Provided by: liblapack-doc_3.11.0-2build1_all bug

NAME

       complexGEeigen - complex

SYNOPSIS

   Functions
       subroutine cgees (JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS, LDVS, WORK, LWORK, RWORK,
           BWORK, INFO)
            CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine cgeesx (JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS, LDVS, RCONDE,
           RCONDV, WORK, LWORK, RWORK, BWORK, INFO)
            CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine cgeev (JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK,
           INFO)
            CGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices
       subroutine cgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL, VR, LDVR, ILO,
           IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, INFO)
            CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices
       subroutine cgges (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL,
           LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)
            CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine cgges3 (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA,
           VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)
            CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices (blocked algorithm)
       subroutine cggesx (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHA,
           BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK,
           BWORK, INFO)
            CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine cggev (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK,
           LWORK, RWORK, INFO)
            CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices
       subroutine cggev3 (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK,
           LWORK, RWORK, INFO)
            CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices (blocked algorithm)
       subroutine cggevx (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)
            CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices

Detailed Description

       This is the group of complex eigenvalue driver functions for GE matrices

Function Documentation

   subroutine cgees (character JOBVS, character SORT, external SELECT, integer N, complex,
       dimension( lda, * ) A, integer LDA, integer SDIM, complex, dimension( * ) W, complex,
       dimension( ldvs, * ) VS, integer LDVS, complex, dimension( * ) WORK, integer LWORK, real,
       dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)
        CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

            CGEES computes for an N-by-N complex nonsymmetric matrix A, the
            eigenvalues, the Schur form T, and, optionally, the matrix of Schur
            vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).

            Optionally, it also orders the eigenvalues on the diagonal of the
            Schur form so that selected eigenvalues are at the top left.
            The leading columns of Z then form an orthonormal basis for the
            invariant subspace corresponding to the selected eigenvalues.

            A complex matrix is in Schur form if it is upper triangular.

       Parameters
           JOBVS

                     JOBVS is CHARACTER*1
                     = 'N': Schur vectors are not computed;
                     = 'V': Schur vectors are computed.

           SORT

                     SORT is CHARACTER*1
                     Specifies whether or not to order the eigenvalues on the
                     diagonal of the Schur form.
                     = 'N': Eigenvalues are not ordered:
                     = 'S': Eigenvalues are ordered (see SELECT).

           SELECT

                     SELECT is a LOGICAL FUNCTION of one COMPLEX argument
                     SELECT must be declared EXTERNAL in the calling subroutine.
                     If SORT = 'S', SELECT is used to select eigenvalues to order
                     to the top left of the Schur form.
                     IF SORT = 'N', SELECT is not referenced.
                     The eigenvalue W(j) is selected if SELECT(W(j)) is true.

           N

                     N is INTEGER
                     The order of the matrix A. N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the N-by-N matrix A.
                     On exit, A has been overwritten by its Schur form T.

           LDA

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

           SDIM

                     SDIM is INTEGER
                     If SORT = 'N', SDIM = 0.
                     If SORT = 'S', SDIM = number of eigenvalues for which
                                    SELECT is true.

           W

                     W is COMPLEX array, dimension (N)
                     W contains the computed eigenvalues, in the same order that
                     they appear on the diagonal of the output Schur form T.

           VS

                     VS is COMPLEX array, dimension (LDVS,N)
                     If JOBVS = 'V', VS contains the unitary matrix Z of Schur
                     vectors.
                     If JOBVS = 'N', VS is not referenced.

           LDVS

                     LDVS is INTEGER
                     The leading dimension of the array VS.  LDVS >= 1; if
                     JOBVS = 'V', LDVS >= N.

           WORK

                     WORK is COMPLEX 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 REAL array, dimension (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.
                     > 0: if INFO = i, and i is
                          <= N:  the QR algorithm failed to compute all the
                                 eigenvalues; elements 1:ILO-1 and i+1:N of W
                                 contain those eigenvalues which have converged;
                                 if JOBVS = 'V', VS contains the matrix which
                                 reduces A to its partially converged Schur form.
                          = N+1: the eigenvalues could not be reordered because
                                 some eigenvalues were too close to separate (the
                                 problem is very ill-conditioned);
                          = N+2: after reordering, roundoff changed values of
                                 some complex eigenvalues so that leading
                                 eigenvalues in the Schur form no longer satisfy
                                 SELECT = .TRUE..  This could also be caused by
                                 underflow due to scaling.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cgeesx (character JOBVS, character SORT, external SELECT, character SENSE, integer
       N, complex, dimension( lda, * ) A, integer LDA, integer SDIM, complex, dimension( * ) W,
       complex, dimension( ldvs, * ) VS, integer LDVS, real RCONDE, real RCONDV, complex,
       dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, logical, dimension( * )
       BWORK, integer INFO)
        CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

            CGEESX computes for an N-by-N complex nonsymmetric matrix A, the
            eigenvalues, the Schur form T, and, optionally, the matrix of Schur
            vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).

            Optionally, it also orders the eigenvalues on the diagonal of the
            Schur form so that selected eigenvalues are at the top left;
            computes a reciprocal condition number for the average of the
            selected eigenvalues (RCONDE); and computes a reciprocal condition
            number for the right invariant subspace corresponding to the
            selected eigenvalues (RCONDV).  The leading columns of Z form an
            orthonormal basis for this invariant subspace.

            For further explanation of the reciprocal condition numbers RCONDE
            and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
            these quantities are called s and sep respectively).

            A complex matrix is in Schur form if it is upper triangular.

       Parameters
           JOBVS

                     JOBVS is CHARACTER*1
                     = 'N': Schur vectors are not computed;
                     = 'V': Schur vectors are computed.

           SORT

                     SORT is CHARACTER*1
                     Specifies whether or not to order the eigenvalues on the
                     diagonal of the Schur form.
                     = 'N': Eigenvalues are not ordered;
                     = 'S': Eigenvalues are ordered (see SELECT).

           SELECT

                     SELECT is a LOGICAL FUNCTION of one COMPLEX argument
                     SELECT must be declared EXTERNAL in the calling subroutine.
                     If SORT = 'S', SELECT is used to select eigenvalues to order
                     to the top left of the Schur form.
                     If SORT = 'N', SELECT is not referenced.
                     An eigenvalue W(j) is selected if SELECT(W(j)) is true.

           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 right invariant subspace only;
                     = 'B': Computed for both.
                     If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

           N

                     N is INTEGER
                     The order of the matrix A. N >= 0.

           A

                     A is COMPLEX array, dimension (LDA, N)
                     On entry, the N-by-N matrix A.
                     On exit, A is overwritten by its Schur form T.

           LDA

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

           SDIM

                     SDIM is INTEGER
                     If SORT = 'N', SDIM = 0.
                     If SORT = 'S', SDIM = number of eigenvalues for which
                                    SELECT is true.

           W

                     W is COMPLEX array, dimension (N)
                     W contains the computed eigenvalues, in the same order
                     that they appear on the diagonal of the output Schur form T.

           VS

                     VS is COMPLEX array, dimension (LDVS,N)
                     If JOBVS = 'V', VS contains the unitary matrix Z of Schur
                     vectors.
                     If JOBVS = 'N', VS is not referenced.

           LDVS

                     LDVS is INTEGER
                     The leading dimension of the array VS.  LDVS >= 1, and if
                     JOBVS = 'V', LDVS >= N.

           RCONDE

                     RCONDE is REAL
                     If SENSE = 'E' or 'B', RCONDE contains the reciprocal
                     condition number for the average of the selected eigenvalues.
                     Not referenced if SENSE = 'N' or 'V'.

           RCONDV

                     RCONDV is REAL
                     If SENSE = 'V' or 'B', RCONDV contains the reciprocal
                     condition number for the selected right invariant subspace.
                     Not referenced if SENSE = 'N' or 'E'.

           WORK

                     WORK is COMPLEX 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).
                     Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
                     where SDIM is the number of selected eigenvalues computed by
                     this routine.  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.
                     For good performance, LWORK must generally be larger.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates upper bound on the optimal size of the
                     array WORK, 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 REAL array, dimension (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.
                     > 0: if INFO = i, and i is
                        <= N: the QR algorithm failed to compute all the
                              eigenvalues; elements 1:ILO-1 and i+1:N of W
                              contain those eigenvalues which have converged; if
                              JOBVS = 'V', VS contains the transformation which
                              reduces A to its partially converged Schur form.
                        = N+1: the eigenvalues could not be reordered because some
                              eigenvalues were too close to separate (the problem
                              is very ill-conditioned);
                        = N+2: after reordering, roundoff changed values of some
                              complex eigenvalues so that leading eigenvalues in
                              the Schur form no longer satisfy SELECT=.TRUE.  This
                              could also be caused by underflow due to scaling.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cgeev (character JOBVL, character JOBVR, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( * ) W, complex, dimension( ldvl, * ) VL, integer LDVL,
       complex, dimension( ldvr, * ) VR, integer LDVR, complex, dimension( * ) WORK, integer
       LWORK, real, dimension( * ) RWORK, integer INFO)
        CGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE
       matrices

       Purpose:

            CGEEV computes for an N-by-N complex nonsymmetric matrix A, the
            eigenvalues and, optionally, the left and/or right eigenvectors.

            The right eigenvector v(j) of A satisfies
                             A * v(j) = lambda(j) * v(j)
            where lambda(j) is its eigenvalue.
            The left eigenvector u(j) of A satisfies
                          u(j)**H * A = lambda(j) * u(j)**H
            where u(j)**H denotes the conjugate transpose of u(j).

            The computed eigenvectors are normalized to have Euclidean norm
            equal to 1 and largest component real.

       Parameters
           JOBVL

                     JOBVL is CHARACTER*1
                     = 'N': left eigenvectors of A are not computed;
                     = 'V': left eigenvectors of are computed.

           JOBVR

                     JOBVR is CHARACTER*1
                     = 'N': right eigenvectors of A are not computed;
                     = 'V': right eigenvectors of A are computed.

           N

                     N is INTEGER
                     The order of the matrix A. N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the N-by-N matrix A.
                     On exit, A has been overwritten.

           LDA

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

           W

                     W is COMPLEX array, dimension (N)
                     W contains the computed eigenvalues.

           VL

                     VL is COMPLEX 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 JOBVL = 'N', VL is not referenced.
                     u(j) = VL(:,j), the j-th column of VL.

           LDVL

                     LDVL is INTEGER
                     The leading dimension of the array VL.  LDVL >= 1; if
                     JOBVL = 'V', LDVL >= N.

           VR

                     VR is COMPLEX 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 JOBVR = 'N', VR is not referenced.
                     v(j) = VR(:,j), the j-th column of VR.

           LDVR

                     LDVR is INTEGER
                     The leading dimension of the array VR.  LDVR >= 1; if
                     JOBVR = 'V', LDVR >= N.

           WORK

                     WORK is COMPLEX 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 REAL array, dimension (2*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  if INFO = i, the QR algorithm failed to compute all the
                           eigenvalues, and no eigenvectors have been computed;
                           elements i+1:N of W contain eigenvalues which have
                           converged.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cgeevx (character BALANC, character JOBVL, character JOBVR, character SENSE,
       integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) W,
       complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer
       LDVR, integer ILO, integer IHI, real, dimension( * ) SCALE, real ABNRM, real, dimension( *
       ) RCONDE, real, dimension( * ) RCONDV, complex, dimension( * ) WORK, integer LWORK, real,
       dimension( * ) RWORK, integer INFO)
        CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       GE matrices

       Purpose:

            CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
            eigenvalues and, optionally, the left and/or right eigenvectors.

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

            The right eigenvector v(j) of A satisfies
                             A * v(j) = lambda(j) * v(j)
            where lambda(j) is its eigenvalue.
            The left eigenvector u(j) of A satisfies
                          u(j)**H * A = lambda(j) * u(j)**H
            where u(j)**H denotes the conjugate transpose of u(j).

            The computed eigenvectors are normalized to have Euclidean norm
            equal to 1 and largest component real.

            Balancing a matrix means permuting the rows and columns to make it
            more nearly upper triangular, and applying a diagonal similarity
            transformation D * A * D**(-1), where D is a diagonal matrix, to
            make its rows and columns closer in norm and the condition numbers
            of its eigenvalues and eigenvectors smaller.  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 4.10.2 of the LAPACK
            Users' Guide.

       Parameters
           BALANC

                     BALANC is CHARACTER*1
                     Indicates how the input matrix should be diagonally scaled
                     and/or permuted to improve the conditioning of its
                     eigenvalues.
                     = 'N': Do not diagonally scale or permute;
                     = 'P': Perform permutations to make the matrix more nearly
                            upper triangular. Do not diagonally scale;
                     = 'S': Diagonally scale the matrix, ie. replace A by
                            D*A*D**(-1), where D is a diagonal matrix chosen
                            to make the rows and columns of A more equal in
                            norm. Do not permute;
                     = 'B': Both diagonally scale and permute A.

                     Computed reciprocal condition numbers will be for the matrix
                     after balancing and/or permuting. Permuting does not change
                     condition numbers (in exact arithmetic), but balancing does.

           JOBVL

                     JOBVL is CHARACTER*1
                     = 'N': left eigenvectors of A are not computed;
                     = 'V': left eigenvectors of A are computed.
                     If SENSE = 'E' or 'B', JOBVL must = 'V'.

           JOBVR

                     JOBVR is CHARACTER*1
                     = 'N': right eigenvectors of A are not computed;
                     = 'V': right eigenvectors of A are computed.
                     If SENSE = 'E' or 'B', JOBVR must = 'V'.

           SENSE

                     SENSE is CHARACTER*1
                     Determines which reciprocal condition numbers are computed.
                     = 'N': None are computed;
                     = 'E': Computed for eigenvalues only;
                     = 'V': Computed for right eigenvectors only;
                     = 'B': Computed for eigenvalues and right eigenvectors.

                     If SENSE = 'E' or 'B', both left and right eigenvectors
                     must also be computed (JOBVL = 'V' and JOBVR = 'V').

           N

                     N is INTEGER
                     The order of the matrix A. N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the N-by-N matrix A.
                     On exit, A has been overwritten.  If JOBVL = 'V' or
                     JOBVR = 'V', A contains the Schur form of the balanced
                     version of the matrix A.

           LDA

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

           W

                     W is COMPLEX array, dimension (N)
                     W contains the computed eigenvalues.

           VL

                     VL is COMPLEX 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 JOBVL = 'N', VL is not referenced.
                     u(j) = VL(:,j), the j-th column of VL.

           LDVL

                     LDVL is INTEGER
                     The leading dimension of the array VL.  LDVL >= 1; if
                     JOBVL = 'V', LDVL >= N.

           VR

                     VR is COMPLEX 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 JOBVR = 'N', VR is not referenced.
                     v(j) = VR(:,j), the j-th column of VR.

           LDVR

                     LDVR is INTEGER
                     The leading dimension of the array VR.  LDVR >= 1; if
                     JOBVR = 'V', LDVR >= N.

           ILO

                     ILO is INTEGER

           IHI

                     IHI is INTEGER
                     ILO and IHI are integer values determined when A was
                     balanced.  The balanced A(i,j) = 0 if I > J and
                     J = 1,...,ILO-1 or I = IHI+1,...,N.

           SCALE

                     SCALE is REAL array, dimension (N)
                     Details of the permutations and scaling factors applied
                     when balancing A.  If P(j) is the index of the row and column
                     interchanged with row and column j, and D(j) is the scaling
                     factor applied to row and column j, then
                     SCALE(J) = P(J),    for J = 1,...,ILO-1
                              = D(J),    for J = ILO,...,IHI
                              = P(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

                     ABNRM is REAL
                     The one-norm of the balanced matrix (the maximum
                     of the sum of absolute values of elements of any column).

           RCONDE

                     RCONDE is REAL array, dimension (N)
                     RCONDE(j) is the reciprocal condition number of the j-th
                     eigenvalue.

           RCONDV

                     RCONDV is REAL array, dimension (N)
                     RCONDV(j) is the reciprocal condition number of the j-th
                     right eigenvector.

           WORK

                     WORK is COMPLEX 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 SENSE = 'N' or 'E',
                     LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
                     LWORK >= N*N+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 REAL array, dimension (2*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  if INFO = i, the QR algorithm failed to compute all the
                           eigenvalues, and no eigenvectors or condition numbers
                           have been computed; elements 1:ILO-1 and i+1:N of W
                           contain eigenvalues which have converged.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cgges (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer
       N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer
       LDB, integer SDIM, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex,
       dimension( ldvsl, * ) VSL, integer LDVSL, complex, dimension( ldvsr, * ) VSR, integer
       LDVSR, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, logical,
       dimension( * ) BWORK, integer INFO)
        CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

            CGGES 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
            CGGEV 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.

       Parameters
           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 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 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 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 array, dimension (N)

           BETA

                     BETA is COMPLEX 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 CGGES. 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 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 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 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 REAL 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 CHGEQZ
                           =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 CTGSEN.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cgges3 (character JOBVSL, character JOBVSR, character SORT, external SELCTG,
       integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B,
       integer LDB, integer SDIM, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA,
       complex, dimension( ldvsl, * ) VSL, integer LDVSL, complex, dimension( ldvsr, * ) VSR,
       integer LDVSR, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK,
       logical, dimension( * ) BWORK, integer INFO)
        CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices (blocked algorithm)

       Purpose:

            CGGES3 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
            CGGEV 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.

       Parameters
           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 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 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 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 array, dimension (N)

           BETA

                     BETA is COMPLEX 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 CGGES3. 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 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 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 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 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 REAL 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 CLAQZ0
                           =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 CTGSEN.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cggesx (character JOBVSL, character JOBVSR, character SORT, external SELCTG,
       character SENSE, integer N, complex, dimension( lda, * ) A, integer LDA, complex,
       dimension( ldb, * ) B, integer LDB, integer SDIM, complex, dimension( * ) ALPHA, complex,
       dimension( * ) BETA, complex, dimension( ldvsl, * ) VSL, integer LDVSL, complex,
       dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( 2 ) RCONDE, real, dimension( 2
       ) RCONDV, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK,
       integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer
       INFO)
        CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

            CGGESX 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.

       Parameters
           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 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 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 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 array, dimension (N)

           BETA

                     BETA is COMPLEX 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 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 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 REAL 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 REAL 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 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 REAL 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 WORK.
                     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 CHGEQZ
                           =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 CTGSEN.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cggev (character JOBVL, character JOBVR, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) ALPHA,
       complex, dimension( * ) BETA, complex, dimension( ldvl, * ) VL, integer LDVL, complex,
       dimension( ldvr, * ) VR, integer LDVR, complex, dimension( * ) WORK, integer LWORK, real,
       dimension( * ) RWORK, integer INFO)
        CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE
       matrices

       Purpose:

            CGGEV 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.

            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 generalized eigenvector v(j) corresponding to the
            generalized eigenvalue lambda(j) of (A,B) satisfies

                         A * v(j) = lambda(j) * B * v(j).

            The left generalized eigenvector u(j) corresponding to the
            generalized eigenvalues 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).

       Parameters
           JOBVL

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

           JOBVR

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

           N

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

           A

                     A is COMPLEX array, dimension (LDA, N)
                     On entry, the matrix A in the pair (A,B).
                     On exit, A has been overwritten.

           LDA

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

           B

                     B is COMPLEX array, dimension (LDB, N)
                     On entry, the matrix B in the pair (A,B).
                     On exit, B has been overwritten.

           LDB

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

           ALPHA

                     ALPHA is COMPLEX array, dimension (N)

           BETA

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

                     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).

           VL

                     VL is COMPLEX 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 is scaled so the largest component has
                     abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVL = 'N'.

           LDVL

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

           VR

                     VR is COMPLEX 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 is scaled so the largest component has
                     abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVR = 'N'.

           LDVR

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

           WORK

                     WORK is COMPLEX 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 REAL array, dimension (8*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.  No eigenvectors have been
                           calculated, but ALPHA(j) and BETA(j) should be
                           correct for j=INFO+1,...,N.
                     > N:  =N+1: other then QZ iteration failed in CHGEQZ,
                           =N+2: error return from CTGEVC.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cggev3 (character JOBVL, character JOBVR, integer N, complex, dimension( lda, * )
       A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * )
       ALPHA, complex, dimension( * ) BETA, complex, dimension( ldvl, * ) VL, integer LDVL,
       complex, dimension( ldvr, * ) VR, integer LDVR, complex, dimension( * ) WORK, integer
       LWORK, real, dimension( * ) RWORK, integer INFO)
        CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       GE matrices (blocked algorithm)

       Purpose:

            CGGEV3 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.

            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 generalized eigenvector v(j) corresponding to the
            generalized eigenvalue lambda(j) of (A,B) satisfies

                         A * v(j) = lambda(j) * B * v(j).

            The left generalized eigenvector u(j) corresponding to the
            generalized eigenvalues 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).

       Parameters
           JOBVL

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

           JOBVR

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

           N

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

           A

                     A is COMPLEX array, dimension (LDA, N)
                     On entry, the matrix A in the pair (A,B).
                     On exit, A has been overwritten.

           LDA

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

           B

                     B is COMPLEX array, dimension (LDB, N)
                     On entry, the matrix B in the pair (A,B).
                     On exit, B has been overwritten.

           LDB

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

           ALPHA

                     ALPHA is COMPLEX array, dimension (N)

           BETA

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

                     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).

           VL

                     VL is COMPLEX 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 is scaled so the largest component has
                     abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVL = 'N'.

           LDVL

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

           VR

                     VR is COMPLEX 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 is scaled so the largest component has
                     abs(real part) + abs(imag. part) = 1.
                     Not referenced if JOBVR = 'N'.

           LDVR

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

           WORK

                     WORK is COMPLEX 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 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 REAL array, dimension (8*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.  No eigenvectors have been
                           calculated, but ALPHA(j) and BETA(j) should be
                           correct for j=INFO+1,...,N.
                     > N:  =N+1: other then QZ iteration failed in CHGEQZ,
                           =N+2: error return from CTGEVC.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cggevx (character BALANC, character JOBVL, character JOBVR, character SENSE,
       integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B,
       integer LDB, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex,
       dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR,
       integer ILO, integer IHI, real, dimension( * ) LSCALE, real, dimension( * ) RSCALE, real
       ABNRM, real BBNRM, real, dimension( * ) RCONDE, real, dimension( * ) RCONDV, complex,
       dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * )
       IWORK, logical, dimension( * ) BWORK, integer INFO)
        CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       GE matrices

       Purpose:

            CGGEVX 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).

       Parameters
           BALANC

                     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

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

           JOBVR

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

           SENSE

                     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

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

           A

                     A is COMPLEX 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

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

           B

                     B is COMPLEX 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

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

           ALPHA

                     ALPHA is COMPLEX array, dimension (N)

           BETA

                     BETA is COMPLEX 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

                     VL is COMPLEX 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

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

           VR

                     VR is COMPLEX 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

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

           ILO

                     ILO is INTEGER

           IHI

                     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

                     LSCALE is REAL 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

                     RSCALE is REAL 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

                     ABNRM is REAL
                     The one-norm of the balanced matrix A.

           BBNRM

                     BBNRM is REAL
                     The one-norm of the balanced matrix B.

           RCONDE

                     RCONDE is REAL 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

                     RCONDV is REAL array, dimension (N)
                     If SENSE = '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

                     WORK is COMPLEX 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).
                     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

                     RWORK is REAL 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

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

           BWORK

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

           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.  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 CHGEQZ.
                           =N+2: error return from CTGEVC.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       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.

Author

       Generated automatically by Doxygen for LAPACK from the source code.