handle and document corner cases of lwork in lapack

[dklyuchinskiy]

Handle and document corner cases of lwork in lapack

This PR contains fixes related to the following issues:

1. GEQRF: document corner case of LWORK #257 - parameter LWORK can't be zero on any case, info checker should know about this
2. GEQRF: document corner case of LWORK #257, x{OR,UN}M{QR,RQ,LQ,QL} optimal workspace size is always > 4096 #546 - parameter LWORK should be small as possible for corner cases M = 0 or N = 0. Final LWORK value should be aligned with quick exit and info checker.
3. Some typos and code style fixes, changes in documentation.

Actually, for many functions in LAPACK these issues are already solved or did not exist at all. This PR handles remaining ones for general, symmetric and generalized problems (double precision only at the moment). Please review. If there are some questions, let's discuss!

• The documentation has been updated.
• If the PR solves a specific issue, it is set to be closed on merge.

[codecov]

Codecov Report

Attention: 1205 lines in your changes are missing coverage. Please review.

Comparison is base (3200980) 0.00% compared to head (280e161) 0.00%.
Report is 2 commits behind head on master.

Files	Patch %	Lines
SRC/clatsqr.f	0.00%	25 Missing ⚠️
SRC/dlatsqr.f	0.00%	25 Missing ⚠️
SRC/slatsqr.f	0.00%	25 Missing ⚠️
SRC/zlatsqr.f	0.00%	25 Missing ⚠️
SRC/dlaswlq.f	0.00%	24 Missing ⚠️
SRC/slaswlq.f	0.00%	24 Missing ⚠️
SRC/zlaswlq.f	0.00%	24 Missing ⚠️
SRC/claswlq.f	0.00%	21 Missing ⚠️
SRC/clamswlq.f	0.00%	20 Missing ⚠️
SRC/clamtsqr.f	0.00%	20 Missing ⚠️
... and 150 more

Additional details and impacted files

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##           master     #942    +/-   ##
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  Coverage    0.00%    0.00%            
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  Files        1930     1930            
  Lines      190055   190413   +358     
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- Misses     190055   190413   +358     

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[dklyuchinskiy]

Extension to all precisions for this scope of functions is in progress.

[dklyuchinskiy]

PR is ready! @langou please review one more time.

I think, several changes should be pointed out in addition to already mentioned issues.

• 1. sytrd_2stage/sytrd_sb2st

• deleted debug code

• removed return of parameter LHOUS into the first element of array HOUS. This array is used for storage of Householder vectors, and cant be overwritten at the end. This issue will be critical, when case VECT=V is implemented.

• 2. sgesvj/dgesvj

• added missed LQUERY mode of the function, as it is already done in cgesvj/zgesvj.

• 3. chesv_aa_2stage, chesv_aa, cggev3 and etc.

• minimal value of parameter LWKOPT should be equal to LWKMIN (if such exist) to guarantee pass of error checker for corresponding parameter LWORK in normal run

• 4. fix of tests for eigensolver evr/evr_2stage and linear solver trf_aa_2stage in order to follow change of corner case of LWORK inside related functions

• 5. added missed usage of functions sroundup_lwork into single and single complex precisions

• 6. fixed output format of function chkher, since it printed only 6 characters of function's name

One can see, that important functions of generation/application of orthogonal matrices are not taken into account in this PR. I believe, this work should be implemented during fix of #546.

Also, thanks to @kleineLi for the help with complex precisions and intermediate review.