zlahef_rk.f

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*> \brief \b ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZLAHEF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
*                             INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, KB, LDA, LDW, N, NB
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       COMPLEX*16         A( LDA, * ), E( * ), W( LDW, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*> ZLAHEF_RK computes a partial factorization of a complex Hermitian
*> 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**H U22**H )
*>
*> A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  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.
*>
*> ZLAHEF_RK is an auxiliary routine called by ZHETRF_RK. 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
*>          Hermitian 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 COMPLEX*16 array, dimension (LDA,N)
*>          On entry, the Hermitian 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 Hermitian 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*>          E is COMPLEX*16 array, dimension (N)
*>          On exit, contains the superdiagonal (or subdiagonal)
*>          elements of the Hermitian 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.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*>          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 Hermitian 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.
*>
*>          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 submatrix A(1:N,N-KB+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,N-KB+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 submatrix A(1:N,N-KB+1:N).
*>                  If -IPIV(k-1) = k-1, no interchange occurred.
*>
*>            c) In both cases a) and b) is 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 submatrix A(1:N,1:KB).
*>               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 submatrix A(1:N,1:KB).
*>                  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 submatrix A(1:N,1:KB).
*>                  If -IPIV(k+1) = k+1, no interchange occurred.
*>
*>            c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
*>
*>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is COMPLEX*16 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, 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.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16HEcomputational
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>  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
*>
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE zlahef_rk( UPLO, N, NB, KB, A, LDA, E, 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( * )
      COMPLEX*16         A( LDA, * ), W( LDW, * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
      COMPLEX*16         CONE
      parameter( cone = ( 1.0d+0, 0.0d+0 ) )
      DOUBLE PRECISION   EIGHT, SEVTEN
      parameter( eight = 8.0d+0, sevten = 17.0d+0 )
      COMPLEX*16         CZERO
      parameter( czero = ( 0.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            DONE
      INTEGER            IMAX, ITEMP, II, J, JB, JJ, JMAX, K, KK, KKW,
     $                   kp, kstep, kw, p
      DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, DTEMP, R1, ROWMAX, T,
     $                   sfmin
      COMPLEX*16         D11, D21, D22, Z
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IZAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           lsame, izamax, dlamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           zcopy, zdscal, zgemm, zgemv, zlacgv, zswap
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dconjg, dimag, max, min, sqrt
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( z ) = abs( dble( z ) ) + abs( dimag( z ) )
*     ..
*     .. Executable Statements ..
*
      info = 0
*
*     Initialize ALPHA for use in choosing pivot block size.
*
      alpha = ( one+sqrt( sevten ) ) / eight
*
*     Compute machine safe minimum
*
      sfmin = dlamch( '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 (note that conjg(W) is actually stored)
*        Initialize the first entry of array E, where superdiagonal
*        elements of D are stored
*
         e( 1 ) = czero
*
*        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
*
         IF( k.GT.1 )
     $      CALL zcopy( k-1, a( 1, k ), 1, w( 1, kw ), 1 )
         w( k, kw ) = dble( a( k, k ) )
         IF( k.LT.n ) THEN
            CALL zgemv( 'No transpose', k, n-k, -cone, a( 1, k+1 ), lda,
     $                  w( k, kw+1 ), ldw, cone, w( 1, kw ), 1 )
            w( k, kw ) = dble( w( k, kw ) )
         END IF
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         absakk = abs( dble( 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 = izamax( k-1, w( 1, kw ), 1 )
            colmax = cabs1( 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
            a( k, k ) = dble( w( k, kw ) )
            IF( k.GT.1 )
     $         CALL zcopy( k-1, w( 1, kw ), 1, a( 1, k ), 1 )
*
*           Set E( K ) to zero
*
            IF( k.GT.1 )
     $         e( k ) = czero
*
         ELSE
*
*           ============================================================
*
*           BEGIN pivot search
*
*           Case(1)
*           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
*
*              Lop until pivot found
*
               done = .false.
*
   12          CONTINUE
*
*                 BEGIN pivot search loop body
*
*
*                 Copy column IMAX to column KW-1 of W and update it
*
                  IF( imax.GT.1 )
     $               CALL zcopy( imax-1, a( 1, imax ), 1, w( 1, kw-1 ),
     $                           1 )
                  w( imax, kw-1 ) = dble( a( imax, imax ) )
*
                  CALL zcopy( k-imax, a( imax, imax+1 ), lda,
     $                        w( imax+1, kw-1 ), 1 )
                  CALL zlacgv( k-imax, w( imax+1, kw-1 ), 1 )
*
                  IF( k.LT.n ) THEN
                     CALL zgemv( 'No transpose', k, n-k, -cone,
     $                           a( 1, k+1 ), lda, w( imax, kw+1 ), ldw,
     $                           cone, w( 1, kw-1 ), 1 )
                     w( imax, kw-1 ) = dble( w( imax, kw-1 ) )
                  END IF
*
*                 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 + izamax( k-imax, w( imax+1, kw-1 ),
     $                                     1 )
                     rowmax = cabs1( w( jmax, kw-1 ) )
                  ELSE
                     rowmax = zero
                  END IF
*
                  IF( imax.GT.1 ) THEN
                     itemp = izamax( imax-1, w( 1, kw-1 ), 1 )
                     dtemp = cabs1( w( itemp, kw-1 ) )
                     IF( dtemp.GT.rowmax ) THEN
                        rowmax = dtemp
                        jmax = itemp
                     END IF
                  END IF
*
*                 Case(2)
*                 Equivalent to testing for
*                 ABS( DBLE( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
*                 (used to handle NaN and Inf)
*
                  IF( .NOT.( abs( dble( 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 zcopy( k, w( 1, kw-1 ), 1, w( 1, kw ), 1 )
*
                     done = .true.
*
*                 Case(3)
*                 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.
*
*                 Case(4)
                  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 zcopy( k, w( 1, kw-1 ), 1, w( 1, kw ), 1 )
*
                  END IF
*
*
*                 END pivot search loop body
*
               IF( .NOT.done ) GOTO 12
*
            END IF
*
*           END pivot search
*
*           ============================================================
*
*           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 P and K.
*           Updated column P is already stored in column KW of W.
*
            IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
*
*              Copy non-updated column K to column P of submatrix A
*              at step K. No need to copy element into columns
*              K and K-1 of A for 2-by-2 pivot, since these columns
*              will be later overwritten.
*
               a( p, p ) = dble( a( k, k ) )
               CALL zcopy( k-1-p, a( p+1, k ), 1, a( p, p+1 ),
     $                     lda )
               CALL zlacgv( k-1-p, a( p, p+1 ), lda )
               IF( p.GT.1 )
     $            CALL zcopy( p-1, a( 1, k ), 1, a( 1, p ), 1 )
*
*              Interchange rows K and P in the last K+1 to N columns of A
*              (columns K and K-1 of A for 2-by-2 pivot will be
*              later overwritten). Interchange rows K and P
*              in last KKW to NB columns of W.
*
               IF( k.LT.n )
     $            CALL zswap( n-k, a( k, k+1 ), lda, a( p, k+1 ),
     $                        lda )
               CALL zswap( n-kk+1, w( k, kkw ), ldw, w( p, kkw ),
     $                     ldw )
            END IF
*
*           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 ) = dble( a( kk, kk ) )
               CALL zcopy( kk-1-kp, a( kp+1, kk ), 1, a( kp, kp+1 ),
     $                     lda )
               CALL zlacgv( kk-1-kp, a( kp, kp+1 ), lda )
               IF( kp.GT.1 )
     $            CALL zcopy( 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 zswap( n-k, a( kk, k+1 ), lda, a( kp, k+1 ),
     $                        lda )
               CALL zswap( 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
*
*              (1) 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)
*
*              (NOTE: No need to use for Hermitian matrix
*              A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
*              element D(k,k) from W (potentially saves only one load))
               CALL zcopy( k, w( 1, kw ), 1, a( 1, k ), 1 )
               IF( k.GT.1 ) THEN
*
*                 (NOTE: No need to check if A(k,k) is NOT ZERO,
*                  since that was ensured earlier in pivot search:
*                  case A(k,k) = 0 falls into 2x2 pivot case(3))
*
*                 Handle division by a small number
*
                  t = dble( a( k, k ) )
                  IF( abs( t ).GE.sfmin ) THEN
                     r1 = one / t
                     CALL zdscal( k-1, r1, a( 1, k ), 1 )
                  ELSE
                     DO 14 ii = 1, k-1
                        a( ii, k ) = a( ii, k ) / t
   14                CONTINUE
                  END IF
*
*                 (2) Conjugate column W(kw)
*
                  CALL zlacgv( k-1, w( 1, kw ), 1 )
*
*                 Store the superdiagonal element of D in array E
*
                  e( k ) = czero
*
               END IF
*
            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
*
*              (1) 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
*
*                 Factor out the columns of the inverse of 2-by-2 pivot
*                 block D, so that each column contains 1, to reduce the
*                 number of FLOPS when we multiply panel
*                 ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
*
*                 D**(-1) = ( d11 cj(d21) )**(-1) =
*                           ( d21    d22 )
*
*                 = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
*                                          ( (-d21) (     d11 ) )
*
*                 = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
*
*                   * ( d21*( d22/d21 ) conj(d21)*(           - 1 ) ) =
*                     (     (      -1 )           ( d11/conj(d21) ) )
*
*                 = 1/(|d21|**2) * 1/(D22*D11-1) *
*
*                   * ( d21*( D11 ) conj(d21)*(  -1 ) ) =
*                     (     (  -1 )           ( D22 ) )
*
*                 = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*(  -1 ) ) =
*                                      (     (  -1 )           ( D22 ) )
*
*                 = ( (T/conj(d21))*( D11 ) (T/d21)*(  -1 ) ) =
*                   (               (  -1 )         ( D22 ) )
*
*                 Handle division by a small number. (NOTE: order of
*                 operations is important)
*
*                 = ( T*(( D11 )/conj(D21)) T*((  -1 )/D21 ) )
*                   (   ((  -1 )          )   (( D22 )     ) ),
*
*                 where D11 = d22/d21,
*                       D22 = d11/conj(d21),
*                       D21 = d21,
*                       T = 1/(D22*D11-1).
*
*                 (NOTE: No need to check for division by ZERO,
*                  since that was ensured earlier in pivot search:
*                  (a) d21 != 0 in 2x2 pivot case(4),
*                      since |d21| should be larger than |d11| and |d22|;
*                  (b) (D22*D11 - 1) != 0, since from (a),
*                      both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
*
                  d21 = w( k-1, kw )
                  d11 = w( k, kw ) / dconjg( d21 )
                  d22 = w( k-1, kw-1 ) / d21
                  t = one / ( dble( d11*d22 )-one )
*
*                 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 ) = t*( ( d11*w( j, kw-1 )-w( j, kw ) ) /
     $                             d21 )
                     a( j, k ) = t*( ( d22*w( j, kw )-w( j, kw-1 ) ) /
     $                           dconjg( d21 ) )
   20             CONTINUE
               END IF
*
*              Copy diagonal elements of D(K) to A,
*              copy superdiagonal element of D(K) to E(K) and
*              ZERO out superdiagonal entry of A
*
               a( k-1, k-1 ) = w( k-1, kw-1 )
               a( k-1, k ) = czero
               a( k, k ) = w( k, kw )
               e( k ) = w( k-1, kw )
               e( k-1 ) = czero
*
*              (2) Conjugate columns W(kw) and W(kw-1)
*
               CALL zlacgv( k-1, w( 1, kw ), 1 )
               CALL zlacgv( k-2, w( 1, kw-1 ), 1 )
*
            END IF
*
*           End column K is nonsingular
*
         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**H = A11 - U12*W**H
*
*        computing blocks of NB columns at a time (note that conjg(W) is
*        actually stored)
*
         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
               a( jj, jj ) = dble( a( jj, jj ) )
               CALL zgemv( 'No transpose', jj-j+1, n-k, -cone,
     $                     a( j, k+1 ), lda, w( jj, kw+1 ), ldw, cone,
     $                     a( j, jj ), 1 )
               a( jj, jj ) = dble( a( jj, jj ) )
   40       CONTINUE
*
*           Update the rectangular superdiagonal block
*
            IF( j.GE.2 )
     $         CALL zgemm( 'No transpose', 'Transpose', j-1, jb, n-k,
     $                     -cone, a( 1, k+1 ), lda, w( j, kw+1 ), ldw,
     $                     cone, a( 1, j ), lda )
   50    CONTINUE
*
*        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 (note that conjg(W) is actually stored)
*
*        Initialize the unused last entry of the subdiagonal array E.
*
         e( n ) = czero
*
*        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 column K of W
*
         w( k, k ) = dble( a( k, k ) )
         IF( k.LT.n )
     $      CALL zcopy( n-k, a( k+1, k ), 1, w( k+1, k ), 1 )
         IF( k.GT.1 ) THEN
            CALL zgemv( 'No transpose', n-k+1, k-1, -cone, a( k, 1 ),
     $                  lda, w( k, 1 ), ldw, cone, w( k, k ), 1 )
            w( k, k ) = dble( w( k, k ) )
         END IF
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         absakk = abs( dble( 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 + izamax( n-k, w( k+1, k ), 1 )
            colmax = cabs1( 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
            a( k, k ) = dble( w( k, k ) )
            IF( k.LT.n )
     $         CALL zcopy( n-k, w( k+1, k ), 1, a( k+1, k ), 1 )
*
*           Set E( K ) to zero
*
            IF( k.LT.n )
     $         e( k ) = czero
*
         ELSE
*
*           ============================================================
*
*           BEGIN pivot search
*
*           Case(1)
*           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 zcopy( imax-k, a( imax, k ), lda, w( k, k+1 ), 1)
                  CALL zlacgv( imax-k, w( k, k+1 ), 1 )
                  w( imax, k+1 ) = dble( a( imax, imax ) )
*
                  IF( imax.LT.n )
     $               CALL zcopy( n-imax, a( imax+1, imax ), 1,
     $                           w( imax+1, k+1 ), 1 )
*
                  IF( k.GT.1 ) THEN
                     CALL zgemv( 'No transpose', n-k+1, k-1, -cone,
     $                            a( k, 1 ), lda, w( imax, 1 ), ldw,
     $                            cone, w( k, k+1 ), 1 )
                     w( imax, k+1 ) = dble( w( imax, k+1 ) )
                  END IF
*
*                 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 + izamax( imax-k, w( k, k+1 ), 1 )
                     rowmax = cabs1( w( jmax, k+1 ) )
                  ELSE
                     rowmax = zero
                  END IF
*
                  IF( imax.LT.n ) THEN
                     itemp = imax + izamax( n-imax, w( imax+1, k+1 ), 1)
                     dtemp = cabs1( w( itemp, k+1 ) )
                     IF( dtemp.GT.rowmax ) THEN
                        rowmax = dtemp
                        jmax = itemp
                     END IF
                  END IF
*
*                 Case(2)
*                 Equivalent to testing for
*                 ABS( DBLE( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX
*                 (used to handle NaN and Inf)
*
                  IF( .NOT.( abs( dble( 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 zcopy( n-k+1, w( k, k+1 ), 1, w( k, k ), 1 )
*
                     done = .true.
*
*                 Case(3)
*                 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.
*
*                 Case(4)
                  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 zcopy( 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
*
*           END pivot search
*
*           ============================================================
*
*           KK is the column of A where pivoting step stopped
*
            kk = k + kstep - 1
*
*           Interchange rows and columns P and K (only for 2-by-2 pivot).
*           Updated column P is already stored in column K of W.
*
            IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
*
*              Copy non-updated column KK-1 to column P of submatrix A
*              at step K. No need to copy element into columns
*              K and K+1 of A for 2-by-2 pivot, since these columns
*              will be later overwritten.
*
               a( p, p ) = dble( a( k, k ) )
               CALL zcopy( p-k-1, a( k+1, k ), 1, a( p, k+1 ), lda )
               CALL zlacgv( p-k-1, a( p, k+1 ), lda )
               IF( p.LT.n )
     $            CALL zcopy( n-p, a( p+1, k ), 1, a( p+1, p ), 1 )
*
*              Interchange rows K and P in first K-1 columns of A
*              (columns K and K+1 of A for 2-by-2 pivot will be
*              later overwritten). Interchange rows K and P
*              in first KK columns of W.
*
               IF( k.GT.1 )
     $            CALL zswap( k-1, a( k, 1 ), lda, a( p, 1 ), lda )
               CALL zswap( kk, w( k, 1 ), ldw, w( p, 1 ), ldw )
            END IF
*
*           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 ) = dble( a( kk, kk ) )
               CALL zcopy( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
     $                     lda )
               CALL zlacgv( kp-kk-1, a( kp, kk+1 ), lda )
               IF( kp.LT.n )
     $            CALL zcopy( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
*
*              Interchange rows KK and KP in first K-1 columns of A
*              (column 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 zswap( k-1, a( kk, 1 ), lda, a( kp, 1 ), lda )
               CALL zswap( 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
*
*              (1) 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)
*
*              (NOTE: No need to use for Hermitian matrix
*              A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
*              element D(k,k) from W (potentially saves only one load))
               CALL zcopy( n-k+1, w( k, k ), 1, a( k, k ), 1 )
               IF( k.LT.n ) THEN
*
*                 (NOTE: No need to check if A(k,k) is NOT ZERO,
*                  since that was ensured earlier in pivot search:
*                  case A(k,k) = 0 falls into 2x2 pivot case(3))
*
*                 Handle division by a small number
*
                  t = dble( a( k, k ) )
                  IF( abs( t ).GE.sfmin ) THEN
                     r1 = one / t
                     CALL zdscal( n-k, r1, a( k+1, k ), 1 )
                  ELSE
                     DO 74 ii = k + 1, n
                        a( ii, k ) = a( ii, k ) / t
   74                CONTINUE
                  END IF
*
*                 (2) Conjugate column W(k)
*
                  CALL zlacgv( n-k, w( k+1, k ), 1 )
*
*                 Store the subdiagonal element of D in array E
*
                  e( k ) = czero
*
               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
*
*              (1) 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
*
*                 Factor out the columns of the inverse of 2-by-2 pivot
*                 block D, so that each column contains 1, to reduce the
*                 number of FLOPS when we multiply panel
*                 ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
*
*                 D**(-1) = ( d11 cj(d21) )**(-1) =
*                           ( d21    d22 )
*
*                 = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
*                                          ( (-d21) (     d11 ) )
*
*                 = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
*
*                   * ( d21*( d22/d21 ) conj(d21)*(           - 1 ) ) =
*                     (     (      -1 )           ( d11/conj(d21) ) )
*
*                 = 1/(|d21|**2) * 1/(D22*D11-1) *
*
*                   * ( d21*( D11 ) conj(d21)*(  -1 ) ) =
*                     (     (  -1 )           ( D22 ) )
*
*                 = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*(  -1 ) ) =
*                                      (     (  -1 )           ( D22 ) )
*
*                 = ( (T/conj(d21))*( D11 ) (T/d21)*(  -1 ) ) =
*                   (               (  -1 )         ( D22 ) )
*
*                 Handle division by a small number. (NOTE: order of
*                 operations is important)
*
*                 = ( T*(( D11 )/conj(D21)) T*((  -1 )/D21 ) )
*                   (   ((  -1 )          )   (( D22 )     ) ),
*
*                 where D11 = d22/d21,
*                       D22 = d11/conj(d21),
*                       D21 = d21,
*                       T = 1/(D22*D11-1).
*
*                 (NOTE: No need to check for division by ZERO,
*                  since that was ensured earlier in pivot search:
*                  (a) d21 != 0 in 2x2 pivot case(4),
*                      since |d21| should be larger than |d11| and |d22|;
*                  (b) (D22*D11 - 1) != 0, since from (a),
*                      both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
*
                  d21 = w( k+1, k )
                  d11 = w( k+1, k+1 ) / d21
                  d22 = w( k, k ) / dconjg( d21 )
                  t = one / ( dble( d11*d22 )-one )
*
*                 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 ) = t*( ( d11*w( j, k )-w( j, k+1 ) ) /
     $                           dconjg( d21 ) )
                     a( j, k+1 ) = t*( ( d22*w( j, k+1 )-w( j, k ) ) /
     $                             d21 )
   80             CONTINUE
               END IF
*
*              Copy diagonal elements of D(K) to A,
*              copy subdiagonal element of D(K) to E(K) and
*              ZERO out subdiagonal entry of A
*
               a( k, k ) = w( k, k )
               a( k+1, k ) = czero
               a( k+1, k+1 ) = w( k+1, k+1 )
               e( k ) = w( k+1, k )
               e( k+1 ) = czero
*
*              (2) Conjugate columns W(k) and W(k+1)
*
               CALL zlacgv( n-k, w( k+1, k ), 1 )
               CALL zlacgv( n-k-1, w( k+2, k+1 ), 1 )
*
            END IF
*
*           End column K is nonsingular
*
         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**H = A22 - L21*W**H
*
*        computing blocks of NB columns at a time (note that conjg(W) is
*        actually stored)
*
         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
               a( jj, jj ) = dble( a( jj, jj ) )
               CALL zgemv( 'No transpose', j+jb-jj, k-1, -cone,
     $                     a( jj, 1 ), lda, w( jj, 1 ), ldw, cone,
     $                     a( jj, jj ), 1 )
               a( jj, jj ) = dble( a( jj, jj ) )
  100       CONTINUE
*
*           Update the rectangular subdiagonal block
*
            IF( j+jb.LE.n )
     $         CALL zgemm( 'No transpose', 'Transpose', n-j-jb+1, jb,
     $                     k-1, -cone, a( j+jb, 1 ), lda, w( j, 1 ),
     $                     ldw, cone, a( j+jb, j ), lda )
  110    CONTINUE
*
*        Set KB to the number of columns factorized
*
         kb = k - 1
*
      END IF
      RETURN
*
*     End of ZLAHEF_RK
*
      END