy the equation.
> Only the last few terms in   a   are satisfactory enough as solutions. 
>
> Is the degree too high?
>
>
>
>
>
>
>

• References:
  • How accurate is the solution for high degree algebraic equation?
    • From: Alexandra <watanabe.junzo@gmail.com>