LAPACK tests not working

[masterleinad]

After updating deal.II, i.e. after dealii/dealii#7687, the LAPACK tests in test_hierarchy were not working (#138). One of the reasons is that the agglomerates might not have boundary conditions and hence the corresponding matrix has a non-trivial kernel shifting the diagonal entries slightly at least allows inverting the matrix (using UMFPACK). I observed this even before updating, i.e. with with the version in the image. After that the LAPACK tests are still failing due to the restriction matrix not being full-rank (possibly due to the spurious eigenvalues).
Just switching from LAPACK to ARPACK gives similar results as with Lanczos. Copying the reference values from the Lanczos tests to the ARPACK tests gives:

1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_16": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.11466298397904537 != 0.1148067738]. Relative difference exceeds tolerance [0.00125402 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = false; reordering = None; mesh_evaluator_type = DealIIMeshEvaluator; eigensolver = arpack; 
1: FE degree: 1
1: Convergence rate: 0.11
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_17": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.11469510272606935 != 0.1148067738]. Relative difference exceeds tolerance [0.000973634 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = false; reordering = None; mesh_evaluator_type = DealIIMeshEvaluator; eigensolver = lanczos; 
1: skip
1: FE degree: 1
1: Convergence rate: 0.30
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_19": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.2995474407510747 != 0.3012330587]. Relative difference exceeds tolerance [0.00562722 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = false; reordering = None; mesh_evaluator_type = DealIIMatrixFreeMeshEvaluator; eigensolver = lanczos; 
1: FE degree: 1
1: Convergence rate: 0.11
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_20": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.11466298397904537 != 0.1148067738]. Relative difference exceeds tolerance [0.00125402 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = false; reordering = Reverse Cuthill_McKee; mesh_evaluator_type = DealIIMeshEvaluator; eigensolver = arpack; 
1: FE degree: 1
1: Convergence rate: 0.11
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_21": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.11469510272606935 != 0.1148067738]. Relative difference exceeds tolerance [0.000973634 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = false; reordering = Reverse Cuthill_McKee; mesh_evaluator_type = DealIIMeshEvaluator; eigensolver = lanczos; 
1: skip
1: FE degree: 1
1: Convergence rate: 0.30
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_23": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.2995474407510747 != 0.3012330587]. Relative difference exceeds tolerance [0.00562722 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = false; reordering = Reverse Cuthill_McKee; mesh_evaluator_type = DealIIMatrixFreeMeshEvaluator; eigensolver = lanczos; 
1: FE degree: 1
1: Convergence rate: 0.10
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_24": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.10243805241099262 != 0.10245448259999999]. Relative difference exceeds tolerance [0.000160391 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = true; reordering = None; mesh_evaluator_type = DealIIMeshEvaluator; eigensolver = arpack; 
1: FE degree: 1
1: Convergence rate: 0.10
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_25": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.10245562595409574 != 0.10245448259999999]. Relative difference exceeds tolerance [1.11596e-05 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = true; reordering = None; mesh_evaluator_type = DealIIMeshEvaluator; eigensolver = lanczos; 
1: skip
1: FE degree: 1
1: Convergence rate: 0.30
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_27": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.29778963080796284 != 0.29779748690000002]. Relative difference exceeds tolerance [2.63813e-05 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = true; reordering = None; mesh_evaluator_type = DealIIMatrixFreeMeshEvaluator; eigensolver = lanczos; 
1: FE degree: 1
1: Convergence rate: 0.10
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_28": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.10243805241099262 != 0.10245448259999999]. Relative difference exceeds tolerance [0.000160391 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = true; reordering = Reverse Cuthill_McKee; mesh_evaluator_type = DealIIMeshEvaluator; eigensolver = arpack; 
1: FE degree: 1
1: Convergence rate: 0.10
1: /tmp/mfmg/tests/test_hierarchy.cc(241): error: in "hierarchy_3d/_29": check conv_rate == ref_solution[std::make_tuple(mesh, distort_random_str, reordering, eigensolver, matrix_free_str)] has failed [0.10245562595409574 != 0.10245448259999999]. Relative difference exceeds tolerance [1.11596e-05 > 1e-06]
1: Failure occurred in a following context:
1:     mesh = hyper_ball; distort_random = true; reordering = Reverse Cuthill_McKee; mesh_evaluator_type = DealIIMeshEvaluator; eigensolver = lanczos;

The Lanczos results are also slightly different due to the minor modification of the agglomerates.

[masterleinad]

We don't hit the problem with LAPACK with

 "number of eigenvectors" 1

instead of

 "number of eigenvectors" 2

[masterleinad]

The restriction matrix appears to be the same for LAPACK and ARPACK...

[masterleinad]

I compared the restriction matrices in more detail and they are different, everything is well-behaved for ARPACK but we have two zero eigenvalues with LAPACK.