doubleGEeigen(3)

NAME
doubleGEeigen - double

SYNOPSIS

  Functions
subroutine dgees (jobvs, sort, select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info)
DGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine dgeesx (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info)
DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine dgeev (jobvl, jobvr, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, work, lwork, info)
DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine dgeevx (balanc, jobvl, jobvr, sense, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, iwork, info)
DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine dgges (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info)
DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine dgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info)
DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)
subroutine dggesx (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)
DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine dggev (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine dggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
DGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)
subroutine dggevx (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)
DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

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

Function Documentation

  subroutine dgees (character jobvs, character sort, external select, integer n, double precision, dimension( lda, * ) a, integer lda, integer sdim, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvs, * ) vs, integer ldvs, double precision, dimension( * ) work, integer lwork, logical, dimension( * ) bwork, integer info)

DGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

     DGEES 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.

Definition at line 214 of file dgees.f.

  subroutine dgeesx (character jobvs, character sort, external select, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, integer sdim, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvs, * ) vs, integer ldvs, double precision rconde, double precision rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

     DGEESX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
              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 DOUBLE PRECISION
              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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  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.

Definition at line 278 of file dgeesx.f.

  subroutine dgeev (character jobvl, character jobvr, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) work, integer lwork, integer info)

DGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

     DGEEV 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  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.

Definition at line 190 of file dgeev.f.

  subroutine dgeevx (character balanc, character jobvl, character jobvr, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, double precision, dimension( * ) scale, double precision abnrm, double precision, dimension( * ) rconde, double precision, dimension( * ) rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

     DGEEVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
              The one-norm of the balanced matrix (the maximum
              of the sum of absolute values of elements of any column).

    RCONDE

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

    RCONDV

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

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.   If 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.

Definition at line 303 of file dgeevx.f.

  subroutine dgges (character jobvsl, character jobvsr, character sort, external selctg, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl, double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( * ) work, integer lwork, logical, dimension( * ) bwork, integer info)

DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

     DGGES 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
     DGGEV 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 DOUBLE PRECISION arguments
              SELCTG must be declared EXTERNAL in the calling subroutine.
              If SORT = 'N', SELCTG is not referenced.
              If SORT = 'S', SELCTG is used to select eigenvalues to sort
              to the top left of the Schur form.
              An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
              SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
              one of a complex conjugate pair of eigenvalues is selected,
              then both complex eigenvalues are selected.
              Note that 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 DOUBLE PRECISION array, dimension (LDA, N)
              On entry, the first of the pair of matrices.
              On exit, A has been overwritten by its generalized Schur
              form S.

    LDA

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

    B

              B is DOUBLE PRECISION array, dimension (LDB, N)
              On entry, the second of the pair of matrices.
              On exit, B has been overwritten by its generalized Schur
              form T.

    LDB

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues (after sorting)
              for which SELCTG is true.  (Complex conjugate pairs for which
              SELCTG is true for either eigenvalue count as 2.)

    ALPHAR

              ALPHAR is DOUBLE PRECISION array, dimension (N)

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
              be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
              and  BETA(j),j=1,...,N are the diagonals of the complex Schur
              form (S,T) that would result if the 2-by-2 diagonal blocks of
              the real Schur form of (A,B) were further reduced to
              triangular form using 2-by-2 complex unitary transformations.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) negative.
              Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
              may easily over- or underflow, and BETA(j) may even be zero.
              Thus, the user should avoid naively computing the ratio.
              However, ALPHAR and ALPHAI will be always less than and
              usually comparable with norm(A) in magnitude, and BETA always
              less than and usually comparable with norm(B).

    VSL

              VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
              If JOBVSL = 'V', VSL will contain the left Schur vectors.
              Not referenced if JOBVSL = 'N'.

    LDVSL

              LDVSL is INTEGER
              The leading dimension of the matrix VSL. LDVSL >=1, and
              if JOBVSL = 'V', LDVSL >= N.

    VSR

              VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
              If JOBVSR = 'V', VSR will contain the right Schur vectors.
              Not referenced if JOBVSR = 'N'.

    LDVSR

              LDVSR is INTEGER
              The leading dimension of the matrix VSR. LDVSR >= 1, and
              if JOBVSR = 'V', LDVSR >= N.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N = 0, LWORK >= 1, else LWORK >= 8*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 DHGEQZ.
                    =N+2: after reordering, roundoff changed values of
                          some complex eigenvalues so that leading
                          eigenvalues in the Generalized Schur form no
                          longer satisfy SELCTG=.TRUE.  This could also
                          be caused due to scaling.
                    =N+3: reordering failed in DTGSEN.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 281 of file dgges.f.

  subroutine dgges3 (character jobvsl, character jobvsr, character sort, external selctg, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl, double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( * ) work, integer lwork, logical, dimension( * ) bwork, integer info)

DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Purpose:

     DGGES3 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
     DGGEV 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 DOUBLE PRECISION arguments
              SELCTG must be declared EXTERNAL in the calling subroutine.
              If SORT = 'N', SELCTG is not referenced.
              If SORT = 'S', SELCTG is used to select eigenvalues to sort
              to the top left of the Schur form.
              An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
              SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
              one of a complex conjugate pair of eigenvalues is selected,
              then both complex eigenvalues are selected.
              Note that 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 DOUBLE PRECISION array, dimension (LDA, N)
              On entry, the first of the pair of matrices.
              On exit, A has been overwritten by its generalized Schur
              form S.

    LDA

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

    B

              B is DOUBLE PRECISION array, dimension (LDB, N)
              On entry, the second of the pair of matrices.
              On exit, B has been overwritten by its generalized Schur
              form T.

    LDB

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues (after sorting)
              for which SELCTG is true.  (Complex conjugate pairs for which
              SELCTG is true for either eigenvalue count as 2.)

    ALPHAR

              ALPHAR is DOUBLE PRECISION array, dimension (N)

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
              be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
              and  BETA(j),j=1,...,N are the diagonals of the complex Schur
              form (S,T) that would result if the 2-by-2 diagonal blocks of
              the real Schur form of (A,B) were further reduced to
              triangular form using 2-by-2 complex unitary transformations.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) negative.
              Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
              may easily over- or underflow, and BETA(j) may even be zero.
              Thus, the user should avoid naively computing the ratio.
              However, ALPHAR and ALPHAI will be always less than and
              usually comparable with norm(A) in magnitude, and BETA always
              less than and usually comparable with norm(B).

    VSL

              VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
              If JOBVSL = 'V', VSL will contain the left Schur vectors.
              Not referenced if JOBVSL = 'N'.

    LDVSL

              LDVSL is INTEGER
              The leading dimension of the matrix VSL. LDVSL >=1, and
              if JOBVSL = 'V', LDVSL >= N.

    VSR

              VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
              If JOBVSR = 'V', VSR will contain the right Schur vectors.
              Not referenced if JOBVSR = 'N'.

    LDVSR

              LDVSR is INTEGER
              The leading dimension of the matrix VSR. LDVSR >= 1, and
              if JOBVSR = 'V', LDVSR >= N.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If 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 DLAQZ0.
                    =N+2: after reordering, roundoff changed values of
                          some complex eigenvalues so that leading
                          eigenvalues in the Generalized Schur form no
                          longer satisfy SELCTG=.TRUE.  This could also
                          be caused due to scaling.
                    =N+3: reordering failed in DTGSEN.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 279 of file dgges3.f.

  subroutine dggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl, double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( 2 ) rconde, double precision, dimension( 2 ) rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

     DGGESX computes for a pair of N-by-N real nonsymmetric matrices
     (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
     optionally, the left and/or right matrices of Schur vectors (VSL and
     VSR).  This gives the generalized Schur factorization
          (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
     Optionally, it also orders the eigenvalues so that a selected cluster
     of eigenvalues appears in the leading diagonal blocks of the upper
     quasi-triangular matrix S and the upper triangular matrix T; computes
     a reciprocal condition number for the average of the selected
     eigenvalues (RCONDE); and computes a reciprocal condition number for
     the right and left deflating subspaces corresponding to the selected
     eigenvalues (RCONDV). The leading columns of VSL and VSR then form
     an orthonormal basis for the corresponding left and right eigenspaces
     (deflating subspaces).
     A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
     or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
     usually represented as the pair (alpha,beta), as there is a
     reasonable interpretation for beta=0 or for both being zero.
     A pair of matrices (S,T) is in generalized real Schur form if T is
     upper triangular with non-negative diagonal and S is block upper
     triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
     to real generalized eigenvalues, while 2-by-2 blocks of S will be
     'standardized' by making the corresponding elements of T have the
     form:
             [  a  0  ]
             [  0  b  ]
     and the pair of corresponding 2-by-2 blocks in S and T will have a
     complex conjugate pair of generalized eigenvalues.

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 DOUBLE PRECISION arguments
              SELCTG must be declared EXTERNAL in the calling subroutine.
              If SORT = 'N', SELCTG is not referenced.
              If SORT = 'S', SELCTG is used to select eigenvalues to sort
              to the top left of the Schur form.
              An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
              SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
              one of a complex conjugate pair of eigenvalues is selected,
              then both complex eigenvalues are selected.
              Note that a selected complex eigenvalue may no longer satisfy
              SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
              since ordering may change the value of complex eigenvalues
              (especially if the eigenvalue is ill-conditioned), in this
              case INFO is set to N+3.

    SENSE

              SENSE is CHARACTER*1
              Determines which reciprocal condition numbers are computed.
              = 'N':  None are computed;
              = 'E':  Computed for average of selected eigenvalues only;
              = 'V':  Computed for selected deflating subspaces only;
              = 'B':  Computed for both.
              If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

    N

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

    A

              A is DOUBLE PRECISION array, dimension (LDA, N)
              On entry, the first of the pair of matrices.
              On exit, A has been overwritten by its generalized Schur
              form S.

    LDA

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

    B

              B is DOUBLE PRECISION array, dimension (LDB, N)
              On entry, the second of the pair of matrices.
              On exit, B has been overwritten by its generalized Schur
              form T.

    LDB

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues (after sorting)
              for which SELCTG is true.  (Complex conjugate pairs for which
              SELCTG is true for either eigenvalue count as 2.)

    ALPHAR

              ALPHAR is DOUBLE PRECISION array, dimension (N)

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
              be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
              and BETA(j),j=1,...,N  are the diagonals of the complex Schur
              form (S,T) that would result if the 2-by-2 diagonal blocks of
              the real Schur form of (A,B) were further reduced to
              triangular form using 2-by-2 complex unitary transformations.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) negative.
              Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
              may easily over- or underflow, and BETA(j) may even be zero.
              Thus, the user should avoid naively computing the ratio.
              However, ALPHAR and ALPHAI will be always less than and
              usually comparable with norm(A) in magnitude, and BETA always
              less than and usually comparable with norm(B).

    VSL

              VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
              If JOBVSL = 'V', VSL will contain the left Schur vectors.
              Not referenced if JOBVSL = 'N'.

    LDVSL

              LDVSL is INTEGER
              The leading dimension of the matrix VSL. LDVSL >=1, and
              if JOBVSL = 'V', LDVSL >= N.

    VSR

              VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
              If JOBVSR = 'V', VSR will contain the right Schur vectors.
              Not referenced if JOBVSR = 'N'.

    LDVSR

              LDVSR is INTEGER
              The leading dimension of the matrix VSR. LDVSR >= 1, and
              if JOBVSR = 'V', LDVSR >= N.

    RCONDE

              RCONDE is DOUBLE PRECISION array, dimension ( 2 )
              If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
              reciprocal condition numbers for the average of the selected
              eigenvalues.
              Not referenced if SENSE = 'N' or 'V'.

    RCONDV

              RCONDV is DOUBLE PRECISION array, dimension ( 2 )
              If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
              reciprocal condition numbers for the selected deflating
              subspaces.
              Not referenced if SENSE = 'N' or 'E'.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
              LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
              LWORK >= max( 8*N, 6*N+16 ).
              Note that 2*SDIM*(N-SDIM) <= N*N/2.
              Note also that an error is only returned if
              LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
              this may not be large enough.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the bound on the optimal size of the WORK
              array and the minimum size of the IWORK array, returns these
              values as the first entries of the WORK and IWORK arrays, and
              no error message related to LWORK or LIWORK is issued by
              XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
              LIWORK >= N+6.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the bound on the optimal size of the
              WORK array and the minimum size of the IWORK array, returns
              these values as the first entries of the WORK and IWORK
              arrays, and no error message related to LWORK or LIWORK is
              issued by XERBLA.

    BWORK

              BWORK is LOGICAL array, dimension (N)
              Not referenced if SORT = 'N'.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              = 1,...,N:
                    The QZ iteration failed.  (A,B) are not in Schur
                    form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                    be correct for j=INFO+1,...,N.
              > N:  =N+1: other than QZ iteration failed in DHGEQZ
                    =N+2: after reordering, roundoff changed values of
                          some complex eigenvalues so that leading
                          eigenvalues in the Generalized Schur form no
                          longer satisfy SELCTG=.TRUE.  This could also
                          be caused due to scaling.
                    =N+3: reordering failed in DTGSEN.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

Definition at line 361 of file dggesx.f.

  subroutine dggev (character jobvl, character jobvr, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) work, integer lwork, integer info)

DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

     DGGEV 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
              be the generalized eigenvalues.  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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  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 DHGEQZ.
                    =N+2: error return from DTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 224 of file dggev.f.

  subroutine dggev3 (character jobvl, character jobvr, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) work, integer lwork, integer info)

DGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

     DGGEV3 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
              be the generalized eigenvalues.  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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DLAQZ0.
                    =N+2: error return from DTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 223 of file dggev3.f.

  subroutine dggevx (character balanc, character jobvl, character jobvr, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, double precision, dimension( * ) lscale, double precision, dimension( * ) rscale, double precision abnrm, double precision bbnrm, double precision, dimension( * ) rconde, double precision, dimension( * ) rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

     DGGEVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
              be the generalized eigenvalues.  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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
              The one-norm of the balanced matrix A.

    BBNRM

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

    RCONDE

              RCONDE is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,2*N).
              If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
              LWORK >= max(1,6*N).
              If SENSE = 'E' or 'B', 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 DHGEQZ.
                    =N+2: error return from DTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      Balancing a matrix pair (A,B) includes, first, permuting rows and
      columns to isolate eigenvalues, second, applying diagonal similarity
      transformation to the rows and columns to make the rows and columns
      as close in norm as possible. The computed reciprocal condition
      numbers correspond to the balanced matrix. Permuting rows and columns
      will not change the condition numbers (in exact arithmetic) but
      diagonal scaling will.  For further explanation of balancing, see
      section 4.11.1.2 of LAPACK Users' Guide.
      An approximate error bound on the chordal distance between the i-th
      computed generalized eigenvalue w and the corresponding exact
      eigenvalue lambda is
           chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
      An approximate error bound for the angle between the i-th computed
      eigenvector VL(i) or VR(i) is given by
           EPS * norm(ABNRM, BBNRM) / DIF(i).
      For further explanation of the reciprocal condition numbers RCONDE
      and RCONDV, see section 4.11 of LAPACK User's Guide.

Definition at line 387 of file dggevx.f.

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