DGGESX computes for a pair of N-by-N real nonsymmetric matrices


 (A,B), the generalized eigenvalues, the real 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)**T, (VSL) T (VSR)**T )


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


 of eigenvalues appears in the leading diagonal blocks of the upper


 quasi-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 real Schur form if T is


 upper triangular with non-negative diagonal and S is block upper


 triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond


 to real generalized eigenvalues, while 2-by-2 blocks of S will be


 "standardized" by making the corresponding elements of T have the


 form:


         [  a  0  ]


         [  0  b  ]


 and the pair of corresponding 2-by-2 blocks in S and T will have a


 complex conjugate pair of generalized eigenvalues.

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 three DOUBLE PRECISION 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 (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if


          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either


          one of a complex conjugate pair of eigenvalues is selected,


          then both complex eigenvalues are selected.


          Note that a selected complex eigenvalue may no longer satisfy


          SELCTG(ALPHAR(j),ALPHAI(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.

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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.  (Complex conjugate pairs for which


          SELCTG is true for either eigenvalue count as 2.)

ALPHAR

          ALPHAR is DOUBLE PRECISION array, dimension (N)

ALPHAI

          ALPHAI is DOUBLE PRECISION array, dimension (N)

BETA

          BETA is DOUBLE PRECISION array, dimension (N)


          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will


          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i


          and BETA(j),j=1,...,N  are the diagonals of the complex Schur


          form (S,T) that would result if the 2-by-2 diagonal blocks of


          the real Schur form of (A,B) were further reduced to


          triangular form using 2-by-2 complex unitary transformations.


          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.


          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).

VSL

          VSL is DOUBLE PRECISION 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 DOUBLE PRECISION 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 numbers for the selected deflating


          subspaces.


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

WORK

          WORK is DOUBLE PRECISION 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( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else


          LWORK >= max( 8*N, 6*N+16 ).


          Note that 2*SDIM*(N-SDIM) <= N*N/2.


          Note also that an error is only returned if


          LWORK < max( 8*N, 6*N+16), 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.

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+6.


          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 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: 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 DTGSEN.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2017

  An approximate (asymptotic) bound on the average absolute error of


  the selected eigenvalues is


       EPS * norm((A, B)) / RCONDE( 1 ).


  An approximate (asymptotic) bound on the maximum angular error in


  the computed deflating subspaces is


       EPS * norm((A, B)) / RCONDV( 2 ).


  See LAPACK User's Guide, section 4.11 for more information.