zgeevx.f

          SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL,
         $                   LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
         $                   RCONDV, 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          BALANC, JOBVL, JOBVR, SENSE
          INTEGER            IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
          DOUBLE PRECISION   ABNRM
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   RCONDE( * ), RCONDV( * ), RWORK( * ),
         $                   SCALE( * )
          COMPLEX*16         A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
         $                   W( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
    *  eigenvalues and, optionally, the left and/or right eigenvectors.
    *
    *  Optionally also, it computes a balancing transformation to improve
    *  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
    *  SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
    *  (RCONDE), and reciprocal condition numbers for the right
    *  eigenvectors (RCONDV).
    *
    *  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.
    *
    *  Balancing a matrix means permuting the rows and columns to make it
    *  more nearly upper triangular, and applying a diagonal similarity
    *  transformation D * A * D**(-1), where D is a diagonal matrix, to
    *  make its rows and columns closer in norm and the condition numbers
    *  of its eigenvalues and eigenvectors smaller.  The computed
    *  reciprocal condition numbers correspond to the balanced matrix.
    *  Permuting rows and columns will not change the condition numbers
    *  (in exact arithmetic) but diagonal scaling will.  For further
    *  explanation of balancing, see section 4.10.2 of the LAPACK
    *  Users' Guide.
    *
    *  Arguments
    *  =========
    *
    *  BALANC  (input) CHARACTER*1
    *          Indicates how the input matrix should be diagonally scaled
    *          and/or permuted to improve the conditioning of its
    *          eigenvalues.
    *          = 'N': Do not diagonally scale or permute;
    *          = 'P': Perform permutations to make the matrix more nearly
    *                 upper triangular. Do not diagonally scale;
    *          = 'S': Diagonally scale the matrix, ie. replace A by
    *                 D*A*D**(-1), where D is a diagonal matrix chosen
    *                 to make the rows and columns of A more equal in
    *                 norm. Do not permute;
    *          = 'B': Both diagonally scale and permute A.
    *
    *          Computed reciprocal condition numbers will be for the matrix
    *          after balancing and/or permuting. Permuting does not change
    *          condition numbers (in exact arithmetic), but balancing does.
    *
    *  JOBVL   (input) CHARACTER*1
    *          = 'N': left eigenvectors of A are not computed;
    *          = 'V': left eigenvectors of A are computed.
    *          If SENSE = 'E' or 'B', JOBVL must = 'V'.
    *
    *  JOBVR   (input) CHARACTER*1
    *          = 'N': right eigenvectors of A are not computed;
    *          = 'V': right eigenvectors of A are computed.
    *          If SENSE = 'E' or 'B', JOBVR must = 'V'.
    *
    *  SENSE   (input) CHARACTER*1
    *          Determines which reciprocal condition numbers are computed.
    *          = 'N': None are computed;
    *          = 'E': Computed for eigenvalues only;
    *          = 'V': Computed for right eigenvectors only;
    *          = 'B': Computed for eigenvalues and right eigenvectors.
    *
    *          If SENSE = 'E' or 'B', both left and right eigenvectors
    *          must also be computed (JOBVL = 'V' and JOBVR = 'V').
    *
    *  N       (input) INTEGER
    *          The order of the matrix A. N >= 0.
    *
    *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
    *          On entry, the N-by-N matrix A.
    *          On exit, A has been overwritten.  If JOBVL = 'V' or
    *          JOBVR = 'V', A contains the Schur form of the balanced
    *          version of the matrix A.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  W       (output) COMPLEX*16 array, dimension (N)
    *          W contains the computed eigenvalues.
    *
    *  VL      (output) 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.
    *
    *  LDVL    (input) INTEGER
    *          The leading dimension of the array VL.  LDVL >= 1; if
    *          JOBVL = 'V', LDVL >= N.
    *
    *  VR      (output) 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.
    *
    *  LDVR    (input) INTEGER
    *          The leading dimension of the array VR.  LDVR >= 1; if
    *          JOBVR = 'V', LDVR >= N.
    *
    *  ILO     (output) INTEGER
    *  IHI     (output) INTEGER
    *          ILO and IHI are integer values determined when A was
    *          balanced.  The balanced A(i,j) = 0 if I > J and
    *          J = 1,...,ILO-1 or I = IHI+1,...,N.
    *
    *  SCALE   (output) DOUBLE PRECISION array, dimension (N)
    *          Details of the permutations and scaling factors applied
    *          when balancing A.  If P(j) is the index of the row and column
    *          interchanged with row and column j, and D(j) is the scaling
    *          factor applied to row and column j, then
    *          SCALE(J) = P(J),    for J = 1,...,ILO-1
    *                   = D(J),    for J = ILO,...,IHI
    *                   = P(J)     for J = IHI+1,...,N.
    *          The order in which the interchanges are made is N to IHI+1,
    *          then 1 to ILO-1.
    *
    *  ABNRM   (output) DOUBLE PRECISION
    *          The one-norm of the balanced matrix (the maximum
    *          of the sum of absolute values of elements of any column).
    *
    *  RCONDE  (output) DOUBLE PRECISION array, dimension (N)
    *          RCONDE(j) is the reciprocal condition number of the j-th
    *          eigenvalue.
    *
    *  RCONDV  (output) DOUBLE PRECISION array, dimension (N)
    *          RCONDV(j) is the reciprocal condition number of the j-th
    *          right eigenvector.
    *
    *  WORK    (workspace/output) COMPLEX*16 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.  If SENSE = 'N' or 'E',
    *          LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
    *          LWORK >= N*N+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) DOUBLE PRECISION 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 or condition numbers
    *                have been computed; elements 1:ILO-1 and i+1:N of W
    *                contain eigenvalues which have converged.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
         $                   WNTSNN, WNTSNV
          CHARACTER          JOB, SIDE
          INTEGER            HSWORK, I, ICOND, IERR, ITAU, IWRK, K, 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, DLASCL, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL,
         $                   ZGEHRD, ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC,
         $                   ZTRSNA, 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, DCONJG, DIMAG, MAX, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input arguments
    *
          INFO = 0
          LQUERY = ( LWORK.EQ.-1 )
          WANTVL = LSAME( JOBVL, 'V' )
          WANTVR = LSAME( JOBVR, 'V' )
          WNTSNN = LSAME( SENSE, 'N' )
          WNTSNE = LSAME( SENSE, 'E' )
          WNTSNV = LSAME( SENSE, 'V' )
          WNTSNB = LSAME( SENSE, 'B' )
          IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' ) .OR.
         $    LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) THEN
             INFO = -1
          ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
             INFO = -2
          ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
             INFO = -3
          ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
         $         ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
         $         WANTVR ) ) ) THEN
             INFO = -4
          ELSE IF( N.LT.0 ) THEN
             INFO = -5
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -7
          ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
             INFO = -10
          ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
             INFO = -12
          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 )
    *
                IF( WANTVL ) THEN
                   CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL,
         $                WORK, -1, INFO )
                ELSE IF( WANTVR ) THEN
                   CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR,
         $                WORK, -1, INFO )
                ELSE
                   IF( WNTSNN ) THEN
                      CALL ZHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR,
         $                WORK, -1, INFO )
                   ELSE
                      CALL ZHSEQR( 'S', 'N', N, 1, N, A, LDA, W, VR, LDVR,
         $                WORK, -1, INFO )
                   END IF
                END IF
                HSWORK = WORK( 1 )
    *
                IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
                   MINWRK = 2*N
                   IF( .NOT.( WNTSNN .OR. WNTSNE ) )
         $            MINWRK = MAX( MINWRK, N*N + 2*N )
                   MAXWRK = MAX( MAXWRK, HSWORK )
                   IF( .NOT.( WNTSNN .OR. WNTSNE ) )
         $            MAXWRK = MAX( MAXWRK, N*N + 2*N )
                ELSE
                   MINWRK = 2*N
                   IF( .NOT.( WNTSNN .OR. WNTSNE ) )
         $            MINWRK = MAX( MINWRK, N*N + 2*N )
                   MAXWRK = MAX( MAXWRK, HSWORK )
                   MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
         $                       ' ', N, 1, N, -1 ) )
                   IF( .NOT.( WNTSNN .OR. WNTSNE ) )
         $            MAXWRK = MAX( MAXWRK, N*N + 2*N )
                   MAXWRK = MAX( MAXWRK, 2*N )
                END IF
                MAXWRK = MAX( MAXWRK, MINWRK )
             END IF
             WORK( 1 ) = MAXWRK
    *
             IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
                INFO = -20
             END IF
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZGEEVX', -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]
    *
          ICOND = 0
          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 and compute ABNRM
    *
          CALL ZGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
          ABNRM = ZLANGE( '1', N, N, A, LDA, DUM )
          IF( SCALEA ) THEN
             DUM( 1 ) = ABNRM
             CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
             ABNRM = DUM( 1 )
          END IF
    *
    *     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
    *        If condition numbers desired, compute Schur form
    *
             IF( WNTSNN ) THEN
                JOB = 'E'
             ELSE
                JOB = 'S'
             END IF
    *
    *        (CWorkspace: need 1, prefer HSWORK (see comments) )
    *        (RWorkspace: none)
    *
             IWRK = ITAU
             CALL ZHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, W, VR, LDVR,
         $                WORK( IWRK ), LWORK-IWRK+1, INFO )
          END IF
    *
    *     If INFO > 0 from ZHSEQR, 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 N)
    *
             CALL ZTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
         $                N, NOUT, WORK( IWRK ), RWORK, IERR )
          END IF
    *
    *     Compute condition numbers if desired
    *     (CWorkspace: need N*N+2*N unless SENSE = 'E')
    *     (RWorkspace: need 2*N unless SENSE = 'E')
    *
          IF( .NOT.WNTSNN ) THEN
             CALL ZTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
         $                RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, RWORK,
         $                ICOND )
          END IF
    *
          IF( WANTVL ) THEN
    *
    *        Undo balancing of left eigenvectors
    *
             CALL ZGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, 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( K ) = DBLE( VL( K, I ) )**2 +
         $                      DIMAG( VL( K, I ) )**2
       10       CONTINUE
                K = IDAMAX( N, RWORK, 1 )
                TMP = DCONJG( VL( K, I ) ) / SQRT( RWORK( K ) )
                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
    *
             CALL ZGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, 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( K ) = DBLE( VR( K, I ) )**2 +
         $                      DIMAG( VR( K, I ) )**2
       30       CONTINUE
                K = IDAMAX( N, RWORK, 1 )
                TMP = DCONJG( VR( K, I ) ) / SQRT( RWORK( K ) )
                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.EQ.0 ) THEN
                IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
         $         CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
         $                      IERR )
             ELSE
                CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
             END IF
          END IF
    *
          WORK( 1 ) = MAXWRK
          RETURN
    *
    *     End of ZGEEVX
    *
          END