New (experimental) determinant function

[JohnAAbbott]

This is the determinant code I have been working on for the last several weeks/months.
It is fragile in that it requires that the input matrix have non-zero determinant -- it won't work if the determinant is zero (I'm not sure exactly what it will do in that case).

[JohnAAbbott]

For reasons that I do not understand the OSCAR det function is significantly faster on random matrices whose entries are uniformly (& symmetrically?) chosen.
The algorithm is "new" in that it has not been published anywhere. At least on some of the matrices which arose during a collaboration with Linda Hoyer the new algorithm is faster than OSCAR's det and also faster than Pauderis-Storjohann's HCOL algorithm. These matrices have many non-trivial Smith invariant factors.

[codecov]

Codecov Report

❌ Patch coverage is 0% with 199 lines in your changes missing coverage. Please review.
✅ Project coverage is 87.78%. Comparing base (e23ad1d) to head (8815d60).
⚠️ Report is 3 commits behind head on master.

Files with missing lines	Patch %	Lines
src/ZZMatrix-linalg.jl	0.00%	199 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2212      +/-   ##
==========================================
- Coverage   88.25%   87.78%   -0.48%     
==========================================
  Files          99       99              
  Lines       37123    37322     +199     
==========================================
- Hits        32764    32763       -1     
- Misses       4359     4559     +200     

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[thofma]

The is the decision tree for the "Oscar det" method in flint:

    if (dim < 5)
        fmpz_mat_det_cofactor(det, A);
    else if (dim < 25)
        fmpz_mat_det_bareiss(det, A);
    else if (dim < 60)
        fmpz_mat_det_modular(det, A, 1);
    else
    {
        slong bits = fmpz_mat_max_bits(A);

        if (dim < FLINT_ABS(bits))
            fmpz_mat_det_modular(det, A, 1);
        else
            fmpz_mat_det_modular_accelerated(det, A, 1);
    }

A random matrix will have few non-trivial Smith divisors, but I think the runtime of modular algorithms should be Smith invariant factor-agnostic.