Use BLAS.trsm! instead of LAPACK.trtrs! in left-triangular solves

[dkarrasch]

We have used that in right-triangular solves already.

[dkarrasch]

@nanosoldier runbenchmarks("linalg"; vs=":master")

[dkarrasch]

In a manual 1000x1000 test, I see slight to no improvement, but certainly no regression.

[dkarrasch]

It turns out that the LAPACK function is calling the BLAS function internally (https://netlib.org/lapack/explore-html/d4/dc1/group__trtrs_gab0b6a7438a7eb98fe2ab28e6c4d84b21.html), but checks for exact singularity upfront, nothing else. It seems we could rather switch to the "augmented" LAPACK functions throughout instead?

@jishnub @andreasnoack @amontoison

[dkarrasch]

Nevermind, there is no triangular right-solve in LAPACK.

[amontoison]

@dkarrasch I was thinking of LBT if we switch the backend, like BLIS. Sometimes, we only have the BLAS routine available.

[dkarrasch]

Ok, we can of course switch, but then we should perhaps add the exact singularity check?

[amontoison]

Yes, and we can do that for both trsv! and trsm! such that it is consistent.
Using trsv! for one right-hand and trtrs! for multiple ones have a different behaviour if the triangular matrix is singular.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 91.86%. Comparing base (e7da19f) to head (f8850f0).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1194      +/-   ##
==========================================
- Coverage   91.89%   91.86%   -0.03%     
==========================================
  Files          34       34              
  Lines       15360    15371      +11     
==========================================
+ Hits        14115    14121       +6     
- Misses       1245     1250       +5     

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[vtjnash]

This appears to have introduced a bug into the benchmarking tests (I cannot vouch for their quality, only that this commit seems to have introduced a regression in them): JuliaLang/julia@c8ed17f#commitcomment-153581870