Update Frobenius norms for better accuracy.

[mgates3]

This fixes issue #261.

I added scombssq and dcombssq to merge two sum-of-squares quantities. These functions are taken from ScaLAPACK (TOOLS/pdtreecomb.f), just the documentation is updated to match LAPACK's style.

Updated norms were tested via the LAPACK++ tester, comparing single to double precision on identical matrices, and results from new code with results from old code.

For instance, original results (output abbreviated):

lapackpp/test> ./test lange --norm fro --dim 4000 --verbose 1 --type s
                                            LAPACK++        Ref.
type     norm       m       n      error    time (s)    time (s)  status
norm_tst = 3.0072395019531250e+03
   s      fro    4000    4000   0.00e+00      0.0443      0.0557  pass

lapackpp/test> ./test lange --norm fro --dim 4000 --verbose 1 --type d
norm_tst = 3.0541497549406699e+03
   d      fro    4000    4000   0.00e+00      0.0694      0.0810  pass

Using double precision (type d) as reference, this gives a relative error for single precision (type s) of 1.5360e-02.

With updated code:

lapackpp/test> ./test lange --norm fro --dim 4000 --verbose 1 --type s
                                            LAPACK++        Ref.
type     norm       m       n      error    time (s)    time (s)  status
norm_tst = 3.0541506347656250e+03
   s      fro    4000    4000   0.00e+00      0.0449      0.0524  pass

lapackpp/test> ./test lange --norm fro --dim 4000 --verbose 1 --type d
norm_tst = 3.0541497549404239e+03
   d      fro    4000    4000   0.00e+00      0.0688      0.0808  pass

Again using double-precision as reference, this gives a much better relative error for single precision of 2.8808e-07. Note also that the double-precision result is close to that generated by the old code.

For reference, in LAPACK++ test I ran:

python ./run_tests.py --aux-norm --norm fro --verbose 1 --medium --large --type s > snorm.txt
python ./run_tests.py --aux-norm --norm fro --verbose 1 --medium --large --type d > dnorm.txt

python ./run_tests.py --aux-norm --norm fro --verbose 1 --medium --large --type c > cnorm.txt
python ./run_tests.py --aux-norm --norm fro --verbose 1 --medium --large --type z > znorm.txt

and compared those files. I modified the LAPACK++ matrix_generator.cc to generate exactly the same matrix for single and double precision, e.g., for all i, j:

    A( i, j ) = rand() / float(RAND_MAX);

instead of using larnv.

I did not address RFP matrix norms (dlansf, ...). Nor tridiagonal matrices (dlangt, ...), which due to having fewer entries (only O(n) instead of O(n^2)), are less affected by the accuracy issue.

[codecov]

Codecov Report

Merging #290 into master will decrease coverage by 0.12%.
The diff coverage is 18.66%.

@@            Coverage Diff             @@
##           master     #290      +/-   ##
==========================================
- Coverage   82.59%   82.46%   -0.13%     
==========================================
  Files        1841     1843       +2     
  Lines      177881   178231     +350     
==========================================
+ Hits       146916   146984      +68     
- Misses      30965    31247     +282

Impacted Files	Coverage Δ
SRC/zlanhp.f	62.88% <0%> (-6.44%)	⬇️
SRC/clantb.f	59.86% <0%> (-5.33%)	⬇️
SRC/dlantb.f	59.86% <0%> (-5.33%)	⬇️
SRC/zlantr.f	59.72% <0%> (-5.43%)	⬇️
SRC/clanhp.f	62.88% <0%> (-6.44%)	⬇️
SRC/clantp.f	51.82% <0%> (-4.1%)	⬇️
SRC/slantr.f	59.72% <0%> (-5.43%)	⬇️
SRC/zlantp.f	51.82% <0%> (-4.1%)	⬇️
SRC/slantp.f	51.82% <0%> (-4.1%)	⬇️
SRC/clanhb.f	62.06% <0%> (-7.17%)	⬇️
... and 36 more

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[julielangou]

Cool! Will close #261
We may need some dedicated testing at some point... but seems the unit testing went fine.
Thanks @mgates3