text: *> \brief \b SLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
  *
  *  =========== DOCUMENTATION ===========
  *
  * Online html documentation available at
  *            http://www.netlib.org/lapack/explore-html/
  *
  *> \htmlonly
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  *> [TGZ]</a>
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  *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slasyf_rook.f">
  *> [TXT]</a>
  *> \endhtmlonly
  *
  *  Definition:
  *  ===========
  *
  *       SUBROUTINE SLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
  *
  *       .. Scalar Arguments ..
  *       CHARACTER          UPLO
  *       INTEGER            INFO, KB, LDA, LDW, N, NB
  *       ..
  *       .. Array Arguments ..
  *       INTEGER            IPIV( * )
  *       REAL               A( LDA, * ), W( LDW, * )
  *       ..
  *
  *
  *> \par Purpose:
  *  =============
  *>
  *> \verbatim
  *>
  *> 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').
  *> \endverbatim
  *
  *  Arguments:
  *  ==========
  *
  *> \param[in] UPLO
  *> \verbatim
  *>          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
  *> \endverbatim
  *>
  *> \param[in] N
  *> \verbatim
  *>          N is INTEGER
  *>          The order of the matrix A.  N >= 0.
  *> \endverbatim
  *>
  *> \param[in] NB
  *> \verbatim
  *>          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.
  *> \endverbatim
  *>
  *> \param[out] KB
  *> \verbatim
  *>          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.
  *> \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, A contains details of the partial factorization.
  *> \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':
  *>             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.
  *> \endverbatim
  *>
  *> \param[out] W
  *> \verbatim
  *>          W is REAL array, dimension (LDW,NB)
  *> \endverbatim
  *>
  *> \param[in] LDW
  *> \verbatim
  *>          LDW is INTEGER
  *>          The leading dimension of the array W.  LDW >= max(1,N).
  *> \endverbatim
  *>
  *> \param[out] INFO
  *> \verbatim
  *>          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.
  *> \endverbatim
  *
  *  Authors:
  *  ========
  *
  *> \author Univ. of Tennessee
  *> \author Univ. of California Berkeley
  *> \author Univ. of Colorado Denver
  *> \author NAG Ltd.
  *
  *> \date November 2013
  *
  *> \ingroup realSYcomputational
  *
  *> \par Contributors:
  *  ==================
  *>
  *> \verbatim
  *>
  *>  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
  *>
  *> \endverbatim
  *
  *  =====================================================================
        SUBROUTINE SLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
       $                        INFO )
  *
  *  -- LAPACK computational routine (version 3.5.0) --
  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  *     November 2013
  *
  *     .. Scalar Arguments ..
        CHARACTER          UPLO
        INTEGER            INFO, KB, LDA, LDW, N, NB
  *     ..
  *     .. Array Arguments ..
        INTEGER            IPIV( * )
        REAL               A( LDA, * ), W( LDW, * )
  *     ..
  *
  *  =====================================================================
  *
  *     .. Parameters ..
        REAL               ZERO, ONE
        PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
        REAL               EIGHT, SEVTEN
        PARAMETER          ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 )
  *     ..
  *     .. Local Scalars ..
        LOGICAL            DONE
        INTEGER            IMAX, ITEMP, J, JB, JJ, JMAX, JP1, JP2, K, KK,
       $                   KW, KKW, KP, KSTEP, P, II
  
        REAL               ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
       $                   STEMP, R1, ROWMAX, T, SFMIN
  *     ..
  *     .. External Functions ..
        LOGICAL            LSAME
        INTEGER            ISAMAX
        REAL               SLAMCH
        EXTERNAL           LSAME, ISAMAX, SLAMCH
  *     ..
  *     .. External Subroutines ..
        EXTERNAL           SCOPY, SGEMM, SGEMV, SSCAL, SSWAP
  *     ..
  *     .. Intrinsic Functions ..
        INTRINSIC          ABS, MAX, MIN, SQRT
  *     ..
  *     .. Executable Statements ..
  *
        INFO = 0
  *
  *     Initialize ALPHA for use in choosing pivot block size.
  *
        ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
  *
  *     Compute machine safe minimum
  *
        SFMIN = SLAMCH( 'S' )
  *
        IF( LSAME( UPLO, 'U' ) ) THEN
  *
  *        Factorize the trailing columns of A using the upper triangle
  *        of A and working backwards, and compute the matrix W = U12*D
  *        for use in updating A11
  *
  *        K is the main loop index, decreasing from N in steps of 1 or 2
  *
           K = N
     10    CONTINUE
  *
  *        KW is the column of W which corresponds to column K of A
  *
           KW = NB + K - N
  *
  *        Exit from loop
  *
           IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
       $      GO TO 30
  *
           KSTEP = 1
           P = K
  *
  *        Copy column K of A to column KW of W and update it
  *
           CALL SCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
           IF( K.LT.N )
       $      CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
       $                  LDA, W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
  *
  *        Determine rows and columns to be interchanged and whether
  *        a 1-by-1 or 2-by-2 pivot block will be used
  *
           ABSAKK = ABS( W( K, KW ) )
  *
  *        IMAX is the row-index of the largest off-diagonal element in
  *        column K, and COLMAX is its absolute value.
  *        Determine both COLMAX and IMAX.
  *
           IF( K.GT.1 ) THEN
              IMAX = ISAMAX( K-1, W( 1, KW ), 1 )
              COLMAX = ABS( W( IMAX, KW ) )
           ELSE
              COLMAX = ZERO
           END IF
  *
           IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
  *
  *           Column K is zero or underflow: set INFO and continue
  *
              IF( INFO.EQ.0 )
       $         INFO = K
              KP = K
              CALL SCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
           ELSE
  *
  *           ============================================================
  *
  *           Test for interchange
  *
  *           Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
  *           (used to handle NaN and Inf)
  *
              IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
  *
  *              no interchange, use 1-by-1 pivot block
  *
                 KP = K
  *
              ELSE
  *
                 DONE = .FALSE.
  *
  *              Loop until pivot found
  *
     12          CONTINUE
  *
  *                 Begin pivot search loop body
  *
  *
  *                 Copy column IMAX to column KW-1 of W and update it
  *
                    CALL SCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
                    CALL SCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
       $                        W( IMAX+1, KW-1 ), 1 )
  *
                    IF( K.LT.N )
       $               CALL SGEMV( 'No transpose', K, N-K, -ONE,
       $                           A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
       $                           ONE, W( 1, KW-1 ), 1 )
  *
  *                 JMAX is the column-index of the largest off-diagonal
  *                 element in row IMAX, and ROWMAX is its absolute value.
  *                 Determine both ROWMAX and JMAX.
  *
                    IF( IMAX.NE.K ) THEN
                       JMAX = IMAX + ISAMAX( K-IMAX, W( IMAX+1, KW-1 ),
       $                                     1 )
                       ROWMAX = ABS( W( JMAX, KW-1 ) )
                    ELSE
                       ROWMAX = ZERO
                    END IF
  *
                    IF( IMAX.GT.1 ) THEN
                       ITEMP = ISAMAX( IMAX-1, W( 1, KW-1 ), 1 )
                       STEMP = ABS( W( ITEMP, KW-1 ) )
                       IF( STEMP.GT.ROWMAX ) THEN
                          ROWMAX = STEMP
                          JMAX = ITEMP
                       END IF
                    END IF
  *
  *                 Equivalent to testing for
  *                 ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
  *                 (used to handle NaN and Inf)
  *
                    IF( .NOT.(ABS( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
       $            THEN
  *
  *                    interchange rows and columns K and IMAX,
  *                    use 1-by-1 pivot block
  *
                       KP = IMAX
  *
  *                    copy column KW-1 of W to column KW of W
  *
                       CALL SCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
  *
                       DONE = .TRUE.
  *
  *                 Equivalent to testing for ROWMAX.EQ.COLMAX,
  *                 (used to handle NaN and Inf)
  *
                    ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
       $            THEN
  *
  *                    interchange rows and columns K-1 and IMAX,
  *                    use 2-by-2 pivot block
  *
                       KP = IMAX
                       KSTEP = 2
                       DONE = .TRUE.
                    ELSE
  *
  *                    Pivot not found: set params and repeat
  *
                       P = IMAX
                       COLMAX = ROWMAX
                       IMAX = JMAX
  *
  *                    Copy updated JMAXth (next IMAXth) column to Kth of W
  *
                       CALL SCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
  *
                    END IF
  *
  *                 End pivot search loop body
  *
                 IF( .NOT. DONE ) GOTO 12
  *
              END IF
  *
  *           ============================================================
  *
              KK = K - KSTEP + 1
  *
  *           KKW is the column of W which corresponds to column KK of A
  *
              KKW = NB + KK - N
  *
              IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
  *
  *              Copy non-updated column K to column P
  *
                 CALL SCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
                 CALL SCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
  *
  *              Interchange rows K and P in last N-K+1 columns of A
  *              and last N-K+2 columns of W
  *
                 CALL SSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
                 CALL SSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
              END IF
  *
  *           Updated column KP is already stored in column KKW of W
  *
              IF( KP.NE.KK ) THEN
  *
  *              Copy non-updated column KK to column KP
  *
                 A( KP, K ) = A( KK, K )
                 CALL SCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
       $                     LDA )
                 CALL SCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
  *
  *              Interchange rows KK and KP in last N-KK+1 columns
  *              of A and W
  *
                 CALL SSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
                 CALL SSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
       $                     LDW )
              END IF
  *
              IF( KSTEP.EQ.1 ) THEN
  *
  *              1-by-1 pivot block D(k): column KW of W now holds
  *
  *              W(k) = U(k)*D(k)
  *
  *              where U(k) is the k-th column of U
  *
  *              Store U(k) in column k of A
  *
                 CALL SCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
                 IF( K.GT.1 ) THEN
                    IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
                       R1 = ONE / A( K, K )
                       CALL SSCAL( K-1, R1, A( 1, K ), 1 )
                    ELSE IF( A( K, K ).NE.ZERO ) THEN
                       DO 14 II = 1, K - 1
                          A( II, K ) = A( II, K ) / A( K, K )
     14                CONTINUE
                    END IF
                 END IF
  *
              ELSE
  *
  *              2-by-2 pivot block D(k): columns KW and KW-1 of W now
  *              hold
  *
  *              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
  *
  *              where U(k) and U(k-1) are the k-th and (k-1)-th columns
  *              of U
  *
                 IF( K.GT.2 ) THEN
  *
  *                 Store U(k) and U(k-1) in columns k and k-1 of A
  *
                    D12 = W( K-1, KW )
                    D11 = W( K, KW ) / D12
                    D22 = W( K-1, KW-1 ) / D12
                    T = ONE / ( D11*D22-ONE )
                    DO 20 J = 1, K - 2
                       A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
       $                             D12 )
                       A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
       $                           D12 )
     20             CONTINUE
                 END IF
  *
  *              Copy D(k) to A
  *
                 A( K-1, K-1 ) = W( K-1, KW-1 )
                 A( K-1, K ) = W( K-1, KW )
                 A( K, K ) = W( K, KW )
              END IF
           END IF
  *
  *        Store details of the interchanges in IPIV
  *
           IF( KSTEP.EQ.1 ) THEN
              IPIV( K ) = KP
           ELSE
              IPIV( K ) = -P
              IPIV( K-1 ) = -KP
           END IF
  *
  *        Decrease K and return to the start of the main loop
  *
           K = K - KSTEP
           GO TO 10
  *
     30    CONTINUE
  *
  *        Update the upper triangle of A11 (= A(1:k,1:k)) as
  *
  *        A11 := A11 - U12*D*U12**T = A11 - U12*W**T
  *
  *        computing blocks of NB columns at a time
  *
           DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
              JB = MIN( NB, K-J+1 )
  *
  *           Update the upper triangle of the diagonal block
  *
              DO 40 JJ = J, J + JB - 1
                 CALL SGEMV( 'No transpose', JJ-J+1, N-K, -ONE,
       $                     A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE,
       $                     A( J, JJ ), 1 )
     40       CONTINUE
  *
  *           Update the rectangular superdiagonal block
  *
              IF( J.GE.2 )
       $         CALL SGEMM( 'No transpose', 'Transpose', J-1, JB,
       $                  N-K, -ONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
       $                  ONE, A( 1, J ), LDA )
     50    CONTINUE
  *
  *        Put U12 in standard form by partially undoing the interchanges
  *        in columns k+1:n
  *
           J = K + 1
     60    CONTINUE
  *
              KSTEP = 1
              JP1 = 1
              JJ = J
              JP2 = IPIV( J )
              IF( JP2.LT.0 ) THEN
                 JP2 = -JP2
                 J = J + 1
                 JP1 = -IPIV( J )
                 KSTEP = 2
              END IF
  *
              J = J + 1
              IF( JP2.NE.JJ .AND. J.LE.N )
       $         CALL SSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
              JJ = J - 1
              IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
       $         CALL SSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
           IF( J.LE.N )
       $      GO TO 60
  *
  *        Set KB to the number of columns factorized
  *
           KB = N - K
  *
        ELSE
  *
  *        Factorize the leading columns of A using the lower triangle
  *        of A and working forwards, and compute the matrix W = L21*D
  *        for use in updating A22
  *
  *        K is the main loop index, increasing from 1 in steps of 1 or 2
  *
           K = 1
     70   CONTINUE
  *
  *        Exit from loop
  *
           IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
       $      GO TO 90
  *
           KSTEP = 1
           P = K
  *
  *        Copy column K of A to column K of W and update it
  *
           CALL SCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
           IF( K.GT.1 )
       $      CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
       $                  LDA, W( K, 1 ), LDW, ONE, W( K, K ), 1 )
  *
  *        Determine rows and columns to be interchanged and whether
  *        a 1-by-1 or 2-by-2 pivot block will be used
  *
           ABSAKK = ABS( W( K, K ) )
  *
  *        IMAX is the row-index of the largest off-diagonal element in
  *        column K, and COLMAX is its absolute value.
  *        Determine both COLMAX and IMAX.
  *
           IF( K.LT.N ) THEN
              IMAX = K + ISAMAX( N-K, W( K+1, K ), 1 )
              COLMAX = ABS( W( IMAX, K ) )
           ELSE
              COLMAX = ZERO
           END IF
  *
           IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
  *
  *           Column K is zero or underflow: set INFO and continue
  *
              IF( INFO.EQ.0 )
       $         INFO = K
              KP = K
              CALL SCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
           ELSE
  *
  *           ============================================================
  *
  *           Test for interchange
  *
  *           Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
  *           (used to handle NaN and Inf)
  *
              IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
  *
  *              no interchange, use 1-by-1 pivot block
  *
                 KP = K
  *
              ELSE
  *
                 DONE = .FALSE.
  *
  *              Loop until pivot found
  *
     72          CONTINUE
  *
  *                 Begin pivot search loop body
  *
  *
  *                 Copy column IMAX to column K+1 of W and update it
  *
                    CALL SCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
                    CALL SCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
       $                        W( IMAX, K+1 ), 1 )
                    IF( K.GT.1 )
       $               CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE,
       $                           A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
       $                           ONE, W( K, K+1 ), 1 )
  *
  *                 JMAX is the column-index of the largest off-diagonal
  *                 element in row IMAX, and ROWMAX is its absolute value.
  *                 Determine both ROWMAX and JMAX.
  *
                    IF( IMAX.NE.K ) THEN
                       JMAX = K - 1 + ISAMAX( IMAX-K, W( K, K+1 ), 1 )
                       ROWMAX = ABS( W( JMAX, K+1 ) )
                    ELSE
                       ROWMAX = ZERO
                    END IF
  *
                    IF( IMAX.LT.N ) THEN
                       ITEMP = IMAX + ISAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
                       STEMP = ABS( W( ITEMP, K+1 ) )
                       IF( STEMP.GT.ROWMAX ) THEN
                          ROWMAX = STEMP
                          JMAX = ITEMP
                       END IF
                    END IF
  *
  *                 Equivalent to testing for
  *                 ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
  *                 (used to handle NaN and Inf)
  *
                    IF( .NOT.( ABS( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
       $            THEN
  *
  *                    interchange rows and columns K and IMAX,
  *                    use 1-by-1 pivot block
  *
                       KP = IMAX
  *
  *                    copy column K+1 of W to column K of W
  *
                       CALL SCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
  *
                       DONE = .TRUE.
  *
  *                 Equivalent to testing for ROWMAX.EQ.COLMAX,
  *                 (used to handle NaN and Inf)
  *
                    ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
       $            THEN
  *
  *                    interchange rows and columns K+1 and IMAX,
  *                    use 2-by-2 pivot block
  *
                       KP = IMAX
                       KSTEP = 2
                       DONE = .TRUE.
                    ELSE
  *
  *                    Pivot not found: set params and repeat
  *
                       P = IMAX
                       COLMAX = ROWMAX
                       IMAX = JMAX
  *
  *                    Copy updated JMAXth (next IMAXth) column to Kth of W
  *
                       CALL SCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
  *
                    END IF
  *
  *                 End pivot search loop body
  *
                 IF( .NOT. DONE ) GOTO 72
  *
              END IF
  *
  *           ============================================================
  *
              KK = K + KSTEP - 1
  *
              IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
  *
  *              Copy non-updated column K to column P
  *
                 CALL SCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
                 CALL SCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
  *
  *              Interchange rows K and P in first K columns of A
  *              and first K+1 columns of W
  *
                 CALL SSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
                 CALL SSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
              END IF
  *
  *           Updated column KP is already stored in column KK of W
  *
              IF( KP.NE.KK ) THEN
  *
  *              Copy non-updated column KK to column KP
  *
                 A( KP, K ) = A( KK, K )
                 CALL SCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
                 CALL SCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
  *
  *              Interchange rows KK and KP in first KK columns of A and W
  *
                 CALL SSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
                 CALL SSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
              END IF
  *
              IF( KSTEP.EQ.1 ) THEN
  *
  *              1-by-1 pivot block D(k): column k of W now holds
  *
  *              W(k) = L(k)*D(k)
  *
  *              where L(k) is the k-th column of L
  *
  *              Store L(k) in column k of A
  *
                 CALL SCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
                 IF( K.LT.N ) THEN
                    IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
                       R1 = ONE / A( K, K )
                       CALL SSCAL( N-K, R1, A( K+1, K ), 1 )
                    ELSE IF( A( K, K ).NE.ZERO ) THEN
                       DO 74 II = K + 1, N
                          A( II, K ) = A( II, K ) / A( K, K )
     74                CONTINUE
                    END IF
                 END IF
  *
              ELSE
  *
  *              2-by-2 pivot block D(k): columns k and k+1 of W now hold
  *
  *              ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
  *
  *              where L(k) and L(k+1) are the k-th and (k+1)-th columns
  *              of L
  *
                 IF( K.LT.N-1 ) THEN
  *
  *                 Store L(k) and L(k+1) in columns k and k+1 of A
  *
                    D21 = W( K+1, K )
                    D11 = W( K+1, K+1 ) / D21
                    D22 = W( K, K ) / D21
                    T = ONE / ( D11*D22-ONE )
                    DO 80 J = K + 2, N
                       A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
       $                           D21 )
                       A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
       $                             D21 )
     80             CONTINUE
                 END IF
  *
  *              Copy D(k) to A
  *
                 A( K, K ) = W( K, K )
                 A( K+1, K ) = W( K+1, K )
                 A( K+1, K+1 ) = W( K+1, K+1 )
              END IF
           END IF
  *
  *        Store details of the interchanges in IPIV
  *
           IF( KSTEP.EQ.1 ) THEN
              IPIV( K ) = KP
           ELSE
              IPIV( K ) = -P
              IPIV( K+1 ) = -KP
           END IF
  *
  *        Increase K and return to the start of the main loop
  *
           K = K + KSTEP
           GO TO 70
  *
     90    CONTINUE
  *
  *        Update the lower triangle of A22 (= A(k:n,k:n)) as
  *
  *        A22 := A22 - L21*D*L21**T = A22 - L21*W**T
  *
  *        computing blocks of NB columns at a time
  *
           DO 110 J = K, N, NB
              JB = MIN( NB, N-J+1 )
  *
  *           Update the lower triangle of the diagonal block
  *
              DO 100 JJ = J, J + JB - 1
                 CALL SGEMV( 'No transpose', J+JB-JJ, K-1, -ONE,
       $                     A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE,
       $                     A( JJ, JJ ), 1 )
    100       CONTINUE
  *
  *           Update the rectangular subdiagonal block
  *
              IF( J+JB.LE.N )
       $         CALL SGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
       $                     K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW,
       $                     ONE, A( J+JB, J ), LDA )
    110    CONTINUE
  *
  *        Put L21 in standard form by partially undoing the interchanges
  *        in columns 1:k-1
  *
           J = K - 1
    120    CONTINUE
  *
              KSTEP = 1
              JP1 = 1
              JJ = J
              JP2 = IPIV( J )
              IF( JP2.LT.0 ) THEN
                 JP2 = -JP2
                 J = J - 1
                 JP1 = -IPIV( J )
                 KSTEP = 2
              END IF
  *
              J = J - 1
              IF( JP2.NE.JJ .AND. J.GE.1 )
       $         CALL SSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
              JJ = J + 1
              IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
       $         CALL SSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
           IF( J.GE.1 )
       $      GO TO 120
  *
  *        Set KB to the number of columns factorized
  *
           KB = K - 1
  *
        END IF
        RETURN
  *
  *     End of SLASYF_ROOK
  *
        END
  