feat(symmetric): support SPD GPU

[JieRen98]

This PR

• Symmetric struct
• Symmetric GPU solver
• Reformat
• Symmetric matching support
• Symmetric equilibration support

Future

• Symmetric MAGMA support
• Symmetric distributed support
• Symmetric Low Rank support

[GitHubbeer]

I observed that the matching process in STRUMPACK relies on the mc64ad algorithm (STRUMPACK ./src/sparse/MC64ad.cpp), which unfortunately does not support symmetric matrices. I found an alternative approach for Cholesky decomposition of symmetric matrices by incorporating the mc64_matching from HSL_MC64 ("matrix type indicates if the matrix is symmetric or not". section 2.5 in https://www.hsl.rl.ac.uk/specs/hsl_mc64_C.pdf). So I think we should implement our own symmetric version or use HSL_MC64 directly. I suggest that we can make it a separate pull request.

[GitHubbeer]

@pghysels Could you please check this PR?

[pghysels]

Can you add a few tests?
You can run testSymmetricPositiveDefinite and testMixedPrecisionSymmetricPositiveDefinite.cpp in test/CMakeLists.txt.
You can download a small SPD matrix in download_mtx.sh.

Maybe you can rename files like testSymmetricPositiveDefinite to testSPD.

[pghysels]

Maybe we should add a routine like set_matrix that takes a symmetric matrix, i.e., only taking the lower/upper triangle.

The current CSRMatrix class (and CompressedSparseMatrix) have a symm_sparse argument. But this flag was only meant to denote whether the sparsity pattern is symmetric. It is not about the value symmetry or positive definiteness. So I feel like we should not confuse these things.

[JieRen98]

How did you format codes? @pghysels

[pghysels]

How did you format codes? @pghysels

I don't use any strict formatting guidelines.
We should probably setup auto formatting using clang-format or something like that.

[JieRen98]

How did you format codes? @pghysels

I don't use any strict formatting guidelines. We should probably setup auto formatting using clang-format or something like that.

Sounds great! I opened a new issue (#114) for this. I think we can do this in the next PR for clarity because I usually follow the philosophy that one PR for one feature.

[GitHubbeer]

"test_SPD_mixedPrecison" and "test_SPD_seq" in test/CMakeList and routine "set_lower_triangle_matrix" for taking a symmetric matrix are ready. Please review. I also changed symm_sparse() to is_symmetric(opts_) to check whether the matrix is symmetric.

[pghysels]

This looks good. I will merge and then clean up a bit myself.

There might be licensing issue with the newer MC64 version.