OpenBLAS/lapack-netlib/SRC/zgeev.f

*> \brief <b> ZGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE ZGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR,
*                         WORK, LWORK, RWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVL, JOBVR
*       INTEGER            INFO, LDA, LDVL, LDVR, LWORK, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   RWORK( * )
*       COMPLEX*16         A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   W( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the
*> eigenvalues and, optionally, the left and/or right eigenvectors.
*>
*> The right eigenvector v(j) of A satisfies
*>                  A * v(j) = lambda(j) * v(j)
*> where lambda(j) is its eigenvalue.
*> The left eigenvector u(j) of A satisfies
*>               u(j)**H * A = lambda(j) * u(j)**H
*> where u(j)**H denotes the conjugate transpose of u(j).
*>
*> The computed eigenvectors are normalized to have Euclidean norm
*> equal to 1 and largest component real.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N': left eigenvectors of A are not computed;
*>          = 'V': left eigenvectors of are computed.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N': right eigenvectors of A are not computed;
*>          = 'V': right eigenvectors of A are computed.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension (LDA,N)
*>          On entry, the N-by-N matrix A.
*>          On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is COMPLEX*16 array, dimension (N)
*>          W contains the computed eigenvalues.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*>          VL is COMPLEX*16 array, dimension (LDVL,N)
*>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*>          after another in the columns of VL, in the same order
*>          as their eigenvalues.
*>          If JOBVL = 'N', VL is not referenced.
*>          u(j) = VL(:,j), the j-th column of VL.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.  LDVL >= 1; if
*>          JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is COMPLEX*16 array, dimension (LDVR,N)
*>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*>          after another in the columns of VR, in the same order
*>          as their eigenvalues.
*>          If JOBVR = 'N', VR is not referenced.
*>          v(j) = VR(:,j), the j-th column of VR.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.  LDVR >= 1; if
*>          JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,2*N).
*>          For good performance, LWORK must generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, the QR algorithm failed to compute all the
*>                eigenvalues, and no eigenvectors have been computed;
*>                elements i+1:N of W contain eigenvalues which have
*>                converged.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*
*  @precisions fortran z -> c
*
*> \ingroup complex16GEeigen
*
*  =====================================================================
      SUBROUTINE ZGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR,
     $                  WORK, LWORK, RWORK, INFO )
      implicit none
*
*  -- LAPACK driver 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          JOBVL, JOBVR
      INTEGER            INFO, LDA, LDVL, LDVR, LWORK, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   RWORK( * )
      COMPLEX*16         A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   W( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR
      CHARACTER          SIDE
      INTEGER            HSWORK, I, IBAL, IERR, IHI, ILO, IRWORK, ITAU,
     $                   IWRK, K, LWORK_TREVC, MAXWRK, MINWRK, NOUT
      DOUBLE PRECISION   ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM
      COMPLEX*16         TMP
*     ..
*     .. Local Arrays ..
      LOGICAL            SELECT( 1 )
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLABAD, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL, ZGEHRD,
     $                   ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC3, ZUNGHR
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX, ILAENV
      DOUBLE PRECISION   DLAMCH, DZNRM2, ZLANGE
      EXTERNAL           LSAME, IDAMAX, ILAENV, DLAMCH, DZNRM2, ZLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, DCMPLX, CONJG, AIMAG, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      WANTVL = LSAME( JOBVL, 'V' )
      WANTVR = LSAME( JOBVR, 'V' )
      IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
         INFO = -8
      ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
         INFO = -10
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       CWorkspace refers to complex workspace, and RWorkspace to real
*       workspace. NB refers to the optimal block size for the
*       immediately following subroutine, as returned by ILAENV.
*       HSWORK refers to the workspace preferred by ZHSEQR, as
*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
*       the worst case.)
*
      IF( INFO.EQ.0 ) THEN
         IF( N.EQ.0 ) THEN
            MINWRK = 1
            MAXWRK = 1
         ELSE
            MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 )
            MINWRK = 2*N
            IF( WANTVL ) THEN
               MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
     $                       ' ', N, 1, N, -1 ) )
               CALL ZTREVC3( 'L', 'B', SELECT, N, A, LDA,
     $                       VL, LDVL, VR, LDVR,
     $                       N, NOUT, WORK, -1, RWORK, -1, IERR )
               LWORK_TREVC = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
               CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL,
     $                      WORK, -1, INFO )
            ELSE IF( WANTVR ) THEN
               MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
     $                       ' ', N, 1, N, -1 ) )
               CALL ZTREVC3( 'R', 'B', SELECT, N, A, LDA,
     $                       VL, LDVL, VR, LDVR,
     $                       N, NOUT, WORK, -1, RWORK, -1, IERR )
               LWORK_TREVC = INT( WORK(1) )
               MAXWRK = MAX( MAXWRK, N + LWORK_TREVC )
               CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR,
     $                      WORK, -1, INFO )
            ELSE
               CALL ZHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR,
     $                      WORK, -1, INFO )
            END IF
            HSWORK = INT( WORK(1) )
            MAXWRK = MAX( MAXWRK, HSWORK, MINWRK )
         END IF
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -12
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'ZGEEV ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      CALL DLABAD( SMLNUM, BIGNUM )
      SMLNUM = SQRT( SMLNUM ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = ZLANGE( 'M', N, N, A, LDA, DUM )
      SCALEA = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = SMLNUM
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         SCALEA = .TRUE.
         CSCALE = BIGNUM
      END IF
      IF( SCALEA )
     $   CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
*     Balance the matrix
*     (CWorkspace: none)
*     (RWorkspace: need N)
*
      IBAL = 1
      CALL ZGEBAL( 'B', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
*
*     Reduce to upper Hessenberg form
*     (CWorkspace: need 2*N, prefer N+N*NB)
*     (RWorkspace: none)
*
      ITAU = 1
      IWRK = ITAU + N
      CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
     $             LWORK-IWRK+1, IERR )
*
      IF( WANTVL ) THEN
*
*        Want left eigenvectors
*        Copy Householder vectors to VL
*
         SIDE = 'L'
         CALL ZLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
*        Generate unitary matrix in VL
*        (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
*        (RWorkspace: none)
*
         CALL ZUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
     $                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VL
*        (CWorkspace: need 1, prefer HSWORK (see comments) )
*        (RWorkspace: none)
*
         IWRK = ITAU
         CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
         IF( WANTVR ) THEN
*
*           Want left and right eigenvectors
*           Copy Schur vectors to VR
*
            SIDE = 'B'
            CALL ZLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
         END IF
*
      ELSE IF( WANTVR ) THEN
*
*        Want right eigenvectors
*        Copy Householder vectors to VR
*
         SIDE = 'R'
         CALL ZLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
*        Generate unitary matrix in VR
*        (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
*        (RWorkspace: none)
*
         CALL ZUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
     $                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VR
*        (CWorkspace: need 1, prefer HSWORK (see comments) )
*        (RWorkspace: none)
*
         IWRK = ITAU
         CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
      ELSE
*
*        Compute eigenvalues only
*        (CWorkspace: need 1, prefer HSWORK (see comments) )
*        (RWorkspace: none)
*
         IWRK = ITAU
         CALL ZHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, W, VR, LDVR,
     $                WORK( IWRK ), LWORK-IWRK+1, INFO )
      END IF
*
*     If INFO .NE. 0 from ZHSEQR, then quit
*
      IF( INFO.NE.0 )
     $   GO TO 50
*
      IF( WANTVL .OR. WANTVR ) THEN
*
*        Compute left and/or right eigenvectors
*        (CWorkspace: need 2*N, prefer N + 2*N*NB)
*        (RWorkspace: need 2*N)
*
         IRWORK = IBAL + N
         CALL ZTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
     $                 N, NOUT, WORK( IWRK ), LWORK-IWRK+1,
     $                 RWORK( IRWORK ), N, IERR )
      END IF
*
      IF( WANTVL ) THEN
*
*        Undo balancing of left eigenvectors
*        (CWorkspace: none)
*        (RWorkspace: need N)
*
         CALL ZGEBAK( 'B', 'L', N, ILO, IHI, RWORK( IBAL ), N, VL, LDVL,
     $                IERR )
*
*        Normalize left eigenvectors and make largest component real
*
         DO 20 I = 1, N
            SCL = ONE / DZNRM2( N, VL( 1, I ), 1 )
            CALL ZDSCAL( N, SCL, VL( 1, I ), 1 )
            DO 10 K = 1, N
               RWORK( IRWORK+K-1 ) = DBLE( VL( K, I ) )**2 +
     $                               AIMAG( VL( K, I ) )**2
   10       CONTINUE
            K = IDAMAX( N, RWORK( IRWORK ), 1 )
            TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
            CALL ZSCAL( N, TMP, VL( 1, I ), 1 )
            VL( K, I ) = DCMPLX( DBLE( VL( K, I ) ), ZERO )
   20    CONTINUE
      END IF
*
      IF( WANTVR ) THEN
*
*        Undo balancing of right eigenvectors
*        (CWorkspace: none)
*        (RWorkspace: need N)
*
         CALL ZGEBAK( 'B', 'R', N, ILO, IHI, RWORK( IBAL ), N, VR, LDVR,
     $                IERR )
*
*        Normalize right eigenvectors and make largest component real
*
         DO 40 I = 1, N
            SCL = ONE / DZNRM2( N, VR( 1, I ), 1 )
            CALL ZDSCAL( N, SCL, VR( 1, I ), 1 )
            DO 30 K = 1, N
               RWORK( IRWORK+K-1 ) = DBLE( VR( K, I ) )**2 +
     $                               AIMAG( VR( K, I ) )**2
   30       CONTINUE
            K = IDAMAX( N, RWORK( IRWORK ), 1 )
            TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
            CALL ZSCAL( N, TMP, VR( 1, I ), 1 )
            VR( K, I ) = DCMPLX( DBLE( VR( K, I ) ), ZERO )
   40    CONTINUE
      END IF
*
*     Undo scaling if necessary
*
   50 CONTINUE
      IF( SCALEA ) THEN
         CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ),
     $                MAX( N-INFO, 1 ), IERR )
         IF( INFO.GT.0 ) THEN
            CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
         END IF
      END IF
*
      WORK( 1 ) = MAXWRK
      RETURN
*
*     End of ZGEEV
*
      END