ggevx(3)

NAME
ggevx - ggevx: eig, expert

SYNOPSIS

  Functions

subroutine cggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)
CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
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
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
subroutine zggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)
ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

Function Documentation

  subroutine cggevx (character balanc, character jobvl, character jobvr, character sense, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real abnrm, real bbnrm, real, dimension( * ) rconde, real, dimension( * ) rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

    !>
    !> CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
    !> (A,B) the generalized eigenvalues, and optionally, the left and/or
    !> right generalized eigenvectors.
    !>
    !> Optionally, it also computes a balancing transformation to improve
    !> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
    !> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
    !> the eigenvalues (RCONDE), and reciprocal condition numbers for the
    !> right eigenvectors (RCONDV).
    !>
    !> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
    !> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
    !> singular. It is usually represented as the pair (alpha,beta), as
    !> there is a reasonable interpretation for beta=0, and even for both
    !> being zero.
    !>
    !> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
    !> of (A,B) satisfies
    !>                  A * v(j) = lambda(j) * B * v(j) .
    !> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
    !> of (A,B) satisfies
    !>                  u(j)**H * A  = lambda(j) * u(j)**H * B.
    !> where u(j)**H is the conjugate-transpose of u(j).
    !>
    !>

Parameters
    BALANC

    !>          BALANC is CHARACTER*1
    !>          Specifies the balance option to be performed:
    !>          = 'N':  do not diagonally scale or permute;
    !>          = 'P':  permute only;
    !>          = 'S':  scale only;
    !>          = 'B':  both permute and scale.
    !>          Computed reciprocal condition numbers will be for the
    !>          matrices after permuting and/or balancing. Permuting does
    !>          not change condition numbers (in exact arithmetic), but
    !>          balancing does.
    !>

    JOBVL

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

    JOBVR

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

    SENSE

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

    N

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

    A

    !>          A is COMPLEX array, dimension (LDA, N)
    !>          On entry, the matrix A in the pair (A,B).
    !>          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
    !>          or both, then A contains the first part of the complex Schur
    !>          form of the  versions of the input A and B.
    !>

    LDA

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

    B

    !>          B is COMPLEX array, dimension (LDB, N)
    !>          On entry, the matrix B in the pair (A,B).
    !>          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
    !>          or both, then B contains the second part of the complex
    !>          Schur form of the  versions of the input A and B.
    !>

    LDB

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

    ALPHA

    !>          ALPHA is COMPLEX array, dimension (N)
    !>

    BETA

    !>          BETA is COMPLEX array, dimension (N)
    !>          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized
    !>          eigenvalues.
    !>
    !>          Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or
    !>          underflow, and BETA(j) may even be zero.  Thus, the user
    !>          should avoid naively computing the ratio ALPHA/BETA.
    !>          However, ALPHA will be always less than and usually
    !>          comparable with norm(A) in magnitude, and BETA always less
    !>          than and usually comparable with norm(B).
    !>

    VL

    !>          VL is COMPLEX array, dimension (LDVL,N)
    !>          If JOBVL = 'V', the left generalized eigenvectors u(j) are
    !>          stored one after another in the columns of VL, in the same
    !>          order as their eigenvalues.
    !>          Each eigenvector will be scaled so the largest component
    !>          will have abs(real part) + abs(imag. part) = 1.
    !>          Not referenced if JOBVL = 'N'.
    !>

    LDVL

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

    VR

    !>          VR is COMPLEX array, dimension (LDVR,N)
    !>          If JOBVR = 'V', the right generalized eigenvectors v(j) are
    !>          stored one after another in the columns of VR, in the same
    !>          order as their eigenvalues.
    !>          Each eigenvector will be scaled so the largest component
    !>          will have abs(real part) + abs(imag. part) = 1.
    !>          Not referenced if JOBVR = 'N'.
    !>

    LDVR

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

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          ILO and IHI are integer values such that on exit
    !>          A(i,j) = 0 and B(i,j) = 0 if i > j and
    !>          j = 1,...,ILO-1 or i = IHI+1,...,N.
    !>          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
    !>

    LSCALE

    !>          LSCALE is REAL array, dimension (N)
    !>          Details of the permutations and scaling factors applied
    !>          to the left side of A and B.  If PL(j) is the index of the
    !>          row interchanged with row j, and DL(j) is the scaling
    !>          factor applied to row j, then
    !>            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
    !>                      = DL(j)  for j = ILO,...,IHI
    !>                      = PL(j)  for j = IHI+1,...,N.
    !>          The order in which the interchanges are made is N to IHI+1,
    !>          then 1 to ILO-1.
    !>

    RSCALE

    !>          RSCALE is REAL array, dimension (N)
    !>          Details of the permutations and scaling factors applied
    !>          to the right side of A and B.  If PR(j) is the index of the
    !>          column interchanged with column j, and DR(j) is the scaling
    !>          factor applied to column j, then
    !>            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
    !>                      = DR(j)  for j = ILO,...,IHI
    !>                      = PR(j)  for j = IHI+1,...,N
    !>          The order in which the interchanges are made is N to IHI+1,
    !>          then 1 to ILO-1.
    !>

    ABNRM

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

    BBNRM

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

    RCONDE

    !>          RCONDE is REAL array, dimension (N)
    !>          If SENSE = 'E' or 'B', the reciprocal condition numbers of
    !>          the eigenvalues, stored in consecutive elements of the array.
    !>          If SENSE = 'N' or 'V', RCONDE is not referenced.
    !>

    RCONDV

    !>          RCONDV is REAL array, dimension (N)
    !>          If SENSE = 'V' or 'B', the estimated reciprocal condition
    !>          numbers of the eigenvectors, stored in consecutive elements
    !>          of the array. If the eigenvalues cannot be reordered to
    !>          compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
    !>          when the true value would be very small anyway.
    !>          If SENSE = 'N' or 'E', RCONDV is not referenced.
    !>

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= max(1,2*N).
    !>          If SENSE = 'E', LWORK >= max(1,4*N).
    !>          If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (lrwork)
    !>          lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',
    !>          and at least max(1,2*N) otherwise.
    !>          Real workspace.
    !>

    IWORK

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

    BWORK

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

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          = 1,...,N:
    !>                The QZ iteration failed.  No eigenvectors have been
    !>                calculated, but ALPHA(j) and BETA(j) should be correct
    !>                for j=INFO+1,...,N.
    !>          > N:  =N+1: other than QZ iteration failed in CHGEQZ.
    !>                =N+2: error return from CTGEVC.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

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

  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  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  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.
    !>

  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  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  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.

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.
    !>

  subroutine zggevx (character balanc, character jobvl, character jobvr, character sense, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, 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, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

    !>
    !> ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
    !> (A,B) the generalized eigenvalues, and optionally, the left and/or
    !> right generalized eigenvectors.
    !>
    !> Optionally, it also computes a balancing transformation to improve
    !> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
    !> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
    !> the eigenvalues (RCONDE), and reciprocal condition numbers for the
    !> right eigenvectors (RCONDV).
    !>
    !> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
    !> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
    !> singular. It is usually represented as the pair (alpha,beta), as
    !> there is a reasonable interpretation for beta=0, and even for both
    !> being zero.
    !>
    !> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
    !> of (A,B) satisfies
    !>                  A * v(j) = lambda(j) * B * v(j) .
    !> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
    !> of (A,B) satisfies
    !>                  u(j)**H * A  = lambda(j) * u(j)**H * B.
    !> where u(j)**H is the conjugate-transpose of u(j).
    !>
    !>

Parameters
    BALANC

    !>          BALANC is CHARACTER*1
    !>          Specifies the balance option to be performed:
    !>          = 'N':  do not diagonally scale or permute;
    !>          = 'P':  permute only;
    !>          = 'S':  scale only;
    !>          = 'B':  both permute and scale.
    !>          Computed reciprocal condition numbers will be for the
    !>          matrices after permuting and/or balancing. Permuting does
    !>          not change condition numbers (in exact arithmetic), but
    !>          balancing does.
    !>

    JOBVL

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

    JOBVR

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

    SENSE

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

    N

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

    A

    !>          A is COMPLEX*16 array, dimension (LDA, N)
    !>          On entry, the matrix A in the pair (A,B).
    !>          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
    !>          or both, then A contains the first part of the complex Schur
    !>          form of the  versions of the input A and B.
    !>

    LDA

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

    B

    !>          B is COMPLEX*16 array, dimension (LDB, N)
    !>          On entry, the matrix B in the pair (A,B).
    !>          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
    !>          or both, then B contains the second part of the complex
    !>          Schur form of the  versions of the input A and B.
    !>

    LDB

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

    ALPHA

    !>          ALPHA is COMPLEX*16 array, dimension (N)
    !>

    BETA

    !>          BETA is COMPLEX*16 array, dimension (N)
    !>          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized
    !>          eigenvalues.
    !>
    !>          Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or
    !>          underflow, and BETA(j) may even be zero.  Thus, the user
    !>          should avoid naively computing the ratio ALPHA/BETA.
    !>          However, ALPHA will be always less than and usually
    !>          comparable with norm(A) in magnitude, and BETA always less
    !>          than and usually comparable with norm(B).
    !>

    VL

    !>          VL is COMPLEX*16 array, dimension (LDVL,N)
    !>          If JOBVL = 'V', the left generalized eigenvectors u(j) are
    !>          stored one after another in the columns of VL, in the same
    !>          order as their eigenvalues.
    !>          Each eigenvector will be scaled so the largest component
    !>          will have abs(real part) + abs(imag. part) = 1.
    !>          Not referenced if JOBVL = 'N'.
    !>

    LDVL

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

    VR

    !>          VR is COMPLEX*16 array, dimension (LDVR,N)
    !>          If JOBVR = 'V', the right generalized eigenvectors v(j) are
    !>          stored one after another in the columns of VR, in the same
    !>          order as their eigenvalues.
    !>          Each eigenvector will be scaled so the largest component
    !>          will have abs(real part) + abs(imag. part) = 1.
    !>          Not referenced if JOBVR = 'N'.
    !>

    LDVR

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

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          ILO and IHI are integer values such that on exit
    !>          A(i,j) = 0 and B(i,j) = 0 if i > j and
    !>          j = 1,...,ILO-1 or i = IHI+1,...,N.
    !>          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
    !>

    LSCALE

    !>          LSCALE is 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.
    !>          If SENSE = 'N' or 'V', RCONDE is not referenced.
    !>

    RCONDV

    !>          RCONDV is DOUBLE PRECISION array, dimension (N)
    !>          If JOB = 'V' or 'B', the estimated reciprocal condition
    !>          numbers of the eigenvectors, stored in consecutive elements
    !>          of the array. If the eigenvalues cannot be reordered to
    !>          compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
    !>          when the true value would be very small anyway.
    !>          If SENSE = 'N' or 'E', RCONDV is not referenced.
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= max(1,2*N).
    !>          If SENSE = 'E', LWORK >= max(1,4*N).
    !>          If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (lrwork)
    !>          lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',
    !>          and at least max(1,2*N) otherwise.
    !>          Real workspace.
    !>

    IWORK

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

    BWORK

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

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          = 1,...,N:
    !>                The QZ iteration failed.  No eigenvectors have been
    !>                calculated, but ALPHA(j) and BETA(j) should be correct
    !>                for j=INFO+1,...,N.
    !>          > N:  =N+1: other than QZ iteration failed in ZHGEQZ.
    !>                =N+2: error return from ZTGEVC.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

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

Author

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