ZGGEVX(3)      LAPACK routine of NEC Numeric Library Collection      ZGGEVX(3)



NAME
       ZGGEVX

SYNOPSIS
       SUBROUTINE ZGGEVX (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
           ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM,
           BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO)



PURPOSE
            ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
            (A,B) the generalized eigenvalues, and optionally, the left and/or
            right generalized eigenvectors.

            Optionally, it also computes a balancing transformation to improve
            the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
            LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
            the eigenvalues (RCONDE), and reciprocal condition numbers for the
            right eigenvectors (RCONDV).

            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
           BALANC    (input)
                     BALANC is CHARACTER*1
                     Specifies the balance option to be performed:
                     = 'N':  do not diagonally scale or permute;
                     = 'P':  permute only;
                     = 'S':  scale only;
                     = 'B':  both permute and scale.
                     Computed reciprocal condition numbers will be for the
                     matrices after permuting and/or balancing. Permuting does
                     not change condition numbers (in exact arithmetic), but
                     balancing does.

           JOBVL     (input)
                     JOBVL is CHARACTER*1
                     = 'N':  do not compute the left generalized eigenvectors;
                     = 'V':  compute the left generalized eigenvectors.

           JOBVR     (input)
                     JOBVR is CHARACTER*1
                     = 'N':  do not compute the right generalized eigenvectors;
                     = 'V':  compute the right generalized eigenvectors.

           SENSE     (input)
                     SENSE is CHARACTER*1
                     Determines which reciprocal condition numbers are computed.
                     = 'N': none are computed;
                     = 'E': computed for eigenvalues only;
                     = 'V': computed for eigenvectors only;
                     = 'B': computed for eigenvalues and eigenvectors.

           N         (input)
                     N is INTEGER
                     The order of the matrices A, B, VL, and VR.  N >= 0.

           A         (input/output)
                     A is COMPLEX*16 array, dimension (LDA, N)
                     On entry, the matrix A in the pair (A,B).
                     On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
                     or both, then A contains the first part of the complex Schur
                     form of the "balanced" versions of the input A and B.

           LDA       (input)
                     LDA is INTEGER
                     The leading dimension of A.  LDA >= max(1,N).

           B         (input/output)
                     B is COMPLEX*16 array, dimension (LDB, N)
                     On entry, the matrix B in the pair (A,B).
                     On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
                     or both, then B contains the second part of the complex
                     Schur form of the "balanced" versions of the input A and B.

           LDB       (input)
                     LDB is INTEGER
                     The leading dimension of B.  LDB >= max(1,N).

           ALPHA     (output)
                     ALPHA is COMPLEX*16 array, dimension (N)

           BETA      (output)
                     BETA is COMPLEX*16 array, dimension (N)
                     On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized
                     eigenvalues.

                     Note: the quotient ALPHA(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, ALPHA 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)
                     VL is COMPLEX*16 array, dimension (LDVL,N)
                     If JOBVL = 'V', the left generalized eigenvectors u(j) are
                     stored one after another in the columns of VL, in the same
                     order as their eigenvalues.
                     Each eigenvector will be scaled so the largest component
                     will have abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVL = 'N'.

           LDVL      (input)
                     LDVL is INTEGER
                     The leading dimension of the matrix VL. LDVL >= 1, and
                     if JOBVL = 'V', LDVL >= N.

           VR        (output)
                     VR is COMPLEX*16 array, dimension (LDVR,N)
                     If JOBVR = 'V', the right generalized eigenvectors v(j) are
                     stored one after another in the columns of VR, in the same
                     order as their eigenvalues.
                     Each eigenvector will be scaled so the largest component
                     will have abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVR = 'N'.

           LDVR      (input)
                     LDVR is INTEGER
                     The leading dimension of the matrix VR. LDVR >= 1, and
                     if JOBVR = 'V', LDVR >= N.

           ILO       (output)
                     ILO is INTEGER

           IHI       (output)
                     IHI is INTEGER
                     ILO and IHI are integer values such that on exit
                     A(i,j) = 0 and B(i,j) = 0 if i > j and
                     j = 1,...,ILO-1 or i = IHI+1,...,N.
                     If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

           LSCALE    (output)
                     LSCALE is DOUBLE PRECISION array, dimension (N)
                     Details of the permutations and scaling factors applied
                     to the left side of A and B.  If PL(j) is the index of the
                     row interchanged with row j, and DL(j) is the scaling
                     factor applied to row j, then
                       LSCALE(j) = PL(j)  for j = 1,...,ILO-1
                                 = DL(j)  for j = ILO,...,IHI
                                 = PL(j)  for j = IHI+1,...,N.
                     The order in which the interchanges are made is N to IHI+1,
                     then 1 to ILO-1.

           RSCALE    (output)
                     RSCALE is DOUBLE PRECISION array, dimension (N)
                     Details of the permutations and scaling factors applied
                     to the right side of A and B.  If PR(j) is the index of the
                     column interchanged with column j, and DR(j) is the scaling
                     factor applied to column j, then
                       RSCALE(j) = PR(j)  for j = 1,...,ILO-1
                                 = DR(j)  for j = ILO,...,IHI
                                 = PR(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)
                     ABNRM is DOUBLE PRECISION
                     The one-norm of the balanced matrix A.

           BBNRM     (output)
                     BBNRM is DOUBLE PRECISION
                     The one-norm of the balanced matrix B.

           RCONDE    (output)
                     RCONDE is DOUBLE PRECISION array, dimension (N)
                     If SENSE = 'E' or 'B', the reciprocal condition numbers of
                     the eigenvalues, stored in consecutive elements of the array.
                     If SENSE = 'N' or 'V', RCONDE is not referenced.

           RCONDV    (output)
                     RCONDV is DOUBLE PRECISION array, dimension (N)
                     If JOB = 'V' or 'B', the estimated reciprocal condition
                     numbers of the eigenvectors, stored in consecutive elements
                     of the array. If the eigenvalues cannot be reordered to
                     compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
                     when the true value would be very small anyway.
                     If SENSE = 'N' or 'E', RCONDV is not referenced.

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK >= max(1,2*N).
                     If SENSE = 'E', LWORK >= max(1,4*N).
                     If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).

                     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     (output)
                     RWORK is DOUBLE PRECISION array, dimension (lrwork)
                     lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',
                     and at least max(1,2*N) otherwise.
                     Real workspace.

           IWORK     (output)
                     IWORK is INTEGER array, dimension (N+2)
                     If SENSE = 'E', IWORK is not referenced.

           BWORK     (output)
                     BWORK is LOGICAL array, dimension (N)
                     If SENSE = 'N', BWORK is not referenced.

           INFO      (output)
                     INFO is 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 ALPHA(j) and BETA(j) should be correct
                           for j=INFO+1,...,N.
                     > N:  =N+1: other than QZ iteration failed in ZHGEQZ.
                           =N+2: error return from ZTGEVC.






FURTHER DETAILS
             Balancing a matrix pair (A,B) includes, first, permuting rows and
             columns to isolate eigenvalues, second, applying diagonal similarity
             transformation to the rows and columns to make the rows and columns
             as close in norm as possible. 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 4.11.1.2 of LAPACK Users' Guide.

             An approximate error bound on the chordal distance between the i-th
             computed generalized eigenvalue w and the corresponding exact
             eigenvalue lambda is

                  chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

             An approximate error bound for the angle between the i-th computed
             eigenvector VL(i) or VR(i) is given by

                  EPS * norm(ABNRM, BBNRM) / DIF(i).

             For further explanation of the reciprocal condition numbers RCONDE
             and RCONDV, see section 4.11 of LAPACK User's Guide.



LAPACK routine                  31 October 2017                      ZGGEVX(3)