An issue regarding Eigenvalues function [closed]

Question

Can anybody please kindly explain to me why the members of the output vector produced by the following example are not equal to zero?

mm = 1000;
Clear[i, j, ss, EIG];
ss = ConstantArray[0, {mm, mm}];
%%% "Generating a random sparse matrix (ss)";
For[i = 1, i < (mm) + 1, i++,
  For[j = 1, j < (mm) + 1, j++,
    If[i == j,
     ss[[i,j]] = (RandomReal[{-Mod[{i + j}, 2][[1]],Mod[{i + j}, 3][[1]]}, 1])[[1]];
     ];
    If[Abs[i - j] == 1,
     ss[[i,j]] = (RandomReal[{-Mod[{i + j}, 2][[1]],Mod[{i + j}, 3][[1]]}, 1])[[1]]+0.2;
     ];
    If[Abs[i - j] == 2,
     ss[[i,j]] = (RandomReal[{-Mod[{i + j}, 2][[1]],Mod[{i + j}, 3][[1]]}, 1])[[1]]-0.1;
     ];
    If[Abs[i - j] == 3,
     ss[[i,j]] = (RandomReal[{-Mod[{i + j}, 2][[1]],Mod[{i + j}, 2][[1]]}, 1])[[1]]+0.1;
     ];
    ];
  ];

EIG = Sort[(Select[Cases[(Eigenvalues[ss, -10]), _Real], # > 0 &]),Less];
EIG - Sort[(Select[Cases[(Eigenvalues[ss, -10]), _Real], # > 0 &]),Less]

For instance, one run of this example leaded to this vector:

{8.41341*10^-16, -2.57433*10^-15, -1.9082*10^-16, -8.32667*10^-17}

I am wondering why the output vector members are not zero! You can test this example on your PC. THNAKS.

Answer 1

[Too long for a comment...]

If I understand correctly, the claim is this. Taking Eigenvalues[...,-10] multiple times, on the same matrix, gives results that differ by an ULP or so. If this is indeed the claim, then (1) the post does a really good jjob of disguising it, and (2) this probably can happen for a couple of reasons.

One is if the algorithm uses randomized starting values for any iteration. Since I do not see this behavior on Linux machines I'll bet against that. The other has to do with certain subtleties of the OS as to when 64 bit vs 80 bit register arithmetic is used. Some of the remarks here might help to explain this.

--- edit ---

By coincidence I was looking through an old bug report. The gist was that in some Intel MKL library code one can get two different numeric eigen-results that are typically close together. It does seem to depend on byte boundary alignment, which I believe affects whether or how 80-bit floats registers get used.

--- end edit ---