dlasyf.f

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*> \brief \b DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASYF( 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( * )
*       DOUBLE PRECISION   A( LDA, * ), W( LDW, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASYF computes a partial factorization of a real symmetric matrix A
*> using the Bunch-Kaufman 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.
*>
*> DLASYF is an auxiliary routine called by DSYTRF. 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 DOUBLE PRECISION 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) = IPIV(k-1) < 0, then rows and columns
*>             k-1 and -IPIV(k) were interchanged and 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) = IPIV(k+1) < 0, then rows and columns
*>             k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
*>             is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION 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.
*
*> \ingroup lahef
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>  November 2013,  Igor Kozachenko,
*>                  Computer Science Division,
*>                  University of California, Berkeley
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE dlasyf( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, 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, KB, LDA, LDW, N, NB
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * ), W( LDW, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
      DOUBLE PRECISION   EIGHT, SEVTEN
      parameter( eight = 8.0d+0, sevten = 17.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP,
     $                   KSTEP, KW
      DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1,
     $                   ROWMAX, T
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      EXTERNAL           lsame, idamax
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dgemm, dgemv, dscal, dswap
*     ..
*     .. 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
*
      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
*
*        KW is the column of W which corresponds to column K of A
*
         k = n
   10    CONTINUE
         kw = nb + k - n
*
*        Exit from loop
*
         IF( ( k.LE.n-nb+1 .AND. nb.LT.n ) .OR. k.LT.1 )
     $      GO TO 30
*
*        Copy column K of A to column KW of W and update it
*
         CALL dcopy( k, a( 1, k ), 1, w( 1, kw ), 1 )
         IF( k.LT.n )
     $      CALL dgemv( 'No transpose', k, n-k, -one, a( 1, k+1 ), lda,
     $                  w( k, kw+1 ), ldw, one, w( 1, kw ), 1 )
*
         kstep = 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 = idamax( 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
         ELSE
            IF( absakk.GE.alpha*colmax ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               kp = k
            ELSE
*
*              Copy column IMAX to column KW-1 of W and update it
*
               CALL dcopy( imax, a( 1, imax ), 1, w( 1, kw-1 ), 1 )
               CALL dcopy( k-imax, a( imax, imax+1 ), lda,
     $                     w( imax+1, kw-1 ), 1 )
               IF( k.LT.n )
     $            CALL dgemv( '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
*
               jmax = imax + idamax( k-imax, w( imax+1, kw-1 ), 1 )
               rowmax = abs( w( jmax, kw-1 ) )
               IF( imax.GT.1 ) THEN
                  jmax = idamax( imax-1, w( 1, kw-1 ), 1 )
                  rowmax = max( rowmax, abs( w( jmax, kw-1 ) ) )
               END IF
*
               IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  kp = k
               ELSE IF( abs( w( imax, kw-1 ) ).GE.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 dcopy( k, w( 1, kw-1 ), 1, w( 1, kw ), 1 )
               ELSE
*
*                 interchange rows and columns K-1 and IMAX, use 2-by-2
*                 pivot block
*
                  kp = imax
                  kstep = 2
               END IF
            END IF
*
*           ============================================================
*
*           KK is the column of A where pivoting step stopped
*
            kk = k - kstep + 1
*
*           KKW is the column of W which corresponds to column KK of A
*
            kkw = nb + kk - n
*
*           Interchange rows and columns KP and KK.
*           Updated column KP is already stored in column KKW of W.
*
            IF( kp.NE.kk ) THEN
*
*              Copy non-updated column KK to column KP of submatrix A
*              at step K. No need to copy element into column K
*              (or K and K-1 for 2-by-2 pivot) of A, since these columns
*              will be later overwritten.
*
               a( kp, kp ) = a( kk, kk )
               CALL dcopy( kk-1-kp, a( kp+1, kk ), 1, a( kp, kp+1 ),
     $                     lda )
               IF( kp.GT.1 )
     $            CALL dcopy( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
*
*              Interchange rows KK and KP in last K+1 to N columns of A
*              (columns K (or K and K-1 for 2-by-2 pivot) of A will be
*              later overwritten). Interchange rows KK and KP
*              in last KKW to NB columns of W.
*
               IF( k.LT.n )
     $            CALL dswap( n-k, a( kk, k+1 ), lda, a( kp, k+1 ),
     $                        lda )
               CALL dswap( 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(kw) = U(k)*D(k),
*
*              where U(k) is the k-th column of U
*
*              Store subdiag. elements of column U(k)
*              and 1-by-1 block D(k) in column k of A.
*              NOTE: Diagonal element U(k,k) is a UNIT element
*              and not stored.
*                 A(k,k) := D(k,k) = W(k,kw)
*                 A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
*
               CALL dcopy( k, w( 1, kw ), 1, a( 1, k ), 1 )
               r1 = one / a( k, k )
               CALL dscal( k-1, r1, a( 1, k ), 1 )
*
            ELSE
*
*              2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
*
*              ( W(kw-1) W(kw) ) = ( 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
*
*              Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
*              block D(k-1:k,k-1:k) in columns k-1 and k of A.
*              NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
*              block and not stored.
*                 A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
*                 A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
*                 = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
*
               IF( k.GT.2 ) THEN
*
*                 Compose the columns of the inverse of 2-by-2 pivot
*                 block D in the following way to reduce the number
*                 of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by
*                 this inverse
*
*                 D**(-1) = ( d11 d21 )**(-1) =
*                           ( d21 d22 )
*
*                 = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
*                                        ( (-d21 ) ( d11 ) )
*
*                 = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
*
*                   * ( ( d22/d21 ) (      -1 ) ) =
*                     ( (      -1 ) ( d11/d21 ) )
*
*                 = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) (  -1 ) ) =
*                                           ( ( -1  ) ( D22 ) )
*
*                 = 1/d21 * T * ( ( D11 ) (  -1 ) )
*                               ( (  -1 ) ( D22 ) )
*
*                 = D21 * ( ( D11 ) (  -1 ) )
*                         ( (  -1 ) ( D22 ) )
*
                  d21 = w( k-1, kw )
                  d11 = w( k, kw ) / d21
                  d22 = w( k-1, kw-1 ) / d21
                  t = one / ( d11*d22-one )
                  d21 = t / d21
*
*                 Update elements in columns A(k-1) and A(k) as
*                 dot products of rows of ( W(kw-1) W(kw) ) and columns
*                 of D**(-1)
*
                  DO 20 j = 1, k - 2
                     a( j, k-1 ) = d21*( d11*w( j, kw-1 )-w( j, kw ) )
                     a( j, k ) = d21*( d22*w( j, kw )-w( j, kw-1 ) )
   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 ) = -kp
            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 dgemv( '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
*
            CALL dgemm( '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 looping backwards from k+1 to n
*
         j = k + 1
   60    CONTINUE
*
*           Undo the interchanges (if any) of rows JJ and JP at each
*           step J
*
*           (Here, J is a diagonal index)
            jj = j
            jp = ipiv( j )
            IF( jp.LT.0 ) THEN
               jp = -jp
*              (Here, J is a diagonal index)
               j = j + 1
            END IF
*           (NOTE: Here, J is used to determine row length. Length N-J+1
*           of the rows to swap back doesn't include diagonal element)
            j = j + 1
            IF( jp.NE.jj .AND. j.LE.n )
     $         CALL dswap( n-j+1, a( jp, j ), lda, a( jj, j ), lda )
         IF( j.LT.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
*
*        Copy column K of A to column K of W and update it
*
         CALL dcopy( n-k+1, a( k, k ), 1, w( k, k ), 1 )
         CALL dgemv( 'No transpose', n-k+1, k-1, -one, a( k, 1 ), lda,
     $               w( k, 1 ), ldw, one, w( k, k ), 1 )
*
         kstep = 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 + idamax( 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
         ELSE
            IF( absakk.GE.alpha*colmax ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               kp = k
            ELSE
*
*              Copy column IMAX to column K+1 of W and update it
*
               CALL dcopy( imax-k, a( imax, k ), lda, w( k, k+1 ), 1 )
               CALL dcopy( n-imax+1, a( imax, imax ), 1, w( imax, k+1 ),
     $                     1 )
               CALL dgemv( '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
*
               jmax = k - 1 + idamax( imax-k, w( k, k+1 ), 1 )
               rowmax = abs( w( jmax, k+1 ) )
               IF( imax.LT.n ) THEN
                  jmax = imax + idamax( n-imax, w( imax+1, k+1 ), 1 )
                  rowmax = max( rowmax, abs( w( jmax, k+1 ) ) )
               END IF
*
               IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  kp = k
               ELSE IF( abs( w( imax, k+1 ) ).GE.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 dcopy( n-k+1, w( k, k+1 ), 1, w( k, k ), 1 )
               ELSE
*
*                 interchange rows and columns K+1 and IMAX, use 2-by-2
*                 pivot block
*
                  kp = imax
                  kstep = 2
               END IF
            END IF
*
*           ============================================================
*
*           KK is the column of A where pivoting step stopped
*
            kk = k + kstep - 1
*
*           Interchange rows and columns KP and KK.
*           Updated column KP is already stored in column KK of W.
*
            IF( kp.NE.kk ) THEN
*
*              Copy non-updated column KK to column KP of submatrix A
*              at step K. No need to copy element into column K
*              (or K and K+1 for 2-by-2 pivot) of A, since these columns
*              will be later overwritten.
*
               a( kp, kp ) = a( kk, kk )
               CALL dcopy( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
     $                     lda )
               IF( kp.LT.n )
     $            CALL dcopy( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
*
*              Interchange rows KK and KP in first K-1 columns of A
*              (columns K (or K and K+1 for 2-by-2 pivot) of A will be
*              later overwritten). Interchange rows KK and KP
*              in first KK columns of W.
*
               IF( k.GT.1 )
     $            CALL dswap( k-1, a( kk, 1 ), lda, a( kp, 1 ), lda )
               CALL dswap( 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 subdiag. elements of column L(k)
*              and 1-by-1 block D(k) in column k of A.
*              (NOTE: Diagonal element L(k,k) is a UNIT element
*              and not stored)
*                 A(k,k) := D(k,k) = W(k,k)
*                 A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
*
               CALL dcopy( n-k+1, w( k, k ), 1, a( k, k ), 1 )
               IF( k.LT.n ) THEN
                  r1 = one / a( k, k )
                  CALL dscal( n-k, r1, a( k+1, k ), 1 )
               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
*
*              Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
*              block D(k:k+1,k:k+1) in columns k and k+1 of A.
*              (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
*              block and not stored)
*                 A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
*                 A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
*                 = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
*
               IF( k.LT.n-1 ) THEN
*
*                 Compose the columns of the inverse of 2-by-2 pivot
*                 block D in the following way to reduce the number
*                 of FLOPS when we myltiply panel ( W(k) W(k+1) ) by
*                 this inverse
*
*                 D**(-1) = ( d11 d21 )**(-1) =
*                           ( d21 d22 )
*
*                 = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
*                                        ( (-d21 ) ( d11 ) )
*
*                 = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
*
*                   * ( ( d22/d21 ) (      -1 ) ) =
*                     ( (      -1 ) ( d11/d21 ) )
*
*                 = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) (  -1 ) ) =
*                                           ( ( -1  ) ( D22 ) )
*
*                 = 1/d21 * T * ( ( D11 ) (  -1 ) )
*                               ( (  -1 ) ( D22 ) )
*
*                 = D21 * ( ( D11 ) (  -1 ) )
*                         ( (  -1 ) ( D22 ) )
*
                  d21 = w( k+1, k )
                  d11 = w( k+1, k+1 ) / d21
                  d22 = w( k, k ) / d21
                  t = one / ( d11*d22-one )
                  d21 = t / d21
*
*                 Update elements in columns A(k) and A(k+1) as
*                 dot products of rows of ( W(k) W(k+1) ) and columns
*                 of D**(-1)
*
                  DO 80 j = k + 2, n
                     a( j, k ) = d21*( d11*w( j, k )-w( j, k+1 ) )
                     a( j, k+1 ) = d21*( d22*w( j, k+1 )-w( j, k ) )
   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 ) = -kp
            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 dgemv( '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 dgemm( '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
*        of rows in columns 1:k-1 looping backwards from k-1 to 1
*
         j = k - 1
  120    CONTINUE
*
*           Undo the interchanges (if any) of rows JJ and JP at each
*           step J
*
*           (Here, J is a diagonal index)
            jj = j
            jp = ipiv( j )
            IF( jp.LT.0 ) THEN
               jp = -jp
*              (Here, J is a diagonal index)
               j = j - 1
            END IF
*           (NOTE: Here, J is used to determine row length. Length J
*           of the rows to swap back doesn't include diagonal element)
            j = j - 1
            IF( jp.NE.jj .AND. j.GE.1 )
     $         CALL dswap( j, a( jp, 1 ), lda, a( jj, 1 ), lda )
         IF( j.GT.1 )
     $      GO TO 120
*
*        Set KB to the number of columns factorized
*
         kb = k - 1
*
      END IF
      RETURN
*
*     End of DLASYF
*
      END