Fix accuracy issues in the Givens rotations

[weslleyspereira]

Solves #629

Description

@sergey-v-kuznetsov highlighted in #629 that the new Givens rotations operations may have lower accuracy than the ones that were in LAPACK up to release 3.9. In fact, I noticed that | c^2 + s^2 - 1 | (which should be 0) is 3x larger with 3.10 than with 3.9, and so this means that the norm of the 2x2 matrix is 3x further away from 1 with 3.10 than with 3.9. This answers the results @sergey-v-kuznetsov obtained with his test.

I kept overall the 3.10 code and changed a few lines to recover the behavior LAPACK 3.9 had. Main changes:

• g2 = ABSSQ( g ); d = sqrt( g2 ) changed to d = abs( g )
• sqrt( one + ( g2/f2 ) ) instead of sqrt( f2 * h2 ) or sqrt( f2 )*sqrt( h2 )

Here is my modified version of @sergey-v-kuznetsov's test: test.zip

Checklist

• The documentation has been updated.
• Improve the tests related to the precision of | c^2 + s^2 - 1 |

[codecov]

Codecov Report

Merging #630 (fa7350f) into master (79aa0f2) will not change coverage.
The diff coverage is 0.00%.

❗ Current head fa7350f differs from pull request most recent head 72e100c. Consider uploading reports for the commit 72e100c to get more accurate results

@@           Coverage Diff           @@
##           master     #630   +/-   ##
=======================================
  Coverage    0.00%    0.00%           
=======================================
  Files        1894     1894           
  Lines      184021   184011   -10     
=======================================
+ Misses     184021   184011   -10     

Impacted Files	Coverage Δ
SRC/clarrv.f	0.00% <0.00%> (ø)
SRC/clartg.f90	0.00% <0.00%> (ø)
SRC/dlarrv.f	0.00% <0.00%> (ø)
SRC/slarrv.f	0.00% <0.00%> (ø)
SRC/zlarrv.f	0.00% <0.00%> (ø)
SRC/zlartg.f90	0.00% <0.00%> (ø)

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[h-vetinari]

Any updates on this? I believe this is one of the issues blocking MKL conformance with lapack 3.10 (which I care about with my conda-forge hat on).

[weslleyspereira]

Any updates on this? I believe this is one of the issues blocking MKL conformance with lapack 3.10 (which I care about with my conda-forge hat on).

Hi @h-vetinari. We made many improvements, and there is a recent new version of the algorithms on #631. I am preparing a short report with what's new and how the algorithms were improved. The algorithms on #631 are ready to review.

[weslleyspereira]

I am closing this PR. I am addressing #629 on a new PR #631.