SLASYF_ROOK computes a partial factorization of a real symmetric
 matrix A using the bounded Bunch-Kaufman ("rook") diagonal
 pivoting method. The partial factorization has the form:

 A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
       ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

 A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L'
       ( L21  I ) (  0  A22 ) (  0       I    )

 where the order of D is at most NB. The actual order is returned in
 the argument KB, and is either NB or NB-1, or N if N <= NB.

 SLASYF_ROOK is an auxiliary routine called by SSYTRF_ROOK. It uses
 blocked code (calling Level 3 BLAS) to update the submatrix
 A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

          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.

          NB is INTEGER
          The maximum number of columns of the matrix A that should be
          factored.  NB should be at least 2 to allow for 2-by-2 pivot
          blocks.

          KB is INTEGER
          The number of columns of A that were actually factored.
          KB is either NB-1 or NB, or N if N <= NB.

          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, A contains details of the partial factorization.

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

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.

          If UPLO = 'U':
             Only the last KB elements of IPIV are set.

             If IPIV(k) > 0, then rows and columns k and IPIV(k) were
             interchanged and D(k,k) is a 1-by-1 diagonal block.

             If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
             columns k and -IPIV(k) were interchanged and rows and
             columns k-1 and -IPIV(k-1) were inerchaged,
             D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

          If UPLO = 'L':
             Only the first KB elements of IPIV are set.

             If IPIV(k) > 0, then rows and columns k and IPIV(k)
             were interchanged and D(k,k) is a 1-by-1 diagonal block.

             If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
             columns k and -IPIV(k) were interchanged and rows and
             columns k+1 and -IPIV(k+1) were inerchaged,
             D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

          W is REAL array, dimension (LDW,NB)

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

          INFO is INTEGER
          = 0: successful exit
          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular.

  November 2013,     Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley

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