Somehow, the next failure is

sol1: {-0.600937,0.0132837 -1.45082 I,0.0132837 +1.45082 I,1.04832}
sol2: {-0.600937,0.0132837 -1.45082 I,0.0132837 +1.45082 I,1.04832}

lets get this in the long form (accents ommited in courtesy of the WC (* Wolfram Community *) Editor)

In[184]:= {-0.6009373760877655, 0.013283715568729561 - 1.4508231140635948 I, 
            0.013283715568729561 + 1.4508231140635948 I, 1.0483249164991404} == 
          {-0.6009373760877655, 0.01328371556872926 - 1.450823114063595 I, 
            0.01328371556872926 + 1.450823114063595 I, 1.0483249164991404}
Out[184]= False

it's false in a strict sense, but: has been computed with precision 16 and is wrong beginning with the 15-th digit only.

So it seems that it could be made correct to any meaningful precision and I tend to believe Daniel and Frank.

I think you need to do the calculation with infinite precision inputs and not numericize (sic) the results.

Ups, sorry. Here's the fix; the problem remains.

MCNumber = 1000; Tally[
 Monitor[Table[Atemp = RandomReal[] - 0.5; 
   Btemp = RandomReal[] - 0.5;
   Ctemp = RandomReal[] - 0.5; Dtemp = RandomReal[] - 0.5;
   Etemp = RandomReal[] - 0.5;
   Sort[Table[
      N[Reduce[
         Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0 /. {Aa -> Atemp, 
           Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, Ee -> Etemp}, 
         x]][[i, 2]], {i, 1, 4}]] == 
    Sort[Table[
      N[Reduce[Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0, 
          x] /. {Aa -> Atemp, Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, 
          Ee -> Etemp}][[i, 2]], {i, 1, 4}]], {k, 1, MCNumber}], 
  ProgressIndicator[k, {0, MCNumber}]]]

results in:

{{True, 996}, {False, 4}}

I'll fix that in the code above not to confuse readers. Thanks a lot for your corrections.

It does not change the general result though.

Cheers,

Marco

I have just checked Frank's observation. It is quite true that Reduce seems to do much better (I ignored some error messages), but it is not always right either.

MCNumber = 10000; Tally[Monitor[Table[
   Atemp = RandomReal[] - 0.5; Btemp = RandomReal[] - 0.5; 
   Ctemp = RandomReal[] - 0.5; Dtemp = RandomReal[] - 0.5; 
   Etemp = RandomReal[] - 0.5; 
   Sort[Table[
      N[Reduce[
         Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0 /. {Aa -> Atemp, 
           Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, Ee -> Etemp}, 
         x]][[i, 2]], {i, 1, 
       Length[N[
         Reduce[Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 
            0 /. {Aa -> Atemp, Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, 
            Ee -> Temp}, x]]]}]] == 
    Sort[Table[
      N[Reduce[Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0, 
          x] /. {Aa -> Atemp, Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, 
          Ee -> Etemp}][[i, 2]], {i, 1, 
       Length[N[
         Reduce[Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0, 
           x] /. {Aa -> Atemp, Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, 
           Ee -> Etemp}]]}]], {k, 1, MCNumber}], 
  ProgressIndicator[k, {0, MCNumber}]]]

gave

 {{True, 9981}, {False, 19}}

Another thing is that Solve is performing about the same as Roots:

MCNumber = 1000; Tally[Monitor[Table[
   Atemp = RandomReal[] - 0.5; Btemp = RandomReal[] - 0.5; 
   Ctemp = RandomReal[] - 0.5; Dtemp = RandomReal[] - 0.5; 
   Etemp = RandomReal[] - 0.5; 
   Sort[Table[
      x /. N[Solve[
          Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0 /. {Aa -> Atemp, 
            Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, Ee -> Etemp}, 
          x]][[i]], {i, 1, 
       Length[N[
         Solve[Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 
            0 /. {Aa -> Atemp, Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, 
            Ee -> Temp}, x]]]}]
     ] == 
    Sort[Table[
      x /. N[Solve[Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0, 
           x] /. {Aa -> Atemp, Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, 
           Ee -> Etemp}][[i]], {i, 1, 
       Length[N[
         Solve[Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0, 
           x] /. {Aa -> Atemp, Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, 
           Ee -> Etemp}]]}]

     ], {k, 1, MCNumber}], ProgressIndicator[k, {0, MCNumber}]]]

gives

{{True, 621}, {False, 379}}

So 38% wrong results. Also, even if Reduce is better, it is curious that Solve and Roots have problems in more than one third of the cases without giving any warning.

Cheers,

Marco

PS: In fact I think that the program is very inefficient. I solve the equation once more for the upper limit of the Table command which is not really required. A 4 there does the trick.

MCNumber = 10000; Tally[Monitor[Table[
   Atemp = RandomReal[] - 0.5; Btemp = RandomReal[] - 0.5; 
   Ctemp = RandomReal[] - 0.5; Dtemp = RandomReal[] - 0.5; 
   Etemp = RandomReal[] - 0.5; 
   Sort[Table[
      N[Reduce[
         Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0 /. {Aa -> Atemp, 
           Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, Ee -> Etemp}, 
         x]][[i, 2]], {i, 1, 4}]] == 
    Sort[Table[
      N[Reduce[Aa x^4 + Bb x^3 + Cc x^2 + Dd x + Ee == 0, 
          x] /. {Aa -> Atemp, Bb -> Btemp, Cc -> Ctemp, Dd -> Dtemp, 
          Ee -> Etemp}][[i, 2]], {i, 1, 4}]], {k, 1, MCNumber}], 
  ProgressIndicator[k, {0, MCNumber}]]]

Update, I have checked the

Quartics-> False 

option and as you say it does work for Solve and Roots- (not quite for Reduce). It does circumvent the idea that Yuriy had originally, but it does seem to work.

Thanks, Marco

Dear Yuriy,

regarding Quartics, you might want to have a look at :

http://reference.wolfram.com/language/ref/Quartics.html

It is about how the roots are represented, implicitly or explicitly. The documentation page explains that quite nicely.

Cheers,

Marco

I am surprised at that behaviour as well. I have filed a report.

Cheers, Marco

Marco, thanks for your contribution. Basically that is exactly what I am surprised at. If the algebraic solution (symbolic) is claimed to be general, then it should work no matter what the sequence of substituting of coefficients is, if I am not mistaken.