gistfile1.txt

This program estimates a rotation under constraints that are naturally stated in angles.
The original version works directly on Euler angles: first an initial solution is
determined using an iterative algorithm, then positions of other local optima are
determined (this is the part that works in angles), then the candidate solutions are
evaluated and passed through the iterative algorithm again to polish the results. Here's
the result of solving a few instances of the problem. Note that the initial solution is
always found as one of the valid solutions (as it should be), but the constrained
minimization part sometimes produces significant amount of error that requires a lot of
extra iterations to correct:

initial it=57 err=0.000095
  refine solution 0 it=17 err=0.003206
  refine solution 1 it=26 err=0.000095
initial it=413 err=0.000034
  refine solution 0 it=1 err=0.000034
initial it=97 err=0.000083
  refine solution 0 it=31 err=0.002072
  refine solution 1 it=1 err=0.000083
initial it=275 err=0.000018
  refine solution 0 it=500 err=0.000136
  refine solution 1 it=117 err=0.000018
initial it=291 err=0.000055
  refine solution 0 it=201 err=0.000327
  refine solution 1 it=146 err=0.000055
initial it=92 err=0.000013
  refine solution 0 it=15 err=0.002535
  refine solution 1 it=1 err=0.000013
initial it=75 err=0.000010
  refine solution 0 it=19 err=0.002742
  refine solution 1 it=38 err=0.000010
initial it=99 err=0.000004
  refine solution 0 it=26 err=0.002084
  refine solution 1 it=52 err=0.000004
initial it=163 err=0.000048
  refine solution 0 it=82 err=0.001408
  refine solution 1 it=76 err=0.000048
initial it=110 err=0.000003
  refine solution 0 it=7 err=0.002033
  refine solution 1 it=1 err=0.000003
initial it=71 err=0.000082
  refine solution 0 it=22 err=0.003111
  refine solution 1 it=33 err=0.000082
initial it=46 err=0.000096
  refine solution 0 it=13 err=0.003676
  refine solution 1 it=21 err=0.000096
initial it=36 err=0.000145
  refine solution 0 it=11 err=0.003889
  refine solution 1 it=16 err=0.000145



The improved version doesn't parametrize the rotation in terms of angles, but instead
represents rotations in the corresponding planes as (unit) complex numbers. The (simple)
constraints are solved using projections. Everything else is the same; the main
difference is avoiding an (unnecessary) path that first takes an atan2(y, x) only to
determine the sine and cosine of the corresponding angle. Here's the results with the
new version. Note how the original solutions immediately stop without further iterations;
this is because the adjusted solution produces an error below the set error threshold of
1e-5. The original implementation perturbed solutions significantly.

initial it= 57 err=0.000095
  refine solution 0 it= 11 err=0.003206
  refine solution 1 it=  1 err=0.000095
initial it=413 err=0.000034
  refine solution 0 it=  1 err=0.000034
initial it= 97 err=0.000083
  refine solution 0 it= 31 err=0.002072
  refine solution 1 it=  1 err=0.000083
initial it=275 err=0.000018
  refine solution 0 it=  1 err=0.000018
initial it=291 err=0.000055
  refine solution 0 it= 52 err=0.000327
  refine solution 1 it=  1 err=0.000055
initial it= 92 err=0.000013
  refine solution 0 it= 15 err=0.002535
  refine solution 1 it=  1 err=0.000013
initial it= 75 err=0.000010
  refine solution 0 it=  6 err=0.002742
  refine solution 1 it=  1 err=0.000010
initial it= 99 err=0.000004
  refine solution 0 it=  5 err=0.002084
  refine solution 1 it=  1 err=0.000004
initial it=163 err=0.000048
  refine solution 0 it= 52 err=0.001408
  refine solution 1 it=  1 err=0.000048
initial it=110 err=0.000003
  refine solution 0 it=  7 err=0.002033
  refine solution 1 it=  1 err=0.000003
initial it= 71 err=0.000082
  refine solution 0 it=  9 err=0.003111
  refine solution 1 it=  1 err=0.000082
initial it= 46 err=0.000096
  refine solution 0 it=  1 err=0.003676
  refine solution 1 it=  1 err=0.000096
initial it= 36 err=0.000145
  refine solution 0 it=  6 err=0.003889
  refine solution 1 it=  1 err=0.000145