clahef_rk.f

*> \brief \b CLAHEF_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 CLAHEF_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            A( LDA, * ), E( * ), W( LDW, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*> CLAHEF_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.
*>
*> CLAHEF_RK is an auxiliary routine called by CHETRF_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 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 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 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 lahef_rk
*
*> \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 CLAHEF_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            A( LDA, * ), W( LDW, * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. 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 )
      COMPLEX            CONE, CZERO
      PARAMETER          ( CONE = ( 1.0E+0, 0.0E+0 ),
     $                   CZERO = ( 0.0E+0, 0.0E+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            DONE
      INTEGER            IMAX, ITEMP, II, J, JMAX, K, KK, KKW,
     $                   KP, KSTEP, KW, P
      REAL               ABSAKK, ALPHA, COLMAX, STEMP, R1, ROWMAX, T,
     $                   SFMIN
      COMPLEX            D11, D21, D22, Z
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ICAMAX
      REAL               SLAMCH
      EXTERNAL           LSAME, ICAMAX, SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           CCOPY, CSSCAL, CGEMM, CGEMV, CLACGV,
     $                   CSWAP
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, CONJG, AIMAG, MAX, MIN, REAL, SQRT
*     ..
*     .. Statement Functions ..
      REAL               CABS1
*     ..
*     .. Statement Function definitions ..
      CABS1( Z ) = ABS( REAL( Z ) ) + ABS( AIMAG( Z ) )
*     ..
*     .. 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 (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 CCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 )
         W( K, KW ) = REAL( A( K, K ) )
         IF( K.LT.N ) THEN
            CALL CGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ),
     $                  LDA,
     $                  W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
            W( K, KW ) = REAL( 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( REAL( 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 = ICAMAX( 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 ) = REAL( W( K, KW ) )
            IF( K.GT.1 )
     $         CALL CCOPY( 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 CCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1,
     $                           KW-1 ),
     $                           1 )
                  W( IMAX, KW-1 ) = REAL( A( IMAX, IMAX ) )
*
                  CALL CCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
     $                        W( IMAX+1, KW-1 ), 1 )
                  CALL CLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
*
                  IF( K.LT.N ) THEN
                     CALL CGEMV( '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 ) = REAL( 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 + ICAMAX( 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 = ICAMAX( IMAX-1, W( 1, KW-1 ), 1 )
                     STEMP = CABS1( W( ITEMP, KW-1 ) )
                     IF( STEMP.GT.ROWMAX ) THEN
                        ROWMAX = STEMP
                        JMAX = ITEMP
                     END IF
                  END IF
*
*                 Case(2)
*                 Equivalent to testing for
*                 ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
*                 (used to handle NaN and Inf)
*
                  IF( .NOT.( ABS( REAL( 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 CCOPY( 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 CCOPY( 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 ) = REAL( A( K, K ) )
               CALL CCOPY( K-1-P, A( P+1, K ), 1, A( P, P+1 ),
     $                     LDA )
               CALL CLACGV( K-1-P, A( P, P+1 ), LDA )
               IF( P.GT.1 )
     $            CALL CCOPY( 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 CSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ),
     $                        LDA )
               CALL CSWAP( 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 ) = REAL( A( KK, KK ) )
               CALL CCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
     $                     LDA )
               CALL CLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
               IF( KP.GT.1 )
     $            CALL CCOPY( 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 CSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
     $                        LDA )
               CALL CSWAP( 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 ) = REAL( W( K, K) ) to separately copy diagonal
*              element D(k,k) from W (potentially saves only one load))
               CALL CCOPY( 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 = REAL( A( K, K ) )
                  IF( ABS( T ).GE.SFMIN ) THEN
                     R1 = ONE / T
                     CALL CSSCAL( 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 CLACGV( 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 ) / CONJG( D21 )
                  D22 = W( K-1, KW-1 ) / D21
                  T = ONE / ( REAL( 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 ) ) /
     $                           CONJG( 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 CLACGV( K-1, W( 1, KW ), 1 )
               CALL CLACGV( 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
*
*        (note that conjg(W) is actually stored)
*
         CALL CGEMMTR( 'Upper', 'No transpose', 'Transpose', K, N-K,
     $                 -CONE, A( 1, K+1 ), LDA, W( 1, KW+1 ), LDW,
     $                 CONE, A( 1, 1 ), LDA )
*
*        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 ) = REAL( A( K, K ) )
         IF( K.LT.N )
     $      CALL CCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
         IF( K.GT.1 ) THEN
            CALL CGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
     $                  LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
            W( K, K ) = REAL( 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( REAL( 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 + ICAMAX( 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 ) = REAL( W( K, K ) )
            IF( K.LT.N )
     $         CALL CCOPY( 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 CCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ),
     $                        1)
                  CALL CLACGV( IMAX-K, W( K, K+1 ), 1 )
                  W( IMAX, K+1 ) = REAL( A( IMAX, IMAX ) )
*
                  IF( IMAX.LT.N )
     $               CALL CCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
     $                           W( IMAX+1, K+1 ), 1 )
*
                  IF( K.GT.1 ) THEN
                     CALL CGEMV( '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 ) = REAL( 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 + ICAMAX( 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 + ICAMAX( N-IMAX, W( IMAX+1, K+1 ),
     $                                      1)
                     STEMP = CABS1( W( ITEMP, K+1 ) )
                     IF( STEMP.GT.ROWMAX ) THEN
                        ROWMAX = STEMP
                        JMAX = ITEMP
                     END IF
                  END IF
*
*                 Case(2)
*                 Equivalent to testing for
*                 ABS( REAL( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX
*                 (used to handle NaN and Inf)
*
                  IF( .NOT.( ABS( REAL( 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 CCOPY( 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 CCOPY( 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 ) = REAL( A( K, K ) )
               CALL CCOPY( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
               CALL CLACGV( P-K-1, A( P, K+1 ), LDA )
               IF( P.LT.N )
     $            CALL CCOPY( 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 CSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
               CALL CSWAP( 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 ) = REAL( A( KK, KK ) )
               CALL CCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
     $                     LDA )
               CALL CLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
               IF( KP.LT.N )
     $            CALL CCOPY( 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 CSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
               CALL CSWAP( 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 ) = REAL( W( K, K) ) to separately copy diagonal
*              element D(k,k) from W (potentially saves only one load))
               CALL CCOPY( 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 = REAL( A( K, K ) )
                  IF( ABS( T ).GE.SFMIN ) THEN
                     R1 = ONE / T
                     CALL CSSCAL( 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 CLACGV( 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 ) / CONJG( D21 )
                  T = ONE / ( REAL( 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 ) ) /
     $                           CONJG( 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 CLACGV( N-K, W( K+1, K ), 1 )
               CALL CLACGV( 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
*
*        (note that conjg(W) is actually stored)
*
         CALL CGEMMTR( 'Lower', 'No transpose', 'Transpose', N-K+1,
     $                 K-1, -CONE, A( K, 1 ), LDA, W( K, 1 ), LDW,
     $                 CONE, A( K, K ), LDA )
*
*        Set KB to the number of columns factorized
*
         KB = K - 1
*
      END IF
      RETURN
*
*     End of CLAHEF_RK
*
      END