*> \brief \b SSYTRF_ROOK
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SSYTRF_ROOK( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       REAL               A( LDA, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SSYTRF_ROOK computes the factorization of a real symmetric matrix A
*> using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
*> The form of the factorization is
*>
*>    A = U*D*U**T  or  A = L*D*L**T
*>
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          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, the block diagonal matrix D and the multipliers used
*>          to obtain the factor U or L (see below for further details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D.
*>
*>          If UPLO = 'U':
*>               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':
*>               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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (MAX(1,LWORK)).
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, D(i,i) is exactly zero.  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.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup realSYcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  If UPLO = 'U', then A = U*D*U**T, where
*>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    v    0   )   k-s
*>     U(k) =  (   0    I    0   )   s
*>             (   0    0    I   )   n-k
*>                k-s   s   n-k
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*>
*>  If UPLO = 'L', then A = L*D*L**T, where
*>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*>
*>             (   I    0     0   )  k-1
*>     L(k) =  (   0    I     0   )  s
*>             (   0    v     I   )  n-k-s+1
*>                k-1   s  n-k-s+1
*>
*>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>   June 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
*>
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE SSYTRF_ROOK( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      REAL               A( LDA, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            LQUERY, UPPER
      INTEGER            IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           SLASYF_ROOK, SSYTF2_ROOK, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
         INFO = -7
      END IF
*
      IF( INFO.EQ.0 ) THEN
*
*        Determine the block size
*
         NB = ILAENV( 1, 'SSYTRF_ROOK', UPLO, N, -1, -1, -1 )
         LWKOPT = MAX( 1, N*NB )
         WORK( 1 ) = LWKOPT
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SSYTRF_ROOK', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
      NBMIN = 2
      LDWORK = N
      IF( NB.GT.1 .AND. NB.LT.N ) THEN
         IWS = LDWORK*NB
         IF( LWORK.LT.IWS ) THEN
            NB = MAX( LWORK / LDWORK, 1 )
            NBMIN = MAX( 2, ILAENV( 2, 'SSYTRF_ROOK',
     $                              UPLO, N, -1, -1, -1 ) )
         END IF
      ELSE
         IWS = 1
      END IF
      IF( NB.LT.NBMIN )
     $   NB = N
*
      IF( UPPER ) THEN
*
*        Factorize A as U*D*U**T using the upper triangle of A
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        KB, where KB is the number of columns factorized by SLASYF_ROOK;
*        KB is either NB or NB-1, or K for the last block
*
         K = N
   10    CONTINUE
*
*        If K < 1, exit from loop
*
         IF( K.LT.1 )
     $      GO TO 40
*
         IF( K.GT.NB ) THEN
*
*           Factorize columns k-kb+1:k of A and use blocked code to
*           update columns 1:k-kb
*
            CALL SLASYF_ROOK( UPLO, K, NB, KB, A, LDA,
     $                        IPIV, WORK, LDWORK, IINFO )
         ELSE
*
*           Use unblocked code to factorize columns 1:k of A
*
            CALL SSYTF2_ROOK( UPLO, K, A, LDA, IPIV, IINFO )
            KB = K
         END IF
*
*        Set INFO on the first occurrence of a zero pivot
*
         IF( INFO.EQ.0 .AND. IINFO.GT.0 )
     $      INFO = IINFO
*
*        No need to adjust IPIV
*
*        Decrease K and return to the start of the main loop
*
         K = K - KB
         GO TO 10
*
      ELSE
*
*        Factorize A as L*D*L**T using the lower triangle of A
*
*        K is the main loop index, increasing from 1 to N in steps of
*        KB, where KB is the number of columns factorized by SLASYF_ROOK;
*        KB is either NB or NB-1, or N-K+1 for the last block
*
         K = 1
   20    CONTINUE
*
*        If K > N, exit from loop
*
         IF( K.GT.N )
     $      GO TO 40
*
         IF( K.LE.N-NB ) THEN
*
*           Factorize columns k:k+kb-1 of A and use blocked code to
*           update columns k+kb:n
*
            CALL SLASYF_ROOK( UPLO, N-K+1, NB, KB, A( K, K ), LDA,
     $                        IPIV( K ), WORK, LDWORK, IINFO )
         ELSE
*
*           Use unblocked code to factorize columns k:n of A
*
            CALL SSYTF2_ROOK( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ),
     $                   IINFO )
            KB = N - K + 1
         END IF
*
*        Set INFO on the first occurrence of a zero pivot
*
         IF( INFO.EQ.0 .AND. IINFO.GT.0 )
     $      INFO = IINFO + K - 1
*
*        Adjust IPIV
*
         DO 30 J = K, K + KB - 1
            IF( IPIV( J ).GT.0 ) THEN
               IPIV( J ) = IPIV( J ) + K - 1
            ELSE
               IPIV( J ) = IPIV( J ) - K + 1
            END IF
   30    CONTINUE
*
*        Increase K and return to the start of the main loop
*
         K = K + KB
         GO TO 20
*
      END IF
*
   40 CONTINUE
      WORK( 1 ) = LWKOPT
      RETURN
*
*     End of SSYTRF_ROOK
*
      END