dggev.f

          SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
         $                  BETA, VL, LDVL, VR, LDVR, WORK, LWORK, 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, LDB, LDVL, LDVR, LWORK, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
         $                   B( LDB, * ), BETA( * ), VL( LDVL, * ),
         $                   VR( LDVR, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
    *  the generalized eigenvalues, and optionally, the left and/or right
    *  generalized eigenvectors.
    *
    *  A generalized eigenvalue for a pair of matrices (A,B) is a scalar
    *  lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
    *  singular. It is usually represented as the pair (alpha,beta), as
    *  there is a reasonable interpretation for beta=0, and even for both
    *  being zero.
    *
    *  The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
    *  of (A,B) satisfies
    *
    *                   A * v(j) = lambda(j) * B * v(j).
    *
    *  The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
    *  of (A,B) satisfies
    *
    *                   u(j)**H * A  = lambda(j) * u(j)**H * B .
    *
    *  where u(j)**H is the conjugate-transpose of u(j).
    *
    *
    *  Arguments
    *  =========
    *
    *  JOBVL   (input) CHARACTER*1
    *          = 'N':  do not compute the left generalized eigenvectors;
    *          = 'V':  compute the left generalized eigenvectors.
    *
    *  JOBVR   (input) CHARACTER*1
    *          = 'N':  do not compute the right generalized eigenvectors;
    *          = 'V':  compute the right generalized eigenvectors.
    *
    *  N       (input) INTEGER
    *          The order of the matrices A, B, VL, and VR.  N >= 0.
    *
    *  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
    *          On entry, the matrix A in the pair (A,B).
    *          On exit, A has been overwritten.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of A.  LDA >= max(1,N).
    *
    *  B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
    *          On entry, the matrix B in the pair (A,B).
    *          On exit, B has been overwritten.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of B.  LDB >= max(1,N).
    *
    *  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
    *  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
    *  BETA    (output) DOUBLE PRECISION array, dimension (N)
    *          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
    *          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
    *          the j-th eigenvalue is real; if positive, then the j-th and
    *          (j+1)-st eigenvalues are a complex conjugate pair, with
    *          ALPHAI(j+1) negative.
    *
    *          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
    *          may easily over- or underflow, and BETA(j) may even be zero.
    *          Thus, the user should avoid naively computing the ratio
    *          alpha/beta.  However, ALPHAR and ALPHAI will be always less
    *          than and usually comparable with norm(A) in magnitude, and
    *          BETA always less than and usually comparable with norm(B).
    *
    *  VL      (output) DOUBLE PRECISION 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 the j-th eigenvalue is real, then
    *          u(j) = VL(:,j), the j-th column of VL. If the j-th and
    *          (j+1)-th eigenvalues form a complex conjugate pair, then
    *          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
    *          Each eigenvector is scaled so the largest component has
    *          abs(real part)+abs(imag. part)=1.
    *          Not referenced if JOBVL = 'N'.
    *
    *  LDVL    (input) INTEGER
    *          The leading dimension of the matrix VL. LDVL >= 1, and
    *          if JOBVL = 'V', LDVL >= N.
    *
    *  VR      (output) DOUBLE PRECISION 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 the j-th eigenvalue is real, then
    *          v(j) = VR(:,j), the j-th column of VR. If the j-th and
    *          (j+1)-th eigenvalues form a complex conjugate pair, then
    *          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
    *          Each eigenvector is scaled so the largest component has
    *          abs(real part)+abs(imag. part)=1.
    *          Not referenced if JOBVR = 'N'.
    *
    *  LDVR    (input) INTEGER
    *          The leading dimension of the matrix VR. LDVR >= 1, and
    *          if JOBVR = 'V', LDVR >= N.
    *
    *  WORK    (workspace/output) DOUBLE PRECISION 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(1,8*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.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *          = 1,...,N:
    *                The QZ iteration failed.  No eigenvectors have been
    *                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
    *                should be correct for j=INFO+1,...,N.
    *          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
    *                =N+2: error return from DTGEVC.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
          CHARACTER          CHTEMP
          INTEGER            ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
         $                   IN, IRIGHT, IROWS, ITAU, IWRK, JC, JR, MAXWRK,
         $                   MINWRK
          DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
         $                   SMLNUM, TEMP
    *     ..
    *     .. Local Arrays ..
          LOGICAL            LDUMMA( 1 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
         $                   DLACPY,DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
         $                   XERBLA
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV
          DOUBLE PRECISION   DLAMCH, DLANGE
          EXTERNAL           LSAME, ILAENV, DLAMCH, DLANGE
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Decode the input arguments
    *
          IF( LSAME( JOBVL, 'N' ) ) THEN
             IJOBVL = 1
             ILVL = .FALSE.
          ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
             IJOBVL = 2
             ILVL = .TRUE.
          ELSE
             IJOBVL = -1
             ILVL = .FALSE.
          END IF
    *
          IF( LSAME( JOBVR, 'N' ) ) THEN
             IJOBVR = 1
             ILVR = .FALSE.
          ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
             IJOBVR = 2
             ILVR = .TRUE.
          ELSE
             IJOBVR = -1
             ILVR = .FALSE.
          END IF
          ILV = ILVL .OR. ILVR
    *
    *     Test the input arguments
    *
          INFO = 0
          LQUERY = ( LWORK.EQ.-1 )
          IF( IJOBVL.LE.0 ) THEN
             INFO = -1
          ELSE IF( IJOBVR.LE.0 ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -5
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -7
          ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
             INFO = -12
          ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
             INFO = -14
          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.
    *       NB refers to the optimal block size for the immediately
    *       following subroutine, as returned by ILAENV. The workspace is
    *       computed assuming ILO = 1 and IHI = N, the worst case.)
    *
          IF( INFO.EQ.0 ) THEN
             MINWRK = MAX( 1, 8*N )
             MAXWRK = MAX( 1, N*( 7 +
         $                 ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) ) )
             MAXWRK = MAX( MAXWRK, N*( 7 +
         $                 ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) ) )
             IF( ILVL ) THEN
                MAXWRK = MAX( MAXWRK, N*( 7 +
         $                 ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) ) )
             END IF
             WORK( 1 ) = MAXWRK
    *
             IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
         $      INFO = -16
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DGGEV ', -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 = DLANGE( 'M', N, N, A, LDA, WORK )
          ILASCL = .FALSE.
          IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
             ANRMTO = SMLNUM
             ILASCL = .TRUE.
          ELSE IF( ANRM.GT.BIGNUM ) THEN
             ANRMTO = BIGNUM
             ILASCL = .TRUE.
          END IF
          IF( ILASCL )
         $   CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
    *
    *     Scale B if max element outside range [SMLNUM,BIGNUM]
    *
          BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
          ILBSCL = .FALSE.
          IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
             BNRMTO = SMLNUM
             ILBSCL = .TRUE.
          ELSE IF( BNRM.GT.BIGNUM ) THEN
             BNRMTO = BIGNUM
             ILBSCL = .TRUE.
          END IF
          IF( ILBSCL )
         $   CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
    *
    *     Permute the matrices A, B to isolate eigenvalues if possible
    *     (Workspace: need 6*N)
    *
          ILEFT = 1
          IRIGHT = N + 1
          IWRK = IRIGHT + N
          CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
         $             WORK( IRIGHT ), WORK( IWRK ), IERR )
    *
    *     Reduce B to triangular form (QR decomposition of B)
    *     (Workspace: need N, prefer N*NB)
    *
          IROWS = IHI + 1 - ILO
          IF( ILV ) THEN
             ICOLS = N + 1 - ILO
          ELSE
             ICOLS = IROWS
          END IF
          ITAU = IWRK
          IWRK = ITAU + IROWS
          CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
         $             WORK( IWRK ), LWORK+1-IWRK, IERR )
    *
    *     Apply the orthogonal transformation to matrix A
    *     (Workspace: need N, prefer N*NB)
    *
          CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
         $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
         $             LWORK+1-IWRK, IERR )
    *
    *     Initialize VL
    *     (Workspace: need N, prefer N*NB)
    *
          IF( ILVL ) THEN
             CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
             IF( IROWS.GT.1 ) THEN
                CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
         $                   VL( ILO+1, ILO ), LDVL )
             END IF
             CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
         $                WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
          END IF
    *
    *     Initialize VR
    *
          IF( ILVR )
         $   CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
    *
    *     Reduce to generalized Hessenberg form
    *     (Workspace: none needed)
    *
          IF( ILV ) THEN
    *
    *        Eigenvectors requested -- work on whole matrix.
    *
             CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
         $                LDVL, VR, LDVR, IERR )
          ELSE
             CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
         $                B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
          END IF
    *
    *     Perform QZ algorithm (Compute eigenvalues, and optionally, the
    *     Schur forms and Schur vectors)
    *     (Workspace: need N)
    *
          IWRK = ITAU
          IF( ILV ) THEN
             CHTEMP = 'S'
          ELSE
             CHTEMP = 'E'
          END IF
          CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
         $             ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
         $             WORK( IWRK ), LWORK+1-IWRK, IERR )
          IF( IERR.NE.0 ) THEN
             IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
                INFO = IERR
             ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
                INFO = IERR - N
             ELSE
                INFO = N + 1
             END IF
             GO TO 110
          END IF
    *
    *     Compute Eigenvectors
    *     (Workspace: need 6*N)
    *
          IF( ILV ) THEN
             IF( ILVL ) THEN
                IF( ILVR ) THEN
                   CHTEMP = 'B'
                ELSE
                   CHTEMP = 'L'
                END IF
             ELSE
                CHTEMP = 'R'
             END IF
             CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
         $                VR, LDVR, N, IN, WORK( IWRK ), IERR )
             IF( IERR.NE.0 ) THEN
                INFO = N + 2
                GO TO 110
             END IF
    *
    *        Undo balancing on VL and VR and normalization
    *        (Workspace: none needed)
    *
             IF( ILVL ) THEN
                CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
         $                   WORK( IRIGHT ), N, VL, LDVL, IERR )
                DO 50 JC = 1, N
                   IF( ALPHAI( JC ).LT.ZERO )
         $            GO TO 50
                   TEMP = ZERO
                   IF( ALPHAI( JC ).EQ.ZERO ) THEN
                      DO 10 JR = 1, N
                         TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
       10             CONTINUE
                   ELSE
                      DO 20 JR = 1, N
                         TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
         $                      ABS( VL( JR, JC+1 ) ) )
       20             CONTINUE
                   END IF
                   IF( TEMP.LT.SMLNUM )
         $            GO TO 50
                   TEMP = ONE / TEMP
                   IF( ALPHAI( JC ).EQ.ZERO ) THEN
                      DO 30 JR = 1, N
                         VL( JR, JC ) = VL( JR, JC )*TEMP
       30             CONTINUE
                   ELSE
                      DO 40 JR = 1, N
                         VL( JR, JC ) = VL( JR, JC )*TEMP
                         VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
       40             CONTINUE
                   END IF
       50       CONTINUE
             END IF
             IF( ILVR ) THEN
                CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
         $                   WORK( IRIGHT ), N, VR, LDVR, IERR )
                DO 100 JC = 1, N
                   IF( ALPHAI( JC ).LT.ZERO )
         $            GO TO 100
                   TEMP = ZERO
                   IF( ALPHAI( JC ).EQ.ZERO ) THEN
                      DO 60 JR = 1, N
                         TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
       60             CONTINUE
                   ELSE
                      DO 70 JR = 1, N
                         TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
         $                      ABS( VR( JR, JC+1 ) ) )
       70             CONTINUE
                   END IF
                   IF( TEMP.LT.SMLNUM )
         $            GO TO 100
                   TEMP = ONE / TEMP
                   IF( ALPHAI( JC ).EQ.ZERO ) THEN
                      DO 80 JR = 1, N
                         VR( JR, JC ) = VR( JR, JC )*TEMP
       80             CONTINUE
                   ELSE
                      DO 90 JR = 1, N
                         VR( JR, JC ) = VR( JR, JC )*TEMP
                         VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
       90             CONTINUE
                   END IF
      100       CONTINUE
             END IF
    *
    *        End of eigenvector calculation
    *
          END IF
    *
    *     Undo scaling if necessary
    *
          IF( ILASCL ) THEN
             CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
             CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
          END IF
    *
          IF( ILBSCL ) THEN
             CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
          END IF
    *
      110 CONTINUE
    *
          WORK( 1 ) = MAXWRK
    *
          RETURN
    *
    *     End of DGGEV
    *
          END