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


 (A,B), the generalized eigenvalues, the generalized complex Schur


 form (S, T), and optionally left and/or right 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. The leading


 columns of VSL and VSR then form an unitary basis for the


 corresponding left and right eigenspaces (deflating subspaces).


 (If only the generalized eigenvalues are needed, use the driver


 ZGGEV instead, which is faster.)


 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, and even for both being zero.


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


 and T are upper triangular and, in addition, the diagonal elements


 of T are non-negative real numbers.

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.


          An eigenvalue ALPHA(j)/BETA(j) is selected if


          SELCTG(ALPHA(j),BETA(j)) is true.


          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+2 (See INFO below).

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), j=1,...,N  and  BETA(j),


          j=1,...,N  are the diagonals of the complex Schur form (A,B)


          output by ZGGES. The  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.

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.  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

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

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016