Provided by: liblapack-doc_3.8.0-2_all bug

NAME

       realGEeigen

SYNOPSIS

   Functions
       subroutine sgees (JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS, LDVS, WORK, LWORK,
           BWORK, INFO)
            SGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine sgeesx (JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI, VS, LDVS, RCONDE,
           RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
            SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine sgeev (JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
            SGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices
       subroutine sgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR,
           ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)
            SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices
       subroutine sgges (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI,
           BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)
            SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine sgges3 (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI,
           BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)
            SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices (blocked algorithm)
       subroutine sggesx (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHAR,
           ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK,
           BWORK, INFO)
            SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine sggev (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR,
           LDVR, WORK, LWORK, INFO)
            SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices
       subroutine sggev3 (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR,
           LDVR, WORK, LWORK, INFO)
            SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices (blocked algorithm)
       subroutine sggevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
           VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK,
           LWORK, IWORK, BWORK, INFO)
            SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices

Detailed Description

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

Function Documentation

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

       Purpose:

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

            Optionally, it also orders the eigenvalues on the diagonal of the
            real 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 matrix is in real Schur form if it is upper quasi-triangular with
            1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
            form
                    [  a  b  ]
                    [  c  a  ]

            where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

       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 two REAL arguments
                     SELECT must be declared EXTERNAL in the calling subroutine.
                     If SORT = 'S', SELECT is used to select eigenvalues to sort
                     to the top left of the Schur form.
                     If SORT = 'N', SELECT is not referenced.
                     An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
                     SELECT(WR(j),WI(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 SELECT(WR(j),WI(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 matrix A. N >= 0.

           A

                     A is REAL array, dimension (LDA,N)
                     On entry, the N-by-N matrix A.
                     On exit, A has been overwritten by its real 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 (after sorting)
                                    for which SELECT is true. (Complex conjugate
                                    pairs for which SELECT is true for either
                                    eigenvalue count as 2.)

           WR

                     WR is REAL array, dimension (N)

           WI

                     WI is REAL array, dimension (N)
                     WR and WI contain the real and imaginary parts,
                     respectively, of the computed eigenvalues in the same order
                     that they appear on the diagonal of the output Schur form T.
                     Complex conjugate pairs of eigenvalues will appear
                     consecutively with the eigenvalue having the positive
                     imaginary part first.

           VS

                     VS is REAL array, dimension (LDVS,N)
                     If JOBVS = 'V', VS contains the orthogonal 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 REAL array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,3*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.

           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 WR and WI
                              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.

       Date:
           June 2017

   subroutine sgeesx (character JOBVS, character SORT, external SELECT, character SENSE, integer
       N, real, dimension( lda, * ) A, integer LDA, integer SDIM, real, dimension( * ) WR, real,
       dimension( * ) WI, real, dimension( ldvs, * ) VS, integer LDVS, real RCONDE, real RCONDV,
       real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK,
       logical, dimension( * ) BWORK, integer INFO)
        SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

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

            Optionally, it also orders the eigenvalues on the diagonal of the
            real 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 real matrix is in real Schur form if it is upper quasi-triangular
            with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
            the form
                      [  a  b  ]
                      [  c  a  ]

            where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

       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 two REAL arguments
                     SELECT must be declared EXTERNAL in the calling subroutine.
                     If SORT = 'S', SELECT is used to select eigenvalues to sort
                     to the top left of the Schur form.
                     If SORT = 'N', SELECT is not referenced.
                     An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
                     SELECT(WR(j),WI(j)) is true; i.e., if either one of a
                     complex conjugate pair of eigenvalues is selected, then both
                     are.  Note that a selected complex eigenvalue may no longer
                     satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
                     ordering may change the value of complex eigenvalues
                     (especially if the eigenvalue is ill-conditioned); in this
                     case INFO may be 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 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 REAL array, dimension (LDA, N)
                     On entry, the N-by-N matrix A.
                     On exit, A is overwritten by its real 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 (after sorting)
                                    for which SELECT is true. (Complex conjugate
                                    pairs for which SELECT is true for either
                                    eigenvalue count as 2.)

           WR

                     WR is REAL array, dimension (N)

           WI

                     WI is REAL array, dimension (N)
                     WR and WI contain the real and imaginary parts, respectively,
                     of the computed eigenvalues, in the same order that they
                     appear on the diagonal of the output Schur form T.  Complex
                     conjugate pairs of eigenvalues appear consecutively with the
                     eigenvalue having the positive imaginary part first.

           VS

                     VS is REAL array, dimension (LDVS,N)
                     If JOBVS = 'V', VS contains the orthogonal 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 REAL 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(1,3*N).
                     Also, if SENSE = 'E' or 'V' or 'B',
                     LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
                     selected eigenvalues computed by this routine.  Note that
                     N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
                     returned if LWORK < max(1,3*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 bounds on the optimal sizes of the
                     arrays WORK and IWORK, returns these values as the first
                     entries of the WORK and IWORK arrays, and no error messages
                     related to LWORK or LIWORK are issued by XERBLA.

           IWORK

                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

           LIWORK

                     LIWORK is INTEGER
                     The dimension of the array IWORK.
                     LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
                     Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
                     only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
                     may not be large enough.

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates upper bounds on the optimal sizes of
                     the arrays WORK and IWORK, returns these values as the first
                     entries of the WORK and IWORK arrays, and no error messages
                     related to LWORK or LIWORK are 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.
                     > 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 WR and WI
                              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.

       Date:
           June 2016

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

       Purpose:

            SGEEV computes for an N-by-N real 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 A 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 REAL 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).

           WR

                     WR is REAL array, dimension (N)

           WI

                     WI is REAL array, dimension (N)
                     WR and WI contain the real and imaginary parts,
                     respectively, of the computed eigenvalues.  Complex
                     conjugate pairs of eigenvalues appear consecutively
                     with the eigenvalue having the positive imaginary part
                     first.

           VL

                     VL is REAL 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.
                     If the j-th eigenvalue is real, then u(j) = VL(:,j),
                     the j-th column of VL.
                     If the j-th and (j+1)-st eigenvalues form a complex
                     conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
                     u(j+1) = VL(:,j) - i*VL(:,j+1).

           LDVL

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

           VR

                     VR is REAL 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.
                     If the j-th eigenvalue is real, then v(j) = VR(:,j),
                     the j-th column of VR.
                     If the j-th and (j+1)-st eigenvalues form a complex
                     conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
                     v(j+1) = VR(:,j) - i*VR(:,j+1).

           LDVR

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

           WORK

                     WORK is REAL 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(1,3*N), and
                     if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*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.

           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 WR and WI contain eigenvalues which
                           have converged.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2016

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

       Purpose:

            SGEEVX computes for an N-by-N real 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, i.e. 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 REAL 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 real Schur form of the balanced
                     version of the input matrix A.

           LDA

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

           WR

                     WR is REAL array, dimension (N)

           WI

                     WI is REAL array, dimension (N)
                     WR and WI contain the real and imaginary parts,
                     respectively, of the computed eigenvalues.  Complex
                     conjugate pairs of eigenvalues will appear consecutively
                     with the eigenvalue having the positive imaginary part
                     first.

           VL

                     VL is REAL 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.
                     If the j-th eigenvalue is real, then u(j) = VL(:,j),
                     the j-th column of VL.
                     If the j-th and (j+1)-st eigenvalues form a complex
                     conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
                     u(j+1) = VL(:,j) - i*VL(:,j+1).

           LDVL

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

           VR

                     VR is REAL 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.
                     If the j-th eigenvalue is real, then v(j) = VR(:,j),
                     the j-th column of VR.
                     If the j-th and (j+1)-st eigenvalues form a complex
                     conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
                     v(j+1) = VR(:,j) - i*VR(:,j+1).

           LDVR

                     LDVR is INTEGER
                     The leading dimension of the array VR.  LDVR >= 1, and 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 REAL 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 JOBVL = 'V' or JOBVR = 'V',
                     LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
                     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.

           IWORK

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

           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 WR
                           and WI contain eigenvalues which have converged.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2016

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

       Purpose:

            SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
            the generalized eigenvalues, the generalized real Schur form (S,T),
            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.The
            leading columns of VSL and VSR then form an orthonormal basis for the
            corresponding left and right eigenspaces (deflating subspaces).

            (If only the generalized eigenvalues are needed, use the driver
            SGGEV 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 or 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.

       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 three REAL 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 in the ill-conditioned case, a selected complex
                     eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
                     BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
                     in this case.

           N

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

           A

                     A is REAL 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 REAL 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 REAL array, dimension (N)

           ALPHAI

                     ALPHAI is REAL array, dimension (N)

           BETA

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

           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 SHGEQZ.
                           =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 STGSEN.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine sgges3 (character JOBVSL, character JOBVSR, character SORT, external SELCTG,
       integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer
       LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real,
       dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension(
       ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, logical,
       dimension( * ) BWORK, integer INFO)
        SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices (blocked algorithm)

       Purpose:

            SGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
            the generalized eigenvalues, the generalized real Schur form (S,T),
            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.The
            leading columns of VSL and VSR then form an orthonormal basis for the
            corresponding left and right eigenspaces (deflating subspaces).

            (If only the generalized eigenvalues are needed, use the driver
            SGGEV 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 or 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.

       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 three REAL 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 in the ill-conditioned case, a selected complex
                     eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
                     BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
                     in this case.

           N

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

           A

                     A is REAL 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 REAL 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 REAL array, dimension (N)

           ALPHAI

                     ALPHAI is REAL array, dimension (N)

           BETA

                     BETA is REAL 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 REAL 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 REAL 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 REAL 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.

           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 SHGEQZ.
                           =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 STGSEN.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           January 2015

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

       Purpose:

            SGGESX 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.

       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 three REAL 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 REAL 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 REAL 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 REAL array, dimension (N)

           ALPHAI

                     ALPHAI is REAL array, dimension (N)

           BETA

                     BETA is REAL 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 REAL 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 REAL 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 numbers for the selected deflating
                     subspaces.
                     Not referenced if SENSE = 'N' or 'E'.

           WORK

                     WORK is REAL 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 SHGEQZ
                           =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 STGSEN.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

       Further Details:

             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.

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

       Purpose:

            SGGEV computes for a pair of N-by-N real 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 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:
           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 REAL 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 REAL 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).

           ALPHAR

                     ALPHAR is REAL array, dimension (N)

           ALPHAI

                     ALPHAI is REAL array, dimension (N)

           BETA

                     BETA is REAL array, dimension (N)
                     On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
                     be the generalized eigenvalues.  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
                     alpha/beta.  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).

           VL

                     VL is REAL 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 the j-th eigenvalue is real, then
                     u(j) = VL(:,j), the j-th column of VL. If the j-th and
                     (j+1)-th eigenvalues form a complex conjugate pair, then
                     u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
                     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 REAL 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 the j-th eigenvalue is real, then
                     v(j) = VR(:,j), the j-th column of VR. If the j-th and
                     (j+1)-th eigenvalues form a complex conjugate pair, then
                     v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
                     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 REAL 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(1,8*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.

           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 ALPHAR(j), ALPHAI(j), and BETA(j)
                           should be correct for j=INFO+1,...,N.
                     > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                           =N+2: error return from STGEVC.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           April 2012

   subroutine sggev3 (character JOBVL, character JOBVR, integer N, real, dimension( lda, * ) A,
       integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHAR, real,
       dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvl, * ) VL, integer
       LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) WORK, integer
       LWORK, integer INFO)
        SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       GE matrices (blocked algorithm)

       Purpose:

            SGGEV3 computes for a pair of N-by-N real 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 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:
           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 REAL 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 REAL 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).

           ALPHAR

                     ALPHAR is REAL array, dimension (N)

           ALPHAI

                     ALPHAI is REAL array, dimension (N)

           BETA

                     BETA is REAL array, dimension (N)
                     On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
                     be the generalized eigenvalues.  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
                     alpha/beta.  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).

           VL

                     VL is REAL 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 the j-th eigenvalue is real, then
                     u(j) = VL(:,j), the j-th column of VL. If the j-th and
                     (j+1)-th eigenvalues form a complex conjugate pair, then
                     u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
                     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 REAL 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 the j-th eigenvalue is real, then
                     v(j) = VR(:,j), the j-th column of VR. If the j-th and
                     (j+1)-th eigenvalues form a complex conjugate pair, then
                     v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
                     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 REAL array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER

                     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.

           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 ALPHAR(j), ALPHAI(j), and BETA(j)
                           should be correct for j=INFO+1,...,N.
                     > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                           =N+2: error return from STGEVC.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           January 2015

   subroutine sggevx (character BALANC, character JOBVL, character JOBVR, character SENSE,
       integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer
       LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA,
       real, dimension( ldvl, * ) VL, integer LDVL, real, 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, real,
       dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, logical, dimension( * )
       BWORK, integer INFO)
        SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       GE matrices

       Purpose:

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

            Optionally also, it 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 REAL 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 real 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 REAL 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 real 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).

           ALPHAR

                     ALPHAR is REAL array, dimension (N)

           ALPHAI

                     ALPHAI is REAL array, dimension (N)

           BETA

                     BETA is REAL array, dimension (N)
                     On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
                     be the generalized eigenvalues.  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
                     ALPHA/BETA. 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).

           VL

                     VL is REAL 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 the j-th eigenvalue is real, then
                     u(j) = VL(:,j), the j-th column of VL. If the j-th and
                     (j+1)-th eigenvalues form a complex conjugate pair, then
                     u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
                     Each eigenvector will be scaled so the largest component 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 REAL 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 the j-th eigenvalue is real, then
                     v(j) = VR(:,j), the j-th column of VR. If the j-th and
                     (j+1)-th eigenvalues form a complex conjugate pair, then
                     v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
                     Each eigenvector will be scaled so the largest component 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.
                     For a complex conjugate pair of eigenvalues two consecutive
                     elements of RCONDE are set to the same value. Thus RCONDE(j),
                     RCONDV(j), and the j-th columns of VL and VR all correspond
                     to the j-th eigenpair.
                     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. For a complex eigenvector two consecutive
                     elements of RCONDV are set to the same value. 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 REAL 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 BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
                     LWORK >= max(1,6*N).
                     If SENSE = 'E', LWORK >= max(1,10*N).
                     If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.

                     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.

           IWORK

                     IWORK is INTEGER array, dimension (N+6)
                     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 ALPHAR(j), ALPHAI(j), and BETA(j)
                           should be correct for j=INFO+1,...,N.
                     > N:  =N+1: other than QZ iteration failed in SHGEQZ.
                           =N+2: error return from STGEVC.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           April 2012

       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.