**Bug introduced after 5.0, in or before 8.0 and persisting through 12.0.**

----------
I notice in the following example that wrong smallest 2 eigenvalues are resulted if calculating from a sparse matrix. But it gives correct result if we   

 - calculate the smallest 3,4,... eigenvalues from the sparse matrix
 - calculate any smallest eigenvalues from the corresponding normal
   matrix

I found many cases with this behavior. Why and any remedy?

In the code below, the 104×104 matrix in the minimal example I found is imported from [Pastebin](https://pastebin.com/PpDfY3EQ) because it exceeds the length limit of a post.

    Import["https://pastebin.com/raw/PpDfY3EQ", "Package"];
    mysparsemat = mymat;
    mymat = Normal[mysparsemat];
    m = 2;
    Reverse@First[Eigensystem[mymat, -m]]
    Reverse@First[Eigensystem[mysparsemat, -m]]
    m = 4;
    Reverse@First[Eigensystem[mymat, -m]]
    Reverse@First[Eigensystem[mysparsemat, -m]]

The result of the above code is 

    {-0.712477 + 1.02863*10^-16 I, 0.712477 + 1.46577*10^-16 I}

    {0.712477 - 1.44294*10^-11 I, 0.712656 - 2.12258*10^-11 I}
    
    {-0.712477 + 1.02863*10^-16 I, 0.712477 + 1.46577*10^-16 I, 
    -0.712656 - 1.05578*10^-16 I, 0.712656 + 6.49144*10^-16 I}
    
    {-0.712477 - 5.10777*10^-10 I, 0.712477 - 5.44863*10^-12 I, 
     0.712656 + 3.10198*10^-11 I, -0.712656 + 3.64677*10^-10 I}
So the resultant wrong 2nd smallest eigenvalue is actually the correct 3rd or 4th smallest eigenvalue. (The eigenvalues should be real and doubly degenerate in absolute value as expected from the original problem's nature). 

**Update**  
As you might have noticed in the comments below, a first guess accusing the Arnoldi algorithm is irrelevant. (And *Matlab* gives the correct result.)

---

A bug report has been filed for this in the Wolfram community.