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Previously we checked ortonormality of eigenvectors by comparing
the matrix S^T * S to identity matrix. The new feature is also
checking just the diagonal of the matrix S^T * S. By that we get
the information, how far are the eigenvectors from having length 1.
If it turns out to be important, we could try to normalize them
at the end of elpa, which is simple (in contrast with enforcing
better orthogonality).

This stupid bug was there since commit ae45bbb3

From now on call make check with TASKS=x, e.g.

make check TASKS=2

for mpi_allreduce in test_check_correctness

of _GEMM and P_GEMM calls

The module is intended to be hidden, thus I moved it into the public
part of the legacy API.

Some test programs use it, thus the test programs have now also access
to ELPA's private modules. This is of course anyway necessary to test
ELPA internals that are not exposed via the public API.

The functions in elpa_utilities are considered "ELPA internal",
i.e. the should not be accessible by the users and thus not
be part of the API.

Should be done in a more systematic way. In this case, in certain
configuration 1stage_analytic test pased, while 2stage_analytic tests
failed due to error larger then tolerance

Introducing new test in which matrix and its eigendecomposition is
known and thus can be easily created and checked directly, without the
need to use scalapack or any other communication (apart from reducing
error).

The test is based on the fact, that if L_A and S_A are eigenvalues and
eigenvectors of matrix A, respectively, and L_B and S_B eigenvalues and
eigenvectors of B, then kron(L_A, L_B) and kron (S_A, S_B) are
eigenvalues and eigenvectors of kron(A, B).
Since it is easy to know exact eigendecomposition of a small matrix (e.g.
2x2), and kron operator has very simple structure, we can construct
arbitrarily large matrix and its eigendecomposition. We only have to
select small matrices such that the resulting matrix has unique and
ordered eigenvalues, so that the checking of the result is than easy.
Each element of matrix, eigenvector matrix and eigenvalue vector can
be quickly computed independently, just using its global coordinates.

The test is currently limited to matrices of size 2^n, but by
storing eigendecompositions of more small matrices (e.g. 3x3 and 5x5) we
could construct any matrix of size 2^n*3^m*5^o, which would probably be
sufficient, since most often used sizes (150, 1000, 5000, 2000, 60000)
are of this form.

The routines "eigenvalues" and "solve_tridiagonal" are now
also tested directly with the new API (and not only via the
legacy API -> new API mapping)

Todo: adapt test generator script to contain some logic