ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices


 (A,B), the generalized eigenvalues, the complex Schur form (S,T),


 and, optionally, the left and/or right matrices of Schur vectors (VSL


 and VSR).  This gives the generalized Schur factorization


      (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )


 where (VSR)**H is the conjugate-transpose of VSR.


 Optionally, it also orders the eigenvalues so that a selected cluster


 of eigenvalues appears in the leading diagonal blocks of the upper


 triangular matrix S and the upper triangular matrix T; computes


 a reciprocal condition number for the average of the selected


 eigenvalues (RCONDE); and computes a reciprocal condition number for


 the right and left deflating subspaces corresponding to the selected


 eigenvalues (RCONDV). The leading columns of VSL and VSR then form


 an orthonormal basis for the corresponding left and right eigenspaces


 (deflating subspaces).


 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w


 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is


 usually represented as the pair (alpha,beta), as there is a


 reasonable interpretation for beta=0 or for both being zero.


 A pair of matrices (S,T) is in generalized complex Schur form if T is


 upper triangular with non-negative diagonal and S is upper


 triangular.

JOBVSL

          JOBVSL is CHARACTER*1


          = 'N':  do not compute the left Schur vectors;


          = 'V':  compute the left Schur vectors.

JOBVSR

          JOBVSR is CHARACTER*1


          = 'N':  do not compute the right Schur vectors;


          = 'V':  compute the right Schur vectors.

SORT

          SORT is CHARACTER*1


          Specifies whether or not to order the eigenvalues on the


          diagonal of the generalized Schur form.


          = 'N':  Eigenvalues are not ordered;


          = 'S':  Eigenvalues are ordered (see SELCTG).

SELCTG

          SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments


          SELCTG must be declared EXTERNAL in the calling subroutine.


          If SORT = 'N', SELCTG is not referenced.


          If SORT = 'S', SELCTG is used to select eigenvalues to sort


          to the top left of the Schur form.


          Note that a selected complex eigenvalue may no longer satisfy


          SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since


          ordering may change the value of complex eigenvalues


          (especially if the eigenvalue is ill-conditioned), in this


          case INFO is set to N+3 see INFO below).

SENSE

          SENSE is CHARACTER*1


          Determines which reciprocal condition numbers are computed.


          = 'N' : None are computed;


          = 'E' : Computed for average of selected eigenvalues only;


          = 'V' : Computed for selected deflating subspaces only;


          = 'B' : Computed for both.


          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

N

          N is INTEGER


          The order of the matrices A, B, VSL, and VSR.  N >= 0.

A

          A is COMPLEX*16 array, dimension (LDA, N)


          On entry, the first of the pair of matrices.


          On exit, A has been overwritten by its generalized Schur


          form S.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX*16 array, dimension (LDB, N)


          On entry, the second of the pair of matrices.


          On exit, B has been overwritten by its generalized Schur


          form T.

LDB

          LDB is INTEGER


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

SDIM

          SDIM is INTEGER


          If SORT = 'N', SDIM = 0.


          If SORT = 'S', SDIM = number of eigenvalues (after sorting)


          for which SELCTG is true.

ALPHA

          ALPHA is COMPLEX*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 array, dimension (N)


          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the


          generalized eigenvalues.  ALPHA(j) and BETA(j),j=1,...,N  are


          the diagonals of the complex Schur form (S,T).  BETA(j) will


          be non-negative real.


          Note: the quotients 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).

VSL

          VSL is COMPLEX*16 array, dimension (LDVSL,N)


          If JOBVSL = 'V', VSL will contain the left Schur vectors.


          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER


          The leading dimension of the matrix VSL. LDVSL >=1, and


          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is COMPLEX*16 array, dimension (LDVSR,N)


          If JOBVSR = 'V', VSR will contain the right Schur vectors.


          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER


          The leading dimension of the matrix VSR. LDVSR >= 1, and


          if JOBVSR = 'V', LDVSR >= N.

RCONDE

          RCONDE is DOUBLE PRECISION array, dimension ( 2 )


          If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the


          reciprocal condition numbers for the average of the selected


          eigenvalues.


          Not referenced if SENSE = 'N' or 'V'.

RCONDV

          RCONDV is DOUBLE PRECISION array, dimension ( 2 )


          If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the


          reciprocal condition number for the selected deflating


          subspaces.


          Not referenced if SENSE = 'N' or 'E'.

WORK

          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',


          LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else


          LWORK >= MAX(1,2*N).  Note that 2*SDIM*(N-SDIM) <= N*N/2.


          Note also that an error is only returned if


          LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may


          not be large enough.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the bound on the optimal size of the WORK


          array and the minimum size of the IWORK array, returns these


          values as the first entries of the WORK and IWORK arrays, and


          no error message related to LWORK or LIWORK is issued by


          XERBLA.

RWORK

          RWORK is DOUBLE PRECISION array, dimension ( 8*N )


          Real workspace.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))


          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

LIWORK

          LIWORK is INTEGER


          The dimension of the array IWORK.


          If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise


          LIWORK >= N+2.


          If LIWORK = -1, then a workspace query is assumed; the


          routine only calculates the bound on the optimal size of the


          WORK array and the minimum size of the IWORK array, returns


          these values as the first entries of the WORK and IWORK


          arrays, and no error message related to LWORK or LIWORK is


          issued by XERBLA.

BWORK

          BWORK is LOGICAL array, dimension (N)


          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          = 1,...,N:


                The QZ iteration failed.  (A,B) are not in Schur


                form, 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: after reordering, roundoff changed values of


                      some complex eigenvalues so that leading


                      eigenvalues in the Generalized Schur form no


                      longer satisfy SELCTG=.TRUE.  This could also


                      be caused due to scaling.


                =N+3: reordering failed in ZTGSEN.

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June 2017