Fix LAPACK.posv! tests

[dkarrasch]

The test failures are, unsurprisingly, related to high condition numbers, as can be seen from running the tests many times in a loop and displaying the condition number. We could set an appropriate relative tolerance in the test below, or find a random seed that works. I tried the second option here.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 91.95%. Comparing base (c9ad828) to head (5c9cebd).
Report is 3 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1192      +/-   ##
==========================================
+ Coverage   91.89%   91.95%   +0.06%     
==========================================
  Files          34       34              
  Lines       15360    15360              
==========================================
+ Hits        14115    14125      +10     
+ Misses       1245     1235      -10     

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[ViralBShah]

The condition number of the randn generated matrices jumps around a fair bit - no doubt because we are squaring. The scaling should greatly help.

[alanedelman]

yehh randn when squared will produce condition numbers of order n^2, von Neumann fell into this trap decades ago, leading to the belief that finite precision could be dreadful on these new machines that were being called computers.
Of course you can generate Q'DQ with any positive diagonal D you like and a random Q. If speed is a concern we can be more clever, and LAPACK probably has such routines, but quick and dirty is to use Q=Matrix(qr(randn(n,n)).Q)
which is O(n^3) (there are O(n^2) ways)

[ViralBShah]

julia> for i=1:10 
            Q=Matrix(qr(randn(n,n)).Q);
            @show cond(Q'*Diagonal(rand(n))*Q)
       end
cond(Q' * Diagonal(rand(n)) * Q) = 54.84525143924938
cond(Q' * Diagonal(rand(n)) * Q) = 68.73534833441128
cond(Q' * Diagonal(rand(n)) * Q) = 96.3033669626725
cond(Q' * Diagonal(rand(n)) * Q) = 87.56679319167324
cond(Q' * Diagonal(rand(n)) * Q) = 66255.80209047809
cond(Q' * Diagonal(rand(n)) * Q) = 41.302179179853
cond(Q' * Diagonal(rand(n)) * Q) = 45.178741706939526
cond(Q' * Diagonal(rand(n)) * Q) = 164.87644137765756
cond(Q' * Diagonal(rand(n)) * Q) = 93.56871249448481
cond(Q' * Diagonal(rand(n)) * Q) = 257.1891863712371