SSYTRF_RK computes the factorization of a real symmetric matrix A
 using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

    A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

 where U (or L) is unit upper (or lower) triangular matrix,
 U**T (or L**T) is the transpose of U (or L), P is a permutation
 matrix, P**T is the transpose of P, and D is symmetric and block
 diagonal with 1-by-1 and 2-by-2 diagonal blocks.

 This is the blocked version of the algorithm, calling Level 3 BLAS.
 For more information see Further Details section.

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

          N is INTEGER
          The order of the matrix A.  N >= 0.

          A is REAL array, dimension (LDA,N)
          On entry, the symmetric matrix A.
            If UPLO = 'U': the leading N-by-N upper triangular part
            of A contains the upper triangular part of the matrix A,
            and the strictly lower triangular part of A is not
            referenced.

            If UPLO = 'L': the leading N-by-N lower triangular part
            of A contains the lower triangular part of the matrix A,
            and the strictly upper triangular part of A is not
            referenced.

          On exit, contains:
            a) ONLY diagonal elements of the symmetric block diagonal
               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
               (superdiagonal (or subdiagonal) elements of D
                are stored on exit in array E), and
            b) If UPLO = 'U': factor U in the superdiagonal part of A.
               If UPLO = 'L': factor L in the subdiagonal part of A.

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

          E is REAL array, dimension (N)
          On exit, contains the superdiagonal (or subdiagonal)
          elements of the symmetric block diagonal matrix D
          with 1-by-1 or 2-by-2 diagonal blocks, where
          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

          NOTE: For 1-by-1 diagonal block D(k), where
          1 <= k <= N, the element E(k) is set to 0 in both
          UPLO = 'U' or UPLO = 'L' cases.

          IPIV is INTEGER array, dimension (N)
          IPIV describes the permutation matrix P in the factorization
          of matrix A as follows. The absolute value of IPIV(k)
          represents the index of row and column that were
          interchanged with the k-th row and column. The value of UPLO
          describes the order in which the interchanges were applied.
          Also, the sign of IPIV represents the block structure of
          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
          diagonal blocks which correspond to 1 or 2 interchanges
          at each factorization step. For more info see Further
          Details section.

          If UPLO = 'U',
          ( in factorization order, k decreases from N to 1 ):
            a) A single positive entry IPIV(k) > 0 means:
               D(k,k) is a 1-by-1 diagonal block.
               If IPIV(k) != k, rows and columns k and IPIV(k) were
               interchanged in the matrix A(1:N,1:N);
               If IPIV(k) = k, no interchange occurred.

            b) A pair of consecutive negative entries
               IPIV(k) < 0 and IPIV(k-1) < 0 means:
               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
               (NOTE: negative entries in IPIV appear ONLY in pairs).
               1) If -IPIV(k) != k, rows and columns
                  k and -IPIV(k) were interchanged
                  in the matrix A(1:N,1:N).
                  If -IPIV(k) = k, no interchange occurred.
               2) If -IPIV(k-1) != k-1, rows and columns
                  k-1 and -IPIV(k-1) were interchanged
                  in the matrix A(1:N,1:N).
                  If -IPIV(k-1) = k-1, no interchange occurred.

            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.

            d) NOTE: Any entry IPIV(k) is always NONZERO on output.

          If UPLO = 'L',
          ( in factorization order, k increases from 1 to N ):
            a) A single positive entry IPIV(k) > 0 means:
               D(k,k) is a 1-by-1 diagonal block.
               If IPIV(k) != k, rows and columns k and IPIV(k) were
               interchanged in the matrix A(1:N,1:N).
               If IPIV(k) = k, no interchange occurred.

            b) A pair of consecutive negative entries
               IPIV(k) < 0 and IPIV(k+1) < 0 means:
               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
               (NOTE: negative entries in IPIV appear ONLY in pairs).
               1) If -IPIV(k) != k, rows and columns
                  k and -IPIV(k) were interchanged
                  in the matrix A(1:N,1:N).
                  If -IPIV(k) = k, no interchange occurred.
               2) If -IPIV(k+1) != k+1, rows and columns
                  k-1 and -IPIV(k-1) were interchanged
                  in the matrix A(1:N,1:N).
                  If -IPIV(k+1) = k+1, no interchange occurred.

            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.

            d) NOTE: Any entry IPIV(k) is always NONZERO on output.

          WORK is REAL array, dimension ( MAX(1,LWORK) ).
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

          LWORK is INTEGER
          The length of WORK.  LWORK >=1.  For best performance
          LWORK >= N*NB, where NB is the block size returned
          by ILAENV.

          If LWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the WORK
          array, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued
          by XERBLA.

          INFO is INTEGER
          = 0: successful exit

          < 0: If INFO = -k, the k-th argument had an illegal value

          > 0: If INFO = k, the matrix A is singular, because:
                 If UPLO = 'U': column k in the upper
                 triangular part of A contains all zeros.
                 If UPLO = 'L': column k in the lower
                 triangular part of A contains all zeros.

               Therefore D(k,k) is exactly zero, and superdiagonal
               elements of column k of U (or subdiagonal elements of
               column k of L ) are all zeros. The factorization has
               been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if
               it is used to solve a system of equations.

               NOTE: INFO only stores the first occurrence of
               a singularity, any subsequent occurrence of singularity
               is not stored in INFO even though the factorization
               always completes.

 TODO: put correct description

  December 2016,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley

  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                  School of Mathematics,
                  University of Manchester