Reduce occurrences of intrinsic stdlib_matmul test failures

[loiseaujc]

Following the discussion in #1086 about the intrinsic test failure, this PR implements the following changes to the test_matmul (intrinsic module) :

• All matrices used in the test have now been normalized to have unit-norm (1-norm).
• The error check has been replaced from element-wise checks to 1-norm norm of the difference.

As far as I can recall, the error estimates are basically correct, although sharper ones could be used but I haven't been able to find my old lecture notes yet about matrix-matrix multiplications and floating point error analysis. In the mean time, this should (hopefully) reduce the number of times the CI fails on these tests. Something similar could be done with the pseudoinverse tests which do fail quite regularly (the single precision mostly I believe).

ping: @perazz, @jvdp1, @jalvesz

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 68.55%. Comparing base (0ede301) to head (c50bf31).
⚠️ Report is 4 commits behind head on master.

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1110   +/-   ##
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  Coverage   68.55%   68.55%           
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  Files         396      396           
  Lines       12746    12746           
  Branches     1376     1376           
=======================================
  Hits         8738     8738           
  Misses       4008     4008           

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[perazz]

From what I understand, we're switching from a max absolute value to an average column absolute value, looks perfectly reasonable to me. LGTM @loiseaujc

[jalvesz]

This should indeed be more robust, LGTM @loiseaujc

[loiseaujc]

From what I understand, we're switching from a max absolute value to an average column absolute value, looks perfectly reasonable to me. LGTM @loiseaujc

Indeed. Basically, for the product of two matrices, it is relatively to show that

$$ \vert\vert AB - \mathrm{fl}(AB) \vert\vert_1 \leq n \cdot \varepsilon \cdot \vert\vert A \vert\vert_1 \cdot \vert\vert B \vert\vert_1 $$

where $\varepsilon$ is the floating-point unit and $\mathrm{fl}( \cdot )$ denotes the actual floating-point operation. The maximum of the absolute value of the difference is not a matrix norm, and I can't say anything about how it should behave in general.

[loiseaujc]

Awesome. Next PR coming will try to fix the pseudoinverse one.