numerical problems in rotation creation

Key: MATH639
URL: https://issues.apache.org/jira/browse/MATH639
Project: Commons Math
Issue Type: Bug
Affects Versions: 2.2
Environment: Linux
Reporter: Luc Maisonobe
Assignee: Luc Maisonobe
Fix For: 3.0
building a rotation from the following vector pairs leads to NaN:
u1 = 4921140.837095533, 2.1512094250440013E7, 890093.279426377
u2 = 2.7238580938724895E9, 2.169664921341876E9, 6.749688708885301E10
v1 = 1, 0, 0
v2 = 0, 0, 1
The constructor first changes the (v1, v2) pair into (v1', v2') ensuring the following scalar
products hold:
<v1'v1'> == <u1u1>
<v2'v2'> == <u2u2>
<u1 u2> == <v1'v2'>
Once the (v1', v2') pair has been computed, we compute the cross product:
k = (v1'  u1)^(v2'  u2)
and the scalar product:
c = <k  (u1^u2)>
By construction, c is positive or null and the quaternion axis we want to build is q = k/[2*sqrt(c)].
c should be null only if some of the vectors are aligned, and this is dealt with later in
the algorithm.
However, there are numerical problems with the vector above with the way these computations
are done, as shown
by the following comparisons, showing the result we get from our Java code and the result
we get from manual
computation with the same formulas but with enhanced precision:
commons math: k = 38514476.5, 84., 1168590144
high precision: k = 38514410.36093388..., 0.374075245201180409222711..., 1168590152.10599715208...
and it becomes worse when computing c because the vectors are almost orthogonal to each other,
hence inducing additional cancellations. We get:
commons math c = 1.2397173627587605E20
high precision: c = 558382746168463196.7079627...
We have lost ALL significant digits in cancellations, and even the sign is wrong!

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