cgeev.f

          SUBROUTINE CGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR,
         $                  WORK, LWORK, RWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          JOBVL, JOBVR
          INTEGER            INFO, LDA, LDVL, LDVR, LWORK, N
    *     ..
    *     .. Array Arguments ..
          REAL               RWORK( * )
          COMPLEX            A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
         $                   W( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CGEEV 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.
    *
    *  Arguments
    *  =========
    *
    *  JOBVL   (input) CHARACTER*1
    *          = 'N': left eigenvectors of A are not computed;
    *          = 'V': left eigenvectors of are computed.
    *
    *  JOBVR   (input) CHARACTER*1
    *          = 'N': right eigenvectors of A are not computed;
    *          = 'V': right eigenvectors of A are computed.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A. N >= 0.
    *
    *  A       (input/output) COMPLEX array, dimension (LDA,N)
    *          On entry, the N-by-N matrix A.
    *          On exit, A has been overwritten.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  W       (output) COMPLEX array, dimension (N)
    *          W contains the computed eigenvalues.
    *
    *  VL      (output) COMPLEX 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.
    *
    *  LDVL    (input) INTEGER
    *          The leading dimension of the array VL.  LDVL >= 1; if
    *          JOBVL = 'V', LDVL >= N.
    *
    *  VR      (output) COMPLEX 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.
    *
    *  LDVR    (input) INTEGER
    *          The leading dimension of the array VR.  LDVR >= 1; if
    *          JOBVR = 'V', LDVR >= N.
    *
    *  WORK    (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) 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.
    *
    *  RWORK   (workspace) REAL array, dimension (2*N)
    *
    *  INFO    (output) 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 and i+1:N of W contain eigenvalues which have
    *                converged.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR
          CHARACTER          SIDE
          INTEGER            HSWORK, I, IBAL, IERR, IHI, ILO, IRWORK, ITAU,
         $                   IWRK, K, MAXWRK, MINWRK, NOUT
          REAL               ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM
          COMPLEX            TMP
    *     ..
    *     .. Local Arrays ..
          LOGICAL            SELECT( 1 )
          REAL               DUM( 1 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY, CLASCL,
         $                   CSCAL, CSSCAL, CTREVC, CUNGHR, SLABAD, XERBLA
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV, ISAMAX
          REAL               CLANGE, SCNRM2, SLAMCH
          EXTERNAL           LSAME, ILAENV, ISAMAX, CLANGE, SCNRM2, SLAMCH
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          AIMAG, CMPLX, CONJG, MAX, REAL, 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 CHSEQR, 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, 'CGEHRD', ' ', N, 1, N, 0 )
                MINWRK = 2*N
                IF( WANTVL ) THEN
                   MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR',
         $                       ' ', N, 1, N, -1 ) )
                   CALL CHSEQR( '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, 'CUNGHR',
         $                       ' ', N, 1, N, -1 ) )
                   CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR,
         $                WORK, -1, INFO )
                ELSE
                   CALL CHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR,
         $                WORK, -1, INFO )
                END IF
                HSWORK = 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( 'CGEEV ', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
    *     Get machine constants
    *
          EPS = SLAMCH( 'P' )
          SMLNUM = SLAMCH( 'S' )
          BIGNUM = ONE / SMLNUM
          CALL SLABAD( SMLNUM, BIGNUM )
          SMLNUM = SQRT( SMLNUM ) / EPS
          BIGNUM = ONE / SMLNUM
    *
    *     Scale A if max element outside range [SMLNUM,BIGNUM]
    *
          ANRM = CLANGE( '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 CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
    *
    *     Balance the matrix
    *     (CWorkspace: none)
    *     (RWorkspace: need N)
    *
          IBAL = 1
          CALL CGEBAL( '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 CGEHRD( 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 CLACPY( '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 CUNGHR( 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 CHSEQR( '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 CLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
             END IF
    *
          ELSE IF( WANTVR ) THEN
    *
    *        Want right eigenvectors
    *        Copy Householder vectors to VR
    *
             SIDE = 'R'
             CALL CLACPY( '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 CUNGHR( 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 CHSEQR( '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 CHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, W, VR, LDVR,
         $                WORK( IWRK ), LWORK-IWRK+1, INFO )
          END IF
    *
    *     If INFO > 0 from CHSEQR, then quit
    *
          IF( INFO.GT.0 )
         $   GO TO 50
    *
          IF( WANTVL .OR. WANTVR ) THEN
    *
    *        Compute left and/or right eigenvectors
    *        (CWorkspace: need 2*N)
    *        (RWorkspace: need 2*N)
    *
             IRWORK = IBAL + N
             CALL CTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
         $                N, NOUT, WORK( IWRK ), RWORK( IRWORK ), IERR )
          END IF
    *
          IF( WANTVL ) THEN
    *
    *        Undo balancing of left eigenvectors
    *        (CWorkspace: none)
    *        (RWorkspace: need N)
    *
             CALL CGEBAK( '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 / SCNRM2( N, VL( 1, I ), 1 )
                CALL CSSCAL( N, SCL, VL( 1, I ), 1 )
                DO 10 K = 1, N
                   RWORK( IRWORK+K-1 ) = REAL( VL( K, I ) )**2 +
         $                               AIMAG( VL( K, I ) )**2
       10       CONTINUE
                K = ISAMAX( N, RWORK( IRWORK ), 1 )
                TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
                CALL CSCAL( N, TMP, VL( 1, I ), 1 )
                VL( K, I ) = CMPLX( REAL( VL( K, I ) ), ZERO )
       20    CONTINUE
          END IF
    *
          IF( WANTVR ) THEN
    *
    *        Undo balancing of right eigenvectors
    *        (CWorkspace: none)
    *        (RWorkspace: need N)
    *
             CALL CGEBAK( '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 / SCNRM2( N, VR( 1, I ), 1 )
                CALL CSSCAL( N, SCL, VR( 1, I ), 1 )
                DO 30 K = 1, N
                   RWORK( IRWORK+K-1 ) = REAL( VR( K, I ) )**2 +
         $                               AIMAG( VR( K, I ) )**2
       30       CONTINUE
                K = ISAMAX( N, RWORK( IRWORK ), 1 )
                TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) )
                CALL CSCAL( N, TMP, VR( 1, I ), 1 )
                VR( K, I ) = CMPLX( REAL( VR( K, I ) ), ZERO )
       40    CONTINUE
          END IF
    *
    *     Undo scaling if necessary
    *
       50 CONTINUE
          IF( SCALEA ) THEN
             CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ),
         $                MAX( N-INFO, 1 ), IERR )
             IF( INFO.GT.0 ) THEN
                CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
             END IF
          END IF
    *
          WORK( 1 ) = MAXWRK
          RETURN
    *
    *     End of CGEEV
    *
          END